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+{"text":"\\section{Introduction}\n\\IEEEPARstart{T}{elecommunication} research so far has tended to become technology-centric and focused excessively on quality of service (QoS) metrics. On the other hand, due to the rapid emergence of end-user controllable and programmable devices in communication networks, quality of experience (QoE) and user experience (UE) have increasingly attracted researchers' attention in both academia and industry in recent years \\cite{kilkki2008quality,6679038,8447199,finley2017does,boz2019mobile}.  Furthermore, the development of human-centric communications in the 6G network requires not only technological factors but also UE to be considered when modeling, analyzing, and optimizing communication systems \\cite{Dang2019FromAH}. This paradigm shift necessitates the interdisciplinary collaboration among telecommunications, economics, and psychology. \n\nThe first attempt to propose an ecosystem incorporating multiple stakeholders and end users is presented in \\cite{kilkki2008quality}, based on which several advanced versions of ecosystems are proposed in \\cite{6178834,dong2014quality,reichl2010charging,8701705}. However, due to the absence of a quantitative basis, existing ecosystems can hardly be utilized to carry out performance analysis for communication systems. To facilitate quantitative analysis, a standard approach is to measure UE via utility functions, such as logarithmic, sigmoid, and exponential functions \\cite{reichl2013logarithmic,6891176,5430142}. However, most QoE measurements are confined to simple quantitative relationships between QoS and QoE metrics. So far, no analytical framework has been proven to be generic and theoretically practicable for modeling UE in a human-centric manner. \n\n\nIn the area of economics, researchers resort to the prospect theory for accurately predicting decision-making behaviors of human beings \\cite{kahneman1979prospect,tversky1992advances}. The prospect theory is a Nobel prize winning theory and the founding pillar of behavioral economics. It is widely perceived as the most satisfactory descriptive theory of quantity perception currently available. Since its proposal in 1979, this theory has been extensively applied in pricing strategy, labor supply, tax policy making, and finance related topics. Due to its generality and versatility, the prospect theory is also introduced as a nexus to connect the disciplines of telecommunications, economics, and psychology. In \\cite{6747282}, the user behavior interference in networking protocols is modeled and analyzed according to the prospect theory. Specific resource allocation strategies using the prospect theory are investigated in \\cite{7560647}. Data pricing problems relying on the prospect theory for licensed and unlicensed communications are investigated in \\cite{7869340} and \\cite{7294655,8746552}, respectively.  The prospect theory has also been applied to provide secure protection mechanisms for communication systems by formulating dynamic defense games \\cite{7842178,7835168}. Incorporating game theory, the prospect theory can also help communication systems equip with better anti-jamming and random access functions \\cite{7036897,6310922}. A generic but simplistic prospect theoretic analytical framework of UE is proposed in \\cite{7346204}. Even though instructive and creative, the inappropriate and oversimplified modeling and assumptions in \\cite{7346204} result in a huge mismatch between the quantitative UE and resource utilization. In these interdisciplinary applications, however, the prospect theory serves only as a replacement for the utility or probability functions formulated in specific problems (e.g., game theoretic communications). This, to some extent, undervalues the prospect theory's implications and application aspects for human-centric communications.  \n\n\nIn light of this, we summarize the contributions of this paper as follows:\n\\begin{itemize}\n\\item We outline five essential attributes of the prospect theory considering user psychology that should be taken into consideration for wireless system modeling.\n\\item We construct a comprehensive analytical framework for modeling UE and perceptual measurements for human-centric communications.\n\\item We detail how modeling and analysis of communication systems will be reshaped under the novel analytical framework with several exemplary cases. The proposed analytical framework can be directly applied to carry out performance analysis for most communication systems when non-technological factors are taken into consideration and can be easily tailored to fit a broader range of communication applications. \n\\end{itemize}\nBy the contributions given in this paper, we aim to provide a guideline for improving communication services and articulate a new interdisciplinary research area for further investigation. \n\nThe remainder of this paper is as follows. In Section \\ref{fpt}, we present the fundamentals of the prospect theory, including its economic background, concepts, intuitions, and five key attributes of human psychology. Based on the five key attributes, we formulate the prospect theoretic analytical framework in Section \\ref{ptbaf}. In Section \\ref{cs}, we utilize several case studies to demonstrate how the analytical framework can be used to evaluate the subjective perception of communication systems. We outline the challenges and promising research directions for further investigations in Section \\ref{fird} and conclude the paper in Section \\ref{c}.\n\n\n\n\n\n\\section{Fundamentals of the Prospect Theory}\\label{fpt}\n\nAs we are entering the 5G\/6G era, the development of human-centric communications is becoming increasingly important but faces tremendous challenges. The paradigm shift from technology-centric to human-centric applications necessitates the interdisciplinary collaboration among telecommunications, economics, and psychology. In the following subsections, we explain step by step how this interdisciplinary collaboration is enabled by the prospect theory.\n\n\\subsection{Existing Problems with Classic Telecommunication Research}\nResearchers employ different approaches to establish linkages between QoS (e.g., bandwidth and loss rate) and QoE. Perceptual measurements such as the mean opinion score (MOS) and the pseudo-subjective quality assessment (PSQA) have been developed to quantify user experience \\cite{streijl2016mean,5070785}. There are also multi-dimensional evaluation systems designed to assess network performance from end-user perspectives \\cite{6178834}. Although the QoE research is gaining strong momentum in recent years, the literature remains scattered, inconsistent, and excessively technology-centric \\cite{kilkki2008quality,reichl2010charging,de2010proposed}. So far, no analytical framework has been proven sufficiently generic and theoretically practicable for modeling UE in a user-oriented manner. \n\nClassic communication research focuses on technology-centric network optimization and bypasses the model of UE. In such settings, researchers have actually made an implicit assumption: the optimization process maximizes both network capacity and user's subjective utility simultaneously \\cite{barakovic2013survey}. Conventionally, for $n$ mutually exclusive outcomes $y_i$ with occurrence probability $p_i$, the subjective utility can be considered as a linear function of probabilistic outcomes: $U=\\sum_{i=1}^{n}p_iy_i$, given $\\sum_{i=1}^{n}p_i=1$. However, the linear utility function is difficult to reconcile with human psychology due to its unrealistic attributes:\n\\begin{itemize}\n\\item Subjective utility is solely determined by the state of the final outcome.\n\\item Marginal utility is constant.\n\\item Quantity perception over gains and losses is symmetric.\n\\item There is indifference between objective and perceived probabilities.\n\\end{itemize}\nUnder the erroneous characterization of human psychology, the optimization process does not necessarily guarantee optimal UE, and the derived theories would be of limited practical usefulness. As an applied discipline, communications become human-centric only if UE can be properly introduced and optimized in the modeling process, which should be the very foundation of communication science. Overall, there is an urgent need to incorporate UE into the modeling of telecommunication theories.\n\n\\subsection{Prospect Theory}\nIn the past decades, the rapid development of behavioral economics has greatly enriched our understanding of human psychology and proposed promising analytical frameworks for modeling UE. As the founding pillar of behavioral economics, the prospect theory is widely perceived as the most satisfactory descriptive theory of quantity perception and decision making currently available \\cite{tversky1992advances,kahneman1979prospect}. The prospect theory is also a Nobel Prize winning theory, and the research article \\cite{kahneman1979prospect} is the second most cited paper in economics. The prospect theory provides a well-established theoretical framework and mathematical tools for modeling real-life UE, and its gist is that the human perception of quantity and probability are non-linear.\n\nIn the current literature, most QoE measurements for telecommunication studies are confined to quantity perception of discrete states, i.e., quantitative relationships between UE and QoS \\cite{5430142,reichl2013logarithmic}. As a major advantage, the prospect theory characterizes two indispensable dimensions of UE, i.e., quantity and probability, and can help to model and analyze UE in the continuous state. Under the prospect theory, the perception of quantity and probability perception can be expressed in Fig. \\ref{lizi}. This figure is believed to be the most important pictorial illustration of the prospect theory and can help with the mapping from QoS to QoE for telecommunication studies. A value function and a probability weighting function are employed to model the human perception, which can be characterized by the two-part functional form of \\cite{tversky1992advances} and the Prelec function given in \\cite{prelec1998probability}, respectively.\n\n\\begin{figure*}[!t]\n    \\centering\n    \\begin{subfigure}[t]{0.5\\textwidth}\n        \\centering\n        \\includegraphics[width=3.5in]{value_perception.eps}\n        \\caption{}\n    \\end{subfigure}%\n~\n    \\begin{subfigure}[t]{0.5\\textwidth}\n        \\centering\n        \\includegraphics[width=3.5in]{prob_perception.eps}\n        \\caption{}\n    \\end{subfigure}\n    \\caption{(a) Perceptual quantity under the prospect theory; (b) Perceptual probability under the prospect theory \\cite{dhami2016foundations}.}\n    \\label{lizi}\n\\end{figure*}\n\n\n\n\n\n\n\\subsection{Five Key Attributes}\nIn this subsection, we introduce the features of the prospect theory in shaping human psychological foundations and discuss the insights into modeling UE. According to Fig. \\ref{lizi}, we can observe five important features of human psychology, captured by the prospect theory as follows.\n\n\\subsubsection{Reference dependence}\nThe prospect theory states that individuals perceived value through changes (i.e., quantity deviations from a reference point) instead of states. There is always a reference point in each dimension of quantity perception. This can be explained by the following thought experiment. Communication operators upgrade Jack's mobile network from 3G to 4G but downgrade Jim's from 5G to 4G. Jack and James now have the same network capacity but markedly different quantity perception (gains versus losses), due to reference dependence. This characterization is consistent with the advanced notion of QoE: users' perceived value can be either positive (delight) or negative (annoyance) \\cite{le2012qualinet}. As a fundamental trait in human psychology, reference dependence casts doubts on optimization approaches that focus merely on the level of final outcomes, which has also attracted research attention in recent QoE literature \\cite{5430142}.\n\\subsubsection{Diminishing marginal utility}\nIndividuals have diminishing sensitivity towards the scale of changes. This means that the subjective difference between the data packages of 50 Mb and 100 Mb is more salient than that between the data packages of 950 Mb and 1000 Mb. Diminishing marginal utility, also known as diminishing sensitivity and utility curvature, is a widely recognized psychological feature shared by living creatures. Under proper assumptions (e.g., exponential or logarithmic utility functions \\cite{5430142,reichl2013logarithmic}), the concept of diminishing sensitivity can be readily incorporated into the modeling process to improve the practical significance of telecommunication theories.\n\\subsubsection{Loss aversion}\nIndividuals are more sensitive to losses than equivalent gains. Ample experimental evidence suggests that the magnitude of loss aversion is context-dependent, but losses are generally twice as significant as equivalent gains in quantity perception \\cite{dhami2016foundations}. The phenomenon of loss aversion is captured by the value function  in Fig. \\ref{lizi}(a), where quantity perception is steeper in the domain of losses. This asymmetry calls for a reconsideration of conventional approaches that weight gains and losses equally.\n\\subsubsection{Asymmetric risk attitudes}\nUnder diminishing marginal utility and loss aversion, individuals are risk seeking in the domain of losses but risk averse in the domain of gains. A prime example is the behavior of compulsive gambling: money-winning gamblers tend to play safely and prefer low-risk-low-return options, whereas money-losing gamblers prefer high-risk-high-return options and expect a grand slam home run. Asymmetric risk attitudes indicate that users have different risk preferences for improvement and deterioration in communication services, which calls for revisions of risk modeling in the QoE research. \n\\subsubsection{Probability distortion}\nIndividuals tend to overweight small probabilities and underweight moderate and high probabilities. Consider the thought experiment as follows:\n\\begin{problem}\n\\textbf{1} \\textit{How much would you pay to reduce the dropping probability from 5\\% to 0\\%.}\n\\end{problem}\n\\begin{problem}\n\\textbf{2} \\textit{How much would you pay to reduce the dropping probability from 55\\% to 50\\%.}\n\\end{problem}\n\n\nThe two problems seem identical because they are about the perceived value of 5\\% risk of dropping call. However, the great majority of people in real life are willing to pay much higher in \\textbf{Problem 1} than in \\textbf{Problem 2}. This phenomenon suggests a non-linear subjective probability weighting, in contrast to the linear probability weighting that equates objective probability with perceptual probability\\footnote{Similarly, one can consider the acceptable amount of compensation if the dropping probability increases by 5\\%. In most cases, people would not allow an increase of dropping probability from 0\\% to 5\\% but might be willing to negotiate over the compensation if the dropping probability increases from 50\\% to 55\\%. Again, the contrast between the increase and decrease of dropping probability is due to loss aversion, i.e., loss is subjectively more significant than equivalent gain.}. In probability perception, probability changes such as 0-5\\% and 95-100\\% are subjectively more salient than other changes of the same magnitude, since they are qualitative changes between non-existence, probabilistic outcome, and certainty. Extensive experimental evidence indicates that people are rather sensitive to the edges of probability interval [0,1], as documented by the inverse S-shaped probability weighting in Fig. \\ref{lizi}. \n\n\\subsection{Summary}\nUnder the prospect theory, the five psychological features transform into the fourfold pattern of risk attitudes in quantity perception: for low probabilities, individuals are risk seeking in the domain of gains but risk averse in the domain of losses; for moderate and high probabilities, individuals are risk averse in the domain of gains but risk seeking in the domain of losses. In summary, the prospect theory formulates the non-linear quantity and probability perception of human psychology, which has profound implications for developing human-centric communications in both academia and industry. \n\n\\section{Prospect Theoretic Analytical Framework}\\label{ptbaf}\nThe basic mathematical formulation based on the prospect theory enables the mapping from objective QoS to subjective QoE, capturing the behavioral, psychological, and contextual factors. In the process of quantity and probability perception, the QoE or UE is dependent on two metrics: the quantity metric and the probability metric. \n\n\n\\subsection{Quantity Metric and Value Function}\nThe quantity metric denotes conventional QoS parameters such as bandwidth and latency. According to the prospect theory, individuals perceived value through state transition (i.e., quantity deviations from a reference point) instead of the current quantity. In various disciplines, utility\/value functions serve as the standard approach for modeling UE. Following the two-part functional form of \\cite{tversky1992advances}, the value function for telecommunications taking the reference dependence, diminishing marginal utility, and loss aversion into consideration can be modeled as\\footnote{Note that, there is a prerequisite for using this value model, the quantity metric of interest must be a \\textit{desirable} metric. A desirable metric is a metric that will be preferable with a larger value, e.g., transmission rate and network coverage. In this paper, we only consider  desirable metrics without special notes, because of the limitation of the prospect theory originally dealing with the monetary benefit that is also a desirable metric.}$^{,}$\\footnote{For simplicity, we mainly analyze the single-metric scenario, in which only a single quantity metric $x$ with its reference point $x_0$ is taken into consideration.}\n\\begin{equation}\\label{valuefunc}\nv(x,x_0)=\\begin{cases}\n\\lambda_1(x-x_0)^{\\alpha_1},~~~~~~x\\geq x_0\\\\\n-\\lambda_2(x_0-x)^{\\alpha_2},~~~~x< x_0,\\\\\n\\end{cases}\n\\end{equation}\nwhere $\\alpha_1$, $\\alpha_2$, $\\lambda_1$, and $\\lambda_2$ are positive and user specific parameters;  $x_0>0$ is the reference point with respect to $x$, which captures the reference dependence. The reference point $x_0$ can be a previous quantity, expected quantity, or contractual quantity for different application scenarios. For generality, we model the process of quantity perception as a function of both $x$ and $x_0$ so as to emphasize the equal importance of the actual quantity $x$ and the reference point $x_0$. Both dimensions are indispensable in determining user perception of quantity metrics. It is worth noting that all the parameters, including the reference point $x_0$ are specific in terms of user preferences and socio-economic contexts and could change over time. Obviously, if $\\alpha_1=\\alpha_2=1$, $\\lambda_1=\\lambda_2=1$, and $x_0=0$, we have $v(x,x_0)=x$, $\\forall~x\\geq 0$, and the formulated analytical framework reduces to the classic QoS analytical framework. \n\n\n\n\nWithout loss of generality, the value model given in (\\ref{valuefunc}) is called the four-parameter value model, which is different from the classic two-parameter value model widely used in behavioral economics that assumes $\\alpha_1=\\alpha_2$ and fixes $\\lambda_1=1$. For the four-parameter value model, it is worth inspecting and discussing the constraints on parameters that should jointly ensure the key attributes retrieved from the prospect theory. It has been summarized in \\cite{dhami2016foundations} that any value function $v(x,x_0)$ complying with the prospect theory must satisfy the following fundamental properties:\n\\begin{itemize}\n\\item $v(x,x_0)$ is continuous and strictly increasing with respect to $x$;\n\\item $v(x_0,x_0)=0$ (reference dependence);\n\\item With respect to $x$, $v(x,x_0)$ is concave when $x\\geq x_0$ and is convex when $x<x_0$ (diminishing marginal utility);\n\\item $v(x_0+\\delta,x_0)<-v(x_0-\\delta,x_0)$, $\\forall~\\delta>0$ (loss aversion).\n\\end{itemize}\nThe first two fundamental properties of $v(x,x_0)$ are axiomatic for the four-parameter value model. To investigate the concavity and convexity, it is straightforward to derive the second-order piecewise partial derivative of $v(x,x_0)$ with respect to $x\\geq x_0$ as $\\frac{\\partial^2v(x,x_0)}{\\partial x^2}\\vert_{x\\geq x_0}=\\alpha_1(\\alpha_1-1)\\lambda_1(x-x_0)^{\\alpha_1-2}$ and $x<x_0$ as $\\frac{\\partial^2v(x,x_0)}{\\partial x^2}\\vert_{x< x_0}=-\\alpha_2(\\alpha_2-1)\\lambda_2(x_0-x)^{\\alpha_2-2}$. Therefore, solving $\\frac{\\partial^2v(x,x_0)}{\\partial x^2}\\vert_{x\\geq x_0}<0$ and $\\frac{\\partial^2v(x,x_0)}{\\partial x^2}\\vert_{x< x_0}>0$ yields $0<\\alpha_1<1$ and $0<\\alpha_2<1$, respectively. For the last property stipulating $v(x_0+\\delta,x_0)<-v(x_0-\\delta,x_0)$, $\\forall~\\delta>0$, we can simplify the inequality to $\\frac{\\lambda_2}{\\lambda_1}\\delta^{\\alpha_2-\\alpha_1}>1$, $\\forall~\\delta>0$. By rigorous analysis, it can be proven that the only approach to ensure the validity of the inequality regardless of the value of $\\delta$ is to let $\\alpha_1=\\alpha_2$ and $\\lambda_1<\\lambda_2$, which reduce the four-parameter model constructed in (\\ref{valuefunc}) to a three-parameter model regulating $\\alpha=\\alpha_1=\\alpha_2\\in(0,1)$ and $\\lambda_1<\\lambda_2$. Rigorous and comprehensive discussions and proofs regarding the parameters of value model in the prospect theory can be found in \\cite{al2008note}. \n\nFor illustration purposes, we plot $v(x,x_0)$ with different sets of parameters in Fig. \\ref{valuefuncplot} by referring to the suggested parameter ranges yielded by empirical evidence given in \\cite{dhami2016foundations}. We also plot Fig. \\ref{referenceplot} to illustrate the effects of reference point on the value function. In these figures, we can confirm that the key attributes related to the value function from the prospect theory hold and inspect how these key attributes and the relevant parameters jointly affect the user perception of quantity metrics.\n\n\n\n \\begin{figure*}\n    \n    \\begin{subfigure}[t]{0.3\\textwidth}\n        \\includegraphics[width=2.5in]{perceivedvalue1.eps}\n        \\caption{}\n        \\label{}\n    \\end{subfigure}\n    ~~\n     \n    \\begin{subfigure}[t]{0.3\\textwidth}\n        \\includegraphics[width=2.5in]{perceivedvalue2.eps}\n        \\caption{}\n        \\label{}\n    \\end{subfigure}\n    ~~\n   \n    \\begin{subfigure}[t]{0.3\\textwidth}\n        \\includegraphics[width=2.5in]{perceivedvalue3.eps}\n        \\caption{}\n        \\label{}\n    \\end{subfigure}\n    \\caption{Perceived values of state transitions with different sets of parameters. }\n    \\label{valuefuncplot}\n\\end{figure*}\n \n \n \n \n \n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=3.5in]{referenceplot.eps}\n\\caption{Perceived value vs. quantity metric with different reference points, given $\\alpha=0.5$, $\\lambda_1=1.0$, and $\\lambda_2=2.0$.}\n\\label{referenceplot}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Probability Metric and Probability Weighting Function}\nThe probability metric in the context of telecommunications denotes the measurement of opportunistic performance, which encompasses outage probability, error probability, collision probability, handover probability, and etc. An objective probability will be distorted when being perceived by end users, corresponding to the last key attribute of the prospect theory: non-linear probability perception. Introducing the Prelec probability weighting function \\cite{prelec1998probability}, we model the perception of a probability metric $0\\leq p\\leq 1$ under psychological distortion as\n\\begin{equation}\\label{vpxeq}\nw(p)=\\mathrm{exp}(-\\gamma(-\\log(p))^\\theta),\n\\end{equation}\nwhere $\\gamma>0$ and $0<\\theta<1$ are user specific parameters used to characterize the subjective perception of probability metric. The ranges $\\gamma$ and $\\theta$ are derived by the necessary conditions for maintaining the basic properties of probability weighting function, especially its inverse S-shape. Detailed discussions on the functionality and property of $\\gamma$ and $\\theta$ can be found in \\cite{dhami2016foundations}. If $\\gamma=1$ and $\\theta=1$, we have $w(p)=p$, $\\forall~p\\in[0,1]$, and thus the formulated analytical framework for probability perception reduces to the classic QoS analytical framework. \n\nIn a general case with a continuous random variable $X$, the probability argument in (\\ref{vpxeq}) could be any cumulative distribution function (CDF) $F_X(s)=\\mathbb{P}\\{X\\leq s\\}$, where $\\mathbb{P}\\{\\cdot\\}$ denotes the objective probability of the random event enclosed. Following the definition of CDF and the attribute of probability distortion, we gives the perceptual CDF (PCDF) by\n\\begin{equation}\n\\tilde{F}_X(s)=w(F_X(s)).\n\\end{equation}\nDenoting $f_X(s)=\\frac{\\mathrm{d}F(s)}{\\mathrm{d}s}$ as the probability density function (PDF) of $X$, we can define the perceptual PDF (PPDF) as \n\\begin{equation}\n\\begin{split}\n\\tilde{f}_X(s)&=\\frac{\\mathrm{d}\\tilde{F}_X(s)}{\\mathrm{d}s}=\\frac{\\mathrm{d}w(F_X(s))}{\\mathrm{d}s}\\\\\n&=\\gamma\\theta w(F_X(s))(-\\log(F_X(s)))^{\\theta-1}f_X(s)\/F_X(s).\n\\end{split}\n\\end{equation}\n\n\n\nFor illustration purposes, we plot $w(p)$, $\\tilde{F}_X(s)$, and $\\tilde{f}_X(s)$ in Fig. \\ref{probfuncplot} by referring to the suggested parameter ranges yielded by empirical evidence given in \\cite{dhami2016foundations}, ditto. In this figure, we can testify that the attribute of probability distortion from the prospect theory hold and inspect how this attribute and the parameters $\\gamma$ and $\\theta$ jointly affect the user perception of probability metrics. \n\n\\begin{figure*}\n    \n    \\begin{subfigure}[t]{0.3\\textwidth}\n        \\includegraphics[width=2.5in]{perceivedprobability.eps}\n        \\caption{}\n        \\label{}\n    \\end{subfigure}\n    ~~\n     \n    \\begin{subfigure}[t]{0.3\\textwidth}\n        \\includegraphics[width=2.5in]{perceivedcdf.eps}\n        \\caption{}\n        \\label{}\n    \\end{subfigure}\n    ~~\n   \n    \\begin{subfigure}[t]{0.3\\textwidth}\n        \\includegraphics[width=2.5in]{perceivedpdf.eps}\n        \\caption{}\n        \\label{}\n    \\end{subfigure}\n    \\caption{Perceived probability, PCDF, and PPDF with different sets of parameters. Here, we take the normalized exponential distribution as an example to illustrate PCDF and PPDF, i.e., $F_X(s)=1-\\mathrm{exp}(-s)$ and $f_X(s)=\\mathrm{exp}(-s)$ for $s\\geq0$.}\n    \\label{probfuncplot}\n\\end{figure*}\n\n\n\\subsection{Summary of the Proposed Analytical Framework}\nHaving obtained $v(x,x_0)$, $w(p)$, $\\tilde{F}_X(s)$, and $\\tilde{f}_X(s)$, we can reevaluate a set of advanced and composite performance metrics for communication systems incorporating non-technological factors. For example, given a composite metric $\\Omega(g)$ that is a function of a random variable $g$, and the PDF of $g$ is denoted as $f_G(g)$, we can define the perceptual utility (PU) of the composite metric by $\\tilde{\\Omega}=\\int_{0}^{\\infty}v(\\Omega(g),\\Omega_0)\\tilde{f}_G(g)\\mathrm{d}g$, where $\\Omega_0$ is the reference point of the composite metric. $\\tilde{\\Omega}$ can be employed to appraise the subjective performance pertaining to the composite variable. Different from objective performance evaluation metrics, the PU based on the prospect theory is allowed to be negative, which implies a negative user impression\/perception of the objective performance provided. Fig. \\ref{implementationsys} depicts the complete implementation process of the prospect theoretic analytical framework for perceptual performance analysis. Specific case studies for applying the analytical framework are given in the next section. \n\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=3.5in]{implementationsys.png}\n\\caption{Implementation process of the prospect theoretic analytical framework for perceptual performance analysis.}\n\\label{implementationsys}\n\\end{figure}\n\n\n\n\n\\section{Case Studies}\\label{cs}\nConsider a simplistic point-to-point (P2P) wireless communication system consisting of a transmitter, a receiver, and a time-variant fading channel obeying the Rayleigh distribution, in which received signals are attenuated by multi-path fading and contaminated by a complex additive white Gaussian noise (AWGN). However, different from the classic communication system model, we add an \\textit{observer} and a \\textit{perceptor} after the receiving module, as shown in Fig. \\ref{sys}\\footnote{Note that, the proposed communication system model with the perceptor is different from the user-in-the-loop (UIL) model given in \\cite{6736762}, which relies on incentives (e.g., dynamic pricing) to change user behaviors and responses.}. The observer is responsible for evaluating received signals on an objective basis, so that the QoS metrics can be retrieved, whereas the perceptor is employed to assess the QoS metrics on a subjective basis to obtain the perceptual metrics.\n\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[width=5.5in]{sys.png}\n\\caption{Communication system model considered in this paper.}\n\\label{sys}\n\\end{figure*}\n\n\n\nIn such a simplistic P2P wireless communication system, the equivalent baseband input-output relation in the time domain can be written as \\cite{proakis2008digital}\n\\begin{equation}\\label{redasdas}\n\\begin{split}\n&R(t)\\\\\n&=\\sqrt{P_t}\\sum_{k=1}^{K}\\left(A_k(t)\\mathrm{exp}(-j2\\pi f_c \\tau_k(t))T(t-\\tau_k(t))\\right)+N(t),\n\\end{split}\n\\end{equation}\nwhere $T(t)$ and $R(t)$ represent the baseband transmitted and received signals at time $t$, respectively; $K$ is the number of propagation paths yielded by direct propagation, reflection, refraction, and scattering; $A_k(t)$ and $\\tau_k(t)$ characterize the attenuation and delay of the $k$th propagation path; $f_c$ is the central carrier frequency of the transmitted signal; $P_t$ is the transmit power; $N(t)$ represents the AWGN sample at the receiver with an average noise power $N_0$. In telecommunications, $\\tau_k(t)$ is generally assumed to be an exponentially distributed random variable and $f_c\\gg 1$. As a result, $\\theta_k(t)=2\\pi f_c\\tau_k(t)$ is approximated to be a uniform distributed random variable between 0 to $2\\pi$ rad. When the transmission is within a signaling interval, we can write the channel coefficient as\n\\begin{equation}\nH(t)=\\sum_{k=1}^{K}A_k(t)\\mathrm{exp}(-j\\theta_k(t))),\n\\end{equation}\nwhere $H(t)$ is a zero-mean complex Gaussian distributed random variable according to the central limit theorem when $K\\rightarrow\\infty$. For simplicity, we can assume that $G(t)=|H(t)|^2$ is exponentially distributed with CDF $F_G(g)=1-\\mathrm{exp}(-g\/\\mu)$ and PDF $f_G(g)=\\mathrm{exp}(-g\/\\mu)\/\\mu$ for $g\\geq 0$, where $\\mu$ is the average channel power gain \\cite{8344837}.\n\n\nBased on this simplistic model, we analyze the perceptual metrics corresponding to three fundamental but important QoS metrics: signal-to-noise ratio (SNR), transmission rate, and outage probability as follows, through the prospect theoretic analytical framework built in the last section.\n\n\n\\subsection{PU of Signal-to-Noise Ratio}\nFirst of all, the instantaneous SNR can be expressed as\n\\begin{equation}\\label{instantsnrexp}\n\\Gamma(t)={P_tG(t)}\/{N_0}.\n\\end{equation}\nTo analyze the average perception of continuous outcomes instead of discrete values, researchers in \\cite{rieger2006cumulative} generalize the original formulation of utility given in \\cite{tversky1992advances} and provide a new utility metric termed the \\textit{subjective utility}. Tailoring the subjective utility proposed by \\cite{rieger2006cumulative}, we define the PU of SNR in the context of the prospect theory as follows:\n\\begin{definition}\n\\textit{The prospect theoretic PU of SNR is the average perceived value of SNR from the user's perspective, complying with the human psychology of non-linear quantity and probability perception.}\n\\end{definition}\nWith the proposed analytical framework and the definition given above, the prospect theoretic PU of SNR can be written as \n\\begin{equation}\\label{snreq}\n\\begin{split}\n\\widetilde{\\Gamma}&=\\int_{0}^{\\infty} v(\\Gamma(t),\\Gamma_0)\\tilde{f}_G(G(t))\\mathrm{d}G(t)\\\\\n&=-\\lambda_2\\int_{0}^{\\frac{N_0\\Gamma_0}{P_t}} \\left(\\Gamma_0-\\Gamma(t)\\right)^{\\alpha}\\tilde{f}_G(G(t))\\mathrm{d}G(t)\\\\\n&~~~~+\\lambda_1\\int_{\\frac{N_0\\Gamma_0}{P_t}}^{\\infty} \\left(\\Gamma(t)-\\Gamma_0\\right)^{\\alpha}\\tilde{f}_G(G(t))\\mathrm{d}G(t),\n\\end{split}\n\\end{equation}\nwhere $\\Gamma_0$ is a predetermined reference point of SNR. The PU of SNR can be used to characterize the QoE\/UE regarding the reliability of communication networks. For illustration purposes, we plot the PU of SNR versus $P_t\/N_0$ with different sets of parameters for the normalized channel configuration (i.e., $\\mu=1$) in Fig. \\ref{pusnr}.\n\n\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=3.5in]{PUSNR.eps}\n\\caption{PU of SNR with different sets of parameters, given $\\mu=1$.}\n\\label{pusnr}\n\\end{figure}\n\n\n\n\n\\subsection{PU of Transmission Rate}\nBased on the formulation of the instantaneous SNR given in (\\ref{instantsnrexp}), the instantaneous transmission rate for the normalized bandwidth is given by \\cite{7809043}\n\\begin{equation}\n\\Psi(t)=\\log_2(1+\\Gamma(t)).\n\\end{equation}\nSimilarly, according to the formulation of the subjective utility, we can define the PU of transmission rate in the context of the prospect theory infra:\n\\begin{definition}\n\\textit{The prospect theoretic PU of transmission rate is the average perceived value of transmission rate from the user's perspective, complying with the human psychology of non-linear quantity and probability perception.}\n\\end{definition}\nThe prospect theoretic PU of transmission rate can be explicitly expressed as\n\\begin{equation}\\label{rateeq}\n\\begin{split}\n\\widetilde{\\Psi}&=\\int_{0}^{\\infty} v(\\Psi(t),\\Psi_0)\\tilde{f}_G(G(t))\\mathrm{d}G(t)\\\\\n&=-\\lambda_2\\int_{0}^{\\frac{N_0\\Psi_0}{P_t}} \\left(\\Psi_0-\\Psi(t)\\right)^{\\alpha}\\tilde{f}_G(G(t))\\mathrm{d}G(t)\\\\\n&~~~~+\\lambda_1\\int_{\\frac{N_0\\Psi_0}{P_t}}^{\\infty} \\left(\\Psi(t)-\\Psi_0\\right)^{\\alpha}\\tilde{f}_G(G(t))\\mathrm{d}G(t),\n\\end{split}\n\\end{equation}\nwhere $\\Psi_0$ is a predetermined reference point of transmission rate. The PU of transmission rate can be used to characterize the QoE\/UE pertaining to the efficiency of communication networks. We plot the PU of transmission rate versus $P_t\/N_0$ with different sets of parameters for normalized channel configuration in Fig. \\ref{putr}. \n\n\n\n\n\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=3.5in]{PUTR.eps}\n\\caption{PU of transmission rate with different sets of parameters, given $\\mu=1$.}\n\\label{putr}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Perceptual Outage Probability}\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=3.5in]{POP.eps}\n\\caption{POP with different sets of parameters, given $\\mu=1$ and $\\epsilon=1$.}\n\\label{pop}\n\\end{figure}\n\n\n\nFollowing the formulation of the instantaneous transmission rate, we model the outage probability infra \\cite{8703169}:\n\\begin{equation}\n\\begin{split}\nP_{\\mathrm{out}}(\\epsilon)&=\\mathbb{P}\\left\\lbrace\\Psi(t)<\\epsilon\\right\\rbrace\\\\\n&=\\mathbb{P}\\left\\lbrace \\Gamma(t)<2^{\\epsilon}-1\\right\\rbrace\\\\\n&=\\mathbb{P}\\left\\lbrace G(t)<N_0(2^{\\epsilon}-1)\/P_t\\right\\rbrace\\\\\n&=F_G(N_0(2^{\\epsilon}-1)\/P_t),\n\\end{split}\n\\end{equation}\nwhere $\\epsilon$ is a predefined outage threshold related to communication system configurations. We can define the perceptual outage probability (POP) in the context of the prospect theory as\n\\begin{definition}\n\\textit{The POP is the perceived probability from the user's perspective, complying with the human psychology of non-linear probability weighting.}\n\\end{definition}\nTherefore, we employ the Prelec probability weighting function to quantify the prospect theoretic POP as follows:\n\\begin{equation}\n\\begin{split}\n\\tilde{P}_{\\mathrm{out}}(\\epsilon)&=w(P_{\\mathrm{out}}(\\epsilon))=w(F_G(N_0(2^{\\epsilon}-1)\/P_t))\\\\\n&=\\tilde{F}_G(N_0(2^{\\epsilon}-1)\/P_t).\n\\end{split}\n\\end{equation}\nThe POP captures the distorted perception of outage probability and can be utilized to appraise the QoE\/UE of outage. Normalizing parameters $\\mu=1$ and $\\epsilon=1$, we plot the POP versus $P_t\/N_0$ with different sets of parameters in Fig. \\ref{pop}. \n\n\\subsection{Discussion}\nAccording to the plotted data in Fig. \\ref{pusnr}, Fig. \\ref{putr}, and Fig. \\ref{pop}, we can observe that the proposed perceptual metrics exhibit favorable mathematical properties, e.g., continuity, differentiability, and convexity\/concavity. They also perform well in both of the low and high SNR regions, which are aligned with the prediction given by the prospect theory. Moreover, compared to other widely adopted value models, e.g., the logarithmic, sigmoid, and exponential value models \\cite{reichl2013logarithmic,6891176,5430142}, the proposed value model relying on the power function is more computationally efficient. Fig. \\ref{pusnr} and Fig. \\ref{putr} have well captured the human psychology of reference dependence, diminishing sensitivity, and loss aversion; Fig. \\ref{pop} demonstrates the phenomenon of subjective probability distortion. \n\nOverall, the plotted data is consistent with the model's economic implications. A smaller $\\gamma$ denotes a wider range of over-weighted possibilities under exponential distribution, which leads to greater probability distortion and a lower prospect theoretic PU. For $0<\\theta<1$, a smaller $\\theta$ is associated with sharper curvature of the inverse S-shape and greater probability distortion. A higher reference point indicates `higher expectation' and leads to a lower or even negative QoE. A smaller $\\alpha$ indicates a higher degree of diminishing utility and a lower overall utility. A larger $\\lambda_2 \/ \\lambda_1$ ratio indicates a higher degree of loss aversion and, thereby, a lower QoE. In summary, UE is significantly and negatively related to the psychological distortion in quantity and probability perception, consistent with the economic intuition that irrationality causes non-optimal outcomes. \n\nAs an intriguing finding, Fig. \\ref{pusnr} and Fig. \\ref{putr} reveal that prospect theoretic users exhibit abnormally high degrees of loss aversion, reference dependence, and diminishing marginal utility in choosing communication services. Specifically, users with reasonable expectations (reference values) suffer a PU of -4 when $P_t\/N_0$ approaches zero, whereas they enjoy an increase in PU of merely 0.3 when $P_t\/N_0$ increases from 400 to 1000. This evidence is in line with the widespread scepticism that users may be insensitive to the exponential increase in communication performance \\cite{kilkki2008quality}. It also lends credits to the core development strategy of 5G\/6G networks that communications should focus more on user experience and stability improvement rather than a blind pursuit of performance metrics. \n\nThe above analysis, though preliminary and tentative, has illustrated the importance of modeling users' psychological foundations in both QoE measurements and human-centric communications.\n\n\n\\section{Challenges and Promising Research Directions}\\label{fird}\nIn this study, we introduce the prospect theoretic analytical framework for modeling UE in communication systems. Although preliminary and immature, the proposed framework  serves as a promising architecture for developing interdisciplinary applications of human-centric communications. In this section, we outline the challenges, potentials, and application aspects for future research on the prospect theoretic framework. \n\n\\subsection{Extend Experimental Evidence to Non-Monetary Contexts}\nHistorically, the prospect theory has been developed based on monetary experiments (e.g., lottery-choice experiments). On the other hand, interdisciplinary applications of the prospect theory are becoming increasingly popular and have hitherto resulted in over a thousand research articles. In these applications, the analytical framework is based on the parameter estimates elicited by monetary experiments, for which an implicit assumption is that users must exhibit similar psychological features such as loss aversion in non-monetary contexts (e.g., the choice of communication services). However, this critical assumption may not hold. For instance, it is widely recognized that individuals have different quantity and probability preferences for monetary and non-monetary stimuli. As suggested by \\cite{kilkki2008quality}, individuals seem less sensitive to QoS parameters (bit rates and delays) than monetary incentives, which indicates a higher degree of utility curvature. In the current stage, the practical significance of the QoE measurements remains unclear due to the lack of concrete empirical support \\cite{saad2016toward}. There is an urgent need to extend the experimental evidence to non-monetary contexts, which can provide a solid foundation and an explicit guideline for future interdisciplinary applications of the prospect theory.  \n\n\\subsection{Upgrade the Proposed Framework Using Advanced Behavioral Theory}\nIn the past decade, the prospect theory has been successfully applied in diverse research topics that involve quantity and probability perception, which verifies the theory's potential in interdisciplinary applications. The prospect theory is now evolving at a rapid pace; the advances such as the third- and fourth-generation prospect theory are readily applicable and begin to attract researchers' attention. Beyond the prospect theory, there is a bigger picture for interdisciplinary applications of behavioral economics. For instance, behavioral game theory has the potential to reshape the research landscape for game theoretic communications (e.g., radio resource management). Behavioral time discounting has enriched our understanding of time preferences and has profound implications for the intertemporal allocation of network resources. Psychological foundations such as endowment effect, other-regarding preference, and judgment heuristics have promising application prospects in pricing strategy and product design in telecommunication industries. Overall, the paradigm shift from technology-centric to human-centric communications still faces tremendous challenges because communication science has long been preoccupied with classic QoS benchmarks. Human-centric communications remains a prospective uncultivated research area in the 5G era and the forthcoming 6G era.\n\n\\subsection{Develop Mathematical Tools}\nUnder the prospect theoretic framework, quantitative analysis of perpetual performance involves sophisticated mathematical manipulations (differentiation, integration, and power series expansion) on the value function $v(x,x_0)$ and the probability weighting function $w(p)$. However, the probability weighting function is generally employed within socioeconomics and lacks the application foundation in telecommunications. Due to the limited understanding of the mathematical properties of probability weighting function (especially the Prelec function), researchers always encounter challenges in solving the integral forms and find it difficult to further process the quantitative analysis \\cite{rieger2006cumulative}. A complete application of the prospect theory in communication science requires further exploration of the mathematical properties and analytical tools for the proposed functional forms.\n\n\n\n\n\n\\subsection{Analyze the Information Theoretic Properties}\nThe information theory constructed by Claude E. Shannon in `A Mathematical Theory of Communication' has been widely acknowledged as a landmark in telecommunications and the mathematical foundation of modern communication systems \\cite{shannon1948mathematical}. Developed from the information theoretic model, most communication system models consist of transmitter, channel, and receiver, in which objective noise, interference, impairments, and imperfection are taken into consideration \\cite{8750780}. For developing human-centric communications, communication system models should incorporate the fundamental mechanism of human perception after the receiving module. We tentatively term this embodiment the \\textit{perceptor} under the prospect theoretic framework. The introduction of \\textit{perceptor} into communication systems can alter the formulation of information entropy and generate intriguing information theoretic properties. \n\n\n\\subsection{Analyze the Perceptual Performance of Advanced Applications}\nIn this paper, the prospect theoretic analytical framework was employed in several simplistic application scenarios for demonstration purposes. However, due to its generality and versatility, the proposed framework can be readily extended to advanced communication applications in beyond 5G networks. Promising application aspects include massive multiple-input and multiple-output (MIMO) systems, terahertz (THz) communication systems, visible light communication (VLC) systems, multi-user communication systems, cooperative relay aided communication systems, hybrid licensed\/unlicensed communication systems, and etc. Overall, interdisciplinary collaborations between telecommunications, economics, and psychology is indispensable for the development of human-centric communications. \n\n\n\n\n\\section{Conclusion}\\label{c}\nIn this paper, we proposed the prospect theoretic analytical framework for human-centric communications. This analytical framework takes non-technological factors and user psychology  into consideration when modeling and analyzing communication systems, which enables quantitative analysis of perceptual performance. We provided several applications in telecommunications to demonstrate how the proposed analytical framework can be employed to conduct performance analysis in a human-centric manner. We also outlined the unresolved problems and promising directions for future research. Overall, this paper provides a guideline for improving communication services and introduces a common platform that unifies the nomenclature and endeavor from different disciplines, including telecommunications, economics, and psychology. Meanwhile, we also aim to use this paper as a fuze to trigger a new interdisciplinary research area for further investigation.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\\section[]{}\n\n\\raggedright\n\\end{document}\n\n\\section{Introduction}\n\nThis paper, one of a set associated with the 2013 data release from the\n\\textit{Planck}\n\\footnote{\\textit{Planck}\\ (\\url{http:\/\/www.esa.int\/Planck}) is a project of the\nEuropean Space Agency (ESA) with instruments provided by two scientific\nconsortia funded by ESA member states (in particular the lead countries France\nand Italy), with contributions from NASA (USA) and telescope reflectors\nprovided by a collaboration between ESA and a scientific consortium led and\nfunded by Denmark.}\nmission \\citep{planck2013-p01}, presents the first results on the\nintegrated Sachs-Wolfe (ISW) effect using \\textit{Planck}\\ data. \nThe ISW effect \\citep{Sachs1967, Rees1968, Martinez1990b, Hu1994} is a secondary anisotropy\nin the cosmic microwave background (CMB), caused by the interaction of CMB\nphotons with the time-evolving potentials from large-scale structure\n(LSS, hereafter). Photons follow a geodesic that is weakly perturbed by the\nNewtonian gravitational potential, $\\Phi$, and experience a fractional shift\nin their temperature given by\n\\begin{equation}\n\\Theta = \\frac{\\Delta T}{T_\\mathrm{CMB}} =\n \\frac{2}{c^3}\\int_{\\eta_\\ast}^{\\eta_0}\\mathrm{d}\\eta\\:\n \\frac{\\partial\\Phi}{\\partial\\eta},\n\\end{equation}\nwhere the integral is expressed in terms of the conformal time $\\eta$,\ndefined differentially by $\\mathrm{d}\\eta\/\\mathrm{d}a = 1\/(a^2H(a))$ with\n$H(a)$ the Hubble function and $a$ the scale factor. \nThe integration limits here go from the recombination time ($\\eta_\\ast$)\nto the present time ($\\eta_0$).\n\nThe sensitivity of the ISW effect to gravitational potentials\n(that can extend over Gpc scales) \nresults in the power of the ISW being concentrated on the largest scales. \nThe largest scales for the CMB have been mapped out by the {\\it Wilkinson\nMicrowave Anisotropy Probe\\\/} (\\textit{WMAP}) to the statistical limit of cosmic\nvariance.  Some systematics (like foreground removal) can have an impact on\nthe reconstruction of the CMB especially at the largest scales where our\nGalaxy can introduce significant residuals on the reconstructed CMB map. The \nsuperior sensitivity of \\textit{Planck}\\ together with its better angular resolution\nand wider frequency coverage allows\n for a better understanding\n(and hence removal) of Galactic and extagalactic foregrounds, therefore \nreducing the possible negative impact of these residuals. \\textit{Planck}\\ allows us to\nimprove on previous measurements by having a better systematic control,\nan improved removal of foregrounds (that permits us to explore the\nachromatic nature of the ISW signal on a wider frequency range), and a better \nunderstanding of systematics affecting tracer catalogues.\n\nFor cosmological models where $\\Omega_{\\rm m}=1$, gravitational potentials\nremain constant during linear structure formation, and the ISW signal is\nnegligible (to first order, although second order nonlinear \nISW is always expected around smaller over- and under-dense regions). \nIn the presence of dark energy, decaying potentials due to the accelerated\nexpansion rate, result in a net ISW effect which is positive when the CMB\nphotons cross overdense regions \nand negative when the CMB photons cross underdense regions.  \nTherefore, the ISW effect is an indicator of either non-zero curvature\n\\citep{Kamionkowski1994}\\footnote{It is worth mentioning that this was one of the arguments that\nsuggested a preference for $\\Lambda$CDM before the discovery of acceleration using SNe.}, any form of dark energy, such as a\ncosmological constant $\\Lambda$ \\citep{Crittenden1996}, modified gravity\n\\citep{Hu2002a}, or a combination of these possibilities. \nBy measuring the rate at which gravitational potentials\nin the LSS decay (up to redshift of around 2), the ISW effect\ncan be used as an independent probe of cosmology and provides\ncomplementary and independent evidence for dark energy.\n\n\nDetection of the ISW effect was first made possible with all-sky CMB maps\nfrom \\textit{WMAP}.  Based on these data, many works can be found in the\nliterature where the authors aim at making, \nand subsequectly improving, the measurement of the ISW effect through \ncorrelations with tracer catalogues: 2MASS \\citep[an infrared\n  catalogue out to low redshifts around 0.1,][]{Afshordi2004a, Rassat2006, \n  Francis2009a, Dupe2011}, \\textit{HEAO} \\citep[an X-ray survey at low\n  redshift, with the first positive claim for\n  detection,][]{Boughn2004}, Sloan Digital Sky Survey\n  \\citep[SDSS, an optical survey at intermediate\n  redshifts,][]{Fosalba2003, Scranton2003, Fosalba2004, Padmanabhan2005, Cabre2006, \n  Giannantonio2006b, Granett2009a, Xia2009, Bielby2010, Lopez2010,\n  Sawangwit2010},\n  the NRAO VLA Sky Survey\n\\citep[NVSS, a radio catalogue with high-redshift sources,]\n[]{Boughn2005b,Vielva2006,Pietrobon2006a,Mcewen2007,Raccanelli2008,Hernandez2010,Massardi2010,Schiavon2012},\nand combined measurements with multiple tracers \n\\citep{Nolta2004,Ho2008,Corasaniti2005,Gaztanaga2006,Giannantonio2008b,Giannantonio2012}. \nThe significance of the ISW detections that can be found in the literature\nrange between $0.9\\,\\sigma$ and $4.7\\,\\sigma$. There are a number of\npeculiarities related to some of the detection claims, \nas noted by \\citet{Hernandez2010} and \\citet{Lopez2010}. \nThey both found lower significance levels than some previous studies\nand pointed out the absence of the signal at low multipoles \nwhere the ISW effect should be most prominent and the presence of point source emission on small scales for radio surveys.\n\nThe main result that is obtained from an ISW detection is a constraint\non the cosmological constant (or dark energy), $\\Omega_\\Lambda$. \nThe general consensus from the variety of ISW analyses is for a\nvalue of $\\Omega_\\Lambda\\simeq0.75$ with an error of about 20\\%,\nwhich provides independent evidence for the existence of dark energy \n\\citep{Fosalba2003, Fosalba2004, Nolta2004,\n  Corasaniti2005, Padmanabhan2005, Cabre2006, Giannantonio2006b,\n  Pietrobon2006b, Rassat2006, Vielva2006, Mcewen2007, Ho2008, Schiavon2012}. All tests\non spatial flatness find an upper limit for $\\Omega_K$ of a few\npercent \\citep{Nolta2004,Gaztanaga2006,Ho2008,Li2010}. Using a prior\non spatial flatness, the dark energy\nequation of state parameter, $w$, was found to be close to $-1$\n\\citep{Giannantonio2006b,Vielva2006, Ho2008} and has been excluded from\nhaving a strong time evolution \\citep{Giannantonio2008b, Li2010}. \n\nThe ISW effect is achromatic, conserving the Planck spectrum of the\nCMB and can be separated from other CMB fluctuations through\ncross-correlations with catalogues which trace the LSS gravitational\npotentials \\citep[see for instance][]{Crittenden1996}. This cross-correlation\ncan be studied in different ways: angular cross-correlations in real\nspace between the CMB and the catalogues tracing the LSS;\nthe corresponding angular cross-power spectrum of the Fourier-transformed maps;\nor through the covariance of wavelet-filtered maps as a function of wavelet\nscale.  The studies using \\textit{WMAP}\\ data mentioned above follow this survey\ncross-correlation techique.\n\nAn alternative approach, similar to the angular cross-correlation in real\nspace, consists of stacking CMB fields centred on known supersclusters or \nsupervoids \\citep{Granett2008a, Granett2008b, Papai2010b}. The advantage of\nthis technique is that it allows for a detailed study of the profile of the\nCMB fluctuations caused by this secondary anisotropy.\n \nA novel and powerful approach takes advantage of the fact that the\nsame potentials that make CMB photons gain or lose energy along their\npath (ISW), create lensing distortions that can be measured from the \nCMB map directly~\\citep[e.g.,][]{Hu2002b}.\nThe interplay between weak gravitational lensing and the ISW effect\ncauses a non-Gaussian contribution to the CMB, which can be measured\nthrough the lensing-induced bispectrum between small and large angular scales.\nThe measurement of the lensing potential requires a large number of\nmodes that could not be measured before the arrival of \\textit{Planck}\\ data.\n\nThis paper presents new measurements of the ISW effect carried out with\n\\textit{Planck}.  Even although our detections are not in every case as\nstrong as some previously claimed significance levels,\nwe believe that our results are an improvement over earlier studies.\nThis is because we can use the additional power enabled by the frequency\ncoverage and sensitivity of \\textit{Planck}.  To establish this we carry out a\ncomprehensive study of all the main approaches which have previously been\ntaken to estimate the ISW signal.  We also present new results in\nrelation to the non-Gaussian structure induced by the ISW effect.\n\nThe paper is organized as follows: In Sect.~\\ref{sec:data} we describe the\ndata used in this work (both for the CMB and large-scale structure).\nThe first ever results on the estimation of the lensing-induced bispectrum\nare presented in Sect.~\\ref{subsec:iswl}. \nCross-correlations with external surveys are investigated in\nSect.~\\ref{subsec:xcor}, and in Sect.~\\ref{subsec:stack} we present the\nresults for the stacking analysis on the temperature maps,\nas well as aperture photometry on super-clusters and super-voids.  \nThe recovery of the ISW all-sky map is described in Sect.~\\ref{subsec:recov}.\nFinally, we discuss our main results and their cosmological implications \nin Sect.~\\ref{sec:discussion}.\n\n\n\\section{Data description}\\label{sec:data}\n\nIn this Section we describe the different data sets used in this paper. This\nincludes \\textit{Planck}\\ data\n\\citep[the CMB temperature and lensing potential maps, see][]{planck2013-p01,\nplanck2013-p02,planck2013-p03,planck2013-p06,planck2013-p12}\nwe refer to the corresponding \\textit{Planck}\\ papers for details) and external\ndata sets (large-scale structure tracers) used in the ISW determination:\nthe radio NVSS catalogue; optical luminous galaxies (CMASS\/LOWZ) and the\nmain galaxy sample from the Sloan Digital Sky Survey (SDSS); as well\nas several superstructure catalogues.\n\n\\input{.\/section_2.0\/section_2.1}\n\n\\input{.\/section_2.0\/section_2.2}\n\n\n\n\\subsubsection{CMB maps}\n\\label{subsubsec:cmbmaps}\n\nFor the present work we have made use of the \\textit{Planck}\\ \nforeground-cleaned CMB maps provided by the\ndata processing centres~\\citep[as described in the \\textit{Planck}\\ component\nseparation paper][]{planck2013-p06}. In particular, to test robustness, some\nof the results are presented for different cleaned CMB maps,\nwhich were constructed using four different component separation techniques:\n{\\tt Commander-Ruler}\\ ({\\tt C-R}, which uses physical parametrization);\n{\\tt NILC}\\ (an internal linear combination technique);\n{\\tt SEVEM}\\ (a template fitting method);\nand {\\tt SMICA}\\ (which uses spectral matching). Since the contribution of the ISW\nsignal is only important on large scales, low resolution\nmaps, with {\\tt HEALPix} \\cite{Gorski2005} parameter $N_\\mathrm{side} = 64$,\nand pixel size of about 55\\,arcmin, have been\nused for most of the analyses. One exception is the study of the\ncorrelation between the ISW and lensing signals, which requires the\nuse of full-resolution maps ($N_\\mathrm{side}$=2048,\npixel size of $1.7$\\,arcmin). The maps are\ndegraded directly from the original full resolution down to the\ncorresponding $N_\\mathrm{side}$.\n\nIn addition, foreground-cleaned maps per frequency (from 44 to 353\\,GHz) at\nresolution $N_\\mathrm{side}=512$ were used for the stacking\nanalysis presented in Sect.~\\ref{subsec:stack}. These maps were\nconstructed by subtracting a linear combination of internal\ntemplates using {\\tt SEVEM}\\ \\citep[see the {\\tt SEVEM}\\ Appendix of][for a detailed description of\nthe method]{planck2013-p06}. \nAs an example the {\\tt SEVEM}\\ CMB map is shown in Fig.~\\ref{fig:cmb_lensing_maps} (left panel).\n\nFinally, to minimize the presence of foreground contamination in the\nmaps, we have used the official mask described in~\\cite{planck2013-p06},\nwhich excludes regions with larger Galactic and point-source\ncontamination (the U73 mask). This mask is given at the full \\textit{Planck}\\ resolution and\nis downgraded to the required levels. The downgrading\nprocedure consists of the following steps:\nthe mask (originally a map with zero and one values) is convolved with a Gaussian beam of\nFWHM three times the characteristic pixel size of the final\n$N_\\mathrm{side}$ resolution; this convolved map is then degraded to\nthe required $N_\\mathrm{side}$, and, afterwards, a threshold of 0.75 is\nimposed (i.e., pixels with a value above this threshold are set to one, whereas\nthe rest are set to zero).\n\n\n\\subsubsection{Lensing potential map}\n\\label{subsub:lens}\nWeak gravitational lensing distorts the CMB temperature anisotropy pattern.\nThis effect is sensitive to the projected matter distribution in the\nlarge-scale structure at high redshifts, where structure growth is linear and\nthe statistics close to Gaussian. Weak lensing causes correlations between\ndifferent multipoles which are proportional to the lensing deflection field.\nThese correlations can be exploited for reconstructing the density field and\nfor measuring its statistical properties \\citep{Hu2002b, Okamoto2003}. The\nlensing effect in the CMB can be estimated by this homogeneity breaking, and\nin this way individual modes of the lensing potential at multipoles\n$\\ell<100$ can be reconstructed with a significance of around $0.5\\,\\sigma$,\nshowing the necessity of a statistical treatment. Nevertheless the overall\neffect of the lensing is measured to better than $25\\,\\sigma$\n\\citep{planck2013-p12}.\nThe additional lensing effect in the temperature power spectrum is\ndetectable with a significance of about $10\\,\\sigma$~\\citep{planck2013-p08}.\n\nWith \\textit{Planck}\\ data, we aim at detecting a correlation between the ISW effect\nand the lensing potential, where the latter is a tracer of the large-scale\nstructure at high redshift. This correlation is restricted to $9\\,\\sigma$,\neven in the ideal case, limited by cosmic variance and the smallness of the\nISW effect in comparison to the primary CMB~\\citep{Lewis2011}.\nThe data products used in this study are the \\textit{Planck}\\ lensing potential\nreconstruction, and specific lensing maps obtained from the component\nseparation pipelines. The lensing potential is available as part of the first\n\\textit{Planck}\\ data release. Its detailed development is described in the \\textit{Planck}\\\nlensing paper~\\citep{planck2013-p12}. In Fig.~\\ref{fig:cmb_lensing_maps} we\nreproduce (right panel) an optimally filtered version of the \\textit{Planck}\\ lensing\nmap, suitable for the ISW-lensing cross-correlation.\n\nIn addition to a direct correlation between the CMB sky and the reconstructed\nlensing map, we measure the bispectrum generated by weak lensing by applying\na range of estimators: the KSW-bispectrum estimator; bispectra binned in\nmultipole intervals; and a modal decomposition of the bispectrum.\nThis meassurement is made possible for the first time thanks to the \\textit{Planck}\\\ndata. In addition, we will use information from the lensing field as a tracer\nfor an ISW map reconstruction at high redshift (see Sect.~\\ref{subsec:recov}).\n\n\\subsection{\\textit{Planck}\\ data}\n\n\\textit{Planck}\\ data and products used in this paper are described in the following\nsections, in particular the foreground-cleaned CMB maps produced by the\n\\textit{Planck}\\ component separation pipelines, and related products, such as\ndedicated component-separated frequency maps \\citep{planck2013-p06}, as well\nas the \\textit{Planck}\\ lensing map~\\citep{planck2013-p12}.\n\n\\input{.\/section_2.0\/section_2.1.1}\n\n\\input{.\/section_2.0\/section_2.1.2}\n\n\\subsubsection{NVSS radio catalogues}\n\\label{subsub:nvss}\n\nLuminous active galactic nuclei (hereafter AGN) are known to be powerful\nradio sources, visible out to high redshifts. These sources are hence able to\nprobe the cosmic density field during the entire\nredshift range from matter domination to accelerated expansion due to dark\nenergy. If AGN are fair tracers of the underlying density field, these\nsources should likewise probe the spatial distribution of the large-scale\npotential wells that decay at late times after the accelerated expansion\nsets in and generates the ISW effect.\n\nWe shall focus on a single radio survey, with the level of sensitivity and\nsky coverage required for ISW studies, namely the NRAO VLA Sky Survey\n\\citep[hereafter NVSS,][]{Condon1998}. This survey was\nconducted using the Very Large Array (VLA) at $1.4$\\,GHz, and covers up to\nan equatorial latitude of $b_{\\rm E}=-40\\ifmmode^\\circ\\else$^\\circ$\\fi$, with an average\nnoise level of $0.45$\\,mJy\\,beam$^{-1}$.  It results in\nroughly $1.4\\times10^6$\nsources above a flux threshold of $2.5$\\,mJy. Fig.~\\ref{fig:surveys_maps}\ndisplays the number density map computed from the NVSS survey (top panel).\nThe AGN population is known to be dominant in radio\ncatalogues at $1.4$\\,GHz in the high flux density regime. \\citet{Condon1998}\nshow that at this frequency, star-forming galaxies (SFGs) contribute\nabout 30\\% of the total number of weighted source counts above\n$1$\\,mJy, but their presence decreases rapidly as higher flux\nthresholds are adopted. The NVSS SFGs are nearby sources ($z<0.01$), and\nhence may distort the ability of our radio template to probe the\nintermediate and high redshift density field. \n\nWe next address the presence of systematic effects in the NVSS\nsurvey. Two different antenna configurations were used while\nconducting the NVSS survey: the D-configuration (for $b_{\\rm E}\\in\n[-10\\ifmmode^\\circ\\else$^\\circ$\\fi,\\ 78\\ifmmode^\\circ\\else$^\\circ$\\fi]$), and the DnC-configuration for large zenith\nangles ($b_{\\rm E} < -10\\ifmmode^\\circ\\else$^\\circ$\\fi,\\ b_{\\rm E} > 78\\ifmmode^\\circ\\else$^\\circ$\\fi$).  This change in the\nantenna configuration is known to introduce changes in the source number\ndensity above $2.5$\\,mJy, as first pointed out by\n\\citet{Blake2002}. \nThe NVSS map at $2.5$\\,mJy is corrected for this\ndeclination systematic using the following procedure: the sky is divided into\nequatorial strips and the mean number of sources in each strip is\nre-normalised to the full sky mean~\\citep[see e.g.,][]{Vielva2006}.\nWith this procedure\nthe average number of sources in the NVSS map is the same as before the\ncorrection, and hence the shot noise level does not change. The number of\nstrips into which the map is divided is $70$, \nbut the results are independent of this choice.\n\nRegarding the galaxy bias,\nin this work we adopt the Gaussian bias evolution model of\n\\citet{Xia2011}. If $n(M,z)$ is the halo mass function and $b(M,z)$ is\nthe bias of halos with comoving mass $M$, then the bias of the survey is\ngiven by a mass-weighted integral,\n\\begin{equation}\nb(z) = \\frac{\\int_{M_{\\rm min}}^\\infty \\mathrm{d}M \\ b(M,z) \\ M \\\nn(M,z)}{\\int_{M_{\\rm min}}^\\infty \\mathrm{d}M \\ M \\ n(M,z)} \\ .\n\\label{eq:biasxia}\n\\end{equation}\nThis model depends on the minimum mass $M_{\\rm min}$ of halos present in\nthe survey. The upper limit in the mass is taken to be infinity\nbecause the effect of the high mass end on the bias is negligible.\n\\citet{MarcosCaballero2013} proposed a theoretical\nmodel for the NVSS angular power spectrum, which also takes into account the\ninformation of the redshift distribution given by CENSORS data\n\\citep{Brookes2008}. The redshift distribution is parametrized by\n\\begin{equation}\n\\frac{\\mathrm{d}n}{\\mathrm{d}z} =\nn_0 \\left( \\frac{z}{z_0} \\right)^\\alpha e^{-\\alpha z \/ z_0} \\ ,\n\\label{eqn:bias}\n\\end{equation}\nwhere $z_0=0.33$ and $\\alpha=0.37$. The parameter $n_0$ is a constant\nin order to have a distribution normalized to unity. This function is\nrepresented in Fig.~\\ref{fig:surveys_dndz}. \nThe bias follows the prescription of Eq.~\\ref{eq:biasxia}, with \n$M_{\\rm min}$ equal to $10^{12.67} {\\rm M}_{\\odot}$, where the\nSheth-Tormen~\\citep{Sheth1999} mass function is adopted. \nHereafter this model will be regarded as our fiducial model for NVSS.\n\n\\subsubsection{SDSS luminous galaxies}\n\nFor this analysis we use the photometric luminous galaxy (LG) catalogue from\nthe Baryonic Oscillation Spectroscopic Survey (BOSS) of the SDSS III.\nThe data used consist of two sub-samples: CMASS; and LOWZ.  Both samples are\ncombined to form a unique LG map (see Fig.~\\ref{fig:surveys_maps},\nsecond panel). Hereafter, these samples will be referred to as SDSS-CMASS,\nSDSS-LOWZ, and SDSS-CMASS\/LOWZ, for the combination.\n\n\\subsubsection*{SDSS-CMASS}\n\nWe use the BOSS targets chosen to have roughly constant stellar mass and known\nas the photometric ``CMASS'' sample. This sample is mostly contained in the\nredshift range $z=0.4$--0.7, with a galaxy number density close to\n$110\\,{\\rm deg}^{-2}$, and is selected after applying the colour cuts\nexplained in \\citet{Ross2011}.\n\nWhile such color selection yields a catalogue of about $1{,}600{\\,},000$\ngalaxies, further cuts needed to be applied in order to account for dust\nextinction (based on the maps by \\citealt{Schlegel1998} with the\ncriterion $E(B-V) < 0.08$), for seeing in the $r$ band (required to be\n$<2.0^{\\prime\\prime}$) and for the presence of bright stars, similar to\n\\citet{Ho2012}. Finally, we neglected all pixels with a mask value inferred\nfrom the footprint below $0.9$ on a {\\tt HEALPix} map of resolution\n$N_\\mathrm{side}$=64. This procedure left about one million sources\n$10{,}500\\,{\\rm deg}^2$.  Photometric redshifts of this sample are\ncalibrated using a selection of about 100{,}000 BOSS spectra as a training\nsample for the photometric catalogue.\nThese LGs are among the most luminous galaxies in the Universe and\ntherefore allow for a good sampling of the largest scales.\nGiven the large number of such sources included in the sample,\nshot-noise does not dominate clustering errors. According to \\citet{Ross2011},\nabout 3.7\\,\\% of these objects are either stars or quasars, and this\nmakes further corrections necessary, as explained at the end of this section.\n\n\\subsubsection*{SDSS-LOWZ}\n\nThe photometric LOWZ sample is one of the two galaxy samples targeted by the\nBOSS of Sloan III. It selectd luminous, highly biased, mostly red galaxies,\nplaced at an average redshift of $\\langle z\\rangle \\sim 0.3$ and below the\nredshifts of the CMASS sample ($z<0.4$). Our selection criteria in terms of\nthe Sloan five model magnitudes $ugriz$ follow those given in Sect.~2 of\n\\citet{Parejkoetal2012}.\nWith a total number of sources close to 600{,}000, this photometric sample\ncontains a higher number density of galaxies in the southern part of the\nfootprint than in the northern one (by more than 3\\,\\%), which seems to be\nat odds with $\\Lambda$CDM predictions. However, most of this effect vanishes\nwhen we subtract the dipole in the effective area under analysis, in such a\nway that the low $\\ell$ range of the auto power spectrum is consistent\nwith a $\\Lambda$CDM model and a constant bias $b\\simeq 2$\n\\citep{Hernandez2013}.\n\nBoth SDSS-CMASS and SDSS-LOWZ samples are further corrected for any\nscaling introduced by possible systematics like stars, mask value, seeing,\nsky emission, airmass and dust extinction. Following exactly the same\nprocedure as in \\citet{Hernandez2013}, we find that both LG samples are\ncontaminated by stars, in the sense that the galaxy number density decreases\nin areas with higher star density, since the latter tend to ``blind''\ngalaxy detection algorithms.\n\n\n\\subsubsection{Main photometric SDSS galaxy sample}\n\nWe use a sample of photometrically-selected galaxies from the SDSS-DR8\ncatalogue, which covers a \ntotal sky area of $14{,}555\\,{\\rm deg}^2$ \\citep{Aihara2011}.\nThe total number of objects labelled as \ngalaxies in this data release is 208 million. From this catalogue, and\nfollowing~\\cite{Cabre2006}, we \ndefine a subsample by selecting only objects within the range $18<r<21$,\nwhere this $r$-band model magnitude corrected for extinction.\nFollowing~\\cite{Giannantonio2008b}, we also restrict our subsample to objects\nwith redshifts $0.1 < z < 0.9$, and with measured \nredshift errors such that $\\sigma_z < 0.5z$. We rely on the photometric\nredshift estimates of the SDSS photo-$z$ primary galaxy table, which\nhave been obtained through a ``kd-tree'' nearest neighbour technique, \nby fitting the spectroscopic objects observed with similar colour and\ninclination angle.  The total number of galaxies in our final sample is \nabout 42 million, with redshifts distributed around a median value of\naround 0.35, as shown in Fig.~\\ref{fig:surveys_dndz}. To avoid possible\nerrors introduced by singularities in the photometric redshifts estimates,\ninstead of using the real observed redshift distribution in our analysis \nwe resort to the analytical function\n\\begin{equation}\n\\frac{dn}{dz}=\\frac{\\beta}{\\Gamma\\left(\\frac{m+1}{\\beta}\\right)}\\frac{z^m}{z_0^{m+1}}e^{-(z\/z_0)^\\beta}~,\n\\label{eq:dndz_xia}\n\\end{equation}\nwhich is fitted to the data, with parameters $m=1.5$, $\\beta=2.3$ and\n$z_0=0.34$, which are identical to those found \nby~\\cite{Giannantonio2012}. For the galactic bias we use the value\n$b=1.2$, which is found by~\\cite{Giannantonio2012} by fitting the\n$\\Lambda$CDM prediction to the observed auto-correlation \nfunction of the galaxies, and we adopt their proposed mask.\n\n\\subsubsection{SDSS, super-structures}\\label{sssect:catal}\n\n\\citet{Granett2008b} produced a sample\\footnote{Available at\n\\url{http:\/\/ifa.hawaii.edu\/cosmowave\/supervoids\/}.} of 50 superclusters and\n50 supervoids identified from the Luminous Red Galaxies (LRGs) in the SDSS\n(sixth data release, DR6, \\citealt{Adelman2008}) that covers an area of\n$7500\\,{\\rm deg}^2$ on the sky. They used publicly available algorithms,\nbased on the Voronoi tessellation, to find \n2836 superclusters \\citep[using VOBOZ, VOronoi BOund Zones,][]{Neyrinck2005}\nand\n631 supervoids \\citep[using ZOBOV, ZOnes Bordering On Voidness,][]{Neyrinck2008}\nabove a 2$\\,\\sigma$ significance level (defined as the probability of\nobtaining, in a uniform Poisson point sample, the same density contrasts as\nthose of clusters and voids). \nThe 50 superclusters and 50 supervoids they\npublished in their catalogue correspond to density contrasts of\nabout $3\\,\\sigma$ and $3.3\\,\\sigma$ respectively. They span a redshift range\nof $0.4<z<0.75$, with a median of around 0.5, and inhabit a volume of about\n5\\,$h^{-3}$\\,Gpc$^3$. These superstructures can potentially produce measurable\nISW signals, as suggested in \\cite{Granett2008a, Granett2008b}. For each\nstructure, the catalogue provides: the position on the sky of the centre;\nthe mean and maximum angular distance between the galaxies in the structure\nand its centre; the physical volume; and three different measures of the\ndensity contrast (calculated from all its Voronoi cells, from only its\nover- or under-dense cells, and from only its most over- under-dense cell).\nFor the present paper, we concentrate on using the supervoid catalogue by\n\\citet{Granett2008b}, as they can be compared with two other, more recent\ncatalogues of voids.\n\nThe second catalogue of cosmic voids that we consider here is published by\n\\cite{Pan2012}\\footnote{Available at \\url{http:\/\/www.physics.drexel.edu\/~pan\/}.}.\nIt has been built from the seventh data release (DR7) of the SDSS. Using the\n{\\tt VoidFinder} algorithm \\citep{Hoyle2002}, they identified 1055 voids with\nredshifts smaller than $z = 0.1$. Each void is listed with its position on\nthe sky, its physical radius (defined as the radius of the maximal sphere\nenclosing the void), an effective radius defined as the radius of a sphere\nof the same volume, its physical distance to us, its volume and mean density\ncontrast. The filling factor of the voids in the sample volume is 62\\,\\%.\nThe largest void is just over 47\\,Mpc in effective radius, while the median\neffective radius of the void sample is roughly 25\\,Mpc. Some of the voids\nare both very close to us and relatively large (larger than 30\\,Mpc in\nradius), which results in large angular sizes of up to $15\\ifmmode^\\circ\\else$^\\circ$\\fi$.\n\nThe third void catalogue that we use has been released by \\cite{Sutter2012}\nand also made publicly available.\\footnote{Available at\n\\url{http:\/\/www.cosmicvoids.net}.} Note that it is being updated regularly,\nand the results reported here are based on the 21 February 2013 version of\nthe catalogue. Using their own modified version of {\\tt ZOBOV}, these authors\nidentified 1495 voids distributed across the 0--0.44 redshift range.\nThey subdivided their catalogue into six subsamples: \\textit{dim1},\n\\textit{dim2}, \\textit{bright1} and \\textit{bright2}, constructed from the\nmain SDSS; and \\textit{lrgdim}, \\textit{lrgbright} built from the SDSS LRG\nsample. For each void, the information provided includes the position of the\ncentre, the redshift, the volume, the effective radius, and the density\ncontrast. \n\n\\subsection{External data sets}\n\\label{subsec:extdata}\n\nAs described in the introduction, the achromatic nature of the ISW effect\nrequires a tracer of the gravitational potentials from the large-scale\nstructure, so that by cross-correlating the CMB temperature map with that\ntracer distribution the fluctuations due to the ISW effect are singled out.\nThe prerequisites for a tracer catalogue to be used in ISW studies are: a\nlarge survey volume; well-understood biasing properties; and low or at least\nwell-modelled systematics. The radio NVSS catalogue and the optical luminous\ngalaxies (SDSS-CMASS\/LOWZ) and main photometric galaxy sample (SDSS-MphG) catalogues\npossess these\nqualities. Table~\\ref{tab:surveys} summarizes some basic properties of these\ncatalogues. In addition, the redshift distributions of these catalogues are shown\nin Fig.~\\ref{fig:surveys_dndz}. Notice that NVSS presents the widest redshift\ncoverage.  The SDSS-CMASS\/LOWZ\\ sample is peaked around $z \\approx 0.5$, whereas the\nSDSS-MphG\\ sample peaks around $z \\approx 0.3$. \n\nFigure~\\ref{fig:surveys_maps} shows the all-sky density projection for these\nmaps, where the grey area indicates regions not observed by these surveys\n(or discarded for having contamination or low galaxy number density, see next\nsubsections for details).\nIn Fig.~\\ref{fig:surveys_cls} we give the  angular power spectra\n(blue points) of the surveys~\\citep[corrected with a procedure similar to\n{\\tt MASTER}, e.g.,][]{Hivon2002}, as well as the theoretical spectra\n(black lines) and their $1\\,\\sigma$ error bars (grey areas), estimated from\nthe {\\tt MASTER} approach as well. \n\n\\begin{table}[tmb]\n\\begingroup\n\\newdimen\\tblskip \\tblskip=5pt\n\\caption{Major characteristics of the galaxy catalogues used as tracers of\nthe gravitational potential. From left to right, the columns indicate: the\nnumber of galaxies per steradian; the fraction of the sky covered by each\nsurvey; the mean bias; and the median redshift. The bias for\nNVSS is not provided, since the assumed model has a bias which depends on\nredshift (see text for details).\n\\label{tab:surveys}}\n\\nointerlineskip\n\\vskip -3mm\n\\footnotesize\n\\setbox\\tablebox=\\vbox{\n   \\newdimen\\digitwidth \n   \\setbox0=\\hbox{\\rm 0} \n   \\digitwidth=\\wd0 \n   \\catcode`*=\\active \n   \\def*{\\kern\\digitwidth}\n   \\newdimen\\signwidth \n   \\setbox0=\\hbox{+} \n   \\signwidth=\\wd0 \n   \\catcode`!=\\active \n   \\def!{\\kern\\signwidth}\n\\halign{\\hfil#\\hfil\\tabskip=0.8cm& \\hfil#\\hfil\\tabskip=0.5cm&\n \\hfil#\\hfil\\tabskip=0.5cm& \\hfil#\\hfil\\tabskip=0.5cm& \\hfil#\\hfil\\tabskip=0.cm\\cr\n\\noalign{\\vskip 3pt\\hrule \\vskip 1.5pt \\hrule \\vskip 5pt}\n \\noalign{\\vskip -2pt}\nGalaxy catalogue & $\\bar{n}$& $f_{\\rm sky}$ & bias & $\\bar{z}$\\cr\n\\noalign{\\vskip 3pt\\hrule\\vskip 5pt}\nNVSS& $1.584\\times10^5$& 0.73& $\\cdots$& 1.17\\cr\nSDSS-CMASS\/LOWZ& $5.558\\times10^5$& 0.22& 2.03& 0.45\\cr\nSDSS-MphG& $9.680\\times10^6$& 0.22& 1.20& 0.32\\cr\n                                   \n\\noalign{\\vskip 5pt\\hrule\\vskip 3pt}}}\n\\endPlancktable                   \n\\endgroup\n\\end{table}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\hsize]{.\/section_2.0\/figs\/dndz.pdf}\n\\caption{\nRedshift distributions of the different surveys used in this work as LSS\ntracers, to be correlated with the \\textit{Planck}\\ CMB maps. For ease of comparison,\nthese distributions have been normalised to unity.}\n\\label{fig:surveys_dndz}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\hsize]{.\/section_2.0\/figs\/lss_maps_n0064.pdf}\n\\caption{\nDensity contrast maps obtained from the galaxy catalogues at\n$N_\\mathrm{side}$ = 64. From top to bottom: NVSS; SDSS-CMASS\/LOWZ; and SDSS-MphG.}\n\\label{fig:surveys_maps}\n\\end{figure}\n\nBesides the cross-correlation between CMB and LSS tracers\n(Sect.~\\ref{subsec:xcor}), we will  present results from a different\nmethodology in Sect.~\\ref{subsec:stack}, where we use catalogues of\nsuper-structures to study the ISW through stacking of the CMB fluctuations on\nthe positions of these super-structures. The relevant catalogues are\ndescribed in Sect.~\\ref{sssect:catal}.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=1.0\\hsize]{.\/section_2.0\/figs\/cls_gg_nvss.pdf}\n\\includegraphics[width=1.0\\hsize]{.\/section_2.0\/figs\/cls_gg_lrg.pdf}\n\\includegraphics[width=1.0\\hsize]{.\/section_2.0\/figs\/cls_gg_main.pdf}\n\\caption{\nAngular power spectra from the maps in Fig.~\\ref{fig:surveys_maps}.\nFrom top to bottom: NVSS; SDSS-CMASS\/LOWZ; and SDSS-MphG. \nThe observed spectra are the red points, whereas the theoretical models are\nrepresented by the black lines (the grey areas correspond to the sampling\nvariance).}\n\\label{fig:surveys_cls}\n\\end{figure}\n\n\\input{.\/section_2.0\/section_2.2.1}\n\n\\input{.\/section_2.0\/section_2.2.2}\n\n\\input{.\/section_2.0\/section_2.2.3}\n\n\\input{.\/section_2.0\/section_2.2.4}\n\n\n\\section{ISW-lensing bispectrum}\n\\label{subsec:iswl}\nThere is an interesting interplay between gravitational lensing of the CMB\nand the ISW effect, which manifests itself as a non-Gaussian feature.\nCMB-lensing can be described by a convolution of the CMB-temperature map\n$T$ with the weak lensing potential $\\phi$ \\citep[e.g.,][]{Lewis2006},\n\\begin{equation}\nT(\\mathbf{\\ell}) \\rightarrow \nT(\\mathbf{\\ell}) - \n\\int\\frac{\\mathrm{d}^2 \\mathbf{\\ell}^\\prime}{2\\pi}\\:\n\\mathbf{\\ell}^\\prime(\\mathbf{\\ell}-\\mathbf{\\ell}^\\prime)\\:\n\\phi(\\mathbf{\\ell}-\\mathbf{\\ell}^\\prime)T(\\mathbf{\\ell}^\\prime).\n\\end{equation}\nThe CMB lensing can be measured by a direct estimate of the CMB bispectrum,\nbecause the bispectrum acquires first order terms proportional to the product\nof two power spectra $\\tilde{C}^{\\rm TT}_\\ell C^{{\\rm T}\\phi}_\\ell$,\nwhere $\\tilde{C}^{\\rm TT}_\\ell$ is the lensed temperature power spectrum\nand $C^{{\\rm T}\\phi}_\\ell$ is the temperature-potential cross-power spectrum.\nThe potential field $\\phi$ and the temperature field $T$ are correlated,\nbecause $\\phi$, which deflects the CMB photons, also gives rise to the\nISW effect in $T$ \\citep{Hu2002b, Seljak1999, Verde2002,Giovi2003}. This\nsecondary bispectrum contains new information about the cosmological redshift,\nbecause it is generated mainly at redshifts larger than unity, and biases\nmeasurements of the primordial bispectrum. The term $C^{{\\rm T}\\phi}_\\ell$\ncorrelates the CMB temperature on small scales with the lensing potential on\nlarge scales, and causes the bispectrum to assume large amplitudes in the\nsqueezed triangles configuration \\citep[see e.g.,][]{Goldberg1999, Seljak1999,\nHu2000, Giovi2003, Okamoto2003, Giovi2005a, Lewis2006, Serra2008,\nMangilli2009a, Hanson2009a, Hanson2010, Smith2011, Lewis2011}. \n\nDue to the rotational invariance of the sky, the CMB angular bispectrum\n$\\langle a_{\\ell_1m_1}a_{\\ell_2m_2}a_{\\ell_3m_3}\\rangle$ can be factorized\nas follows:\n\\begin{equation}\n\\langle a_{\\ell_1m_1}a_{\\ell_2m_2}a_{\\ell_3m_3}\\rangle\n = {\\cal G}_{\\ell_1\\ell_2\\ell_3}^{m_1m_2m_3}       \n b_{\\ell_1\\ell_2\\ell_3},\n\\label{the_bispectrum_estimator}\n\\end{equation}\nwhere ${\\cal G}_{\\ell_1\\ell_2\\ell_3}^{m_1m_2m_3}\\equiv\n \\int \\mathrm{d}^\\Omega\\ Y_{\\ell_1}^{m_1}(\\hat{\\vec{n}})Y_{\\ell_2}^{m_2}\n (\\hat{\\vec{n}})Y_{l_3}^{m_3}(\\hat{\\vec{n}})$ is the Gaunt-integral and\n$b_{\\ell_1\\ell_2\\ell_3}$ is the so-called reduced bispectrum. \nIn the case where the bispectral signal on the sky is due to the ISW-lensing \neffect, $b_{\\ell_1\\ell_2\\ell_3} =\n A^{{\\rm T} \\phi} b_{\\ell_1\\ell_2\\ell_3}^{\\rm lens-ISW}$, where \n$A^{{\\rm T} \\phi}$ parametrizes the amplitude of the effect and \n\\begin{eqnarray}\n b_{\\ell_1 \\ell_2 \\ell_3}^{\\rm lens-ISW}\n & = &  \n\\Big \\{\n\\frac{\\ell_1 (\\ell_1 + 1) - \\ell_2 (\\ell_2 + 1) + \\ell_3 (\\ell_3 +\n  1)}{2} C_{\\ell_1}^{{\\rm T}\\phi} \\tilde{C}_{\\ell_3}^{\\rm TT}  \\\\\n & &  \n + (5~\\mathrm{permutations}) \\Big \\}. \\nonumber\n\\label{thebl1l2l3}\n\\end{eqnarray}\nA more general expression for intensity and polarization can be found in\n\\citet{Lewis2011}. Estimation of the bispectrum then yields a measurement\nof $A^{{\\rm T}\\phi}$. \n\nWe can also define an alternative rotationally-invariant reduced bispectrum \n$B_{\\ell_1\\ell_2\\ell_3}$ as \n$B_{\\ell _1 \\ell _2 \\ell _3} = h^2_{\\ell _1 \\ell _2 \\ell _3}\n b_{\\ell _1 \\ell _2 \\ell _3}$,\nwhere\n\\begin{eqnarray}\n h^2_{\\ell_1\\ell_2\\ell_3}&\\equiv&\\sum_{m_1 m_2 m_3}\\left({\\cal G}_{\\ell_1\\ell_2\\ell_3}^{m_1m_2m_3}\\right)^2 \\nonumber\\\\\n &&=\\frac{(2\\ell_1 +1)(2\\ell_2 +1)(2\\ell_3 +1)}{4\\pi} \\left(\\begin{array}{ccc} \\ell_1 & \\ell_2 & \\ell_3 \\\\ 0 & 0 & 0 \\end{array}\\right)^2.\n\\end{eqnarray}\nThe interest in $B_{\\ell_1\\ell_2\\ell_3}$ is that it can be directly\nestimated from the observed map using the expression\n\\begin{equation}\nB^\\mathrm{obs}_{\\ell _1 \\ell _2 \\ell _3} = \\int \\mathrm{d}\\Omega T_{\\ell_1} (\\Omega) T_{\\ell_2}(\\Omega) T_{\\ell_3} (\\Omega), \\label{integral_Bobs}\n\\end{equation}\nwhere the filtered maps $T_\\ell(\\Omega)$ are defined as\n\\begin{equation}\nT_{\\ell}(\\Omega) \\equiv \\sum_m a_{\\ell m} Y_{\\ell m}(\\Omega). \\label{def_filtmaps_Tl}\n\\end{equation}\n\nBy basically combining the single-$\\ell$ estimates $B^\\mathrm{obs}\/B^\\mathrm{lens-ISW}$ for $A^{{\\rm T}\\phi}$ using inverse variance weighting, the ISW-lensing bispectrum estimator can be written as  \n\\citep[see][for more details]{planck2013-p09a}\n\\begin{equation}\n{\\hat A}^{{\\rm T}\\phi} = \\frac{\\langle B^{\\rm lens-ISW},\n (B^\\mathrm{obs} - B^\\mathrm{lin}) \\rangle}\n {\\langle B^{\\rm lens-ISW}, B^{\\rm lens-ISW} \\rangle}, \\label{fNL_estim_optim}\n\\end{equation}\nwhere the inner product is defined by \n\\begin{equation}\n\\langle B^i, B^j \\rangle \\equiv\n\\sum_{\\ell _1 \\leq \\ell _2 \\leq \\ell_3} \n\\frac{ \nB^i_{\\ell _1 \\ell _2 \\ell _3} \nB^j_{\\ell _1 \\ell _2 \\ell _3} \n}{ \nV_{\\ell _1 \\ell _2 \\ell _3}\n}. \\label{Binnerprod}\n\\end{equation}\nHere,  $B^\\mathrm{lin}$ is a linear \ncorrection that has zero average but reduces the variance of the estimator\nin the presence of anisotropic noise and a mask. \nFurthermore, $V_{\\ell _1 \\ell _2 \\ell _3} = g_{\\ell _1 \\ell _2 \\ell _3} \nh^2_{\\ell _1 \\ell _2 \\ell _3} C_{\\ell_1} C_{\\ell_2} C_{\\ell_3}$,\nwith $g$ being a simple permutation factor ($g=6$ when all $\\ell$ are equal,\n$g=2$ when two $\\ell$ are equal and $g=1$ otherwise).  \nAs in all expressions in this section,\nwe have implicitly taken the beam \nand noise of the experiment into accout, e.g., $C_\\ell$ should actually be \n$b_\\ell^2 C_\\ell + N_\\ell$ with $b_\\ell$ the beam transfer function\nand $N_\\ell$ the noise power spectrum. \n\nIn Eq.~\\ref{fNL_estim_optim} we have also used the fact that, as discussed in \ndetail in \\citet{planck2013-p09a}, the full inverse covariance weighting can be\nreplaced by a diagonal covariance term, \n$(C^{-1} a)_{\\ell m} \\rightarrow {a_{\\ell m}\/C_\\ell}$, without loss of\noptimality, if the masked regions of the map\nare filled in with a simple diffusive inpainting scheme.\n\nThe normalization of the lensing-ISW estimator in the denominator of\nEq.~\\ref{fNL_estim_optim} can be replaced by \\citep[see e.g.][]{Lewis2011}\n\\begin{equation}\nF = \\sum_{\\ell} \\bigg( F_{\\ell}^{-1} + \\frac{1+r_{\\ell}^{-2}}{2 \\ell + 1}\n \\bigg)^{-1},\n\\label{iswlensing_fisher_T}\n\\end{equation}\nwhere $r_{\\ell} \\equiv C_{\\ell}^{{\\rm T}\\phi}\/\n \\sqrt{\\tilde{C}^{\\rm TT}_{\\ell}\\:C^{\\phi\\phi}_{\\ell}}$ parameterizes\nthe deviation from the Cauchy-Schwarz relation and $F_{\\ell}$ is given in\nterms of the ISW-lensing bispectrum \\citep[see for example][]{Lewis2011}.\nThe first term in Eq.~\\ref{iswlensing_fisher_T} corresponds to the Fisher\nerrors assuming Gaussian $a_{\\ell m}$. However, contrary to the null\nhypothesis that is assumed, for example, in the primordial bispectra\n(Gaussianity), there is an actual non-Gaussian signal already present in\nthe ISW-lensing bispectrum. This guarantees a larger variance for the\nestimators than are included in the additional terms present in the previous\nequations.\n\nAn important issue is the impact of the ISW-lensing bispectrum on estimates\nof the primordial non-Gaussianity. Assuming weak levels of non-Gaussianity\nand considering both the primordial bispectrum\n$B^{\\rm prim}_{\\ell_1 \\ell_2 \\ell_3}$ and the ISW-lensing bispectrum\n$B^{\\rm lens-ISW}_{\\ell_1 \\ell_2 \\ell_3}$, one can compute the expected bias $\\Delta$ induced in the primordial bispectrum using the formula:\n\\begin{equation}\n\\Delta^{\\rm prim} = \n\\frac{\\langle B^\\mathrm{lens-ISW}, B^\\mathrm{\\rm prim} \\rangle}\n{\\langle B^\\mathrm{\\rm prim}, B^\\mathrm{\\rm prim} \\rangle}\n\\label{fnl_bias_iswl}\n\\end{equation}\nwith the inner product defined in Eq.~\\ref{Binnerprod}.\nPredictions of this bias on the primordial $f_\\mathrm{NL}$ for \\textit{Planck}\\\nresolution can be seen for example in \\citet{Hanson2009a},\n\\citet{Mangilli2009a}, \\citet{Smith2011}, and \\citet{Lewis2011}. The most\nimportant bias is introduced to the local shape and, considering\n$\\ell_\\mathrm{max}\\sim 2000$, is expected to be\n$\\Delta^{\\rm local} \\sim 7$ \\citep{planck2013-p09a}.\n\n\\input{.\/section_3.0\/section_3.1}\n\n\\input{.\/section_3.0\/section_3.2}\n\n\n\\subsubsection{Lensing potential reconstruction}\n\n\\label{subsubsec:theory_lensing}\n\nThe estimator given in Eq.~\\ref{the_bispectrum_estimator} can be written in\nterms of the lensing potential amplitude reconstruction $\\hat{\\phi}$ as\n\\begin{equation}\n\\hat{A}^{{\\rm T}\\phi} \\equiv \\hat{S} = \\frac{1}{N}\n \\sum_{\\ell m}C_{\\ell}^{{\\rm T}\\phi}\n \\frac{\\tilde{T}_{\\ell m}}{\\tilde{C}^{\\rm TT}_{\\ell}}\n \\frac{\\hat{\\phi}^*_{\\ell m}}{N_{\\ell}^{\\phi\\phi}},\n\\label{eqn:cltp_estimator}\n\\end{equation}\nwhere $\\hat{\\phi}^*_{\\ell m}$ can be estimated using a quadratic estimator\n\\citep{Okamoto2003} and $N_{\\ell}^{(0)}$ is given in terms of the\nISW-lensing bispectrum \\citep{Lewis2011}. Therefore, this estimator\nquantifes the amount of cross-correlation between the temperature map and\nthe reconstruction of the lensing signal, and most of the correlation is\nfound at multipoles below 100.\n\n\\subsubsection{KSW-estimator}\n\\label{subsubsec:ksw_ext}\n\nThe KSW bispectrum estimator~\\citep{Komatsu2003} for the\nISW-lensing signal can be written as\n\\begin{equation}\n\\label{eq:KSW}\n\\hat{A}^{{\\rm T}\\phi}=(F^{-1}) \\hat{S},\n\\end{equation}\nwhere $ \\hat{S}$ can be computed from data as\n\\begin{eqnarray}\n \\hat{S} &\\equiv &\n\\frac16\\sum_{\\ell_1 m_1}\\sum_{\\ell_2 m_2}\\sum_{\\ell_3 m_3}\n{\\cal G}_{\\ell_1\\ell_2\\ell_3}^{m_1m_2m_3}b_{\\ell_1\\ell_2\\ell_3}^{\\rm lens-ISW}\n \\times \\\\\n& & \\big[ (C^{-1}a)_{\\ell_1m_1}(C^{-1}a)_{\\ell_2m_2}(C^{-1}a)_{\\ell_3m_3}\n - \\nonumber \\\\\n& & 3(C^{-1})_{\\ell_1m_1,\\ell_2m_2}(C^{-1}a)_{\\ell_3m_3} \\big].\\nonumber\n\\end{eqnarray}\nand $(F^{-1})$ is the inverse of the ISW-lensing Fisher matrix\n$F$ of Eq.~\\ref{iswlensing_fisher_T}.\nDetails on the implementation of the KSW estimator for the ISW-lensing\nsignal can be found in~\\citet{Mangilli2013}.\nIn particular, Eq.~\\ref{eq:KSW} takes the form\n\\begin{equation}\n\\hat{A}^{{\\rm T}\\phi}=(F^{-1}) (\\hat{S}_{\\rm cubic}+\\hat{S}_{\\rm linear}),\n\\end{equation}\nwhere $\\hat{S}_{\\rm cubic}$ is the term that extracts the amplitude\ninformation from the data contained in the bispectrum, while\n$\\hat{S}_{\\rm linear}$ \nis a zero-mean term that reduces estimator variance when the experimental\nsetup breaks rotational invariance, i.e., in the presence of sky cut and\nanisotropic noise.\nTo estimate $\\hat{A}^{{\\rm T}\\phi}$ we used the KSW estimator with an\nimplementation of the linear term truncated at $\\ell_{\\rm max}$ as described \nin~\\cite{Munshi2009} and~\\cite{planck2013-p09a}.\n\n\\subsubsection{Binned bispectrum}\n\\label{subsubsec:binned_ext}\n\nThe binned bispectrum estimator \\citep{Bucher2010} achieves the required\ncomputational reduction in determining $A^{{\\rm T}\\phi}$ by binning \nEq.~\\ref{fNL_estim_optim}. In particular, the maximally filtered \nmaps in Eq.~\\ref{def_filtmaps_Tl} are replaced by \n\\begin{equation}\nT_i(\\Omega) = \\sum_{\\ell\\in\\Delta_i} \\sum_{m=-\\ell}^{+\\ell}\na_{\\ell m} Y_{\\ell m}(\\Omega),\n\\end{equation}\nwhere the $\\Delta_i$ are suitably chosen intervals (bins) of multipole values\n(chosen in such a way as to minimize the variance of the quantities to\nbe estimated).\nThese maps are then used in Eq.~\\ref{integral_Bobs} to obtain the binned\nobserved bispectrum, and analogously for $B^\\mathrm{lin}$. The bispectrum\ntemplate $B^\\mathrm{lens-ISW}$ and inverse-variance weights $V$ are also binned\nby summing them over all $\\ell$ values in the bin. Finally these binned\nquantities are inserted in the general expression for $A^{{\\rm T}\\phi}$\n(Eq.~\\ref{fNL_estim_optim}), with the sum over $\\ell$ replaced by a sum over \nbin indices $i$. Since most bispectrum shapes change rather slowly\n(with features on the scale of the acoustic peaks, like the power spectrum),\nthe binned estimator works very well, increasing the variance only\nslightly, while achieving an enormous computational reduction (from\nabout 2000 multipoles in each of the three directions to only about 50\nbins).\n\n\\subsubsection{Modal bispectra}\n\\label{subsubsec:theory_modal}\nModal decomposition of bispectra has been introduced by \\citet{Fergusson2010}\nas a way to compute reduced bispectra that uses a diagonalization ansatz such\nthat the shape function in Fourier space can be separated, which reduces the\ndimensionality of the integration. At the same time it greatly reduces the\ncomplexity of estimating bispectra from data. The separation of the\nbispectrum shape function into coefficients\n$q_p^\\ell(x)$ allows the derivation of a filtered map $M_p(\\hat{n},x)$,\n\\begin{equation}\nM_p(\\hat{n},x) = \\sum_{\\ell m}\n \\frac{q_p^\\ell(x)a_{\\ell m}}{C_\\ell}Y_{\\ell m}(\\hat{n}),\n\\end{equation}\nfrom the coefficients $a_{\\ell m}$ of the temperature map. With that\nexpression, one can obtain a mode expansion coefficient $\\beta$,\n\\begin{equation}\n\\beta_{prs} = \\int\\mathrm{d}\\Omega\\:\\int x^2\\mathrm{d} x\\:\n M_p(\\hat{n},x)M_r(\\hat{n},x)M_s(\\hat{n},x).\n\\end{equation}\nWith that decomposition, the estimator of the bispectrum assumes a\nparticularly simple diagonal shape,\n\\begin{equation}\n\\hat{S} = \\frac{6}{N}\\Delta_\\Phi^2\\sum_{prs}\\alpha_{prs}\\beta_{prs},\n\\end{equation}\nwhere the $\\alpha_{prs}$ are the equivalent coefficients obtained by\nperforming the modal decomposition of the theoretical bispectrum shape\nfunction. The relation between modal bispectra and wavelet bispectra is\nderived by \\citet{Regan2013}.\n\n\\subsection{ISW-lensing estimators}\n\\label{subsub:iswl_est}\n\n\nThere are several implementations of the optimal estimator given in Eq. \\ref{fNL_estim_optim}. For\ntheir \ndetailed description in the context of \\textit{Planck}\\ see \\citet{planck2013-p09a,planck2013-p12}. \nWe have applied four of these implementations to \\textit{Planck}\\ data in order to constrain the ISW-lensing bispectrum. Three of them represent a direct bispectrum estimation: the KSW estimator \\citep{Komatsu2005, Creminelli2006}, the binned bispectrum \\citep{Bucher2010}, and the modal decomposition \\citep{Fergusson2010}.The remaining approach is based on a previous estimation of the gravitational lensing potential field \\cite{Lewis2011}. \nThese estimators differ in the implementation and approximations that are used in order to compute the expression given in Eq.~\\ref{fNL_estim_optim}, the direct computation of which is out of reach of current computing facilities.\nThey will be reviewed in the next subsections.\n\n\\input{.\/section_3.0\/section_3.1.1}\n\n\\input{.\/section_3.0\/section_3.1.2}\n\n\\input{.\/section_3.0\/section_3.1.3}\n\n\\input{.\/section_3.0\/section_3.1.4}\n\n\n\n\\subsection{Results}\n\\label{subsubsec:iswl_res}\n\\begin{table*}[tmb]\n\\begingroup\n\\newdimen\\tblskip \\tblskip=5pt\n\\caption{Amplitudes $A^{{\\rm T}\\phi}$, errors $\\sigma_{A}$ and significance\nlevels (SNR) of the non-Gaussianity due to the ISW effect, for all component\nseparation algorithms ({\\tt C-R}, {\\tt NILC}, {\\tt SEVEM}, and {\\tt SMICA}) and all the\nestimators (potential reconstruction, KSW, binned, and modal). For the\npotential reconstruction case, an additional minimum variance (MV) map has\nbeen considered (see \\citealt{planck2013-p12} for details).\n\\label{table:isw-lensing}}\n\\nointerlineskip\n\\vskip -3mm\n\\footnotesize\n\\setbox\\tablebox=\\vbox{\n   \\newdimen\\digitwidth \n   \\setbox0=\\hbox{\\rm 0} \n   \\digitwidth=\\wd0 \n   \\catcode`*=\\active \n   \\def*{\\kern\\digitwidth}\n   \\newdimen\\signwidth \n   \\setbox0=\\hbox{+} \n   \\signwidth=\\wd0 \n   \\catcode`!=\\active \n   \\def!{\\kern\\signwidth}\n\\halign{#\\hfil\\tabskip=0.4cm & \n \\hfil#\\hfil\\tabskip=0.4cm& \\hfil#\\hfil\\tabskip=0.4cm& \\hfil#\\hfil\\tabskip=0.4cm& \\hfil#\\hfil\\tabskip=0.4cm& \\hfil#\\hfil\\tabskip=0.4cm& \\hfil#\\hfil\\tabskip=0.4cm& \\hfil#\\hfil\\tabskip=0.4cm& \\hfil#\\hfil\\tabskip=0.4cm& \\hfil#\\hfil\\tabskip=0.4cm& \\hfil#\\hfil\\tabskip=0cm\\cr \n\\noalign{\\vskip 3pt\\hrule \\vskip 1.5pt \\hrule \\vskip 5pt}\n\\noalign{\\vskip -2pt}\nEstimator & {{\\tt C-R}}& & {{\\tt NILC}}& & {{\\tt SEVEM}}& & {{\\tt SMICA}}& & MV\\cr \n\\noalign{\\vskip 2pt\\hrule\\vskip 3pt} \n& $A^{{\\rm T}\\phi}$ & SNR & $A^{{\\rm T}\\phi}$ & SNR & $A^{{\\rm T}\\phi}$ & SNR & $A^{{\\rm T}\\phi}$ & SNR & $A^{{\\rm T}\\phi}$ & SNR\\cr\n\\noalign{\\vskip 3pt\\hrule\\vskip 5pt}                                    \n${\\rm T}\\phi \\quad \\ell \\geq 10$ & 0.52 $\\pm$ 0.33& 1.5& 0.72 $\\pm$ 0.30& 2.4& 0.58 $\\pm$ 0.31& 1.9& 0.68 $\\pm$ 0.30& 2.3& 0.78 $\\pm$ 0.32& 2.4\\cr\n$\\rm{T }\\phi \\quad \\ell \\geq 2$ & 0.52 $\\pm$ 0.32& 1.6& 0.75 $\\pm$ 0.28& 2.7& 0.62 $\\pm$ 0.29& 2.1& 0.70 $\\pm$ 0.28& 2.5& & \\cr\n\\noalign{\\vskip 3pt\\hrule\\vskip 5pt}KSW & 0.75 $\\pm$ 0.32& 2.3& 0.85 $\\pm$ 0.32& 2.7& 0.68 $\\pm$ 0.32& 2.1& 0.81 $\\pm$ 0.31& 2.6& & \\cr\n\\noalign{\\vskip 3pt\\hrule\\vskip 5pt}Binned & 0.80 $\\pm$ 0.40& 2.0& 1.03 $\\pm$ 0.37& 2.8& 0.83 $\\pm$ 0.39& 2.1& 0.91 $\\pm$ 0.37& 2.5& & \\cr\n\\noalign{\\vskip 3pt\\hrule\\vskip 5pt}Modal & 0.68 $\\pm$ 0.39& 1.7& 0.93 $\\pm$ 0.37& 2.5& 0.60 $\\pm$ 0.37& 1.6& 0.77 $\\pm$ 0.37& 2.1& & \\cr\n                                   \n\\noalign{\\vskip 3pt\\hrule\\vskip 3pt}}}\n\\endPlancktablewide                \n\\endgroup\n\\end{table*}                       \n\n\nThe detection of the ISW effect via the non-Gaussian signal induced by the\ngravitational lensing potential is summarized in Table~\\ref{table:isw-lensing}.\nWe provide the estimates of the ISW-lensing amplitude $A^{{\\rm T}\\phi}$, its\nuncertainty $\\sigma_{A}$ and the signal-to-noise obtained with the different\nestimator pipelines  described in Sec.~\\ref{subsub:iswl_est}. The estimators\nhave been applied to the official \\textit{Planck}\\ CMB maps made using\n{\\tt C-R}, {\\tt NILC}, {\\tt SEVEM}, and {\\tt SMICA}~\\citep{planck2013-p06}.\nThe quantity $\\sigma_{A}$ is obtained from 200 simulations representative of\nthe analysed CMB data maps. These Monte Carlo\nsimulations~\\citep[FFP-6, see][]{planck2013-p01} account for the expected\nnon-Gaussian ISW-lensing signal, according to the \\textit{Planck}\\ best-fit model,\nand have been passed through the different component separation pipelines, as\ndescribed in~\\cite{planck2013-p06}.\nlensed simulations can be found in \\cite{planck2013-p12}.  The mask\nused in the analysis is the combined Galactic and point source common\nmask~\\citep[U73,][]{planck2013-p01} with sky fraction $f_{\\rm sky}=0.73$.\n\nThe KSW and the ${\\rm T}\\phi$ estimators show similar sensitivity, finding,\nrespectively, $A^{{\\rm T}\\phi}=0.81 \\pm 0.31$\nand $A^{{\\rm T}\\phi}=0.70 \\pm 0.28$ from the {\\tt SMICA}\\ CMB map, which\ncorresponds to a significance at about the $2.5\\,\\sigma$ level. The modal and\nbinned estimators are slightly less optimal, but give consistent results,\nwhich is consistent with the imperfect overlap of the modal estimator\ntemplates with the ISW-lensing signal; the ISW-lensing bispectrum has a\nrapidly oscillating shape in the squeezed limit and both, binned and modal\nestimates, are better suited (and originally implemented) to deal with smooth\nbispectra of the kind predicted by primordial inflationary theories. Since\nthe correlation coefficient of the binned and modal ISW-lensing\ntemplates relative to the actual ISW-lensing bispectrum (Eq.~\\ref{thebl1l2l3})\nis generally $0.8 < r < 0.9$~\\citep[to be compared with $r=0.99$, achieved by\nboth estimators for local, equilateral and orthogonal primordial\ntemplates,][]{planck2013-p09a}, the corresponding estimator's weights are\nexpected to be\nabout 20 \\% suboptimal, consistent with observations.\n\n\nThe ${\\rm T}\\phi$ estimator has also been applied to the specific \\textit{Planck}\\ lensing\nbaseline, i.e., the MV map, which is a noise-weighted combination of the\n217\\,GHz and 143\\,GHz channel maps, previously cleaned from infrared\ncontamination through subtraction of the 857\\,GHz map, taken as\na dust template. From this map the lensing potential is recovered and then\ncorrelated with that potential field in order to estimate the amplitude\n$A^{{\\rm T}\\phi}$. The official baseline adopts a more conservative high-pass\nfiltering, such that as only multipoles $\\ell \\geq 10$ are considered, and the\nmask with $f_{\\rm sky}=0.7$ is used. In this case, the ISW-lensing estimate\nis $0.78 \\pm 0.32$ (a $2.4\\,\\sigma$ detection, where the \nerror bars are obtained from 1000 simulations), as reported in the first\nsub-row for ${\\rm T}\\phi$ in Table~\\ref{table:isw-lensing}. \nThe full multipole range is considered in the second sub-row, obtaining about\n7\\% better sensitivity. \n\nNotice that, according to all the estimators, the {\\tt C-R}\\ CMB map provides\nlower significance for ISW-lensing, since its resolution is slightly lower\nthan that of the other maps. {\\tt NILC}\\ and {\\tt SMICA}\\ exhibit a somewhat\nlarger detection of the ISW signal, since they are the least noisy maps. \n\nIn order to explore the agreement among the different estimators, we performed\na validation test based on 200 lensed simulations processed through the\n{\\tt SMICA}\\ pipeline. \nThe results are summarized in Table~\\ref{tab:Aisw_compared}.  For each pair\nof statistics, we provide the difference in amplitudes\nestimated for the data ($\\Delta A^{{\\rm T}\\phi}$), the dispersion of the\ndifference of amplitudes obtained from the simulations ($s_{A}$),\nthe ratio between this dispersion and the largest of the corresponding\nsensitivities ($\\eta$, according to Table~\\ref{table:isw-lensing}), and the\ncorrelation coefficient ($\\rho$). As can be seen from the Table, the\nagreement among estimators is good and the discrepancies are only\naround $0.5\\,\\sigma$, which is the expected scatter, given the correlation\nbetween the weights of different\nestimators discussed above.\nOverall, the bispectrum estimators provide a larger value of the amplitude\n$A^{{\\rm T}\\phi}$, as compared to the ${\\rm T}\\phi$ estimator.\n \nWe have also explored the joint estimation of the two bispectra that are\nexpected to be found in the data: the ISW-lensing; and the residual point\nsources. A detailed description of the non-Gaussian signal coming from\npoint sources can be found in \\cite{planck2013-p09a}. The joint analysis of\nthese two signals performed with the KSW estimator, and the binned, and modal\nestimators has shown that the ISW-lensing amplitude estimation can be\nconsidered almost completely independent of the non-Gaussian signal induced by\nthe residual sources, and that the two bispectra are nearly perfectly\nuncorrelated.\n\nThere is not a unique way of extracting a single signal-to-noise value from\nTable~\\ref{table:isw-lensing}.  However, all the estimators show evidence\nof ISW-lensing at about the $2.5\\,\\sigma$ level.\n\nFinally, we estimate that the bias introduced by the ISW-lensing signal on\nthe estimation of the primordial local shape bispectrum\n(Eq.~\\ref{fnl_bias_iswl}) is $\\Delta^{\\rm prim}\\simeq 7$, corresponding to\nthe theoretical expectation, as described in detail in~\\cite{planck2013-p09a}. \n\n\\begin{table}[tmb]\n\\begingroup\n\\newdimen\\tblskip \\tblskip=5pt\n\\caption{For each pair of estimators, we give the mean difference among\nthe amplitudes estimated from the data ($\\Delta A^{{\\rm T}\\phi}$), the\ndispersion of the differences between the amplitudes estimated from the\nsimulations ($s_{A}$), the ratio of this dispersion to the larger of the\ncorresponding sensitivities ($\\eta$), and the correlation coefficient ($\\rho$).\n\\label{tab:Aisw_compared}}\n\\nointerlineskip\n\\vskip -3mm\n\\footnotesize\n\\setbox\\tablebox=\\vbox{\n   \\newdimen\\digitwidth \n   \\setbox0=\\hbox{\\rm 0} \n   \\digitwidth=\\wd0 \n   \\catcode`*=\\active \n   \\def*{\\kern\\digitwidth}\n   \\newdimen\\signwidth \n   \\setbox0=\\hbox{+} \n   \\signwidth=\\wd0 \n   \\catcode`!=\\active \n   \\def!{\\kern\\signwidth}\n\\halign{#\\hfil\\tabskip=0.2cm& \\hfil#\\hfil\\tabskip=0.2cm&\n \\hfil#\\hfil\\tabskip=0.2cm& \\hfil#\\hfil\\tabskip=0.2cm& \\hfil#\\hfil\\tabskip=0.cm\\cr\n\\noalign{\\vskip 3pt\\hrule \\vskip 1.5pt \\hrule \\vskip 5pt}\n \\noalign{\\vskip -2pt}\n&&KSW&Binned&Modal\\cr\n\\noalign{\\vskip 3pt\\hrule\\vskip 5pt}\n& $\\Delta A\\pm s_A$& $-0.11\\pm0.10$& $-0.21\\pm0.21$&  $-0.07\\pm0.21$\\cr\n${\\rm T}\\phi$\t& $\\eta$& $0.32$& $0.56$& $0.56$\\cr\n& $\\rho$& $0.95$&$0.84$& $0.84$\\cr\n\\noalign{\\vskip 3pt\\hrule\\vskip 5pt}\n& $\\Delta A\\pm s_A$&\t&$-0.10\\pm0.19$&!$0.04\\pm0.19$\\cr\nKSW& $\\eta$& &$0.52$& $0.51$\\cr\n& $\\rho$& &$0.86$& $0.87$\\cr\n\\noalign{\\vskip 3pt\\hrule\\vskip 5pt}\n& $\\Delta A\\pm s_A$&& &!$0.14\\pm0.15$\\cr\nBinned& $\\eta$&\t& &$0.41$\\cr\n& $\\rho$& & & $0.92$\\cr\n                                   \n\\noalign{\\vskip 1pt\\hrule\\vskip 3pt}}}\n\\endPlancktable                   \n\\endgroup\n\\end{table}\n\n\\section{Cross-correlation with surveys}\n\\label{subsec:xcor}\n\n\nThe ISW effect can be probed through several different approaches. Among the\nones already explored in the literature, the classical test is to study the\ncross-correlation of the CMB temperature fluctuations with a tracer of the\nmatter distribution, typically a galaxy or cluster catalogue. \nAs mentioned in the introduction, the correlation of the CMB with LSS tracers\nwas first proposed by~\\cite{Crittenden1996} as \na natural way to amplify the ISW signal, otherwise very much subdominant with\nrespect to the primordial CMB fluctuations.\nIndeed, this technique led to the first reported detection of the ISW\neffect \\citep{Boughn2004}.\n\nSeveral methods have been proposed in the literature to study statistically\nthe cross-correlation of the CMB fluctuations with LSS tracers, and, they\ncan be divided into: real space statistics (e.g., the cross-correlation\nfunction, hereinafter CCF); harmonic space statistics (e.g., the cross-angular\npower spectrum, hereinafter CAPS); and wavelet space statistics (e.g., the\ncovariance of the Spherical Mexican Hat Wavelet coefficients, or SMHWcov from\nnow on). These statistics are equivalent (in the sense of the significance of\nthe ISW detection) under ideal conditions. However, ISW data analysis presents\nseveral problematic issues (incomplete sky coverage, selection biases in the\nLSS catalogues, foreground residuals in the CMB map, etc.).\nHence, the use of several different statistical approaches provides a more\nrobust framework for studying the ISW-LSS cross-correlation, since different\nstatistics may have different sensitivity to these systematic effects,\nThe individual methods are described in more detail in\nSect.~\\ref{subsubsec:methods}.\n\nBesides the choice of specific statistical tool, the ISW cross-correlation can\nbe studied from two different (and complementary) perspectives. On the one\nhand, we can determine the amplitude of the ISW signal, as well as the\ncorresponding signal-to-noise  ratio, by comparing the observed\ncross-correlation to the expected one. On the other hand, we can postulate a\nnull hypothesis (i.e., that there is no correlation between the CMB and the\nLSS tracer) and study the probability of obtaining the observed\ncross-correlation. Whereas the former answers a question regarding the\ncompatibility of the data with the ISW hypothesis (and provides an estimation\nof the signal-to-noise associated with the observed signal), the latter tells\nus how incompatible the measured signal is with the no-correlation hypothesis,\ni.e., against the presence of dark energy (assuming that the Universe is\nspatially flat). Obviously, both approaches can be extended to account for\nthe cross-correlation signal obtained from several surveys at the same time.\nThese two complementary tests are described in detail in\nSect.~\\ref{subsubsec:xcorrdef}, with the results presented in\nSect.~\\ref{subsubsec:results}.\n\n\\input{.\/section_4.0\/section_4.1}\n\n\\input{.\/section_4.0\/section_4.2}\n\n\\input{.\/section_4.0\/section_4.3}\n\n\n\\subsubsection{Angular cross-power spectrum}\n\\label{subsub:caps}\n\nThe angular cross-power spectrum (CAPS) is a natural tool for studying\nthe cross-correlation of the CMB fluctuations and tracers of the\nLSS. Under certain conditions, it provides a statistical tool with\nuncorrelated (full-sky coverage) or nearly uncorrelated (binned\nspectrum for incomplete sky coverage) components.  Even the unbinned\nCAPS, estimated on incomplete signals, can be easily worked out, since\nthe correlations are mostly related to the geometry of the mask. This\nis the case for the CAPS obtained through {\\tt MASTER}\napproach~\\citep[e.g.,][]{Hivon2002,Hinshaw2003}. \nAnother approach is to work in the map domain, making use of a quadratic \nmaximum likelihood (QML henceforth) estimator \\citep{Tegmark1997c} \nfor the CAPS \\citep{Padmanabhan2005,Schiavon2012}. Such approach \nis optimal, i.e., leads to unbiased estimates for the CAPS with  \nminimum error bars.\n\n\\subsubsection*{Pseudo angular power spectrum}\n\nLet us denote the CAPS between the CMB field $T\\left(p\\right)$ and an LSS\ntracer $G\\left(p\\right)$ map (where $p =\\left(\\theta, \\phi \\right)$\nrepresents a given pixel) as: $C_{\\ell}^{\\mathrm{TG}}$ (i.e., $\\xi_a^{xy}\n\\equiv C_{\\ell}^{\\mathrm{TG}}$ for this cross-correlation estimator). In the\nfull-sky case, an optimal estimator of the CAPS is given by:\n\\begin{equation}\n\\hat{C}_{\\ell}^{\\mathrm{TG}} = \\frac{1}{2\\ell + 1}\n \\sum_{m=-\\ell}^{+\\ell} t_{\\ell m} g_{\\ell m}^*,\n\\label{eq:opt_caps}\n\\end{equation}\nwhere $t_{\\ell m}$ and $g_{\\ell m}$ are the spherical harmonic\ncoefficients of the CMB and the LSS maps, respectively. This CAPS can\nbe seen as a vector with $\\ell_\\mathrm{max}$ components, where\n$\\ell_\\mathrm{max}$ is the maximum multipole considered in the\nanalysis. Here we adopt $3N_{\\rm side} - 1$,\nwhich suffices for ISW analysis, since it is know that most of the ISW signal\nis contained within $\\ell \\lesssim 80$ \\citep{Afshordi2004b,Hernandez2008}.\nWhen a mask $\\Pi\\left(p\\right)$ is applied to\nthe maps, it acts as a weight that modifies the underlying harmonic\ncoefficients. Now, we have $\\tilde{t}_{\\ell m}$ and $\\tilde{g}_{\\ell\n  m}$, where\n\\begin{eqnarray}\n\\tilde{t}_{\\ell m} & = & \\int \\int \\mathrm{d}\\left(\\cos{\\theta} \\right)\n \\mathrm{d}\\phi\\, T\\left(\\theta, \\phi \\right)\\Pi\\left(\\theta, \\phi \\right)\n Y_{\\ell m}^* \\left(\\theta, \\phi \\right), \\\\ \\nonumber\n\\tilde{g}_{\\ell m} & = & \\int \\int \\mathrm{d}\\left( \\cos{\\theta} \\right)\n \\mathrm{d}\\phi\\, G\\left(\\theta, \\phi \\right)\\Pi\\left(\\theta, \\phi \\right)\n Y_{\\ell m}^* \\left(\\theta, \\phi \\right), \n\\label{eq:masked_harm}\n\\end{eqnarray}\nand $Y_{\\ell m} \\left(\\theta, \\phi \\right)$ are the spherical harmonic\nfunctions. In these circumstances, the estimator in Eq.~\\ref{eq:opt_caps} is\nnot longer optimal, and is referred to as pseudo-CAPS. A nearly optimal\nestimator is given by decoupling the masked CAPS (denoted by\n$\\tilde{C}_{\\ell}^{\\mathrm{TG}}$) through the masking kernel\n$\\tens{B}$~\\citep[e.g.,][]{Xia2011}:\n\\begin{equation}\n\\label{eq:pcaps}\n\\hat{C}_{\\ell}^{\\mathrm{TG}} = \\tens{B}^{-1}\\tilde{C}_{\\ell}^{\\mathrm{TG}},\n\\end{equation}\nwhere\n\\begin{equation}\nB_{\\ell \\ell', {\\rm G}} = \\frac{2\\ell+1}{4\\pi}\\sum_{\\ell''} J_{\\ell''}^{\\rm G}\n\\left(\\begin{array}{ccc}\n      \\ell & \\ell' & \\ell'' \\\\\n      0 & 0 & 0\n      \\end{array}\n\\right)^2,\n\\label{eq:master1}\n\\end{equation}\nwith $J_{\\ell''}^G$ the cross-angular power spectrum of the $T$ and $G$ masks.\n\nThe estimator in Eq.~\\ref{eq:pcaps} is nearly optimal because\n$\\tilde{C}_{\\ell}^{\\mathrm{TG}}$ has to be understood as the mean value over\nan ensemble average of skies.\nLet us point out that when more than a single CAPS is considered, for instance\nwhen one is interested in the cross-correlation of the \\textit{Planck}\\ CMB map with\nmore than one LSS tracer map, the CAPS estimator can be seen as a single vector\nwith $N\\ell_{\\rm max}$ components, with $N$ being the number of surveys.\n\nIt can be shown that the element $C_{\\ell \\ell', i j}$ of the covariance\nmatrix of the CAPS estimator in Eq.~\\ref{eq:pcaps} (for the case of a masked\nsky and for $N$ surveys) is given by\n\\begin{equation}\nC_{\\ell \\ell', i j} = K_{\\ell, i j} K_{\\ell', i j}\n \\frac{\\left(M_{ij}\\right)_{\\ell \\ell'}^{-1}}{(2\\ell' + 1)} ,\n\\label{eq:cov_caps1}\n\\end{equation}\nwhere\n\\begin{equation}\nK_{\\ell, i j} = \\left[ C_{\\ell}^{\\mathrm{TG}_i}C_{\\ell}^{\\mathrm{TG}_j}\n + C _{\\ell}^\\mathrm{T}\\left(C_{\\ell}^{\\mathrm{G}_i \\mathrm{G}_j}\n + N_{\\ell}^{\\mathrm{G}_i \\mathrm{G}_j}\\delta_{ij} \\right) \\right]^{1\/2},\n\\label{eq:cov_caps2}\n\\end{equation}\nand $\\left(M_{ij}\\right)_{\\ell \\ell'}^{-1}$ is the\n$\\left(\\ell,\\ell'\\right)$ element of the inverse matrix of $\\tens{M}$\nfor surveys $i$ and $j$ fixed, such as\n\\begin{equation}\nM_{\\ell \\ell', ij} = \\frac{2\\ell+1}{4\\pi}\\sum_{\\ell''} H_{\\ell''}^{ij}\n\\left(\\begin{array}{ccc}\n      \\ell & \\ell' & \\ell'' \\\\\n      0 & 0 & 0\n      \\end{array}.\n\\right)^2,\\label{eq:master2}\n\\end{equation}\nHere $H_{\\ell''}^{ij}$ is the angular cross-power spectrum of the two joint\nmasks, i.e., the masks resulting from the multiplication of the \n$T$ with $G_i$ and $G_j$, respectively.\nThe quantities $C_{\\ell}^{xy}$ are expected or fiducial spectra,\n$N_{\\ell}^{yy}$ is the Poissonian noise of the $y$ survey\n(deconvolved by any beam or pixel filter), and $\\delta_{ij}$ is the Kronecker\ndelta. In Eq.~\\ref{eq:cov_caps2}, the instrumental noise associated with\nthe CMB data has been\nignored, since the \\textit{Planck}\\ sensitivity is such that the noise contribution on\nthe scales of interest is negligible.\nWhen more than one survey has poor sky coverage, then the complexity of the\ncorrelations is not well reflected by the previous expression. Therefore, in\nthis paper, we will compute $C_{\\ell \\ell', i j}$ from coherent CMB and LSS\nMonte Carlo simulations. \nFor each simulation, we generate four independent, Gaussian, and white\nrealizations (at $N_\\mathrm{nside}=64$), which are afterwards properly\ncorrelated using the expected auto- and cross-correlations of the signals.\nCorresponding Poissonian shot noise realizations are added to each survey map.\nThe resulting four maps are masked with the corresponding masks (i.e., one for\nthe CMB and one for mask for each survey).\n\nThe computation of the CAPS in Eq.~\\ref{eq:pcaps} is\nextremely fast (especially for the resolutions used in the study of the ISW).\nHowever, as stated above, it is only a {\\it nearly\\\/} optimal estimator of the\nunderlying CAPS. Moreover, its departure from\noptimality is largest at the smallest\nmultipoles (largest scales), where the ISW signal is more important. \n\n\\subsubsection*{The QML angular power spectrum}\n\nThe QML method for the power spectrum estimation \nof temperature CMB anisotropies was introduced \nby~\\cite{Tegmark1997c} and later extended to polarization by\n\\cite{Tegmark2001}. For an application to temperature and polarization to\n{\\it WMAP\\\/} data see~\\cite{Gruppuso2009} and \\cite{Paci2013}.\nThe same method was employed to measure the cross-correlation between \nthe CMB and LSS in~\\cite{Padmanabhan2005},\n\\cite{Ho2008}, and~\\cite{Schiavon2012}.\nThe QML method is usually stated to be optimal, since it provides unbiased\nestimates and the smallest error bars allowed by the Fisher-Cram{\\'e}r-Rao\ninequality.  As a drawback, from the computational point of view, \nthe QML is a very expensive approach. Let us denote \nthe QML estimator of the CAPS between the CMB map $T$ \nand an LSS tracer $G$ (at multipole $\\ell$) by\n$Q_{\\ell}^{\\mathrm{TG}}$ (i.e., $\\xi_a^{xy} \\equiv Q_{\\ell}^{\\mathrm{TG}}$\nfor this cross-correlation estimator). \n\nA detailed description of the algebra of the QML is given \nin~\\cite{Schiavon2012}. We briefly recall here the\nbasics of the CAPS estimator, which is given by:\n\\begin{equation}\n\\label{eq:qmlcaps}\n\\hat{Q}_\\ell^{\\mathrm{TG}} = \\sum_{\\ell' X'}\n (F^{-1})^{{\\mathrm{TG}} ~ X'}_{\\ell\\ell'}\n \\left[ \\vec{x}^{\\rm T} \\tens{E}_{\\ell'}^{X'} \\vec{x}\n -{\\rm Tr}(\\tens{N}\\tens{E}_{\\ell'}^{X'}) \\right],\n\\end{equation}\nwhere $X$ represents any of the following spectra:\n$X = \\left\\lbrace \\mathrm{T}, \\mathrm{TG}, \\mathrm{G} \\right\\rbrace$.\nThe vector $\\vec{x}$ has $2N_{\\mathrm{pix}}$ elements (with $N_{\\mathrm{pix}}$\nbeing the total number of pixels allowed by the joint CMB and LSS mask): the\nfirst set of $N_{\\mathrm{pix}}$ corresponds to the CMB map, and the second one\naccounts for the LSS map. The $F^{{\\mathrm{TG}} ~ X'}_{\\ell \\ell '}$ is the\nFisher matrix defined as\n\\begin{equation}\n\\label{eq:fisher}\nF_{\\ell\\ell'}^{\\mathrm{TG}~ X'}=\\frac{1}{2}{\\rm Tr}\n \\Big[\\tens{C}^{-1}\\frac{\\partial {\\tens{C}}}{\\partial\n C_\\ell^{\\mathrm{TG}}}\\tens{C}^{-1}\\frac{\\partial\n \\tens{C}}{\\partial C_{\\ell'}^{X'}}\\Big],\n\\end{equation}\nand the $\\tens{E}$ matrix is given by\n\\begin{equation}\n\\label{eq:Elle}\n\\tens{E}_\\ell^X=\\frac{1}{2}\\tens{C}^{-1}\\frac{\\partial \\tens{C}}{\\partial\n  C_\\ell^X}\\tens{C}^{-1}.\n\\end{equation}\nThe object $\\tens{C} =\\tens{S}(C_{\\ell}^{X})+\\tens{N}$ is the global\ncovariance matrix, including the signal $\\tens{S}$ and noise $\\tens{N}$\ncontributions, with $C_\\ell^X$ being the fiducial theoretical angular power\nspectrum. \nThe uncertainties in the QML estimates are given by the inverse of the \nFisher matrix, which includes the correlation among different multipoles.\nThe error associated with the shot noise of the galaxy surveys is modelled\nin the galaxy submatrix of $\\tens{N}$.\n\nThe results presented on this paper are based on\n$\\hat{C}_{\\ell}^{\\mathrm{TG}}$, whereas $\\hat{Q}_{\\ell}^{\\mathrm{TG}}$\nis used as a cross-check, applied to a lower resolution version of the maps of $N_\\mathrm{side} = 32$.\nIn addition the maximum multipole considered in this case is $\\ell_\\mathrm{max} = 2N_\\mathrm{side}$, which\nhas been already verified as a conservative limit for the QML.\n\n\\subsubsection{Cross-correlation function}\n\\label{subsub:ccf}\n\nThe cross-correlation function (CCF) is a suitable tool for studying the\nISW effect via cross-correlation of the\nCMB fluctuations and tracers of the LSS, and it has been one of the most\nextensively used in this\ncontext~\\citep[e.g.,][]{Boughn2002b,Giannantonio2008b,Xia2009b}. On the one\nhand, the signal only appears at very large scales and, therefore,\nit is sufficient to work at resolutions at which the low performance of the\nCCF (in terms of computational time) is not\na serious handicap. On the other hand, neither the CMB nor the LSS data are\navailable with full sky coverage and, in some cases,\nthe geometry of the masks is very complicated: the CCF adapts perfectly to the\neffects of partial sky coverage, since it is defined in real\nspace. As a drawback, the Poissonian noise of the galaxy tracer appears at\nthe smallest angular scales, where the signal-to-noise of the ISW effect is\nhigher for this estimator. Therefore, a proper characterization of the shot\nnoise is especially important for the CCF, in order to obtain a good\nestimation of the uncertainties.\n\nLet us denote the CCF between the CMB map $T$ and an LSS tracer $G$\n(at an angular distance of $\\theta$) as\n$C^{\\mathrm{TG}}(\\theta)$ (i.e., $\\xi_a^{xy} \\equiv C^{\\mathrm{TG}}(\\theta)$\nfor this cross-correlation statistic).\nThe CCF estimator is defined as\n\\begin{equation}\n\\hat{C}^{\\mathrm{TG}}(\\theta) = \\frac{1}{N_\\theta}\\sum_{i,j}T_i G_j,\n\\label{eq:ccf}\n\\end{equation}\nwhere the sum runs over all pixels with a given angular separation.\nFor each angular bin centred around $\\theta$, $N_\\theta$ is the\nnumber of pixel pairs separated by an angle within the bin. Only the pixels\nallowed by the joint CMB and LSS mask are considered.\nThe angular bins used in this work are:\n$\\theta_1 \\in \\left[0,1\\right]^\\circ$; $\\theta_2 \\in \\left(1,3\\right]^\\circ$;\n$\\theta_3 \\in \\left(3,5\\right]^\\circ$; \\ldots; and $\\theta_{61} \\in\n\\left(119,121\\right]^\\circ$. The choice of binning does not\naffect the results significantly.\n\nThe covariance of the CCF estimator can be easily derived from the one\nalready computed for the CAPS in Eq.~\\ref{eq:cov_caps1}. It is sufficient\nto know that the CCF can be expressed in terms of the CAPS as\n\\begin{equation}\nC^{\\mathrm{TG}}\\left(\\theta\\right) =\n \\sum_{\\ell = 0}^{\\ell_\\mathrm{max}}\n \\frac{2\\ell + 1}{4\\pi}C_{\\ell}^{\\mathrm{TG}}P_\\ell\\left(\\cos{\\theta}\\right),\n\\label{eq:ccf_in_caps}\n\\end{equation}\nwhere $P_\\ell\\left(\\cos{\\theta}\\right)$ are the Legendre polynomials.\nHence, it is straightforward to prove that the covariance between the\n$\\theta$ and $\\theta'$ components of the CCF for the surveys $i$ and $j$,\nrespectively, is given by\n\\begin{equation}\nC_{\\theta \\theta', i j} = \\sum_{\\ell} \\sum_{\\ell'}\n \\frac{\\left(2\\ell + 1\\right)}{4\\pi}\n \\frac{\\left(2\\ell' + 1\\right)}{4\\pi}\n P_\\ell\\left(\\cos{\\theta}\\right) P_{\\ell'}\\left(\\cos{\\theta}'\\right)\n C_{\\ell \\ell', i j}.\n\\label{eq:ccf_cov}\n\\end{equation}\n\n\\subsubsection{Wavelet covariance}\n\nWavelets provide an interesting alternative to more traditional tools\n(e.g., CCF or CAPS) for studying the CMB-LSS correlation. They exploit the\nfact that the ISW signal is mostly concentrated at scales of a few\ndegrees~\\citep[e.g.,][]{Afshordi2004b}. Wavelets are ideal kernels to enhance\nfeatures with a characteristic size, since the wavelet analysis at an\nappropriate scale $R$ amplifies those features over the background. Therefore,\nwavelets could provide most of the signal-to-noise of the ISW effect by just\nanalysing a narrow range of scales.\nThey were first proposed for the ISW detection by~\\cite{Vielva2006}, where the\nSpherical Mexican Hat Wavelet~\\citep[SMHW,][]{Martinez2002} was proposed as\nthe filtering kernel.\nThe basic idea of this approach is to estimate the covariance of the SMHW\ncoefficients (SMHWcov) as a function of the wavelet\nscale~\\citep[see e.g.,][for details]{Vielva2006}. Other wavelet kernels can\nbe considered, such as needlets \\citep{Pietrobon2006a}, directional\nwavelets \\citep{Mcewen2007}, or steerable wavelets \\citep{McEwen2008b}.\n\nLet us denote the SMHWcov between the CMB map $T$ and a LSS tracer $G$\n(at a wavelet scale $R$) as\n$\\Omega^{\\mathrm{TG}}\\left(R\\right)$, i.e.,\n$\\xi_a^{xy} \\equiv \\Omega^{\\mathrm{TG}}\\left(R\\right)$\nfor this cross-correlation statistic).\nThe SMHWcov estimator is defined as\n\\begin{equation}\n\\hat{\\Omega}^{\\mathrm{TG}}\\left(R\\right) =\n \\frac{1}{N_\\mathrm{pix}}\\sum_{i}\\omega_{{\\rm T}_i}\\left(R\\right)\n \\omega_{{\\rm G}_i}\\left(R\\right),\n\\label{eq:smhwcov}\n\\end{equation}\nwhere $\\omega_{T}\\left(R\\right)$ and $\\omega_{G}\\left(R\\right)$ are the SMHW\ncoefficients for the CMB and the LSS at scale $R$, respectively\n(note that wavelet coefficients are forced to have zero mean on the observed\nsky). The scales considered in our study are\n$R = \\left\\lbrace  60, 90, 120, 150, 200, 250, 300, 350, 400, 500, 600\n \\right\\rbrace$ in arcminutes.\n\nAs for the CCF, the covariance of the SMHWcov estimator can be easily derived\nfrom the one already computed for the CAPS in Eq.~\\ref{eq:cov_caps1}.\nIt is sufficient to know that the SMHWcov can be expressed in terms of the\nCAPS as\n\\begin{equation}\n\\Omega^{\\mathrm{TG}}\\left(R\\right)\n = \\sum_{\\ell = 0}^{\\ell_\\mathrm{max}}\\frac{2\\ell + 1}{4\\pi}\n C_{\\ell}^{\\mathrm{TG}}\\omega_{\\ell}^2\\left(R\\right),\n\\label{eq:ccf_in_caps}\n\\end{equation}\nwhere $\\omega_\\ell\\left(R\\right)$ is the SMHW window function at the scale\n$R$. Hence, it is straightforward to prove that the covariance between the\n$R$ and $R'$ components of the SMWHcov for surveys $i$ and $j$, respectively,\nis given by:\n\\begin{equation}\nC_{R R', i j} = \\sum_{\\ell} \\sum_{\\ell'} \\frac{\\left(2\\ell + 1\\right)}{4\\pi}\n \\frac{\\left(2\\ell' + 1\\right)}{4\\pi} \\omega_{\\ell}^2\\left(R\\right)\n \\omega_{\\ell'}^2\\left(R'\\right) C_{\\ell \\ell', i j}.\n\\label{eq:ccf_cov}\n\\end{equation}\n\n\\subsection{Cross-correlation statistics}\n\\label{subsubsec:methods}\n\n\nLet us denote the expected cross-correlation of two signals ($x$ and $y$) by $\\xi_a^{xy}$, where $a$ stands for a distance measure (e.g., the angular distance $\\theta$ between two points in the sky, the multipole $\\ell$ of the harmonic transformation, or the wavelet scale $R$). For simplicity, we assume that the two signals are given in terms of a fluctuation field (i.e., with zero mean and dimensionless). \n\nThis cross-correlation could represent either the CCF, the CAPS or the SMHWcov. It has to be understood as a vector of $a_\\mathrm{max}$ components, where $a_\\mathrm{max}$ is the maximum number of considered \\emph{distances}. Obviously, when $x \\equiv y$, $\\xi_a^{xy}$ represents an auto-correlation. \nThe specific forms for $\\xi_a^{xy}$ and $\\mathbf{C}_{\\xi^{xy}}$ for the different cross-correlation statistics (CAPS, CCF, and SMHWcov) are given below. \n\n\n\\input{.\/section_4.0\/section_4.1.1}\n\n\\input{.\/section_4.0\/section_4.1.2}\n\n\\input{.\/section_4.0\/section_4.1.3}\n\n\n\n\n\\subsection{Cross-correlation tests}\n\\label{subsubsec:xcorrdef}\n\nFor any of the cross-correlation estimators described above, we aim two\ndifferent statistical tests.  First,\nif the observed cross-correlation is given by $\\hat{\\xi}_a^{xy}$, then,\na simple $\\chi^2$ can be proposed to estimate the amplitude $A$, such that\n$A\\xi_a^{xy}$ is the closest solution to $\\hat{\\xi}_a^{xy}$:\n\\begin{equation}\n\\chi^2\\left(A\\right) = \\left[\\hat{\\xi}_a^{xy} - A\\xi_a^{xy} \\right]^{\\rm T}\n \\tens{C}_{\\xi^{xy}}^{-1} \\left[\\hat{\\xi}_a^{xy} - A\\xi_a^{xy}\\right],\n\\label{eq:xcorr_fit}\n\\end{equation}\nwhere $\\tens{C}_{\\xi^{xy}}$ is the covariance matrix (of dimension\n$a_\\mathrm{max} \\times a_\\mathrm{max}$) of the expected cross-correlation\n$\\xi_a^{xy}$, i.e.,\n$\\mathbf{C}_{{\\xi^{xy}}_{i,j}} \\equiv\n \\left\\langle \\xi_{a_i}^{xy} \\xi_{a_j}^{xy} \\right\\rangle$.\nIt is straightforward to show that the best-fit amplitude $A$ and its\ndispersion are given by\n\\begin{eqnarray}\n\\label{eq:fit}\nA&=&\\left[\\hat{\\xi}_a^{xy}\\right]^t \\tens{C}_{\\xi^{xy}}^{-1} \\xi_a^{xy}\n \\left[\\left[\\xi_a^{xy}\\right]^t\n \\tens{C}_{\\xi^{xy}}^{-1} \\xi_a^{xy} \\right]^{-1}, \\\\\n \\sigma_A&=&\\left[\\left[\\xi_a^{xy}\\right]^t\n \\tens{C}_{\\xi^{xy}}^{-1} \\xi_a^{xy} \\right]^{-1\/2} \\nonumber.\n\\end{eqnarray}\nAn analogy with Eq.~\\ref{eq:xcorr_fit} can be defined for the null hypothesis\ncase:\n\\begin{equation}\n{\\chi^2}_{\\mathrm{null}} = \\left[\\hat{\\xi}_a^{xy}\\right]^{\\rm T}\n \\tens{D}_{\\xi^{xy}}^{-1} \\hat{\\xi}_a^{xy},\n\\label{eq:xcorr_null}\n\\end{equation}\nwhere $\\tens{D}_{\\xi^{xy}}$ is the covariance of the cross-correlation of the\ntwo signals, in the absence of an intrinsic dependence, i.e.,\nwhen $\\xi_a^{xy} \\equiv 0$. The ISW signal is very weak and, therefore it\nis a good approximation to assume that that\n$\\tens{D}_{\\xi^{xy}} \\approx \\tens{C}_{\\xi^{xy}}$. \n\nFor Gaussian statistics, ${\\chi^2}_{\\mathrm{null}}$ already provides the direct\nprobability of the observed cross-correlation $\\hat{\\xi}_a^{xy}$ under the\nnull hypothesis. However, several non-idealities (sky coverage, systematics,\nforegrounds residuals, etc.) forces is to use alternative approaches to\nestimate the probability. One of the most common options is to perform the\ncross-correlation of survey signal $y$ with realistic simulations of the CMB,\n$x$, and compute a joint statistics (e.g., ${\\chi^2}_{\\mathrm{null}}$) for\neach simulation. The probabiltiy value associated with the data will come then\nthen be the fraction of simulations having a value of\n${\\chi^2}_{\\mathrm{null}}$ equal to or larger than the one obtained for\nthe data.\nBoth, $\\tens{C}_{\\xi^{xy}}$ and $\\tens{D}_{\\xi^{xy}}$ can be derived either\nanalytically or numerically (via simulations). \n\nThe latter approach is computationally expensive, but, in some cases,\ncould provide a more accurate defence against certain systematics, in\nparticular the incomplete sky coverage. There are several options to perform\nsuch kind of simulations. The standard approach is the one\nmentioned above, i.e., cross-correlation of the LSS map with CMB simulations.\nThis is a very robust approach, since it is usually\nhard to reproduce the systematics present in the LSS tracers,\nbut incomplete because the LSS is fixed, resulting in a lack of randomness.\nAn alternative method is to use a jack-knife test, which unfortunately tends\nto underestimate the errors.\nFinally, one can produce simulations of both the CMB and the LSS, assuming\nperfect knowledge of the properties of both signals, in particular of the\nLSS field (which, as mentioned above, is almost never the case).\nComprehensive discussions of these approaches are given\nin~\\cite{Cabre2007} and~\\cite{Giannantonio2008b}.\n\n\\subsection{Results}\n\\label{subsubsec:results}\n\n\n\\begin{table}[tmb]\n\\begingroup\n\\newdimen\\tblskip \\tblskip=5pt\n\\caption{Expected significance $A\/\\,\\sigma_A$ of the CMB-LSS\ncross-correlation. Values obtained from each survey independently, as well as\njointly, are given for all the estimators (CAPS, CCF, and\nSMHWcov).\\label{tab:s2n_expected}\n\\label{tab:surveys}}\n\\nointerlineskip\n\\vskip -3mm\n\\footnotesize\n\\setbox\\tablebox=\\vbox{\n   \\newdimen\\digitwidth \n   \\setbox0=\\hbox{\\rm 0} \n   \\digitwidth=\\wd0 \n   \\catcode`*=\\active \n   \\def*{\\kern\\digitwidth}\n   \\newdimen\\signwidth \n   \\setbox0=\\hbox{+} \n   \\signwidth=\\wd0 \n   \\catcode`!=\\active \n   \\def!{\\kern\\signwidth}\n\\halign{\\hfil#\\hfil\\tabskip=0.4cm& \\hfil#\\hfil\\tabskip=0.4cm&\n \\hfil#\\hfil\\tabskip=0.4cm& \\hfil#\\hfil\\tabskip=0.4cm& \\hfil#\\hfil\\tabskip=0cm\\cr\n\\noalign{\\vskip 3pt\\hrule \\vskip 1.5pt \\hrule \\vskip 5pt}\n \\noalign{\\vskip -2pt}\n$\\hat{\\xi}_a^{xy}$&NVSS&SDSS&SDSS&All\\cr\n& &CMASS\/LOWZ&MphG&\\cr\n\\noalign{\\vskip 3pt\\hrule\\vskip 5pt}\nCAPS\t\t& 3.0& 1.9& 0.6& 3.2\\cr\nCCF\t\t& 3.0& 1.9& 0.6& 3.1\\cr\nSMMHWcov\t& 3.0& 1.9& 0.5& 3.1\\cr\n                                   \n\\noalign{\\vskip 3pt\\hrule\\vskip 3pt}}}\n\\endPlancktable                   \n\\endgroup\n\\end{table}\n\nIn this section we present the results obtained from the cross-correlation\nof the galaxy catalogues described in Sect.~\\ref{subsec:extdata}\n(NVSS, SDSS-CMASS\/LOWZ and SDSS-MphG) with the four \\textit{Planck}\\ CMB maps presented in \nSect.~\\ref{subsubsec:cmbmaps} ({\\tt C-R}, {\\tt NILC}, {\\tt SEVEM}, and {\\tt SMICA}).\nAll these maps are analysed at a {\\tt HEALPix} resolution of\n$N_\\mathrm{side} = 64$. The cross-correlation estimators described in the\nprevious section are applied to all cases. This comprehensive analysis\nwill help to achieve a robust estimation of the ISW.\n\nAs already mentioned the covariance among all the components of the estimators\nare obtained from coherent Gaussian simulations of the CMB and the three\ngalaxy catalogues. Since we are only considering large-scale effects (above\nabout $1^\\circ$), the same set of CMB simulations are equally valid for the\nfour CMB maps, since they are nearly identical on such\nscales~\\citep[see][for details]{planck2013-p06}.\nWe have used 70{,}000 coherent Monte Carlo simulation sets (as described in\nSect.~\\ref{subsub:caps}) to compute the correlations; this is enough to\ncharacterize the covariance.\n\nThe expected signal-to-noise ratio for the ISW effect detection is summarized\nin Table~\\ref{tab:s2n_expected}. Values for all the cross-correlation\nestimators are given. We consider the case of a survey-by-survey detection,\nas well as the joint analysis of all the surveys.\nA signal-to-noise of about $3\\,\\sigma$ is expected for the joint analysis,\nwhich is actually dominated by the NVSS cross-correlation. This is expected,\nsince, firstly, NVSS covers a much larger fraction of the sky compared to\nother surveys, and secondly, it extends over a redshift interval ideal for\nthe detection of the ISW signal~\\citep[e.g.][]{Afshordi2004b}.\n\nThe differences among estimators are not significant, indicating that none of\nthem is clearly optimal compared with the others.\nTo explore this agreement further, we have analysed an extra set of\n1{,}000 CMB and LSS clustered simulations, and have compared, simulation by\nsimulation, the ISW amplitude estimation derived for each cross-correlation\nestimator ($C_\\ell^{\\rm TG}$, $C^{\\rm TG}\\left(\\theta\\right)$ and\n$\\Omega^{\\rm TG}\\left(R\\right)$). \nIn Table~\\ref{tab:A_compared} we summarize the comparison. We only report\nvalues for the joint fit to the ISW amplitude for the three surveys.\nSimilar results are found survey by survey. For each pair of estimators,\nwe provide the mean difference among the amplitude estimations ($\\Delta A$),\nthe dispersion of these differences ($s_A$), the ratio ($\\eta$) of this\ndispersion to the expected sensitivity (i.e., the inverse of the\nsignal-to-noise numbers given in the last column of\nTable~\\ref{tab:s2n_expected}), and the correlation coefficient ($\\rho$).\nIt is clear that the agreement between estimators is very high and that\ndifferences are, on average, lower than half the statistical uncertainty\nimposed by the sampling variance. \n\n\\begin{table}[tmb]\n\\begingroup\n\\newdimen\\tblskip \\tblskip=5pt\n\\caption{For each pair of estimators, we give the mean difference among\nthe amplitude estimations ($\\Delta A$), the dispersion of these differences\n($s_A$), the ratio ($\\eta$) of this dispersion to the expected sensitivity\n(i.e., the inverse of the signal-to-noise numbers given in the last column\nof Table~\\ref{tab:s2n_expected}), and the correlation coefficient\n($\\rho$).\\label{tab:A_compared}}\n\\nointerlineskip\n\\vskip -3mm\n\\footnotesize\n\\setbox\\tablebox=\\vbox{\n   \\newdimen\\digitwidth \n   \\setbox0=\\hbox{\\rm 0} \n   \\digitwidth=\\wd0 \n   \\catcode`*=\\active \n   \\def*{\\kern\\digitwidth}\n   \\newdimen\\signwidth \n   \\setbox0=\\hbox{+} \n   \\signwidth=\\wd0 \n   \\catcode`!=\\active \n   \\def!{\\kern\\signwidth}\n\\halign{\\hfil#\\hfil\\tabskip=0.2cm& \\hfil#\\hfil\\tabskip=0.2cm&\n \\hfil#\\hfil\\tabskip=0.2cm& \\hfil#\\hfil\\tabskip=0.cm\\cr\n\\noalign{\\vskip 3pt\\hrule \\vskip 1.5pt \\hrule \\vskip 5pt}\n \\noalign{\\vskip -2pt}\n&&CCF&SMHWcov\\cr\n\\noalign{\\vskip 3pt\\hrule\\vskip 5pt}\n& $\\Delta A\\pm s_A$& $-0.01\\pm0.12$& $0.06\\pm0.07$\\cr\nCAPS& $\\eta$& $0.36$& $0.21$\\cr\n& $\\rho$& $0.93$& $0.98$\\cr\n\\noalign{\\vskip 3pt\\hrule\\vskip 5pt}\n& $\\Delta A\\pm s_A$&& $0.08\\pm0.14$\\cr\nCAPS& $\\eta$& & $0.42$\\cr\n& $\\rho$& & $0.92$\\cr\n                                   \n\\noalign{\\vskip 3pt\\hrule\\vskip 3pt}}}\n\\endPlancktable                   \n\\endgroup\n\\end{table}\n\n\n\n\n\\begin{table*}[tmb]\n\\begingroup\n\\newdimen\\tblskip \\tblskip=5pt\n\\caption{Amplitudes $A$, errors $\\sigma_A$, and significance\nlevels $A\/\\,\\sigma_A$ of the CMB-LSS cross-correlation (survey by survey and\nall together) due to the ISW effect, for all component separation algorithms\nfor the different estimators.\\label{tab:s2n_data}}\n\\nointerlineskip\n\\vskip -3mm\n\\footnotesize\n\\setbox\\tablebox=\\vbox{\n   \\newdimen\\digitwidth \n   \\setbox0=\\hbox{\\rm 0} \n   \\digitwidth=\\wd0 \n   \\catcode`*=\\active \n   \\def*{\\kern\\digitwidth}\n   \\newdimen\\signwidth \n   \\setbox0=\\hbox{+} \n   \\signwidth=\\wd0 \n   \\catcode`!=\\active \n   \\def!{\\kern\\signwidth}\n\\halign{#\\hfil\\tabskip=0.3cm& \\hfil#\\hfil\\tabskip=0.3cm&\n \\hfil#\\hfil\\tabskip=0.3cm& \\hfil#\\hfil\\tabskip=0.3cm&\n \\hfil#\\hfil\\tabskip=0.3cm& \\hfil#\\hfil\\tabskip=0.3cm&\n \\hfil#\\hfil\\tabskip=0.3cm& \\hfil#\\hfil\\tabskip=0.3cm&\n \\hfil#\\hfil\\tabskip=0.3cm& \\hfil#\\hfil\\tabskip=0.cm\\cr\n\\noalign{\\vskip 3pt\\hrule \\vskip 1.5pt \\hrule \\vskip 5pt}\n \\noalign{\\vskip -2pt}\nLSS data& $\\hat{\\xi}_a^{xy}$& {\\tt C-R}&& {\\tt NILC}\\hfil&& {\\tt SEVEM}&& {\\tt SMICA}\\hfil&\\cr\n\\noalign{\\vskip 2pt\\hrule\\vskip 3pt} \n& & $A^{{\\rm T}\\phi}$ & SNR & $A^{{\\rm T}\\phi}$ & SNR & $A^{{\\rm T}\\phi}$ & SNR & $A^{{\\rm T}\\phi}$ & SNR\\cr\n\\noalign{\\vskip 3pt\\hrule\\vskip 5pt}\nNVSS\t\t& CAPS \t\t& $0.86\\pm0.33$& 2.6& $0.91\\pm0.33$& 2.8& $0.90\\pm0.33$& 2.7& $0.91\\pm0.33$& 2.7\\cr \n\t\t\t& CCF \t\t& $0.80\\pm0.33$& 2.4& $0.84\\pm0.33$& 2.5& $0.83\\pm0.33$& 2.5& $0.84\\pm0.33$& 2.5\\cr \n\t\t\t& SMHWcov \t& $0.89\\pm0.34$& 2.6& $0.93\\pm0.34$& 2.8& $0.89\\pm0.34$& 2.6& $0.92\\pm0.34$& 2.7\\cr \n\\noalign{\\vskip 3pt\\hrule\\vskip 5pt}\nSDSS-CMASS\/LOWZ\t\t\t& CAPS \t\t& $0.98\\pm0.52$& 1.9& $1.09\\pm0.52$& 2.1& $1.06\\pm0.52$& 2.0& $1.09\\pm0.52$& 2.1\\cr \n\t\t& CCF \t\t& $0.81\\pm0.52$& 1.6& $0.91\\pm0.52$& 1.8& $0.89\\pm0.52$& 1.7& $0.90\\pm0.52$& 1.7\\cr \n\t\t\t& SMHWcov \t& $0.80\\pm0.53$& 1.5& $0.89\\pm0.53$& 1.9& $0.87\\pm0.53$& 1.6& $0.88\\pm0.53$& 1.7\\cr \n\\noalign{\\vskip 3pt\\hrule\\vskip 5pt}\nSDSS-MphG\t\t\t& CAPS \t\t& $1.31\\pm0.57$& 2.3& $1.43\\pm0.57$& 2.5& $1.35\\pm0.57$& 2.4& $1.42\\pm0.57$& 2.5\\cr \n\t\t\t& CCF \t\t& $1.00\\pm0.57$& 1.8& $1.11\\pm0.57$& 2.0& $1.10\\pm0.57$& 1.9& $1.10\\pm0.57$& 1.9\\cr \n\t\t\t& SMHWcov \t& $1.03\\pm0.59$& 1.8& $1.18\\pm0.59$& 2.0& $1.15\\pm0.59$& 2.0& $1.17\\pm0.59$& 2.0\\cr \n\\noalign{\\vskip 3pt\\hrule\\vskip 5pt}\nAll\t\t\t\t& CAPS \t\t& $0.84\\pm0.31$& 2.7& $0.91\\pm0.31$& 2.9& $0.88\\pm0.31$& 2.0& $0.90\\pm0.31$& 2.9\\cr \n\t\t& CCF \t\t& $0.77\\pm0.31$& 2.5& $0.83\\pm0.31$& 2.7& $0.82\\pm0.31$& 2.6& $0.82\\pm0.31$& 2.7\\cr \n\t\t\t& SMHWcov \t& $0.86\\pm0.32$& 2.7& $0.92\\pm0.32$& 2.9& $0.89\\pm0.32$& 2.8& $0.91\\pm0.32$& 2.9\\cr \n                                   \n\\noalign{\\vskip 5pt\\hrule\\vskip 3pt}}}\n\\endPlancktablewide                   \n\\endgroup\n\\end{table*}\n\nWe have fitted the observed cross-correlations to the expected ISW signal\n($C_\\ell^{\\rm TG}$, $C^{\\rm TG}\\left(\\theta\\right)$, and\n$\\Omega^{\\rm TG}\\left(R\\right)$, see Fig.~\\ref{fig:xcorr}), following\nEq.~\\ref{eq:fit}, i.e., allowing for a free amplitude of the expected signal. \nResults are summarized in Table~\\ref{tab:s2n_data}. \nNotice that the plots only show partial information on the total\ndetection level of the ISW signal, specifically coming from\nthe diagonal of the covariance matrix $\\tens{C}_{\\xi^{xy}}$.\nThe level of correlation among the components (i.e., the multipoles, angles\nor scales) varies considerably for the different estimators, and, therefore\nthe whole picture is only\nobtained when the full covariance is considered, as in Eq.~\\ref{eq:fit}.\n\nOverall, the ISW detection is at about the $3\\,\\sigma$ level and, as expected,\nit is clearly dominated by the NVSS signal. There are only small differences\namong estimators and CMB maps (as expected from the above discussion),\nindicating that this is a robust result.\nNotice that all the estimated amplitudes are compatible with unity, within the\nerror bars (especially for NVSS and SDSS-CMASS\/LOWZ). This is an additional validation of\nhow CMB and LSS are modelled. Values of $A$ deviating significantly from unity\nwould indicate some tension between the observed cross-correlation and the\nmodel (in particular on the LSS modelling, which is more complex).\nThe CAPS-QML, applied to the {\\tt SEVEM}\\ and NVSS (i.e., the survey with the\nhighest signal-to-noise), yields a value of $A = 0.73\\pm0.33$, which is\ncompatible with the CAPS, when applied to the same $N_\\mathrm{side}=32$\nand $\\ell_\\mathrm{max} = 2N_\\mathrm{side}$ resolution ($A = 0.84\\pm0.34$). \nPreliminary tests indicate that running the\nCAPS-QML at $N_\\mathrm{side}=64$ resolution could increase the sensitivity\nfor detecting the ISW effect with NVSS by $\\approx 20\\%$.\n\nOur results indicate a somewhat smaller signal-to-noise with respect to some\nprevious analyses on \\textit{WMAP}\\ data, where several (and in some case quite\nsimilar) surveys were also considered. For instance,~\\cite{Ho2008}\nand~\\cite{Giannantonio2012} found $3.7\\,\\sigma$ and $4.4\\,\\sigma$ detections,\nrespectively. Compatibility with the former is below $1\\,\\sigma$, whereas\nthere is more tension (around $1.5\\,\\sigma$) with the latter.\nA fraction of around $0.3\\,\\sigma$ of these differences could be explained in terms\nof the comological parameters adopted to defined the theoretical expectations.\nIn particular, the lower values of $H_0$ and $\\Omega_\\Lambda$ found by \\textit{Planck}~\\citep{planck2013-p11}\nwith respect to \\textit{WMAP}~\\citep[e.g.,][]{Larson2011}, imply a sensitivity for the ISW $\\approx 10\\%$\nsmaller.  \nThe rest of the differences come either from the LSS side, or \nfrom the error characterization, which depends on the presence of a correlated\nsignal between CMB and LSS\nsimulations~\\citep[see for instance][for a discussion]{Cabre2007}. \nSurvey modelling is another important aspect: besides systematic errors\nassociated with the galaxy identification and redshift estimation procedures,\nthere are complicated aspects, such as the bias characterization. \nAs was mentioned already, a strong point of our results, is the excellent\ncompatibility between the ISW amplitude estimates with respect to the\nexpected value. Whereas our estimation \ndeviates by about $0.5\\,\\sigma$ from the expected value,\nthe~\\cite{Giannantonio2012} result exceeds it by about $1\\,\\sigma$\nand~\\cite{Ho2008} is around $2\\,\\sigma$ above.\n\nNevertheless, the value of the ISW effect that we measure by means of NVSS\n(that because of to its large sky coverage, \nredshift range, and density of galaxies is probably the best current\ncatalogue for studying the ISW effect) is significant, and in agreement with\npreviously published results using \\textit{WMAP}.\n\n\\begin{figure*}[!ht]\n\\begin{center}\n\\includegraphics[width=6cm]{.\/section_4.0\/figs\/cls_tg_nvss.pdf}\n\\includegraphics[width=6cm]{.\/section_4.0\/figs\/ccf_tg_nvss.pdf}\n\\includegraphics[width=6cm]{.\/section_4.0\/figs\/wave_tg_nvss.pdf}\n\\includegraphics[width=6cm]{.\/section_4.0\/figs\/cls_tg_lrg.pdf}\n\\includegraphics[width=6cm]{.\/section_4.0\/figs\/ccf_tg_lrg.pdf}\n\\includegraphics[width=6cm]{.\/section_4.0\/figs\/wave_tg_lrg.pdf}\n\\includegraphics[width=6cm]{.\/section_4.0\/figs\/cls_tg_main.pdf}\n\\includegraphics[width=6cm]{.\/section_4.0\/figs\/ccf_tg_main.pdf}\n\\includegraphics[width=6cm]{.\/section_4.0\/figs\/wave_tg_main.pdf}\n\\end{center}\n\\caption{\\label{fig:xcorr}Observed and expected cross-correlation\nsignal versus multipole $\\ell$, for several surveys and different\ncross-correlation estimators.  Columns from left to right correspond to: CAPS;\nCCF; and SMHWcov.\nRows from top to bottom represent: NVSS; SDSS-CMASS\/LOWZ; and SDSS-MphG.  On each panel we\nshow the expected cross-correlation (black line) and the $\\pm1\\,\\sigma$\nregion (grey area).} \n\\end{figure*}\n\nWe have also studied the ISW signal from the point of view of its\ncompatibility with the null hypothesis. We have considered in this\nanalysis only the NVSS catalogue, since it provides the largest\ndetection of the ISW effect and, therefore, is the best of the existing\nsurveys for challenging the null hypothesis.  Probability values are\nsummarized in Table~\\ref{tab:s2n_pvalue}. As mentioned in\nSect.~\\ref{subsubsec:xcorrdef}, there is not a unique way of computing\nthe null hypothesis. Our approach follows Eq.~\\ref{eq:xcorr_null},\nwhere $\\tens{D}_{\\xi^{xy}}$ was computed out of 90{,}000 CMB\nsimulations that have been cross-correlated with the LSS data.\nThis matrix is used to\ncompute $\\chi_\\mathrm{null}^2$ from the data. This value is then\ncompared to its distribution for the null hypothesis, obtained from\n1{,}000 realistic CMB simulations (FFP-6) uncorrelated with NVSS,\nwhich have been processed in the same way as the \\textit{Planck}\\ data set.\nCAPS provides the smallest probability value, but the null hypothesis is\nrejected at about 10\\% only; this result is not unexpected, since\nan expected result since the ISW effect is weak.\n\nThe fact that the CAPS statistic provides tighter limits with respect to the\nCCF and SMHWcov could have been anticipated.\nIn our implementation, the CAPS explores the maximum angular range allowed\nfor a given map, whereas the CCF and the\nSMHWcov approaches are only evaluated at certain angles\/scales. This\nlimitation is not an issue in the analysis devoted to estimating the ISW\namplitude, since these angles\/scales are suitable for detecting the ISW.\nHowever, in order to discard the null hypothesis, the larger the number of\n``distances'' the better.\n\nThe previous approach is a frequentist one.  However, there is an\nalternative way of addressing the null hypothesis compatibility, in the\nframework of a hypotheses test.  In particular, we can study the ratio of\nBayesian evidence for both scenarios, comparing the alternative\n(there {\\it is\\\/} ISW signal) and the null (there is no correlation\nbetween the CMB and the LSS) hypotheses.  To compute the Bayesian evidence we\njust need the likelihoods and the priors associated with each hypothesis,\nwhich are already available.  The likelihood for the alternative hypothesis is\nobtained from Eq.~\\ref{eq:xcorr_fit}, peaked at the best-fit value for the ISW\namplitude, and its prior could be described as a Gaussian probability peaked\nat $A=1$ and with a dispersion given by $\\sigma_A$ in Eq.~\\ref{eq:fit}.\nConversely, the likelihood for the null hypothesis is the function given by\nEq.~\\ref{eq:xcorr_null}, and its prior would be as the one for the alternative\nhypothesis, but peaked at $A\\equiv0$ (notice that this is justified, since,\nas explained in the previous section,\n$\\tens{D}_{\\xi^{xy}} \\approx \\tens{C}_{\\xi^{xy}}$). Following this approach,\nwe obtain a Bayesian evidence ratio of around 30, which provides strong\nsupport for the presence of the ISW signal.\n\n\n\\begin{table}[tmb]\n\\begingroup\n\\newdimen\\tblskip \\tblskip=5pt\n\\caption{Probability values of the CMB-LSS cross-correlation for the\nNVSS survey under the null hypothesis, for the four component separation methods\nand for the different\nestimators.\\label{tab:s2n_pvalue}\n\\label{tab:surveys}}\n\\nointerlineskip\n\\vskip -3mm\n\\footnotesize\n\\setbox\\tablebox=\\vbox{\n   \\newdimen\\digitwidth \n   \\setbox0=\\hbox{\\rm 0} \n   \\digitwidth=\\wd0 \n   \\catcode`*=\\active \n   \\def*{\\kern\\digitwidth}\n   \\newdimen\\signwidth \n   \\setbox0=\\hbox{+} \n   \\signwidth=\\wd0 \n   \\catcode`!=\\active \n   \\def!{\\kern\\signwidth}\n\\halign{#\\hfil\\tabskip=0.2cm& \\hfil#\\hfil\\tabskip=0.2cm&\n\\hfil#\\hfil\\tabskip=0.2cm& \\hfil#\\hfil\\tabskip=0.2cm& \\hfil#\\hfil\\tabskip=0.2cm& \\hfil#\\hfil\\tabskip=0.cm\\cr\n\\noalign{\\vskip 3pt\\hrule \\vskip 1.5pt \\hrule \\vskip 5pt}\n\\noalign{\\vskip -2pt}\nLSS data& $\\hat{\\xi}_a^{xy}$& {\\tt C-R}& {\\tt NILC}& {\\tt SEVEM}& {\\tt SMICA}\\cr\n\\noalign{\\vskip 3pt\\hrule\\vskip 5pt}\nNVSS & CAPS& 0.09& 0.10& 0.10& 0.09\\cr\n& CCF& 0.33& 0.34& 0.40& 0.33\\cr\n & SMHWcov& 0.20& 0.23& 0.27& 0.19\\cr\n                                   \n\\noalign{\\vskip 3pt\\hrule\\vskip 3pt}}}\n\\endPlancktable                   \n\\endgroup\n\\end{table}\n\n\\section{Stacking of large-scale structures}\n\\label{subsec:stack}\n\nAn alternative approach for measuring the ISW effect in \\textit{Planck}\\ maps is to\nlook for an ISW signal directly at the positions of positive and\/or negative\npeaks in the potential.  Since the expected (and observed) signal is very weak,\nfor individual structures, a stacking technique needs to be applied.\nUsing the {\\it WMAP\\\/} data, it has been shown that CMB maps show hot spots\nand cold spots in the direction of superclusters and supervoids, respectively\n\\citep[][GR08 hereafter]{Granett2008a,Granett2008b}, which appear to be barely\nconsistent with the predictions of standard $\\Lambda$CDM\n\\citep[see also][]{Hernandez2012}. These structures, which are not yet\nvirialized, are evolving while the CMB photons travel across them and this\nshould contribute to the ISW effect.  We apply here the same approach to the\ndifferent \\textit{Planck}\\ maps, using the catalogues of superstructures introduced\nin Sect.~\\ref{sssect:catal}, and we test for the robustness of our findings.\nWe first discuss our method and the results obtained using the\ncatalogue provided by GR08, and then present the results obtained with the\nother catalogues.\n\n\\subsection{Method}\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.95\\hsize]{.\/section_5.0\/figs\/profile_freq_gr08_clusters.pdf}\n\\includegraphics[width=0.95\\hsize]{.\/section_5.0\/figs\/profile_freq_gr08_voids.pdf}\n\\end{center}\n\\caption{\\label{fig3_stack} Independence of the signal on the {\\tt SEVEM}\\\nfrequency channels.\nThe aperture photometry profiles measured in the stacked patches centred\non superclusters (top) and supervoids (bottom) are virtually identical for all\nfrequencies.}\n\\end{figure}\n\nOur analysis is performed on the {\\tt SMICA}\\ CMB map, although we have checked\nthat results are compatible for the other three \\textit{Planck}\\ maps.\nWe have also used the cleaned frequency maps from {\\tt SEVEM}\\\n(see Sect.~\\ref{subsubsec:cmbmaps}) for some of the tests.\nWe first remove the monopole and dipole of the maps (outside the U73 mask), and\nthen apply a compact source mask based on the Planck Legacy Point Source\nCatalogue~\\citep{planck2013-p06} to remove the contamination from individual\npoint sources. \n\nFor the purpose of comparison with the results of GR08, we smooth the\nCMB maps with a common Gaussian kernel of 30\\,\\ifmmode {^{\\scriptstyle\\prime}\\ FWHM. We then project them\nonto patches around each position in the supervoid and supercluster catalogue.\nThe GR08 structures have a relatively small size on the sky (a few degrees),\nbut the other two catalogues considered here contain many larger and closer\nvoids, covering larger angular sizes. Thus we work with $30\\ifmmode^\\circ\\else$^\\circ$\\fi\\times30\\ifmmode^\\circ\\else$^\\circ$\\fi$\nCMB patches and choose the pixel size to be 6\\,\\ifmmode {^{\\scriptstyle\\prime}, so that all voids\nconsidered are fully enclosed. We then co-add (stack) the maps, taking into\naccount the mask used.\nOn the stacked images, we calculate both the radial temperature profile and\nthe aperture photometry, to characterize the signal around density structures.\nThe temperature profile is obtained by computing the mean of the pixels in\nrings of fixed width and increasing angular radius; in practice, it is\ncalculated for 150 radii between $0\\ifmmode^\\circ\\else$^\\circ$\\fi$ and $15\\ifmmode^\\circ\\else$^\\circ$\\fi$, with a width of\n$\\Delta \\theta = 0\\pdeg1$. The photometry profile is obtained by applying a\ncompensated filter that subtracts the average temperature of a ring from\nthe average temperature within the disk whose radius $\\theta$ is the inner\nradius ring, and where the outer radius is chosen to be $\\theta\\sqrt{2}$, so\nthat the disc and ring have the same area.  This should enhance\nfluctuations of typical angular size $\\theta$ against\nfluctuations at smaller or larger scales.  Aperture photometry results are\nalso provide for at 150 angles, this time\nbetween $0\\ifmmode^\\circ\\else$^\\circ$\\fi$ and $15\/\\sqrt{2} \\approx 10\\pdeg6$.\nIn addition to the monopole and dipole, we also removed from the CMB maps\nthe contribution of large scale angular modes, namely $\\ell=2$--10.\nThese modes correspond to angular scales much larger than those of the\nstructures under investigation, and for our purposes their only effect\nis to introduce gradients in the stacked images; the high-pass filter\nessentially stops such gradients getting into the stacked map (which is\nequivalent to removing gradients at the end).\nThe contribution of the large-scale angular modes has no impact on the aperture\nphotometry profiles, and introduces only an offset in the temperature\nprofiles \\citep{Ilic2013}. \n\nIn order to estimate the significance of the resulting photometry and\ntemperature profiles, we follow a Monte Carlo approach based on stacked CMB\nimages chosen at random positions. In detail, we compute the photometry and\nthe temperature profiles for 16\\,000 sets of 50 CMB patches randomly\ndistributed over the SDSS area. We then compare the profiles obtained from\nthe stacking at the location of the GR08 superstructures to\nthese random profiles, in order to compute their signal-to-noise ratio.\n\n\\subsection{Results}\n\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=\\hsize]{.\/section_5.0\/figs\/fig_sutter_combined.pdf}\n\\end{center}\n\\caption{\\label{fig4_stack} Stacked images (left) and photometry profiles\n(right) obtained for the voids of~\\cite{Sutter2012} after rescaling. From top to bottom, we\nshow results obtained from stacking the largest 231, 589, and 936 voids.\nThe black circles superimposed indicate the angular radius, in terms of the effective radius after rescaling, at which the signal\nis maximal. The statistical dispersion is reduced as we stack more voids.\nHowever, the amplitude of the ``cold spot'' at about 0.5 times the effective\nvoid radius is mostly due to the surrounding ``hot shell'', which is easiest\nto see in the bottom panel.  This is further demonstrated by the $3\\,\\sigma$\nsignal detected using aperture photometry, seen at radii above 1.2 times the\neffective radius (for the 936 void case). Dashed and dotted lines in the\nright-hand panels represent positive and negative error bars, respectively,\nfrom $1$ to $4\\,\\sigma$.}\n\\end{figure}\n\nWe show in Fig.~\\ref{fig1_stack} the stacked images of the 50 supervoids and\n50 superclusters of GR08 in the \\textit{Planck}\\ map.\nThe corresponding temperature and photometry profiles, along with their\nsignificance levels, are shown in Fig.~\\ref{fig2_stack}. \nThe first thing to say is that, although the signatures are fairly weak,\nthe sign of the effect certainly seems to be correct.\nUsing the same catalogue and the \\textit{Planck}\\ CMB map, we find reasonable\nagreement with GR08. The maximal photometric decrement, $ -10.8\\,\\mu$K\n(essentially identical with the $-11.3\\,\\mu$K found by GR08),\ninduced by supervoids is obtained for a preferred scale of about $3\\pdeg5$\n($4\\ifmmode^\\circ\\else$^\\circ$\\fi$ in GR08) and a signal-to-noise of 3.3 ($3.7\\,\\sigma$ in GR08),\nas shown in Fig.~\\ref{fig2_stack}. Superclusters produce a photometric\nincrement of about $8.5\\,\\mu$K (slightly above the $7\\,\\mu$K in GR08),\nwith a significance of $3.0\\,\\sigma$ (compared with $2.6\\,\\sigma$ in GR08)\nat a slightly larger angle of $4\\pdeg7$. Finally, the stack of the combined\nsample (clusters minus voids) gives a temperature deviation of $8.7\\,\\mu$K,\nwith a signal strength of $4.0\\,\\sigma$ at $4\\pdeg1$, which is consistent with\nthe values reported in GR08.  The values of statistical\nsignificance for our aperture aperture photometry results are closely\nrelated to those for the temperature profiles. Indeed, as shown in the top\npanel of Fig.~\\ref{fig2_stack}, the temperature profile for the void stack\nshows a roughly $2\\,\\sigma$ deficit at small angular radii and a roughly\n$2\\,\\sigma$ excess extending to large radii.  Since \nthe aperture photometry is essentially an integral of the temperature profile\nwith a compensated filter, it picks up enhanced significance because of the\nshape of the temperature profile.\n\nAs noted previously by several authors~\\citep[e.g.,][]{Hernandez2012}, the \namplitude and shape of the \nphotometric profile found for voids and clusters is in tension (around $2\\,\\sigma)$ with the values\nexpected from pure ISW within $\\Lambda$CDM.  However, it is not\nstraightforward to associate this entire signal with a pure ISW effect.\nAs seen in the Fig.~\\ref{fig1_stack}, many small-scale structures -- both\ncold and hot -- are present around the region delineated by the angular radius\nat which the signal-to-noise ratio of the aperture photometry is maximal.\nThis small-scale structure contributes to the amplitude of the photometric\ndecrement, but at a few tens of $\\mu$K, which is incompatible with the\n$\\Lambda$CDM predictions for the ISW effect. These are rather simply\nbackground CMB fluctuations, with their lingering presence due to the small\nnumber (50) of patches which are used to produce the stack.\n\nIt is intriguing that the angular sizes of the catalogued superstructures\nare smaller than the angular sizes suggested by the photometry profiles.\nThis result is more apparent when we repeat the stacking analysis after\nrescaling each CMB patch by the effective radius of the structure it contains.\nSince the voids and super clusters identified by GR08 are roughly the same size,\nthe photometric results are similar after rescaling ($-10\\,\\mu$K for voids\nand $7.9\\,\\mu$K for superclusters). However, the deviations have significance\nlevels of $3.3\\,\\sigma$ and $2.7\\,\\sigma$ for supervoids and superclusters,\nrespectively, at angular sizes of $0.9$ (voids) and $2.6$ (clusters) times the\neffective radius of the structures. This mismatch, especially for clusters, could be a result of\nunderestimation of the structure extent the {\\tt ZOBOV} and {\\tt VOBOZ}\nalgorithms (as already suggested by GR08) or because larger potential hills\nand valleys underlie the detected superstructures. Since structure in\nthe potential is related to the density field through the Poisson equation,\ngravitational potential features are expected to cover larger scales than\nstructures in the density field.  Nevertheless, the factor of $2.6$ for\nthe case of superclusters seems large.  It is also true that\nthe GR08 superstructures were identified in the LRG subsample of the SDSS, and\nLRGs are known to be biased tracers of the matter density field\n\\citep[e.g.,][]{Tegmark2006}.  This bias could help explain why structures\nare larger than the scales identified by the {\\it VOBOZ} algorithm, although\nthe argument would go in the opposite sense for the voids.\nAnother way of stating this is \\citep{Hunt2010} that the relatively\nlarge effect decrement found for the GR08 voids can be only be\nattributed to the ISW effect only if the underdensities have been\nsignificantly underestimated or if the LRGs are under-biased.\n\nIt is therefore difficult to be confident that the signal is\ndue entirely to the ISW effect. We know, however, that the ISW signal generated\nby superstructures is expected to be achromatic, since it generates a\nfractional perturbation of the CMB temperature. In order to check if the\nsignal we measure is indeed independent of frequency, we applied the same\ntechnique to \\textit{Planck}\\ individual {\\tt SEVEM}\\ cleaned frequency maps from 44 to\n353\\,GHz. Lower (higher) frequency maps may be contaminated by radio (IR)\nsignals coming from our Galaxy and may thus introduce a bias in the\nmeasurement. Figure~\\ref{fig3_stack} shows the photometry profiles of\nsupercluster- and supervoid-stacked maps at 44, 70, 100, 143, 217, and\n353\\,GHz. The flux measured appears to be quite constant, which supports\nthe idea that the signal is due to the ISW effect induced by structures.\nIn the remainder of this section, we therefore apply our analysis only to\nthe {\\tt SMICA}\\ CMB map.\n \n\\subsection{Discussion and alternative catalogues}\n\nIt should be remembered that although the GR08 structures\nare considered to be good tracers of the cosmic matter distribution on scales\nlarger than $10\\, h^{-1}$\\,Mpc, they are also known for their sparsity at\nthese redshifts ($z\\approx0.4$--0.7). This sparsity could lead\nto biased estimates of the properties of the reconstructed voids, in\nparticular their sizes and depths could be biased. Moreover, some of the\nstructures overlap on the sky, which could lead to systematic effects in the\nstacking analysis.\n\nWe thus turn to other samples, for example\nthe catalogue of \\citet{Pan2012}, introduced in Sect.~\\ref{sssect:catal}.\nThe 1054 statistically significant voids it contains are larger than\n$10\\,h^{-1}$\\,Mpc in radius and, with redshifts lower than 0.1, they are\nmuch closer to us than the structures of GR08.  Direct stacking gives only a\nweak signal at about the $1\\,\\sigma$ level, which is difficult to reconcile\nwith the previous results. This may be due to the inclusion of a large number\nof small voids that could dilute the signal. Also, unlike the voids of GR08,\nthe voids of \\cite{Pan2012} have a large scatter in angular sizes on the sky,\nfrom about $2\\ifmmode^\\circ\\else$^\\circ$\\fi$ to $20\\ifmmode^\\circ\\else$^\\circ$\\fi$ \\citep[e.g.,][]{Ilic2013}.\nIn order to try to enhance the signal, we repeated the stacking after rescaling\nthe voids to their effective sizes. We also subdivided the catalogue into\nsub-samples based on redshift, radius, and\/or angular size.\nHowever, none of these attempts yielded any statistically significant result,\nin agreement with \\cite{Ilic2013}. \n\n\nFinally, we applied our procedure to the catalogue of voids published by\n\\citet[][]{Sutter2012}.  These cover a rather extended range of angular\nscales (about 2--$10\\ifmmode^\\circ\\else$^\\circ$\\fi$), and so we rescaled all the CMB patches by the\neffective radius of each void.  Stacking\nsubsample by subsample (\\textit{dim1}, \\textit{dim2}, \\textit{bright1},\n\\textit{bright2}, \\textit{lrgdim}, \\textit{lrgbright}), does not yield\nany significant signal. Similarly, when stacked together, the entire catalogue\ndoes not yield an ISW detection. However, since the ISW signal is expected to\nbe stronger for the largest and closest voids \\citep[e.g.,][]{Flender2013}\nwe tried starting from the largest void and adding them one by one, looking\nfor the optimal number of voids, i.e., that for which the signal-to-noise\nratio is maximal (see Fig.~\\ref{fig5_stack}). We found that stacking 231,\n589 or 936 voids gives roughly the same signal-to-noise ($2.5\\,\\sigma$,\n$2.0\\,\\sigma$ and $2.2\\,\\sigma$, respectively). However, the more voids we\nstack, the smaller the amplitude of the photometry signal\n(see Fig.~\\ref{fig4_stack}, this being about $-2.0\\,\\mu$K for 936 voids,\n$-2.1\\,\\mu$K for 589 voids and $-4.1\\,\\mu$K for 231 voids, at an angular size\nof about 0.5 times the common rescaled radius. These amplitudes are lower than\nthose found with the 50 GR08 voids, although above what is expected from\nnumerical simulations \\citep[see e.g., ][for higher\nredshift and larger voids]{Hernandez2012}, but see also \\cite{Cai2013}.\n\nThe apparent angular size detected (about $0.5$ times the effective void\nradius) in the photometry profile is smaller than that for the GR08 voids,\nbut in agreement with expectations from simulations \\citep{Cai2013}, and\nconsistent with the sizes detected using {\\it WMAP\\\/} data \\citep{Ilic2013}.\n\nThe profiles in Fig.~\\ref{fig4_stack} show \\modif{intriguing} hints (significance about\n$2\\,\\sigma$) of a positive excess below about 0.2 times the effective void\nradius.  \\modif{This is somewhat surprising, since the centre of each void of\nthe \\citet{Sutter2012} catalogue has been defined as the volume-weighted\nbarycentre of all the galaxies contained in the void volume.  This definition\nin principle removes most of the bias that may arise in other methods, such as\nchoosing the most underdense galaxy position, which would lead to a slight\noverdensity at the centre.}\nFig.~\\ref{fig4_stack} also shows positive excess for larger\napertures, partly caused by the large ``hot ring'' surrounding the cold\nfeature in the stacked images, which raises the mean temperature of the\nstacked image for discs of radii around 0.8--1.2 times the void radius.\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.95\\hsize]{.\/section_5.0\/figs\/number_patchs_sutt.pdf}\n\\end{center}\n\\caption{\\label{fig5_stack} Estimate of the optimal number of patches\/voids\nto stack using the catalogue of \\citet{Sutter2012}. Starting from the largest\nvoid and adding one CMB patch at a time to the stack, we find at each step\n$N$ the minimum of the aperture photometry profile, and we multiply this value\nby $\\sqrt{N}$ to find the largest signal-to-noise, assuming that the noise\nscales roughly as $1\/\\sqrt{N}$. The vertical axis has been normalized to the\nbest signal-to-noise, obtained for 231 voids.}\n\\end{figure}\n\n\n\\section{ISW map recovery}\n\\label{subsec:recov}\n\nIn recent years, some effort has been invested, not only to\nobtain the statistical cross-correlation signal between the CMB and LSS\ndata, but also to recover a map of the ISW signal itself\n\\citep{Barreiro2008,Barreiro2013,Francis2010,Dupe2011}.\nIn particular, assuming the existence of a correlation between the CMB and the\ngravitational potential, it is possible to recover a map of the\nISW fluctuations using a filtering method, given a tracer of the\ngravitational potential (e.g., the galaxy catalogues described in\nSect.~\\ref{subsec:extdata}) and the CMB fluctuations. Given the\nweakness of the signal, the main objective\nof this section is to provide a qualitative image of the ISW\nfluctuations for visual inspection, and  an additional\nconsistency test of the validity of the assumed fiducial model, by\ncomparing the statistical properties of the recovered and expected\nsignals. In addition, this secondary anisotropy map could also be\nused to study the large-scale properties of the CMB, and its possible\nrelation to some possible large-angle anomalies found in the \\textit{Planck}\\ data\n\\citep{planck2013-p09}. \n\n\n\\subsection{Method}\n\nWe have followed the methodology of \\cite{Barreiro2008}, which applies\na linear filter to the CMB and to a gravitational potential tracer map,\nin order to reconstruct an ISW map, assuming that the cross- and\nauto-spectra of the signals are known. This technique has been\nrecently applied to reconstruct the ISW map from the {\\it WMAP\\\/} data and\nNVSS galaxy map~\\citep{Barreiro2013}. The filter is implemented in\nharmonic space and the estimated ISW map $\\hat{s}_{\\ell m}$ at each\nharmonic mode is given by (see \\citealt{Barreiro2008} for details)\n\\begin{equation}\n\\label{eq:rec}\n\\hat{s}_{\\ell m}=\\frac{L_{12}(\\ell)}{L_{11}(\\ell)}g_{\\ell m} + \\frac{L_{22}^2(\\ell)}{L_{22}^2(\\ell)+C_{\\ell}^\\mathrm{n}}\\left( d_{\\ell m}-\\frac{L_{12}(\\ell)}{L_{11}(\\ell)}g_{\\ell m}\\right),\n\\end{equation}\nwhere $L(\\ell)$ corresponds to the Cholesky decomposition of the\ncovariance matrix between the considered tracer of the potential and\nthe ISW signal, at each multipole, which satisfies\n$C(\\ell)=L(\\ell)L^{\\rm T}(\\ell)$.\nHere $d_{\\ell m}$ and $g_{\\ell m}$ are the CMB\ndata and the gravitational potential tracer map, respectively, and\n$C_{\\ell}^\\mathrm{n}$ is the power spectrum of the CMB signal without including\nthe ISW effect. If full-sky coverage is not available, the covariance matrix is\nobtained from the corresponding pseudo-spectra. It can be shown that\nthe expected value of the power spectrum for the reconstructed signal\nis given by\n\\begin{equation}\n\\label{eq:cl_lcb}\n\\left< C_\\ell^{\\hat{\\mathrm{s}}} \\right>= \\frac{(C_\\ell^\\mathrm{gs})^2\\left(\\left|C(\\ell)\\right|+C_\\ell^\\mathrm{g}\nC_\\ell^\\mathrm{n}\\right)+\\left|\nC(\\ell)\\right|^2}{C_\\ell^\\mathrm{g}\\left(\\left|C(\\ell)\\right|+C_\\ell^\\mathrm{g} C_\\ell^\\mathrm{n}\\right)},\n\\end{equation}\nwhere $|C(\\ell)|$ is the determinant of the tracer-ISW covariance\nmatrix at each multipole, and $C_\\ell^\\mathrm{gs}$ and\n$C_\\ell^\\mathrm{g}$ are the\nassumed cross-spectrum and gravitational potential tracer\nspectra, respectively. Note that the recovered\nISW power spectrum will not contain the full ISW signal, since it can\nonly account for the part of the ISW signal probed by the tracer being\nconsidered.  It is also worth noting that in detail the expected\ncross-correlation depends on the assumed model.  However, in practice,\ngiven the weakness of the signal, it would be difficult to distinguish\nbetween two mild variants of the standard $\\Lambda$CDM model.\nNevertheless this approach still provides a useful consistency check.\n\n\\subsection{Results}\n\\label{subsec:iswmapresults}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=17.cm]{.\/section_6.0\/figs\/rec_isw_nvss_lensing_masked_n0064.pdf}\n\\caption[fig:nvsssky]{Reconstructed ISW map from the \\textit{Planck}\\ CMB\n  and NVSS data (left) and from the \\textit{Planck}\\ CMB and lensing potential maps\n  (right). Note that the maps are not expected to look exactly the same,\n  since each of them provides a partial reconstruction of the noisy ISW\n  signal (see Sect.~\\ref{subsec:iswmapresults} for details).  }\n\\label{fig:iswmap}\n\\end{figure*}\nWe have applied the filter described above to two different cases:\ncombining information from the CMB and the NVSS\ngalaxy catalogue; and applying the filter to\nthe CMB and the recovered lensing potential map described in\n\\ref{subsub:lens}. Results have been obtained for the four \\textit{Planck}\\ maps,\n{\\tt C-R}, {\\tt NILC}, {\\tt SEVEM}, and {\\tt SMICA}. For simplicity, we show the\nreconstructions only for the {\\tt SEVEM}\\\nCMB map, since the four methods give very similar results. The\nresolution considered for both analyses is $N_\\mathrm{side}=64$.\n\nFor the first case, we are using the \\textit{Planck}\\ fiducial\nmodel for the CMB and cross-power spectrum, while for the NVSS map we\nassume the model described in Sect.~\\ref{subsub:nvss}. We also take\ninto account the presence of Poissonian noise. We have excluded\nthe area obtained from combining the CMB mask at\n$N_\\mathrm{side}=64$ (described in Sect.~\\ref{subsubsec:cmbmaps})\nas well as the\narea which has not been observed by NVSS. The final mask keeps around\n62\\% of the sky. Since the filter is constructed in harmonic space, we\nhave used an apodized version of the mask in order to reduce the\nmask-induced correlations.  In any case, the degradation\nintroduced by the presence of a mask is small \\citep{Barreiro2008}.\n\nFor the second case, the lensing map involved\napplying a high-pass filter, which removed all\nmultipoles with $\\ell < 10$. This filtering was done in harmonic\nspace with the presence of a mask. To take this effect into\naccount we used a direct estimation of the pseudo-power spectrum of these\ndata for the power spectrum of the lensing map,\nafter applying the corresponding apodized mask.\nWe used the \\textit{Planck}\\ fiducial model for the other\npower spectra involved, but setting to zero the cross-power for $\\ell < 10$.\nA mask has been constructed by combining the CMB mask plus that\nprovided for the lensing potential map (described in\n\\citealt{planck2013-p12}), which keeps around 67\\% of the sky. The\ncorresponding apodized version of this mask was applied before\nreconstructing the ISW map. Note that the map given in\nFig.~\\ref{fig:cmb_lensing_maps} (right panel) corresponds, to a good\napproximation, to the first term of the right hand side of\nEq.~\\ref{eq:rec}.\n\nFigure~\\ref{fig:iswmap} shows the reconstructed ISW map using the \\textit{Planck}\\ CMB\nmap and NVSS (left panel) and that obtained combining the CMB with the\nlensing potential map (right panel).\nThere are similar structures present in both maps, but they are\nnot expected to look exactly the same, since each of them provides only a\npartial reconstruction of the ISW signal.\nThis is due to the fact that the reconstruction accounts for\nthe part of the ISW effect probed by the considered tracer, which is\ndifferent (although correlated) for each case. Moreover,\ndue to the high-pass filter applied to the lensing potential map, the\npower at $\\ell < 10$ for this case corresponds to the Wiener-filtered\nmap of the CMB (to which the filter given by Eq.~\\ref{eq:rec} defaults,\nif the cross-correlation is set to zero, as in this case), without\nadditional information from the considered tracer.\n\nFor both cases, we have tested that the power spectrum of the\nrecovered ISW signal, as well as that of the cross-power between the\nreconstructed ISW and the considered gravitational potential tracer,\nare consistent with the corresponding expected values. This indicates\nthe compatibility between the assumed fiducial model and the\nunderlying statistical properties of the data.\n\n\\section{Conclusions}\\label{sec:discussion}\n\nThis paper presents the first study of the ISW effect using \\textit{Planck}\\ data.\nWe derived results based on three different approaches: the detection of the\ninterplay between weak lensing of the CMB and the ISW effect, by looking at\nnon-Gaussian signatures; the conventional cross-correlations with tracers of\nlarge-scale structure; and aperture photometry on stacks of the CMB field at\nthe positions of known superstructures.  A reconstruction of the ISW map\ninferred from the CMB and LSS tracers was also provided.\n\nThe correlation with lensing allows, for the first time, the detection of the\nISW effect using only CMB data. This is an effective approach, because the\ngravitational potential responsible for deflecting CMB photons also generates\nISW temperature perturbations. Using different estimators, we investigated\nthe correlation of the \\textit{Planck}\\ temperature map with a reconstruction of the\nlensing potential on the one hand, and the estimation of the ISW-lensing\ngenerated non-Gaussian signature on the other. \nWe found that the signal strength is close to $2.5\\,\\sigma$, for several\ncombinations of estimator implementation and foreground-cleaned CMB maps.\n\nWe computed cross-correlations between the \\textit{Planck}\\ CMB temperature map,\nand tracers of large-scale structure, namely: the NVSS survey of radio\nsources; and the SDSS-CMASS\/LOWZ, and SDSS-MphG\\ galaxy samples. As estimators we considered\nthe angular cross-correlation function, the angular cross-spectra,\nand the variance of wavelet coefficients as a function of angular scale. \nWe performed a comparison on different component-separation maps, where we\nconsidered {\\tt C-R}, {\\tt NILC}, {\\tt SEVEM}, and {\\tt SMICA}, and found remarkable agreement\nbetween the results, indicating that the the low multipoles are robustly\nreconstructed. Covariance matrices between the cross-correlation quantities\nwere estimated for a set of Gaussian realizations of the CMB for the \\textit{Planck}\\\nfiducial model. For the ISW effect, we report detection significance levels\nof $2.9\\,\\sigma$ (NVSS), $1.7\\,\\sigma$ (SDSS-CMASS\/LOWZ), and $2.0\\,\\sigma$ (SDSS-MphG), which\nare consistent among the different estimators considered.  Although these\nnumbers are compatible with previous claims which used \\textit{WMAP}\\ data, they are\ngenerally smaller. We believe that this discrepancy is mainly due to the\ndifferent characterization of the surveys and treatment of uncertainties,\nsince the measurement of the CMB fluctuations at the scales which contribute\nto the ISW detection are very similar for \\textit{Planck}\\ and \\textit{WMAP}. Only a fraction\nof these differences (around $0.3\\,\\sigma$) could be understood in terms of the different cosmological models used by each experiment -- in particular, the\nlower values of $H_0$ and $\\Omega_\\Lambda$ reported by \\textit{Planck}\\ compared with\n{\\it WMAP}.\n\nA strength of our new study lies in the fact that the amplitudes derived for\nthe expected signals are largely consistent with unity (i.e., the model\nexpectation), which indicates good modelling of the surveys.\nThe CMB and LSS cross-correlation has also been tested against the null\nhypothesis, i.e., whether the observed signal is compatible with a null\ncorrelation. As expected for such a weak signal, there is no strong evidence\nof incompatibility with the lack of correlation. In this respect, the CAPS\napproach seems to provide better constraints than the other estimators\ninvestigated here (CCF and the SMHWcov).\n\nWe explored the aperture photometry of stacked CMB patches at the positions\nof superstructures identified in the SDSS galaxy distribution. Our analysis\nof the \\citet{Granett2008a} catalogue (50 supervoids and 50 superstructures)\nreproduced previous results, with similarly strong amplitude and significance\nlevels (somewhat above and below $3\\,\\sigma$ for voids and clusters,\nrespectively). While the most plausible source of this signal is the ISW\neffect associated with these structures, it shows some tension with\nexpectations, both in terms of amplitude and scale. \nThe same type of analysis was carried on the latest and much larger void\ncatalogues of \\citet{Sutter2012} (about $1\\,500$ voids) and \\citet{Pan2012}\n(about $1\\,000$ voids). The results range from negligible to evidence at the\n2--2.5\\,$\\sigma$ level, with a more moderate amplitude and a smaller scale,\nin better agreement with theoretical predictions found in the literature.\nThe broad spectral coverage of \\textit{Planck}\\ allows us to confirm the achromatic\nnature of these signals over the 44 to 353\\,GHz range, supporting their\ncosmological origin. \n\nWe reconstructed maps of the ISW effect using a linear filter, by combining\nthe \\textit{Planck}\\ CMB and a gravitational potential tracers. In particular, we\nconsidered both the NVSS catalogue and the reconstructed CMB lensing map as\nLSS tracers. Again we found good agreement between different component\nseparation methods, as well as consistency between the expected and\nreconstructed auto- and cross-power spectra for the recovered ISW map.\n\nWe conclude that the ISW effect is present in \\textit{Planck}\\ data\nat the level expected for $\\Lambda$CDM-cosmologies,\nusing a range of measurement methods, although there is\na possible tension with the results from stacking of CMB fields centred on\nsuperstructures.  Generally, our results are more conservative than previous\nclaims using \\textit{WMAP}\\ data, but the agreement with the expected signal is better.\nFuture \\textit{Planck}\\ data releases, including polarization information, as well as\nimproved understanding of foregrounds, could improve on these results, in\nparticular for ISW-lensing correlation and ISW-lensing map reconstruction.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{\\label{sec:Adaptive-Channel-Recommendation}Adaptive Channel Recommendation\nScheme}\n\nWe first study the optimal channel recommendation in the homogeneous channel environment, i.e., each channel $m$ has the same data rate $B_{m}=B$ and identical channel state changing probabilities $p_{m}=p,q_{m}=q$. The generalization to the heterogeneous channel setting will be discussed in Section \\ref{sec:Adaptive-Channel-RecommendationII}. To find the optimal adaptive spectrum access strategy, we formulate the system as a Markov Decision Process\n(MDP). For the sake of simplicity, we assume that the recommendation\nbuffer size $W=1$, i.e., users only consider the recommendations received in the last time slot. Our method also applies to the case when $W>1$ by using\na high-order MDP formulation, although the analysis is more involved.\n\n\\begin{table*}[tt]\n\\begin{align}\nP{}_{R,R'}^{P_{rec}}= & \\sum_{m_{r}+m_{u}=R'\\ }\\sum_{R\\geq\\bar{m}_{r}\\geq m_{r},M-R\\geq\\bar{m}_{u}\\geq m_{u}\\ }\\sum_{n_{r}+n_{u}=N,n_{r}\\geq\\bar{m}_{r},n_{u}\\geq\\bar{m}_{u}}\\left(\\begin{array}{c}\nN\\\\\nn_{r}\\end{array}\\right)P_{rec}^{n_{r}}(1-P_{rec})^{n_{u}} \\nonumber \\\\\n & \\cdot\\left(\\begin{array}{c}\n\\bar{m}_{r}\\\\\nm_{r}\\end{array}\\right)(1-q)^{m_{r}}q^{\\bar{m}_{r}-m_{r}}\\frac{R!}{(R-\\bar{m}_{r})!}\\left(\\begin{array}{c}\nn_{r}-1\\\\\n\\bar{m}_{r}-1\\end{array}\\right)R{}^{-n_{r}} \\nonumber \\\\\n & \\cdot\\left(\\begin{array}{c}\n\\bar{m}_{u}\\\\\nm_{u}\\end{array}\\right)(\\frac{p}{p+q})^{m_{u}}(\\frac{q}{p+q})^{\\bar{m}_{u}-m_{u}}\\frac{(M-R)!}{(M-R-\\bar{m}_{u})!}\\left(\\begin{array}{c}\nn_{u}-1\\\\\n\\bar{m}_{u}-1\\end{array}\\right)(M-R){}^{-n_{u}}. \\label{eq:2565} \\end{align}\n\\hrule\n\\end{table*}\n\n\\subsection{MDP Formulation For Adaptive Channel Recommendation}\nWe model the system as a MDP as follows:\n\\begin{itemize}\n\\item \\emph{System state}: $R\\in\\mathcal{R}\\triangleq\\{0,1,...,\\min\\{M,N\\}\\}$ denotes the number of recommended channels at the end of time slot $t$. Since we assume that all channels are statistically identical, then there is no need to keep track of the recommended channel IDs\\footnote{Users need to know the IDs of the recommended channels in order to access them. However, the IDs are not important in terms of MDP analysis.}.\n\\item \\emph{Action}: $P_{rec}\\in\\mathcal{P}\\triangleq(0,1)$ denotes the branching probability of choosing the set of recommended channels.\n\\item \\emph{Transition probability}: The probability that action $P_{rec}$ in system state $R$ in time slot $t$ will lead to system state $R'$ in the next time\nslot  is\n\\[P{}_{R,R'}^{P_{rec}}=Pr\\{R(t+1)=R'|R(t)=R,P_{rec}(t)=P_{rec}\\}.\\]\nWe can compute this probability as in (\\ref{eq:2565}), with detailed derivations given in Appendix \\ref{Derivation}.\n\\item \\emph{Reward}: $U(R,P_{rec})$ is the expected system throughput in the next time slot when the action $P_{rec}$ is taken under the current system state $R$, i.e.,\n\\[\nU(R,P_{rec})=\\sum_{R\\in\\mathcal{R'}}P{}_{R,R'}^{P_{rec}}U_{R'},\\]\nwhere $U_{R'}$ is the system throughput in state $R'$. If $R'$ idle channels are utilized by the secondary users in a time slot, then these $R'$ channels will be recommended at the end of the time slot. Thus, we have \\[\nU_{R'}=R'B.\\]\nRecall that $B$ is the data rate that a single user can obtain on an\nidle channel.\n\\item \\emph{Stationary Policy:} $\\pi\\in\\Omega\\triangleq\\mathcal{P}{}^{|\\mathcal{R}|}$ maps each state $R$ to an action $P_{rec}$, i.e., $\\pi(R)$ is the action $P_{rec}$ taken when the system is in state $R$. \\rev{The mapping is stationary and does not depend on time $t$.}\n\\end{itemize}\n\nGiven a stationary policy $\\pi$ and the initial state $R_{0}\\in\\mathcal{R}$,\nwe define the network's value function as the time average system throughput,\ni.e. \\[\n\\Phi_{\\pi}(R_{0})=\\lim_{T\\rightarrow\\infty}\\frac{1}{T}E_{\\pi}\\left[\\sum_{t=0}^{T-1}U(R(t),\\pi(R(t)))\\right].\\]\nWe want to find an optimal stationary policy $\\pi^{*}$ that maximizes the value function $\\Phi_{\\pi}(R_{0})$  for any initial state $R_{0}$, i.e.\\[\n\\pi^{*}=\\arg\\max_{\\pi}\\Phi_{\\pi}(R_{0}),\\forall R_{0}\\in\\mathcal{R}.\\]\n\\rev{Notice that this is a system wide optimization, although the optimal solution can be implemented in a distributed fashion. This is because every user knows the number of recommended channels $R$, and it can determine the same optimal access probability locally.} For example, each user can calculate the optimal spectrum access policy off-line, and determine the real-time optimal channel access probability $P_{rec}$ locally by observing the number of recommended channels $R$ after entering the network.\n\n\n\\subsection{Existence of Optimal Stationary Policy}\n\nMDP formulation above is an average reward based MDP. We can prove that an optimal stationary policy that is independent of initial system state always exists in our MDP formulation. The proof relies on the following lemma from \\cite{key-4}.\n\\begin{lem} \\label{lemmaA}\nIf the state space is finite and every stationary policy leads\nto an irreducible Markov chain, then there exists a stationary policy\nthat is optimal for the average reward based MDP.\n\\end{lem}\nThe irreducibility of Markov chain means that it is possible to get to any state from any state. For the adaptive channel recommendation scheme, we have\n\\begin{lem}\\label{lemmaB}\nGiven a stationary policy $\\pi$ for the adaptive channel recommendation\nMDP, the resulting Markov chain is irreducible.\n\\end{lem}\n\\begin{proof}\nWe consider the following two cases:\n\nCase I, when $0<q<1$: since $0<P_{rec}<1$, $0<p\\leq1$, and $0<q<1$, we can verify that given any state $R$, the transition probability $P{}_{R,R'}^{P_{rec}}>0$ for all $R'\\in\\mathcal{R}$. Thus, any two states communicate with each other.\n\nCase II, when $q=1$: for all $R\\in\\mathcal{R}$, the transition probability $P{}_{R,R'}^{P_{rec}}>0$ if $R'\\in\\{0,...,\\min\\{M-R,N\\}\\}$. It follows that the state $R'=0$ is accessible from any other state\n$R\\in\\mathcal{R}$. By setting $R=0$, we\nsee that $P{}_{R,R'}^{P_{rec}}>0$,  for all $R'\\in\\{0,...,\\min\\{M,N\\}\\}$. That\nis, any other state $R'\\in\\mathcal{R}$ is also accessible from the state\n$R=0$. Thus, any two states communicate with each other.\n\nSince any two states communicate with each other in all cases and\nthe number of system state $|\\mathcal{R}|$ is finite, the resulting\nMarkov chain is irreducible.\n\\end{proof}\nCombining Lemmas \\ref{lemmaA} and \\ref{lemmaB}, we have\n\\begin{thm}\nThere exists an optimal stationary policy for the adaptive channel\nrecommendation MDP.\n\\end{thm}\n\nFurthermore, the irreducibility of the adaptive channel\nrecommendation MDP also implies that the optimal stationary policy $\\pi^{*}$ is independent of the initial state $R_{0}$ \\cite{key-4}, i.e.\n\\[\n\\Phi_{\\pi^{*}}(R_{0})=\\Phi_{\\pi^{*}}, \\forall R_{0}\\in\\mathcal{R},\n\\]where $\\Phi_{\\pi^{*}}$ is the maximum time average system throughput.\nIn the rest of the paper, we will just use \\textquotedblleft{}optimal policy\\textquotedblright{} to refer \\textquotedblleft{}optimal stationary policy that is independent of the initial system state\\textquotedblright{}.\n\n\n\n\\subsection{Structure of Optimal Stationary Policy}\n\nNext we characterize the structure of the optimal policy without using the closed-form expressions of the policy (which is generally hard to achieve). The key idea is to treat the average\nreward based MDPs as the limit of a sequence of discounted reward MDPs with discounted\nfactors going to one. Under the irreducibility condition, the average reward based MDP thus inherits the\nstructure property from the corresponding discounted reward MDP \\cite{key-4}.\nWe can write down the Bellman equations of the discounted version of our MDP problem as:\n\\begin{equation}\nV_{t}(R)=\\max_{P_{rec}\\in\\mathcal{P}}\\sum_{R'\\in\\mathcal{R}}P{}_{R,R'}^{P_{rec}}[U_{R'}+\\beta V_{t+1}(R')],\\forall R\\in\\mathcal{R},\\label{eq:5647}\\end{equation}\nwhere $V_{t}(R)$ is the discounted maximum expected system throughput starting from time\nslot $t$ when the system in state $R$.\n\nDue to the combinatorial complexity of the transition probability\n$P{}_{R,R'}^{P_{rec}}$ in (\\ref{eq:2565}), it is difficult to obtain\nthe structure results for the general case. We further limit our attention to the following two asymptotic cases.\n\n\n\\subsubsection{Case One, the number of channels $M$ goes to infinity while the number of users $N$ stays finite}\n\nIn this case, the number of channels is much larger than the number of secondary users, and thus heavy congestion rarely happens on any channel. Thus it is safe to emphasizing on accessing the recommended channels. Before proving the main result of Case One in Theorem \\ref{thmM>N}, let us first characterize the property of discounted maximum expected system payoff $V_t(R)$.\n\\begin{proposition}\n\\label{lem:When-,-i.e.II}When $M=\\infty$ and $N<\\infty$\n, the value function $V_{t}(R)$ for the discounted adaptive\nchannel recommendation MDP is nondecreasing in $R$ .\\end{proposition}\n\nThe proof of Proposition \\ref{lem:When-,-i.e.II} is given in the Appendix.\nBased on the monotone property of the value function $V_{t}(R)$, we prove the following main result.\n\\begin{thm}\n\\label{thmM>N}\nWhen $M=\\infty$ and $N<\\infty$, for the adaptive channel\nrecommendation MDP, the optimal stationary policy $\\pi^{*}$ is monotone,\nthat is, $\\pi^{*}(R)$ is nondecreasing on $R\\in\\mathcal{R}$.\\end{thm}\n\\begin{proof}\nFor the ease of discussion, we define \\[\nQ_{t}(R,P_{rec})=\\sum_{R'\\in\\mathcal{R}}P{}_{R,R'}^{P_{rec}}[U_{R'}+\\beta V_{t+1}(R')],\\]\nwith the partial cross derivative being \\begin{eqnarray*}\n\\frac{\\partial^{2}Q_{t}(R,P_{rec})}{\\partial R\\partial P_{rec}} & = & \\frac{\\partial\\sum_{R'\\in\\mathcal{R}}P{}_{R+1,R'}^{P_{rec}}[U_{R'}+\\beta V_{t+1}(R')]}{\\partial P_{rec}}\\\\\n &  & -\\frac{\\partial\\sum_{R'\\in\\mathcal{R}}P{}_{R,R'}^{P_{rec}}[U_{R'}+\\beta V_{t+1}(R')]}{\\partial P_{rec}}.\\end{eqnarray*}\nBy Lemma $6$ in the Appendix, we know the reverse cumulative distribution\nfunction $\\sum_{R'\\in\\mathcal{R}}P{}_{R,R'}^{P_{rec}}$ is supermodular on $\\mathcal{R}\\times\\mathcal{P}$.\nIt implies \\[\n\\frac{\\partial\\sum_{R'\\in\\mathcal{R}}P{}_{R+1,R'}^{P_{rec}}}{\\partial P_{rec}}-\\frac{\\partial\\sum_{R'\\in\\mathcal{R}}P{}_{R,R'}^{P_{rec}}}{\\partial P_{rec}}\\geq0.\\]\nSince $V_{t+1}(R')$ is nondecreasing in $R'$ by Proposition \\ref{lem:When-,-i.e.II} and  $U_{R'}=R'B$,   we know that $U_{R'}+\\beta V_{t+1}(R')$ is also nondecreasing in $R'$.\nThen we have\\begin{eqnarray*}\n &  & \\frac{\\partial\\sum_{R'\\in\\mathcal{R}}P{}_{R+1,R'}^{P_{rec}}[U_{R'}+\\beta V_{t+1}(R')]}{\\partial P_{rec}}\\\\\n & \\geq & \\frac{\\partial\\sum_{R'\\in\\mathcal{R}}P{}_{R,R'}^{P_{rec}}[U_{R'}+\\beta V_{t+1}(R')]}{\\partial P_{rec}},\\end{eqnarray*}\ni.e.,\\[\n\\frac{\\partial^{2}Q_{t}(R,P_{rec})}{\\partial R\\partial P_{rec}}\\geq0,\\]\nwhich implies that $Q_{t}(R,P_{rec})$ is supermodular on $\\mathcal{R}\\times\\mathcal{P}$.\nSince \\[\n\\pi^{*}(R)=\\arg\\max_{P_{rec}}Q_{t}(R,P_{rec}),\\]\nby the property of super-modularity, the optimal policy $\\pi^{*}(R)$\nis nondecreasing on $R$ for the discounted MDP above. Since the average\nreward based MDP inherits its structure property, this result is also\ntrue for the adaptive channel recommendation MDP.\n\\end{proof}\n\n\\subsubsection{Case Two, the number of users $N$ goes to infinity while the number of channels $M$ stays finite}\nIn this case, the number of secondary users is much larger than the number of channels, and thus congestion becomes a major concern. However, since there are infinitely many secondary users, all the idle channels at each time slot can be utilized as long as users have positive probabilities to access all channels. From the system's point of view, the cognitive radio network operates in the saturation state. Formally, we show that\n\\begin{thm}\\label{InfN}\nWhen $N=\\infty$ and $M<\\infty$, for the adaptive channel\nchannel recommendation MDP, any stationary policy $\\pi$ satisfying\\[\n0<\\pi(R)<1,\\forall R\\in\\mathcal{R},\\]\nis optimal. \\end{thm}\n\\begin{proof}\nWe first define the sets of policies $\\Delta\\triangleq\\{\\pi:0<\\pi(R)<1,\\forall R\\in\\mathcal{R}\\}$\nand $\\Delta^{c}=\\Omega\\backslash\\Delta$.  Recall that the value of $\\pi(R)$ equals the probability of choosing the set of recommended channels, i.e., $P_{rec}$.\n\nThen it is easy to check that the probability of accessing an arbitrary channel\n$m$ is positive under any policy $\\pi\\in\\Delta$. Since\nthe number of secondary users $N=\\infty$, it implies that all the\nchannels will be accessed by the secondary users. In this case, the\ntransition probability from a system state $R$ to $R'$ of the resulting\nMarkov chain is given by\\begin{eqnarray}\n &  & P_{R,R'}^{\\pi(R)}\\nonumber \\\\\n & = & \\sum_{m_{r}+m_{u}=R',m_{r}\\le R,m_{u}\\le M-R}\\left(\\begin{array}{c}\nR\\\\\nm_{r}\\end{array}\\right)(1-q)^{m_{r}}q^{R-m_{r}}\\nonumber\\\\\n &  & \\cdot\\left(\\begin{array}{c}\nM-R\\\\\nm_{u}\\end{array}\\right)(\\frac{p}{p+q})^{m_{u}}(\\frac{q}{p+q})^{M-R-m_{u}},\\label{eq:215.1}\\end{eqnarray}\nwhich is independent of the branching probability $\\pi(R)$. It implies\nthat any policy $\\pi\\in\\Delta$ leads to a Markov chain with\nthe same transition probabilities $P_{R,R'}^{P_{rec}}$. Thus, any\npolicy $\\pi\\in\\Delta$ offers the same time average system throughput.\n\nWe next show that any policy $\\pi'\\in\\Delta^{c}$  leads to a payoff no better than the payoff of a policy $\\pi\\in\\Delta$. For a policy $\\pi'$ where there exists some states $\\bar{R}$\nsuch that $\\pi'(\\bar{R})=0$, the transition probability from\nthe system state $\\bar{R}$ to $R'$ is\\begin{eqnarray*}\nP_{\\bar{R},R'}^{\\pi'(\\bar{R})} & = & \\begin{cases}\n\\left(\\begin{array}{c}\nM-\\bar{R}\\\\\nR'\\end{array}\\right)(\\frac{p}{p+q})^{R'}(\\frac{q}{p+q})^{M-\\bar{R}-R'}\\\\\n\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\mbox{If}\\ R'\\le M-\\bar{R},\\\\\n0\\mbox{\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ If}\\ R'>M-\\bar{R}.\n\\end{cases}\\end{eqnarray*}\nIf there exists some states $\\hat{R}$ such that $\\pi'(\\hat{R})=1$,\nwe have the transition probability as\\begin{eqnarray*}\nP_{\\hat{R},R'}^{\\pi'(\\hat{R})} & = & \\begin{cases}\n\\left(\\begin{array}{c}\n\\hat{R}\\\\\nR'\\end{array}\\right)(1-q)^{R'}q^{\\hat{R}-R'} & \\mbox{If\\ }R'\\le\\hat{R},\\\\\n0 & \\mbox{If\\ }R'>\\hat{R}.\\end{cases}\\end{eqnarray*}\nSince \\begin{eqnarray*}\n &  & \\left(\\begin{array}{c}\nM-\\bar{R}\\\\\nR'\\end{array}\\right)(\\frac{p}{p+q})^{R'}(\\frac{q}{p+q})^{M-\\bar{R}-R'}\\\\\n & = & \\sum_{j=0}^{\\bar{R}}\\left(\\begin{array}{c}\n\\bar{R}\\\\\nj\\end{array}\\right)(1-q)^{j}q^{\\bar{R}-j}\\\\\n &  & \\cdot\\left(\\begin{array}{c}\nM-\\bar{R}\\\\\nR'\\end{array}\\right)(\\frac{p}{p+q})^{R'}(\\frac{q}{p+q})^{M-\\bar{R}-R'},\\end{eqnarray*}\nand\\begin{eqnarray*}\n &  & \\left(\\begin{array}{c}\n\\hat{R}\\\\\nR'\\end{array}\\right)(1-q)^{R'}q^{\\hat{R}-R'}\\\\\n & = & \\sum_{j=0}^{M-\\hat{R}}\\left(\\begin{array}{c}\nM-\\hat{R}\\\\\nj\\end{array}\\right)(\\frac{p}{p+q})^{j}(\\frac{q}{p+q})^{M-\\hat{R}-j}\\\\\n &  & \\cdot\\left(\\begin{array}{c}\n\\hat{R}\\\\\nR'\\end{array}\\right)(1-q)^{R'}q^{\\hat{R}-R'},\\end{eqnarray*}\ncompared with (\\ref{eq:215.1}), we have\\[\n\\sum_{R'=i}^{M}P_{R,R'}^{\\pi(R)}\\geq\\sum_{R'=i}^{M}R_{R,R'}^{\\pi'(R)},\\forall i,R\\in\\mathcal{R},\\pi\\in\\Delta,\\pi'\\in\\Delta^{c}.\\]\n\n\nSuppose that the time horizon consists of any $T$ time slots, and $V_{t}^{\\pi}(R)$\ndenotes the expected system throughput under the policy $\\pi$ by starting from\ntime slot $t$ when the system in state $R$.\n\nWhen $t=T$, \\begin{eqnarray*}\nV_{T}^{\\pi}(R) & = & V_{T}^{\\pi'}(R)\\\\\n & = & U_{R}\\\\\n & = & RB,\\forall R\\in\\mathcal{R},\\pi\\in\\Delta,\\pi'\\in\\Delta^{c}.\\end{eqnarray*}\nIt follows that $U_{R}+\\beta V_{T}^{\\pi}(R)=U_{R}+\\beta V_{T}^{\\pi'}(R)$,\nand hence \\begin{eqnarray*}\n &  & \\sum_{R'=0}^{M}P_{R,R'}^{\\pi(R)}[U(R)+\\beta V_{T}^{\\pi}(R)]\\\\\n & \\geq & \\sum_{R'=0}^{M}R_{R,R'}^{\\pi'(R)}[U(R)+\\beta V_{T}^{\\pi'}(R)],\\end{eqnarray*}\ni.e., \\[\nV_{T-1}^{\\pi}(R)\\geq V_{T-1}^{\\pi'}(R),\\forall R\\in\\mathcal{R},\\pi\\in\\Delta,\\pi'\\in\\Delta^{c}.\\]\nRecursively, for any time slots $t\\leq T$, we can show that\\[\nV_{t}^{\\pi}(R)\\geq V_{t}^{\\pi'}(R),\\forall R\\in\\mathcal{R},\\pi\\in\\Delta,\\pi'\\in\\Delta^{c}.\\]\nThus, if there exists a policy $\\pi'\\in\\Delta^{c}$ that is optimal, then\nall the policies $\\pi\\in\\Delta$ is also optimal. If there does not\nexist such a policy $\\pi'$, then we conclude that only the policy\n$\\pi\\in\\Delta$ is optimal.\n\\end{proof}\n\n\n\n\\section{\\label{sec:Adaptive-Channel-RecommendationII}Adaptive Channel Recommendation With Channel Heterogeneity}\n\\rev{\nWe now generalize the adaptive channel recommendation to the heterogeneous\nchannel setting. Recall that the system state $R$ in the homogeneous\nchannel case only keeps track of how many\nchannels are recommended. In a heterogeneous channel environment,\neach channel has different a data rate $B_{m}$ and channel state changing\nprobabilities $p_{m}$ and $q_{m}$. Keeping track of the number of recommend channels is not enough for optimal decision. Intuitively, if a channel\nwith higher data rate $B_{m}$ is recommended, users should choose this channel with a higher weight. The new system state for the heterogeneous channel case should be defined as a vector $\\vec{R}\\triangleq(I_{1},...,I_{M})$,\nwhere $I_{m}=1$ if channel $m$ is recommended and $I_{m}=0$ otherwise.\nThe objective of the heterogeneous channel recommendation MDP is then\nto find the optimal channel access probabilities $\\{P_{m}(\\vec{R})\\}_{m=1}^{M}$\nfor each system state $\\vec{R}$ where $P_{m}(\\vec{R})$ is the probability\nof selecting channel $m$. \n\nSimilarly with the homogeneous channel\ncase, we can apply the MRAS method to obtain the optimal\nsolutions with the new formulation. However, the number of decision variables $\\{P_{m}(\\vec{R})\\}_{m=1}^{M}$\nin the heterogeneous channel model equals to $M2^{M}$,\nwhich causes exponential blow up in the computational complexity. We next\nfocus on developing a low complexity efficient heuristic algorithm  to solve the MDP.\n\nRecall that in the heuristic algorithm in Lemma \\ref{lemma12s} for the homogeneous\nchannel recommendation, the weight of selecting each recommended channel\nis $\\frac{1}{N}$ and total weights of choosing recommended channels\nare $R\\frac{1}{N}$. Similarly, we can design a low complexity heuristic\nalgorithm for the heterogeneous channel recommendation. More specifically,\nwe set the weight of selecting channel $m$ is $P_{1}^{m}$ ($P_{0}^{m}$, respectively)\nwhen the channel is recommended (the channel is not recommended, respectively).\nGiven the system is in state $\\vec{R}$, the probability of choosing\nchannel $m$ is proportional to its weight of its state $I_{m}$,\ni.e.,\\begin{equation}\nP_{m}(\\vec{R})=\\frac{P_{I_{m}}^{m}}{\\sum_{m'=1}^{M}P_{I_{m'}}^{m}}.\\label{eq:hcp}\\end{equation}\nIn this case, the total number of decision variables $P_{I_{m}}^{m}$ is\nreduced to $2M$, which grows linearly in the number of channels $M$. Let $\\vec{\\pi}=\\{(P_{1}^{m},P_{0}^{m})\\}_{m=1}^{M}\\in(0,1)^{2M}$denote\nthe set of corresponding decision variables. Our objective is to find\nthe optimal $\\vec{\\pi}$ that maximizes the time average throughput\n$\\Phi_{\\vec{\\pi}}$. We can again apply the MRAS method to find the optimal\nsolution, which is given in Algorithm \\ref{alg:MRAS-Method-ForII}.\nThe procedures of derivation is very similar with the MRAS method\nfor the homogeneous channel recommendation; we omit the details due to\nspace limit. \n\nNote that the optimal policy $\\vec{\\pi}^{*}$ for the\nheuristic heterogeneous channel recommendation is also a feasible policy for\nthe heterogeneous channel recommendation MDP. The performance of the\noptimal policy for the heterogeneous channel recommendation MDP thus\ndominates the heuristic heterogeneous channel recommendation. However,\nnumerical results show that the heuristic heterogeneous channel recommendation\nhas a small performance loss comparing to the optimal policy while gaining a significant computation complexity\nreduction.\n}\n\n\\begin{algorithm}[tt]\n\\begin{algorithmic}[1]\n\\State \\textbf{initialize} parameters for the elite ratio $\\rho$, Gaussian distributions $\\boldsymbol{\\mu}(0)=\\{(\\mu_{1}^{m}(0),\\mu_{0}^{m}(0))\\}_{m=1}^{M},\\boldsymbol{\\sigma}(0)=\\{(\\sigma_{1}^{m}(0),\\sigma_{0}^{m}(0))\\}_{m=1}^{M}$,\nand the stopping criterion $\\xi$. Set initial elite threshold $\\gamma_{0}=0$ and iteration index $k=0$.\n\\Repeat{:}\n\\State \\textbf{increase} iteration index $k$ by 1.\n\\State \\textbf{generate} $L$ candidate policies $\\vec{\\pi}_{1},...,\\vec{\\pi}_{L}$\nfrom the random policy generation mechanism $f(\\vec{\\pi},\\boldsymbol{\\mu}(k-1),\\boldsymbol{\\sigma}(k-1))$.\n\\State \\textbf{select} elite policies by setting the elite threshold $\\gamma_{k}=\\max\\{\\Phi_{\\hat{\\vec{\\pi}}_{\\lceil(1-\\rho)L\\rceil}},\\gamma_{k-1}\\}.$\n\\State \\textbf{update} the random policy generation mechanism by (for any $I_{m}\\in\\{0,1\\},m\\in\\mathcal{M}$) \\begin{align}\n\\mu_{I_{m}}^{m}(k) & =  \\frac{\\sum_{i=1}^{L}e^{(k-1)\\Phi_{\\vec{\\pi}}}I_{\\{\\Phi_{\\vec{\\pi}_{i}}\\geq\\gamma_{k}\\}}P_{I_{m}}^{m}}{\\sum_{i=1}^{L}e^{(k-1)\\Phi_{\\vec{\\pi}}}I_{\\{\\Phi_{\\vec{\\pi}_{i}}\\geq\\gamma_{k}\\}}},\\label{eq:525-1}\\\\\n\\sigma_{I_{m}}^{m}(k) & = \\left(\\frac{\\sum_{i=1}^{L}e^{(k-1)\\Phi_{\\vec{\\pi}}}I_{\\{\\Phi_{\\vec{\\pi}_{i}}\\geq\\gamma_{k}\\}}(P_{I_{m}}^{m}-\\mu_{I_{m}}^{m}(k))^{2}}{\\sum_{i=1}^{L}e^{(k-1)\\Phi_{\\vec{\\pi}}}I_{\\{\\Phi_{\\vec{\\pi}_{i}}\\geq\\gamma_{k}\\}}}\\right)^{\\frac{1}{2}}.\\label{eq:525-2}\\end{align}\n\\Until{$\\max_{I_{m}\\in\\{0,1\\},m\\in\\mathcal{M}}\\sigma_{I_{m}}^{m}(k)<\\xi.$}\n\\end{algorithmic}\n\\caption{\\label{alg:MRAS-Method-ForII}MRAS-based Algorithm For Optimizing Heuristic Heterogeneous Channel Recommendation}\n\\end{algorithm} \n\\subsection{Proof of Lemma \\ref{lemma11s}}\\label{ProofLemma1}\nWhen $S_{m}(t)=0$, this trivially holds. We focus on the case that\n$S_{m}(t)=1$.\n\nLet $\\mathcal{K}_{m}=\\{1,...,k_{m}(t)\\}$ be the set of secondary\nusers accessing the channel $m$, $\\tau_{m}^{i}$ be the backoff time\nbe generated by secondary user $i$ and $\\tau_{m}^{(1)}=\\min\\{\\tau_{m}^{i}|i\\neq n,i\\in\\mathcal{K}_{m}\\}$.\nThe probability that the user $n$ captures the channel $m$ is given\nas\\begin{eqnarray*}\nPr_{n,m} & = & P\\{\\tau_{m}^{(1)}>\\tau_{m}^{n}\\}\\\\\n & = & (1-\\frac{\\tau_{m}^{n}}{\\tau_{max}})^{k_{m}(t)-1}.\\end{eqnarray*}\nThus, the expected throughput of user $n$ is\\begin{eqnarray*}\nu_{n}(t) & = & \\int_{0}^{\\tau_{max}}BPr_{n,m}\\frac{1}{\\tau_{max}}d\\tau_{m}^{n}\\\\\n & = & \\int_{0}^{\\tau_{max}}B(1-\\frac{\\tau_{m}^{n}}{\\tau_{max}})^{k_{m}(t)-1}\\frac{1}{\\tau_{max}}d\\tau_{m}^{n}\\\\\n & = & \\frac{B}{k_{m}(t)}.\\\\\n & = & \\frac{BS_{m}(t)}{k_{m}(t)}.\\end{eqnarray*} \\qed \n \n \n\\subsection{Proof of Lemma \\ref{lemma12s}}\\label{ProofLemma2}\nLet $\\Lambda_{C}$ denote the event that $C$ secondary users choose\nthe recommended channels, and $Pr(c_{1},...,c_{R})$ denote probability\nmass function that the number of secondary users on these $R$ recommended\nchannels equal to $c_{1},...,c_{R}$ respectively. Given the event\n$\\Lambda_{C}$, we have \\[\nPr(c_{1},...,c_{R}|\\mbox{\\ensuremath{\\Lambda_{C}}})=\\left(\\begin{array}{c}\nn\\\\\nc_{1},...,c_{R}\\end{array}\\right)R^{-C},\\]\nwhich is a Multinomial mass function. By the property of Multinomial\ndistribution, we have\\[\nE[c_{m}|\\Lambda_{C}]=\\frac{C}{R}.\\]\nIt follows that the expected number of users choosing a recommended\nchannel $m$ is\\begin{eqnarray*}\nE[c_{m}] & = & \\sum_{C=0}^{N}E[c_{m}|\\Lambda_{C}]Pr(\\Lambda_{C})\\\\\n & = & \\sum_{C=0}^{N}\\frac{C}{R}\\left(\\begin{array}{c}\nN\\\\\nC\\end{array}\\right)P_{rec}^{C}(1-P_{rec})^{N-C}\\\\\n & = & \\frac{P_{rec}N}{R}.\\end{eqnarray*}\nThen $E[c_{m}]=1$ requires that\\[\nP_{rec}=\\frac{R}{N}.\\]  \\qed\n\n\\subsection{Derivation of Transition Probability}\\label{Derivation}\nWhen the system state transits from $R$ to $R'$, we assume that\n$m_{r}$ and $m_{u}$ recommendations, out of $R'$ recommendations,\nare channels that have been recommended and have not been recommended\nat time slot $t$ respectively. Obviously, $m_{r}+m_{u}=R'$. We assume\nthat $\\bar{m}_{r}$ recommended channels and $\\bar{m}_{u}$ unrecommended\nchannels have been accessed by the secondary users at time slot $t+1$.\nWe thus have $R\\geq\\bar{m}_{r}\\geq m_{r}$ and $M-R\\geq\\bar{m}_{u}\\geq m_{u}$.\nWe also assume that there are $n_{r}$ secondary users have accessed\nthese $\\bar{m}_{r}$ recommended channels and $n_{u}$ secondary users\nhave accessed those $\\bar{m}_{u}$ unrecommended channels at time\nslot $t+1$. Obviously, we have $n_{r}+n_{u}=N$ , $n_{r}\\geq\\bar{m}_{r}$\nand $n_{u}\\geq\\bar{m}_{u}$.\n\nFor the first term, the probability that the user distribution $(n_{r},n_{u})$\nhappens follows the Binomial distribution as $\\left(\\begin{array}{c}\nN\\\\\nn_{r}\\end{array}\\right)P_{rec}^{n_{r}}(1-P_{rec})^{n_{u}}$.\n\nFor the second term, when $\\bar{m}_{r}\\ge1$, it is easy to check\nthat there are $\\left(\\begin{array}{c}\nn_{r}-1\\\\\n\\bar{m}_{r}-1\\end{array}\\right)$ ways for $n_{r}$ secondary users to choose $\\bar{m}_{r}$ recommended\nchannels and there are $\\frac{R!}{(R-\\bar{m}_{r})!}$ possibilities\nfor these $\\bar{m}_{r}$ recommended channels out of the $R$ recommended\nchannels, each of which has probability $(\\frac{1}{R})^{n_{r}}$.\nAmong these $\\bar{m}_{r}$ recommended channels that have been accessed\nby the secondary users, the probability that $m_{r}$ channels turn\nout to be idle is given as $\\left(\\begin{array}{c}\n\\bar{m}_{r}\\\\\nm_{r}\\end{array}\\right)(1-q)^{m_{r}}q^{\\bar{m}_{r}-m_{r}}$. When $\\bar{m}_{r}=0$, it requires that $u_{r}=0$. Thus, we define\n\\[\n\\left(\\begin{array}{c}\nn_{r}-1\\\\\n-1\\end{array}\\right)=\\begin{cases}\n1 & \\mbox{If \\ensuremath{n_{r}}=0,}\\\\\n0 & \\mbox{Otherwise.}\\end{cases}\\]\n\n\nSimilarly, we can obtain the third term for the unrecommended channels\ncase.\n\n\\subsection{Lemma 5}\\label{proof_for_lemma1}\nSince the operation $\\sum_{R'\\in\\mathcal{R}}P{}_{R,R'}^{P_{rec}}[\\cdot]$\nplays a key role in the Bellman equation, to facilitate the study,\nwe first define the following function \\begin{eqnarray*}\nf_{r}(R,P_{rec}) & \\triangleq & \\sum_{i=r}^{\\min\\{M,N\\}}P{}_{R,i}^{P_{rec}},\\forall r\\in\\mathcal{R}.\\end{eqnarray*}\nSince\\begin{eqnarray*}\n &  & f_{r}(R,P_{rec})\\\\\n &  & =Pr(R(t+1)\\geq r|R(t)=R,P_{rec}(t)=P_{rec})\\\\\n &  & =1-Pr(R(t+1)<r|R(t)=R,P_{rec}(t)=P_{rec}),\\end{eqnarray*}\nWe call the function $f_{r}(R,P_{rec})$ as \\emph{the reverse cumulative distribution\nfunction} in the sequel.\n\n\\begin{lem}\n\\label{lem:nondecreaseing} When $M=\\infty$ and $N<\\infty$,\nthe reverse cumulative distribution function $f_{r}(R,P_{rec})$ is\nnondecreasing in $R$ for all $r,R\\in\\mathcal{R}$, $P_{rec}\\in\\mathcal{P}$.\\end{lem}\n\n\\emph{proof:} We prove the result by induction argument. In abuse of notation, we\ndenote the transition probability $P{}_{R,R'}^{P_{rec}}$ and the\nreverse cumulative distribution function $f_{r}(R,P_{rec})$ when\nthe number of users $N=k$ as $P_{R,R'}^{P_{rec}}(k)$ and $f_{r}^{k}(R,P_{rec})$\nrespectively.\n\nWhen $N=2$, from (\\ref{eq:2565}), we have\\begin{eqnarray*}\nP_{0,0}^{P_{rec}}(2) & = & P_{rec}^{2}+(1-P_{rec})^{2}(\\frac{q}{p+q})^{2}\\\\\n &  & + 2P_{rec}(1-P_{rec})\\frac{q}{p+q},\\end{eqnarray*}\n\\[\nP_{0,1}^{P_{rec}}(2)=(1-P_{rec})^{2}\\frac{2pq}{(p+q)^{2}}+2P_{rec}(1-P_{rec})\\frac{p}{p+q},\\]\n\\[\nP_{0,2}^{P_{rec}}(2)=(1-P_{rec})^{2}(\\frac{p}{p+q})^{2},\\]\n\\begin{eqnarray*}\nP_{1,0}^{P_{rec}}(2) & = & P_{rec}^{2}q+(1-P_{rec})^{2}(\\frac{q}{p+q})^{2}\\\\\n & & + 2P_{rec}(1-P_{rec})\\frac{q^{2}}{p+q},\\end{eqnarray*}\n\\begin{eqnarray*}\nP_{1,1}^{P_{rec}}(2) & = & P_{rec}^{2}(1-q)+(1-P_{rec})^{2}\\frac{2pq}{(p+q)^{2}}\\\\\n & & + 2P_{rec}(1-P_{rec})\\frac{(1-q)q+pq}{p+q},\\end{eqnarray*}\n\\[\nP_{1,2}^{P_{rec}}(2)=(1-P_{rec})^{2}(\\frac{p}{p+q})^{2}+2P_{rec}(1-P_{rec})\\frac{(1-q)p}{p+q},\\]\n\\begin{eqnarray*}\nP_{2,0}^{P_{rec}}(2) & = & P_{rec}^{2}\\frac{q+q^{2}}{2}+(1-P_{rec})^{2}(\\frac{q}{p+q})^{2}\\\\\n & & + 2P_{rec}(1-P_{rec})\\frac{q^{2}}{p+q},\\end{eqnarray*}\n\\begin{eqnarray*}\nP_{2,1}^{P_{rec}}(2) & = & P_{rec}^{2}\\frac{1-q+(1-q)q}{2}+(1-P_{rec})^{2}\\frac{2pq}{(p+q)^{2}}\\\\\n & & + 2P_{rec}(1-P_{rec})\\frac{(1-q)q+pq}{p+q},\\end{eqnarray*}\n\\begin{eqnarray*}\nP_{2,2}^{P_{rec}}(2) & = & P_{rec}^{2}\\frac{(1-q)^{2}}{2}+(1-P_{rec})^{2}(\\frac{p}{p+q})^{2}\\\\\n & & + 2P_{rec}(1-P_{rec})\\frac{(1-q)p}{p+q}.\\end{eqnarray*}\nIt is easy to check the following holds\\begin{eqnarray*}\nP_{0,0}^{P_{rec}}(2) & \\geq & P_{1,0}^{P_{rec}}(2)\\geq P_{2,0}^{P_{rec}}(2),\\\\\nP_{2,2}^{P_{rec}}(2) & \\geq & P_{1,2}^{P_{rec}}(2)\\geq P_{0,2}^{P_{rec}}(2).\\end{eqnarray*}\nSince \\[\n\\sum_{i=0}^{2}P_{0,i}^{P_{rec}}(2)=\\sum_{i=0}^{2}P_{1,i}^{P_{rec}}(2)=\\sum_{i=0}^{2}P_{2,i}^{P_{rec}}(2)=1,\\]\nwe thus obtain\\[\nf_{r}^{2}(R+1,P_{rec})\\geq f_{r}^{2}(R,P_{rec}),\\forall R,r\\in\\mathcal{R},P_{rec}\\in\\mathcal{P},\\]\ni.e. $f_{r}(R,P_{rec})$ is nondecreasing in $R$ for the case $N=2$.\n\nWe then assume that $f_{r}(R,P_{rec})$ is nondecreasing in $R$ for\nall $R\\in\\mathcal{R}$, $P_{rec}\\in\\mathcal{P}$ for the case that\n$N=k\\geq2$ i.e.\\[\nf_{r}^{k}(R+1,P_{rec})\\geq f_{r}^{k}(R,P_{rec}),\\forall R,r\\in\\mathcal{R},P_{rec}\\in\\mathcal{P}.\\]\nWe next prove that $f_{r}(R,P_{rec})$ is nondecreasing for the case\nthe $N=k+1$ under this hypothesis.\n\nLet $\\psi$ denote the event that one arbitrary user out of these\n$k+1$ users, does not generate a recommendation at time slot $t+1$.\nObviously,\\[\nPr(\\psi)=P_{rec}q+(1-P_{rec})\\frac{q}{p+q},\\]\nwhich depends on $P_{rec}$ and the channel environment only. By conditioning\non the event $\\varphi$, we have\\begin{eqnarray}\nP{}_{R+1,i}^{P_{rec}}(k+1) & = & P{}_{R+1,i-1}^{P_{rec}}(k)[1-Pr(\\psi)]\\nonumber \\\\\n & & + P{}_{R+1,i}^{P_{rec}}(k)Pr(\\psi),\\label{eq:259}\\\\\nP{}_{R,i}^{P_{rec}}(k+1) & = & P{}_{R,i-1}^{P_{rec}}(k)[1-Pr(\\psi)].\\nonumber \\\\\n & & + P{}_{R,i}^{P_{rec}}(k)Pr(\\psi)\\label{eq:259-1}\\end{eqnarray}\nThus, \\begin{eqnarray}\n &  & f_{r}^{k+1}(R+1,P_{rec})-f_{r}^{k+1}(R,P_{rec})\\nonumber \\\\\n & = & \\sum_{i=r}^{k+1}P{}_{R+1,i}^{P_{rec}}(k+1)-\\sum_{i=r}^{k+1}P{}_{R,i}^{P_{rec}}(k+1)\\nonumber \\\\\n & = & [\\sum_{i=r}^{k+1}P{}_{R+1,i-1}^{P_{rec}}(k)-\\sum_{i=r}^{k+1}P{}_{R,i-1}^{P_{rec}}(k)][1-Pr(\\psi)]\\nonumber \\\\\n & & + [\\sum_{i=r}^{k}P{}_{R+1,i}^{P_{rec}}(k)-\\sum_{i=r}^{k}P{}_{R,i}^{P_{rec}}(k)]Pr(\\psi)\\nonumber \\\\\n & = & [\\sum_{j=r}^{k}P{}_{R+1,j}^{P_{rec}}(k)-\\sum_{j=r}^{k}P{}_{R,j}^{P_{rec}}(k)][1-Pr(\\psi)]\\nonumber \\\\\n & & + [\\sum_{i=r}^{k}P{}_{R+1,i}^{P_{rec}}(k)-\\sum_{i=r}^{k}P{}_{R,i}^{P_{rec}}(k)]Pr(\\psi)\\nonumber \\\\\n & = & [f_{r-1}^{k}(R+1,P_{rec})-f_{r-1}^{k}(R,P_{rec})][1-Pr(\\psi)]\\nonumber \\\\\n & = & [f_{r}^{k}(R+1,P_{rec})-f_{r}^{k}(R,P_{rec})]Pr(\\psi)\\nonumber \\\\\n & \\geq & 0.\\label{eq:259-2}\\end{eqnarray}\ni.e. $f_{r}(R,P_{rec})$ is also nondecreasing for the case the $N=k+1$.\nBy the induction argument, the result holds for the case that $N\\geq2$. \\qed\n\n\\subsection{Lemma 6}\\label{proof_for_lemma22}\n\n\\begin{lem}\n\\label{lem:When-,-i.e.}When $M=+\\infty$ and $N<+\\infty$,\nthe reverse cumulative distribution function $f_{r}(R,P_{rec})$ is\nsupermodular on $\\mathcal{R}\\times\\mathcal{P}$.\\end{lem}\n\n\\emph{proof:} To show $f_{r}(R,P_{rec})$ is supermodular on $\\mathcal{R}\\times\\mathcal{P}$\nis equivalent to proving the following is true:\\begin{equation}\n\\frac{\\partial^{2}f_{r}(R,P_{rec})}{\\partial P_{rec}\\partial R}\\ge0.\\label{eq:2588}\\end{equation}\nSince $R$ is an integral variable, (\\ref{eq:2588}) is equivalent\nto \\[\n\\frac{\\partial f_{r}(R+1,P_{rec})}{\\partial P_{rec}}-\\frac{\\partial f_{r}(R,P_{rec})}{\\partial P_{rec}}\\geq0.\\]\nThat is, it is equivalent to showing $\\frac{\\partial f_{r}(R,P_{rec})}{\\partial P_{rec}}$\nis nondecreasing in $R$. By the similar procedure in proof of Lemma\n\\ref{lem:nondecreaseing}, we show this holds. \\qed\n\n\n\\subsection{Proof of Proposition \\ref{lem:When-,-i.e.II}}\n\nWe prove the proposition by induction. Suppose that the time horizon\nconsists of any $T$ time slots.\n\nWhen $t=T$, $V_{T}(R)=U_{R}= RB$,\nand the proposition is trivially true.\n\nNow, we assume it also holds for $V_{t}(R)$ when $t=k+1,k+2,...,T.$\nLet $\\hat{R}$ be a system state such that $\\hat{R}\\geq R$. By the\nhypothesis, we have $V_{k+1}(\\hat{R})\\geq V_{k+1}(R)$.\nLet $\\pi^{*}$ be the optimal policy. From the Bellman equation in\n(\\ref{eq:5647}), we have%\n\\begin{equation}\nV_{k}(R)=\\sum_{R'=0}^{\\min\\{M,N\\}}P{}_{R,R'}^{\\pi^{*}(R)}[U_{R'}+\\beta V_{k+1}(R')],\\forall R\\in\\mathcal{R}.\\label{eq:111-1}\n\\end{equation}\nBy defining a new system state $-1$ such that\n$U_{-1}+\\beta V_{k+1}(-1)=0$,\nwe can rewrite the equation in (\\ref{eq:111-1}) as\\begin{eqnarray*}\nV_{k}(R) & = & \\sum_{R'=0}^{\\min\\{M,N\\}}P{}_{R,R'}^{\\pi^{*}(R)}\\sum_{i=0}^{R'}\\{[U_{i}+\\beta V_{k+1}(i)]\\\\\n & & -[U_{i-1}+\\beta V_{k+1}(i-1)]\\}\\\\\n & = & \\sum_{R'=0}^{\\min\\{M,N\\}}\\{[U_{R'}+\\beta V_{k+1}(R')]\\\\\n & & - [U_{R'-1}+\\beta V_{k+1}(R'-1)]\\}\\sum_{i=R'}^{\\min\\{M,N\\}}P{}_{R,i}^{\\pi^{*}(R)}.\\end{eqnarray*}\nBy lemma $5$ in the Appendix, we have \\[\n\\sum_{i=R'}^{\\min\\{M,N\\}}P{}_{\\hat{R},i}^{\\pi^{*}(R)}\\geq\\sum_{i=R'}^{\\min\\{M,N\\}}P{}_{R,i}^{\\pi^{*}(R)},\\forall R'\\in\\mathcal{R}.\\]\nThen\\begin{eqnarray*}\nV_{k}(R) & \\le & \\sum_{R'=0}^{\\min\\{M,N\\}}\\{[U_{R'}+\\beta V_{k+1}(R')]\\\\\n &  & -[U_{R'-1}+\\beta V_{k+1}(R'-1)]\\}\\sum_{i=R'}^{\\min\\{M,N\\}}P{}_{\\hat{R},i}^{\\pi^{*}(R)}\\\\\n \\end{eqnarray*}\n \\begin{eqnarray*}\n & = & \\sum_{R'=0}^{\\min\\{M,N\\}}P{}_{\\hat{R},R'}^{\\pi^{*}(R)}[U_{R'}+\\beta V_{k+1}(R')]\\\\\n & \\le & \\max_{P_{rec}\\in\\mathcal{P}}\\sum_{R'\\in\\mathcal{R}}P{}_{\\hat{R},R'}^{P_{rec}}[U_{R'}+\\beta V_{t+1}(R')]\\\\\n & = & \\sum_{R'=0}^{\\min\\{M,N\\}}P{}_{\\hat{R},R'}^{\\pi^{*}(\\hat{R})}[U_{R'}+\\beta V_{k+1}(R')]\\\\\n & = & V_{k}(\\hat{R}),\\end{eqnarray*}\ni.e., for $t=k$,\n$V_{k}(\\hat{R})\\geq V_{k}(R)$\nalso holds. This completes the proof. \\qed \n\n\n\\subsection{Proof of Theorem \\ref{theorem1}}\\label{proof_for_theorem1}\n\nWe first show that\nunder the reference distribution, the optimal policy is attainable.\n\\begin{lem}\\label{thm33}\nFor the MRAS algorithm, the policy $\\pi$ generated by the sequence\nof reference distributions $\\{g_{k}\\}$ converges point-wisely to\nthe optimal spectrum access policy $\\pi^{*}$ for the adaptive channel\nrecommendation MDP, i.e. \\begin{eqnarray}\n\\lim_{k\\rightarrow\\infty}E_{g_{k}}[\\pi(R)] & = & \\pi(R)^{*},\\forall R\\in\\mathcal{R},\\label{eq:lemma11-1}\\\\\n\\lim_{k\\rightarrow\\infty}Var_{g_{k}}[\\pi(R)] & = & 0,\\forall R\\in\\mathcal{R}.\\label{eq:lemma11-2}\\end{eqnarray}\n\\end{lem}\n\n\\emph{proof:} The proof is developed on the basis of the results in \\cite{key-5}.\n\nFirst, from the MRAS algorithm, we have \\[\n\\gamma_{k}\\le\\gamma_{k+1},\\]\ni.e. the sequence $\\{\\gamma_{k}\\}$ is monotone. Since $0\\le\\gamma_{k}\\le\\Phi_{\\pi^{*}}$\nis bounded, there must exist a finite $K$ such that $\\gamma_{k+1}=\\gamma_{k},\\forall k\\geq K.$\n\n\nWhen $\\gamma_{K}=\\Phi_{\\pi^{*}}$, we have \\[\n\\lim_{k\\rightarrow\\infty}E_{g_{k}}[e^{\\Phi_{\\pi}}I_{\\{\\Phi_{\\pi}\\geq\\gamma_{k}\\}}]=e^{\\Phi_{\\pi^{*}}}.\\]\nholds.\n\nWhen $\\gamma_{K}<\\Phi_{\\pi^{*}}$, from (\\ref{eq:2354}), we know that\\[\nE_{g_{k}}[e^{\\Phi_{\\pi}}I_{\\{\\Phi_{\\pi}\\geq\\gamma_{k}\\}}]\\geq E_{g_{k-1}}[e^{\\Phi_{\\pi}}I_{\\{\\Phi_{\\pi}\\geq\\gamma_{k}\\}}],\\forall k\\geq K.\\]\nThat is, the sequence $\\{E_{g_{k}}[e^{\\Phi_{\\pi}}I_{\\{\\Phi_{\\pi}\\geq\\gamma_{k}\\}}]\\}$\nis monotone and hence converges. We then show that the limit of this sequence must be $e^{\\Phi_{\\pi^{*}}}$\nby contradiction.\n\nSuppose that\\[\n\\lim_{k\\rightarrow\\infty}E_{g_{k}}[e^{\\Phi_{\\pi}}I_{\\{\\Phi_{\\pi}\\geq\\gamma_{k}\\}}]=e^{\\Phi_{*}}<e^{\\Phi_{\\pi^{*}}}.\\]\nDefine the set\\[\n\\Theta=\\{\\pi:\\Phi_{\\pi}\\geq\\max\\{\\gamma_{K},\\ln\\frac{e^{\\Phi_{*}}+e^{\\Phi_{\\pi^{*}}}}{2}\\}\\}.\\]\nSince $\\gamma_{K}<\\Phi_{\\pi^{*}}$, the set $\\Theta$ is not empty\nby the continuous property over the policy space of MDP \\cite{key-4}.\nNote that\\[\ng_{k}(\\pi)=\\prod_{i=1}^{k}\\frac{e^{\\Phi_{\\pi}}I_{\\{\\Phi_{\\pi}\\geq\\gamma_{i}\\}}g_{k-1}(\\pi)}{E_{g_{i}}[e^{\\Phi_{\\pi}}I_{\\{\\Phi_{\\pi}\\geq\\gamma_{i}\\}}]}g_{1}(\\pi),\\]\nand \\[\n\\lim_{k\\rightarrow\\infty}\\frac{e^{\\Phi_{\\pi}}I_{\\{\\Phi_{\\pi}\\geq\\gamma_{_{k}}\\}}}{E_{g_{k}}[e^{\\Phi_{\\pi}}I_{\\{\\Phi_{\\pi}\\geq\\gamma_{_{k}}\\}}]}=\\frac{e^{\\Phi_{\\pi}}I_{\\{\\Phi_{\\pi}\\geq\\gamma_{_{K}}\\}}}{e^{\\Phi_{*}}}>1,\\forall\\pi\\in\\Theta,\\]\nwe thus have\\[\n\\lim_{k\\rightarrow\\infty}g_{k}(\\pi)=\\infty,\\forall\\pi\\in\\Theta.\\]\nBy Fatou's lemma, we have\\begin{eqnarray*}\n &  & \\lim_{k\\rightarrow\\infty}\\inf\\int_{\\pi\\in\\Omega}g_{k}(\\pi)d\\pi\\\\\n & = & 1\\\\\n & \\geq & \\lim_{k\\rightarrow\\infty}\\inf\\int_{\\pi\\in\\Theta}g_{k}(\\pi)d\\pi\\\\\n & \\geq & \\int_{\\pi\\in\\Theta}\\lim_{k\\rightarrow\\infty}\\inf g_{k}(\\pi)d\\pi\\\\\n & = & \\infty,\\end{eqnarray*}\nwhich forms a contradiction. Hence, we have\\[\n\\lim_{k\\rightarrow\\infty}E_{g_{k}}[e^{\\Phi_{\\pi}}I_{\\{\\Phi_{\\pi}\\geq\\gamma_{k}\\}}]=e^{\\Phi_{\\pi^{*}}}.\\]\n\nSince $e^{\\Phi_{\\pi}}I_{\\{\\Phi_{\\pi}\\geq\\gamma\\}}$ is a monotone\nfunction of $\\Phi_{\\pi}$ and one-to-one map over the field $\\{\\pi:\\Phi_{\\pi}\\geq\\gamma\\}$,\nthe result above implies that\\begin{eqnarray}\n\\lim_{k\\rightarrow\\infty}E_{g_{k}}[\\pi] & = & \\pi^{*},\\label{eq:lemma11-1}\\\\\n\\lim_{k\\rightarrow\\infty}Var_{g_{k}}[\\pi] & = & \\boldsymbol{0}.\\label{eq:lemma11-2}\\end{eqnarray} \\qed\n\nTo complete the proof of the theorem, we next show that \\[\nE_{g_{k}}[\\pi(R)]=E_{f(\\pi,\\boldsymbol{\\mu},\\boldsymbol{\\sigma})}[\\pi(R)],\\forall R\\in\\mathcal{R},\\]\n\\[E_{g_{k}}[\\pi^{2}(R)]=E_{f(\\pi,\\boldsymbol{\\mu},\\boldsymbol{\\sigma})}[\\pi^{2}(R)],\\forall R\\in\\mathcal{R}.\\]\nFor the sake of simplicity, we first define a function \\[\nH(\\boldsymbol{\\mu},\\boldsymbol{\\sigma},\\gamma_{k})\\triangleq\\int_{\\pi\\in\\Omega}e^{(k-1)\\Phi_{\\pi}}I_{\\{\\Phi_{\\pi}\\geq\\gamma_{k}\\}}\\ln f(\\pi,\\boldsymbol{\\mu},\\boldsymbol{\\sigma})d\\pi.\\]\nSince \\begin{eqnarray*}\nf(\\pi,\\boldsymbol{\\mu},\\boldsymbol{\\sigma}) & = & \\prod_{R=0}^{\\min\\{M,N\\}}f(\\pi(R),\\mu_{R},\\sigma_{R})\\\\\n & = & \\prod_{R=0}^{\\min\\{M,N\\}}\\frac{1}{\\sqrt{2pi\\sigma_{R}^{2}}}e^{-\\frac{(\\pi(R)-\\mu_{R})^{2}}{2\\sigma_{R}^{2}}},\\\\\n & = & \\prod_{R=0}^{\\min\\{M,N\\}}e^{\\frac{\\mu_{R}\\pi(R)}{\\sigma_{R}}-\\frac{\\mu_{R}^{2}}{2\\sigma_{R}}}\\frac{1}{\\sqrt{2pi\\sigma_{R}^{2}}}e^{-\\frac{\\pi(R)^{2}}{2\\sigma_{R}^{2}}}\\\\\n & = & \\prod_{R=0}^{\\min\\{M,N\\}}e^{\\frac{\\mu_{R}\\pi(R)}{\\sigma_{R}}-\\frac{\\mu_{R}^{2}}{2\\sigma_{R}}}f(\\pi(R),0,\\sigma_{R})\\\\\n & = & \\prod_{R=0}^{\\min\\{M,N\\}}[e^{\\frac{\\mu_{R}\\pi(R)}{\\sigma_{R}}}f(\\pi(R),0,\\sigma_{R})\\\\\n &  & \\cdot\\int_{\\pi(R)\\in\\mathcal{P}}\\frac{\\mu_{R}\\pi(R)}{\\sigma_{R}}f(\\pi(R),0,\\sigma_{R})d\\pi(R)],\\end{eqnarray*}\nwe then obtain\\begin{eqnarray*}\n &  & H(\\boldsymbol{\\mu},\\boldsymbol{\\sigma},\\gamma_{k})\\\\\n & = & \\sum_{R=0}^{\\min\\{M,N\\}}\\int_{\\pi\\in\\Omega}e^{(k-1)\\Phi_{\\pi}}I_{\\{\\Phi_{\\pi}\\geq\\gamma_{k}\\}}\\frac{\\mu_{R}\\pi(R)}{\\sigma_{R}}d\\pi\\\\\n & & + \\sum_{R=0}^{\\min\\{M,N\\}}\\int_{\\pi\\in\\Omega}e^{(k-1)\\Phi_{\\pi}}I_{\\{\\Phi_{\\pi}\\geq\\gamma_{k}\\}}\\ln f(\\pi(R),0,\\sigma_{R})d\\pi\\\\\n & & - \\sum_{R=0}^{\\min\\{M,N\\}}\\{\\int_{\\pi\\in\\Omega}e^{(k-1)\\Phi_{\\pi}}I_{\\{\\Phi_{\\pi}\\geq\\gamma_{k}\\}}\\\\\n &  & \\cdot\\ln[\\int_{\\pi(R)\\in\\mathcal{P}}\\frac{\\mu_{R}\\pi(R)}{\\sigma_{R}}f(\\pi(R),0,\\sigma_{R})d\\pi(R)]d\\pi\\}.\\end{eqnarray*}\nSince the optimization problem in (\\ref{eq:235-1}) is to solve\\[\n\\max_{\\boldsymbol{\\mu},\\boldsymbol{\\sigma}}H(\\boldsymbol{\\mu},\\boldsymbol{\\sigma},\\gamma_{k}),\\]\nthe updated parameters ($\\boldsymbol{\\mu}_{k},\\boldsymbol{\\sigma}_{k}$)\nthus maximizes $H(\\boldsymbol{\\mu},\\boldsymbol{\\sigma},\\gamma_{k})$.\nIt means that \\[\n\\nabla H(\\boldsymbol{\\mu}_{k},\\boldsymbol{\\sigma}_{k},\\gamma_{k})=0.\\]\nThat is \\begin{eqnarray*}\n &  & \\nabla H(\\boldsymbol{\\mu},\\boldsymbol{\\sigma},\\gamma_{k})\\\\\n & = & \\frac{\\int_{\\pi(R)\\in\\mathcal{P}}e^{\\frac{\\mu_{R}\\pi(R)}{\\sigma_{R}}}f(\\pi(R),0,\\sigma_{R})\\frac{\\pi(R)}{\\sigma_{R}^{2}}d\\pi(R)}{\\int_{\\pi(R)\\in\\mathcal{P}}e^{\\frac{\\mu_{R}\\pi(R)}{\\sigma_{R}}}f(\\pi(R),0,\\sigma_{R})d\\pi(R)}\\\\\n &  & \\cdot\\int_{\\pi\\in\\Omega}e^{(k-1)\\Phi_{\\pi}}I_{\\{\\Phi_{\\pi}\\geq\\gamma_{k}\\}}d\\pi\\\\\n & & - \\int_{\\pi\\in\\Omega}e^{(k-1)\\Phi_{\\pi}}I_{\\{\\Phi_{\\pi}\\geq\\gamma_{k}\\}}\\frac{\\pi(R)}{\\sigma_{R}^{2}}d\\pi,\\\\\n & = & 0.\\end{eqnarray*}\nIt follows that\\begin{eqnarray*}\n &  & \\frac{\\int_{\\pi\\in\\Omega}e^{(k-1)\\Phi_{\\pi}}I_{\\{\\Phi_{\\pi}\\geq\\gamma_{k}\\}}\\pi(R)d\\pi}{\\int_{\\pi\\in\\Omega}e^{(k-1)\\Phi_{\\pi}}I_{\\{\\Phi_{\\pi}\\geq\\gamma_{k}\\}}d\\pi}\\\\\n & =\\\\\n &  & \\frac{\\int_{\\pi(R)\\in\\mathcal{P}}e^{\\frac{\\mu_{R}\\pi(R)}{\\sigma_{R}}}f(\\pi(R),0,\\sigma_{R})\\pi(R)d\\pi(R)}{\\int_{\\pi(R)\\in\\mathcal{P}}e^{\\frac{\\mu_{R}\\pi(R)}{\\sigma_{R}}}f(\\pi(R),0,\\sigma_{R})d\\pi(R)},\\forall R\\in\\mathcal{R}.\\end{eqnarray*}\nBy multiplying the same constant on the numerator and denominator\nof the terms on both sides, we have\\begin{eqnarray*}\n &  & \\frac{\\int_{\\pi\\in\\Omega}\\frac{e^{(k-1)\\Phi_{\\pi}}I_{\\{\\Phi_{\\pi}\\geq\\gamma_{k}\\}}g_{k-1}(\\pi)}{E_{g_{k-1}}[e^{\\Phi_{\\pi}}I_{\\{\\Phi_{\\pi}\\geq\\gamma\\}}]}\\pi(R)d\\pi}{\\int_{\\pi\\in\\Omega}\\frac{e^{(k-1)\\Phi_{\\pi}}I_{\\{\\Phi_{\\pi}\\geq\\gamma_{k}\\}}g_{k-1}(\\pi)}{E_{g_{k-1}}[e^{\\Phi_{\\pi}}I_{\\{\\Phi_{\\pi}\\geq\\gamma\\}}]}d\\pi}\\\\\n & =\\\\\n &  & \\frac{\\int_{\\pi(R)\\in\\mathcal{P}}f(\\pi(R),\\mu_{R},\\sigma_{R})\\pi(R)d\\pi(R)}{\\int_{\\pi(R)\\in\\mathcal{P}}f(\\pi(R),\\mu_{R},\\sigma_{R})d\\pi(R)},\\forall R\\in\\mathcal{R},\\end{eqnarray*}\nSince \\begin{eqnarray*}\n &  & \\int_{\\pi(R)\\in\\mathcal{P}}f(\\pi(R),\\mu_{R},\\sigma_{R})d\\pi(R)\\\\\n & = & \\int_{\\pi\\in\\Omega}\\frac{e^{(k-1)\\Phi_{\\pi}}I_{\\{\\Phi_{\\pi}\\geq\\gamma_{k}\\}}g_{k-1}(\\pi)}{E_{g_{k-1}}[e^{\\Phi_{\\pi}}I_{\\{\\Phi_{\\pi}\\geq\\gamma\\}}]}d\\pi\\\\\n & = & 1,\\end{eqnarray*}\nwe obtain\\begin{eqnarray*}\n &  & \\int_{\\pi\\in\\Omega}\\frac{e^{(k-1)\\Phi_{\\pi}}I_{\\{\\Phi_{\\pi}\\geq\\gamma_{k}\\}}g_{k-1}(\\pi)}{E_{g_{k-1}}[e^{\\Phi_{\\pi}}I_{\\{\\Phi_{\\pi}\\geq\\gamma\\}}]}\\pi(R)d\\pi\\\\\n & = & \\int_{\\pi(R)\\in\\mathcal{P}}f(\\pi(R),\\mu_{R},\\sigma_{R})\\pi(R)d\\pi(R),\\forall R\\in\\mathcal{R},\\end{eqnarray*}\ni.e.\\[\nE_{g_{k}}[\\pi(R)]=E_{f(\\pi,\\boldsymbol{\\mu},\\boldsymbol{\\sigma})}[\\pi(R)],\\forall R\\in\\mathcal{R}.\\]\n\nSimilarly, we can show that\\[\nE_{g_{k}}[\\pi^{2}(R)]=E_{f(\\pi,\\boldsymbol{\\mu},\\boldsymbol{\\sigma})}[\\pi^{2}(R)],\\forall R\\in\\mathcal{R}.\\]\n\nFrom (\\ref{eq:lemma11-1}), it follows that\\begin{eqnarray*}\n\\lim_{k\\rightarrow\\infty}E_{f(\\pi,\\boldsymbol{\\mu}_{k},\\boldsymbol{\\sigma}_{k})}[\\pi] & = & \\lim_{k\\rightarrow\\infty}E_{g_{k}}[\\pi]\\\\\n & = & \\pi^{*}.\\end{eqnarray*}\nand,\n\\begin{eqnarray*}\n &   & \\lim_{k\\rightarrow\\infty}Var_{f(\\pi,\\boldsymbol{\\mu},\\boldsymbol{\\sigma})}[\\pi(R)]\\\\\n & = & \\lim_{k\\rightarrow\\infty}\\{E_{f(\\pi,\\boldsymbol{\\mu},\\boldsymbol{\\sigma})}[\\pi^{2}(R)]-E_{f(\\pi,\\boldsymbol{\\mu},\\boldsymbol{\\sigma})}[\\pi(R)]^{2}\\}\\\\\n & = & \\lim_{k\\rightarrow\\infty}\\{E_{g_{k}}[\\pi^{2}(R)]-E_{g_{k}}[\\pi(R)]^{2}\\}\\\\\n & = & \\lim_{k\\rightarrow\\infty}Var_{g_{k}}[\\pi(R)]\\\\\n & = & 0.\\end{eqnarray*} \\qed\n\\section{\\label{sec:Conclusion}Conclusion}\nIn this paper, we propose an adaptive channel recommendation scheme for efficient spectrum sharing. We formulate the problem as an average reward based Markov decision process. We first prove the existence of the optimal stationary spectrum access policy, and then characterize the structure of the optimal policy in two asymptotic cases. Furthermore, we propose a novel MRAS-based algorithm that is provably convergent to the optimal policy.  Numerical results show that our proposed algorithm outperforms the static approach in the literature by up to $18\\%$ and the Q-learning method by up to $10\\%$ in terms of system throughput. Our algorithm is also more robust to the channel dynamics compared to the static counterpart.\n\nIn terms of future work, we are currently extending the analysis by taking the heterogeneity of channels into consideration. We also plan to consider the case where the secondary users are selfish. Designing an incentive-compatible channel recommendation mechanism for that case will be very interesting and challenging. \n\\section{Introduction}\nCognitive radio technology enables unlicensed secondary wireless users to opportunistically share the spectrum with licensed primary users, and thus offers a promising solution to address the spectrum under-utilization problem \\cite{key-6}. Designing an efficient spectrum access mechanism for cognitive radio networks, however, is challenging for several reasons: (1) \\emph{time-variation}: spectrum opportunities available for secondary users are often time-varying due to primary users' stochastic activities \\cite{key-6}; and (2) \\emph{limited observations}: each secondary user often has a limited view of the spectrum opportunities due to the limited spectrum sensing capability \\cite{key-7}. Several characteristics of the wireless channels, on the other hand, turn out to be useful for designing efficient spectrum access mechanisms: (1) \\emph{temporal correlations}: spectrum availabilities are correlated in time, and thus observations in the past can be useful in the near future \\cite{key-19}; and (2) \\emph{spatial correlation}: secondary users close to one another may experience similar spectrum availabilities \\cite{key-1}. In this paper, we shall explore the time and space correlations and propose a recommendation-based collaborative spectrum access algorithm, which achieves good communication performances for the secondary users.\n\nOur algorithm design is directly inspired by the recommendation system in the electronic commerce industry. For example, existing owners of  various products can provide recommendations (reviews) on Amazon.com, so that other potential customers can pick the products that best suit their needs. Motivated by this, Li in \\cite{key-2} proposed a static channel recommendation scheme, where secondary users recommend the channels they have\nsuccessfully accessed to nearby secondary users. Since each secondary user originally only has a limited view of spectrum availability, such information exchange enables secondary users to take advantages of the correlations in time and space, make more informed decisions, and achieve a high total transmission rate.\n\nThe recommendation scheme in \\cite{key-2}, however, is rather static and does not dynamically change with network conditions. In particular, the static scheme ignores two important characteristics of cognitive radios. The first one is the \\emph{time variability} we mentioned before. The second one is the \\emph{congestion effect}. As depicted in Figure \\ref{ChannelRec}, too many users accessing the same good channel leads to congestion and a reduced rate for everyone.\n\n\\begin{figure}[tt]\n\\begin{center}\n\\includegraphics[scale=0.5]{channel_scheme.eps}\n\\caption{\\label{ChannelRec}Illustration of the channel recommendation scheme. User D recommends channel 4 to other users. As a result, both user A and user C access the same channel 4, and thus lead to congestion and a reduced rate for both users.}\n\\end{center}\n\\end{figure}\n\n\nTo address the shortcomings of the static recommendation scheme, in this paper we propose an adaptive channel recommendation\nscheme, which adaptively changes the spectrum access probabilities based on users' latest channel recommendations. We formulate and analyze the system as a Markov decision process (MDP), and propose a numerical algorithm that always converges to the optimal spectrum access policy.\n\nThe main results and contributions of this paper  include:\n\\begin{itemize}\n\\item \\emph{Markov decision process formulation}: we formulate and analyze the optimal recommendation-based spectrum access as an average reward MDP.\n\\item \\emph{Existence and structure of the optimal policy}: we show that there always exists a stationary optimal spectrum access policy, which requires only the channel recommendation information of the most recent time slot. We also explicitly characterize the structure of the optimal stationary policy in two asymptotic cases (either the number of users or the number of users goes to infinity).\n\\item \\emph{Novel algorithm for finding the optimal policy}: we propose\nan algorithm based on the recently developed Model Reference Adaptive Search method \\cite{key-5} to find the optimal stationary spectrum access policy.\nThe algorithm has a low complexity even when dealing with a continuous action space of the MDP. We also show that it always converges to the optimal stationary policy.\n\\item \\rev{\\emph{Superior Performance}: we show that the proposed algorithm achieves up to $18\\%$ performance improvement than the static channel recommendation scheme  and $10\\%$ performance improvement than the Q-learning method, and is also robust to channel dynamics.}\n\n\\end{itemize}\n\nThe rest of the paper is organized as follows. We introduce the system\nmodel and the static channel recommendation scheme in Sections \\ref{sec:System-Model} and \\ref{sec:Static-Channel-Recommendation}, respectively. We then discuss the motivation for designing an adaptive channel recommendation scheme in Section \\ref{sec:Motivation-Adaptive}. The Markov decision\nprocess formulation and the structure results of the optimal policy\nare presented in Section \\ref{sec:Adaptive-Channel-Recommendation},\nfollowed by the Model Reference Adaptive Search based algorithm in Section\n\\ref{sec:Model-Reference-Adaptive}. We illustrate the performance of the algorithm through numerical results in\nSection \\ref{sec:Numerical-Results}. We discuss the related\nwork in Section \\ref{sec:Related-Work} and conclude in Section \\ref{sec:Conclusion}. \n\\section{\\label{sec:Model-Reference-Adaptive}Model Reference Adaptive Search\nFor Optimal Spectrum Access Policy}\nNext we will design an algorithm that can converge to the optimal policy under general system parameters (not limiting to the two asymptotic cases).\nSince the action space of the adaptive channel recommendation MDP\nis continuous (i.e., choosing a probability $P_{rec}$ in $(0,1)$), the traditional method of discretizing the action space\nfollowed by the policy, value iteration, or Q-learning cannot guarantee to converge to the optimal policy. To overcome this difficulty, we propose a new algorithm developed from the Model Reference Adaptive Search method, which was recently developed in the Operations Research community \\cite{key-5}. We will show that the proposed algorithm is easy to implement and is provably convergent to the optimal policy.\n\n\\subsection{Model Reference Adaptive Search Method}\nWe first introduce the basic idea of the Model Reference Adaptive Search (MRAS) method. Later on, we will show how the method can be used to obtain optimal spectrum access policy for our problem.\n\n\nThe MRAS method is a new randomized\nmethod for global optimization \\cite{key-5}. The key idea is to randomize the original optimization problem over the feasible region according\nto a specified probabilistic model. The method then generates candidate solutions\nand updates the probabilistic model on the basis of elite solutions\nand a reference model, so that to guide the future search toward better solutions.\n\n\nFormally, let $J(x)$ be the objective function to maximize. The\nMRAS method is an iterative algorithm, and it includes three phases in each iteration $k$:\n\\begin{itemize}\n\\item \\emph{Random solution generation}: generate a set of random solutions \\rev{$\\{x\\}$ in the feasible set $\\chi$} according to a parameterized probabilistic model $f(x,v_{k})$, which is a probability density function (pdf) with parameter $v_{k}$. \\rev{The number of solutions to generate is a fixed system parameter.}\n\\item \\emph{Reference distribution construction}: select elite solutions among the randomly generated set in the previous phase, such that the chosen ones satisfy $J(x)\\geq\\gamma$. Construct a reference probability distribution as\\begin{eqnarray}\ng_{k}(x) & =\\begin{cases}\n\\frac{I_{\\{J(x)\\geq\\gamma\\}}}{E_{f(x,v_{0})}[\\frac{I_{\\{J(x)\\geq\\gamma\\}}}{f(x,v_{0})}]} & k=1,\\\\\n\\frac{e^{J(x)}I_{\\{J(x)\\geq\\gamma\\}}g_{k-1}(x)}{E_{g_{k-1}}[e^{J(x)}I_{\\{J(x)\\geq\\gamma\\}}]} & k\\geq2,\\end{cases}\\label{eq:2546-1}\\end{eqnarray}\nwhere $I_{\\{\\varpi\\}}$ is an indicator function, which equals $1$ if the event $\\varpi$ is true and zero otherwise. Parameter $v_{0}$ is the initial parameter for the probabilistic model (used during the first iteration, i.e., $k=1$),\nand $g_{k-1}(x)$ is the reference distribution in the previous iteration (used when $k\\geq 2$).\n\\item \\emph{Probabilistic model update}: update the parameter $v$ of the probabilistic\nmodel $f(x,v)$ by minimizing the Kullback-Leibler divergence between\n$g_{k}(x)$ and $f(x,v)$, i.e.\\begin{equation}\nv_{k+1}=\\arg\\min_{v}E_{g_{k}}\\left[\\ln\\frac{g_{k}(x)}{f(x,v)}\\right].\\label{eq:12567}\\end{equation}\n\n\\end{itemize}\n\nBy constructing the reference distribution according to\n(\\ref{eq:2546-1}), the expected performance of random elite solutions can be improved under the new reference distribution, i.e.,\\begin{eqnarray}\nE_{g_{k}}[e^{J(x)}I_{\\{J(x)\\geq\\gamma\\}}] & = & \\frac{\\int_{x\\in\\chi}e^{2J(x)}I_{\\{J(x)\\geq\\gamma\\}}g_{k-1}(x)dx}{E_{g_{k-1}}[e^{J(x)}I_{\\{J(x)\\geq\\gamma\\}}]}\\nonumber \\\\\n & = & \\frac{E_{g_{k-1}}[e^{2J(x)}I_{\\{J(x)\\geq\\gamma\\}}]}{E_{g_{k-1}}[e^{J(x)}I_{\\{J(x)\\geq\\gamma\\}}]}\\nonumber \\\\\n & \\geq & E_{g_{k-1}}[e^{J(x)}I_{\\{J(x)\\geq\\gamma\\}}].\\label{eq:2354}\\end{eqnarray}\nTo find a better solution to the optimization problem, it is natural to update the probabilistic model (from which random solution  are generated in the first stage) as close to the new reference probability as possible, as done in the third stage.\n\n\\subsection{Model Reference Adaptive Search For Optimal Spectrum Access Policy}\n\nIn this section, we design an algorithm based on the MRAS method to find the optimal\nspectrum access policy. Here we treat the adaptive channel recommendation MDP as a global optimization problem over the policy space. The key challenge is the choice of proper probabilistic model $f(\\cdot)$, which is crucial for the convergence of the MRAS algorithm.\n\n\n\\subsubsection{Random Policy Generation}\n\nTo apply the MRAS method, we first need to set up a random policy generation\nmechanism. Since the action space of the channel recommendation MDP\nis continuous, we use the Gaussian distributions.\nSpecifically, we generate sample actions $\\pi(R)$ from a Gaussian\ndistribution for each system state $R\\in\\mathcal{R}$ independently,\ni.e. $\\pi(R)\\sim\\mathcal{N}(\\mu_{R},\\sigma_{R}^{2})$.\\footnote{Note that the Gaussian distribution has a support over $(-\\infty,+\\infty)$, which is larger than the feasible region of $\\pi(R)$. This issue will be handled in Section \\ref{STE}.} In this case,\na candidate policy $\\pi$ can be generated from the joint distribution\nof $|\\mathcal{R}|$ independent Gaussian distributions, i.e.,\\begin{eqnarray*}\n(\\pi(0),...,\\pi(\\min\\{M,N\\})) & \\sim & \\mathcal{N}(\\mu_{0},\\sigma_{0}^{2})\\times\\cdots\\\\\n &  & \\times\\mathcal{N}(\\mu_{\\min\\{M,N\\}},\\sigma_{\\min\\{M,N\\}}^{2}).\\end{eqnarray*}\nAs shown later, Gaussian distribution has nice analytical and convergent\nproperties for the MRAS method.\n\nFor the sake of brevity, we denote $f(\\pi(R),\\mu_{R},\\sigma_{R})$\nas the pdf of the Gaussian distribution $\\mathcal{N}(\\mu_{R},\\sigma_{R}^{2})$,\nand $f(\\pi,\\boldsymbol{\\mu},\\boldsymbol{\\sigma})$ as random policy\ngeneration mechanism with parameters $\\boldsymbol{\\mu}\\triangleq(\\mu_{0},...,\\mu_{\\min\\{M,N\\}})$ and $\\boldsymbol{\\sigma}\\triangleq(\\sigma_{0},...,\\sigma_{\\min\\{M,N\\}})$,\ni.e.,\\begin{eqnarray*}\nf(\\pi,\\boldsymbol{\\mu},\\boldsymbol{\\sigma}) & = & \\prod_{R=0}^{\\min\\{M,N\\}}f(\\pi(R),\\mu_{R},\\sigma_{R})\\\\\n & = & \\prod_{R=0}^{\\min\\{M,N\\}}\\frac{1}{\\sqrt{2\\varphi\\sigma_{R}^{2}}}e^{-\\frac{(\\pi(R)-\\mu_{R})^{2}}{2\\sigma_{R}^{2}}},\\end{eqnarray*}\nwhere $\\varphi$ is the circumference-to-diameter ratio. \\com{change $pi$ to a single Greek letter.}\n\n\n\\subsubsection{System Throughput Evaluation}\\label{STE}\n\nGiven a candidate policy $\\pi$ randomly generated based on $f(\\pi,\\boldsymbol{\\mu},\\boldsymbol{\\sigma})$, we need to evaluate the expected system\nthroughput $\\Phi_{\\pi}$. From (\\ref{eq:2565}), we obtain the\ntransition probabilities $P{}_{R,R'}^{\\pi(R)}$ for any system state\n$R,R'\\in\\mathcal{R}$. Since a policy $\\pi$ leads to a finitely\nirreducible Markov chain, we can obtain its stationary distribution.\nLet us denote the transition matrix of the Markov chain as $Q\\triangleq[P{}_{R,R'}^{\\pi(R)}]_{|\\mathcal{R}|\\times|\\mathcal{R}|}$\nand the stationary distribution as $\\boldsymbol{p}=(Pr(0),...,Pr(\\min\\{M,N\\}))$.\nObviously, the stationary distribution can be obtained by solving\nthe following equation\\[\n\\boldsymbol{p}Q=\\boldsymbol{p}.\\]\nWe then calculate the expected system throughput $\\Phi_{\\pi}$ by\n\\[\n\\Phi_{\\pi}=\\sum_{R\\in\\mathcal{R}}Pr(R)U_{R}.\\]\n\n\nNote that in the discussion above, we assume that $\\pi\\in\\Omega$\nimplicitly, where $\\Omega$ is the feasible policy space. Since Gaussian distribution has a support over\n$(-\\infty,+\\infty)$, we thus extend the definition of expected system throughput $\\Phi_{\\pi}$ over $(-\\infty,+\\infty)^{|\\mathcal{R}|}$\nas\\[\n\\Phi_{\\pi}=\\begin{cases}\n\\sum_{R\\in\\mathcal{R}}Pr(R)U_{R} & \\pi\\in\\Omega,\\\\\n-\\infty & \\mbox{Otherwise.}\\end{cases}\\]\nIn this case, whenever any generated policy $\\pi$ is not feasible, we have $\\Phi_{\\pi}=-\\infty$.  \\rev{As a result, such policy $\\pi$ will not be selected as an elite sample (discussed next) and will not used for probability updating.} Hence the search of MRAS algorithm will not bias towards any unfeasible policy space.\n\n\\subsubsection{Reference Distribution Construction}\n\nTo construct the reference distribution, we first need to select the\nelite policies. Suppose $L$ candidate policies, $\\pi_{1},\\pi_{2},...,\\pi_{L}$,\nare generated at each iteration. We order them based on an increasing order of the expected system throughputs\n$\\Phi_{\\pi}$, i.e., $\\Phi_{\\hat{\\pi}_{1}}\\le\\Phi_{\\hat{\\pi}_{2}}\\le...\\le\\Phi_{\\hat{\\pi}_{L}}$, and set the elite threshold as\n\\begin{equation*}\n\\gamma=\\Phi_{\\hat{\\pi}_{\\lceil(1-\\rho)L\\rceil}},\n\\end{equation*}\nwhere $0<\\rho<1$ is the elite ratio. \\rev{For example, when $L=100$ and $\\rho=0.4$, then $\\gamma=\\Phi_{\\hat{\\pi}_{60}}$ and the last $40$ samples in the sequence will be selected as elite samples.}\nNote that as long as $L$ is sufficiently large, we shall have $\\gamma<\\infty$ and hence only feasible policies $\\pi$ are selected. According to (\\ref{eq:2546-1}),\nwe then construct the reference distribution as\n\n\\begin{eqnarray}\ng_{k}(\\pi) & =\\begin{cases}\n\\frac{I_{\\{\\Phi_{\\pi}\\geq\\gamma\\}}}{E_{f(\\pi,\\boldsymbol{\\mu}_{0},\\boldsymbol{\\sigma}_{0})}[\\frac{_{I_{\\{\\Phi_{\\pi}\\geq\\gamma\\}}}}{f(\\pi,\\boldsymbol{\\mu}_{0},\\boldsymbol{\\sigma}_{0})}]} & k=1,\\\\\n\\frac{e^{\\Phi_{\\pi}}I_{\\{\\Phi_{\\pi}\\geq\\gamma\\}}g_{k-1}(\\pi)}{E_{g_{k-1}}[e^{\\Phi_{\\pi}}I_{\\{\\Phi_{\\pi}\\geq\\gamma\\}}]} & k\\geq2.\\end{cases}\\label{eq:8975}\\end{eqnarray}\n\n\n\n\\subsubsection{Policy Generation Update}\n\nFor the MRAS algorithm, the critical issue is the updating of random\npolicy generation mechanism $f(\\pi,\\boldsymbol{\\mu},\\boldsymbol{\\sigma})$,\nor solving the problem in (\\ref{eq:12567}). The optimal update rule is described as follow.\n\\begin{thm}\nThe optimal parameter $(\\boldsymbol{\\mu},\\boldsymbol{\\sigma})$ that\nminimizes the Kullback-Leibler divergence between the reference distribution\n$g_{k}(\\pi)$ in (\\ref{eq:8975}) and the new policy generation mechanism\n$f(\\pi,\\boldsymbol{\\mu},\\boldsymbol{\\sigma})$ is \\begin{eqnarray}\n\\mu_{R} & = & \\frac{\\int_{\\pi\\in\\Omega}e^{(k-1)\\Phi_{\\pi}}I_{\\{\\Phi_{\\pi}\\geq\\gamma\\}}\\pi(R)d\\pi}{\\int_{\\pi\\in\\Omega}e^{(k-1)\\Phi_{\\pi}}I_{\\{\\Phi_{\\pi}\\geq\\gamma\\}}d\\pi},\\forall R\\in\\mathcal{R},\\label{eq:524-1}\\\\\n\\sigma_{R}^{2} & = & \\frac{\\int_{\\pi\\in\\Omega}e^{(k-1)\\Phi_{\\pi}}I_{\\{\\Phi_{\\pi}\\geq\\gamma\\}}[\\pi(R)-\\mu_{R}]^{2}d\\pi}{\\int_{\\pi\\in\\Omega}e^{(k-1)\\Phi_{\\pi}}I_{\\{\\Phi_{\\pi}\\geq\\gamma\\}}d\\pi},\\forall R\\in\\mathcal{R}.\\nonumber \\\\ \\nonumber\n\\label{eq:524-2}\\\\\\end{eqnarray}\n\\end{thm}\n\\linespread{0.9}\n\\begin{proof}\nFirst, from (\\ref{eq:8975}), we have\\begin{eqnarray*}\ng_{1}(\\pi) & = & \\frac{I_{\\{\\Phi_{\\pi}\\geq\\gamma\\}}}{E_{f(\\pi,\\boldsymbol{\\mu}_{0},\\boldsymbol{\\sigma}_{0})}[\\frac{_{I_{\\{\\Phi_{\\pi}\\geq\\gamma\\}}}}{f(\\pi,\\boldsymbol{\\mu}_{0},\\boldsymbol{\\sigma}_{0})}]}\\\\\n & = & \\frac{I_{\\{\\Phi_{\\pi}\\geq\\gamma\\}}}{\\int_{\\pi\\in\\Omega}I_{\\{\\Phi_{\\pi}\\geq\\gamma\\}}d\\pi},\\end{eqnarray*}\nand,\n\\begin{eqnarray*}\ng_{2}(\\pi) & = & \\frac{e^{\\Phi_{\\pi}}I_{\\{\\Phi_{\\pi}\\geq\\gamma\\}}g_{1}(\\pi)}{E_{g_{1}}[e^{\\Phi_{\\pi}}I_{\\{\\Phi_{\\pi}\\geq\\gamma\\}}]}\\\\\n & = & \\frac{e^{\\Phi_{\\pi}}I_{\\{\\Phi_{\\pi}\\geq\\gamma\\}}I_{\\{\\Phi_{\\pi}\\geq\\gamma\\}}}{E_{g_{1}}[e^{\\Phi_{\\pi}}I_{\\{\\Phi_{\\pi}\\geq\\gamma\\}}]\\int_{\\pi\\in\\Omega}I_{\\{\\Phi_{\\pi}\\geq\\gamma\\}}d\\pi}\\\\\n & = & \\frac{e^{\\Phi_{\\pi}}I_{\\{\\Phi_{\\pi}\\geq\\gamma\\}}I_{\\{\\Phi_{\\pi}\\geq\\gamma\\}}}{\\int_{\\pi\\in\\Omega}e^{\\Phi_{\\pi}}I_{\\{\\Phi_{\\pi}\\geq\\gamma\\}}\\frac{I_{\\{\\Phi_{\\pi}\\geq\\gamma\\}}}{\\int_{\\pi\\in\\Omega}I_{\\{\\Phi_{\\pi}\\geq\\gamma\\}}d\\pi}d\\pi\\int_{\\pi\\in\\Omega}I_{\\{\\Phi_{\\pi}\\geq\\gamma\\}}d\\pi}\\\\\n & = & \\frac{e^{\\Phi_{\\pi}}I_{\\{\\Phi_{\\pi}\\geq\\gamma\\}}}{\\int_{\\pi\\in\\Omega}e^{\\Phi_{\\pi}}I_{\\{\\Phi_{\\pi}\\geq\\gamma\\}}d\\pi}.\\end{eqnarray*}\nRepeat the above computation iteratively, we have%\n\\begin{equation}\ng_{k}(\\pi)=\\frac{e^{(k-1)\\Phi_{\\pi}}I_{\\{\\Phi_{\\pi}\\geq\\gamma\\}}}{\\int_{\\pi\\in\\Omega}e^{(k-1)\\Phi_{\\pi}}I_{\\{\\Phi_{\\pi}\\geq\\gamma\\}}d\\pi},k\\geq1.\\label{eq:23589}\\end{equation}\nThen, the problem in (\\ref{eq:12567}) is equivalent to solving \\begin{eqnarray}\n\\max_{\\boldsymbol{\\mu},\\boldsymbol{\\sigma}} & \\int_{\\pi\\in\\Omega}g_{k}(\\pi)\\ln f(\\pi,\\boldsymbol{\\mu},\\boldsymbol{\\sigma})d\\pi,\\label{eq:235}\\\\\n\\mbox{subject to} & \\boldsymbol{\\mu},\\boldsymbol{\\sigma}\\succeq0,\\nonumber\\end{eqnarray}\nSubstituting (\\ref{eq:23589}) into (\\ref{eq:235}), we have\\begin{eqnarray}\n\\max_{\\boldsymbol{\\mu},\\boldsymbol{\\sigma}} & \\int_{\\pi\\in\\Omega}e^{(k-1)\\Phi_{\\pi}}I_{\\{\\Phi_{\\pi}\\geq\\gamma\\}}\\ln f(\\pi,\\boldsymbol{\\mu},\\boldsymbol{\\sigma})d\\pi,\\label{eq:235-1}\\\\\n\\mbox{subject to} & \\boldsymbol{\\mu},\\boldsymbol{\\sigma}\\succeq0,\\nonumber\\end{eqnarray}\n\nFunction $f(\\pi(R),\\mu_{R},\\sigma_{R})$ is log-concave, since it is the pdf of the Gaussian distribution. Since the log-concavity is closed under multiplication,\nthen $f(\\pi,\\boldsymbol{\\mu},\\boldsymbol{\\sigma})=\\prod_{R=0}^{\\min\\{M,N\\}}f(\\pi(R),\\mu_{R},\\sigma_{R})$\nis also log-concave. It implies the problem in (\\ref{eq:235}) is\na concave optimization problem. Solving by the first order condition,\nwe have \\begin{eqnarray*}\n\\frac{\\partial\\int_{\\pi\\in\\Omega}e^{(k-1)\\Phi_{\\pi}}I_{\\{\\Phi_{\\pi}\\geq\\gamma\\}}\\ln f(\\pi,\\boldsymbol{\\mu},\\boldsymbol{\\sigma})d\\pi}{\\partial\\mu_{R}} & = & 0,\\forall R\\in\\mathcal{R},\\\\\n\\frac{\\partial\\int_{\\pi\\in\\Omega}e^{(k-1)\\Phi_{\\pi}}I_{\\{\\Phi_{\\pi}\\geq\\gamma\\}}\\ln f(\\pi,\\boldsymbol{\\mu},\\boldsymbol{\\sigma})d\\pi}{\\partial\\sigma_{R}} & = & 0,\\forall R\\in\\mathcal{R},\\end{eqnarray*}\nwhich leads to (\\ref{eq:524-1}) and (\\ref{eq:524-2}).\nDue to the concavity of the optimization problem in (\\ref{eq:235}),\nthe solution is also the global optimum for the random policy generation updating.\n\\end{proof}\n\\linespread{1.0}\n\n\\subsubsection{MARS Algorithm For Optimal Spectrum Access Policy}\n\nBased on the MARS algorithm, we generate $L$ candidate polices\nat each iteration. Then the updates in (\\ref{eq:524-1}) and\n(\\ref{eq:524-2}) are replaced by the sample average version in (\\ref{eq:525-1})\nand (\\ref{eq:525-2}), respectively. As a summary, we describe the MARS-based\nalgorithm for finding the optimal spectrum access policy of adaptive\nchannel recommendation MDP in Algorithm \\ref{alg:MRAS-Method-For}.\n\n\n\n\\subsection{Convergence of Model Reference Adaptive Search}\n\nIn this part, we discuss the convergence property of the MRAS-based\noptimal spectrum access policy. For ease of exposition, we\nassume that the adaptive channel recommendation MDP has a unique global\noptimal policy. Numerical studies in \\cite{key-5} show that the MRAS\nmethod also converges where there are multiple global optimal solutions.\nWe shall show that the\nrandom policy generation mechanism $f(\\pi,\\boldsymbol{\\mu}_{k},\\boldsymbol{\\sigma}_{k})$ will eventually generate the optimal policy.\n\\begin{thm}\n\\label{theorem1}\nFor the MRAS algorithm, \\rev{the limiting point of the policy sequence}  $\\{\\pi_{k}\\}$ generated by the sequence\nof random policy generation mechanism $\\{f(\\pi,\\boldsymbol{\\mu}_{k},\\boldsymbol{\\sigma}_{k})\\}$ converges point-wisely to\nthe optimal spectrum access policy $\\pi^{*}$ for the adaptive channel\nrecommendation MDP, i.e., \\begin{eqnarray}\n\\lim_{k\\rightarrow\\infty}E_{f(\\pi,\\boldsymbol{\\mu}_{k},\\boldsymbol{\\sigma}_{k})}[\\pi(R)] & = & \\pi^{*}(R),\\forall R\\in\\mathcal{R},\\label{eq:lemma12}\\\\\n\\lim_{k\\rightarrow\\infty}Var_{f(\\pi,\\boldsymbol{\\mu}_{k},\\boldsymbol{\\sigma}_{k})}[\\pi(R)] & = & 0,\\forall R\\in\\mathcal{R}.\\label{eq:lemma12-2}\\end{eqnarray}\n\\end{thm}\nThe proof is given in the Appendix.\n\nFrom Theorem \\ref{theorem1}, we see that parameter $(\\mu_{R,k},\\sigma_{R,k})$\nfor updating in (\\ref{eq:525-1}) and (\\ref{eq:525-2}) also converges, i.e.,\\begin{eqnarray*}\n\\lim_{k\\rightarrow\\infty}\\mu_{R,k} & = & \\pi^{*}(R),\\forall R\\in\\mathcal{R},\\\\\n\\lim_{k\\rightarrow\\infty}\\sigma_{R,k} & = & 0,\\forall R\\in\\mathcal{R}.\\end{eqnarray*}\nThus, we can use $\\max_{R\\in\\mathcal{R}}\\sigma_{R,k}<\\xi$ as the stopping\ncriterion in Algorithm \\ref{alg:MRAS-Method-For}.\n\n\\begin{algorithm}[tt]\n\\begin{algorithmic}[1]\n\\State \\textbf{Initialize} parameters for Gaussian distributions $(\\boldsymbol{\\mu}_{0},\\boldsymbol{\\sigma}_{0})$,\nthe elite ratio $\\rho$, and the stopping criterion $\\xi$. Set initial elite threshold $\\gamma_{0}=0$ and iteration index $k=0$.\n\\Repeat{:}\n\\State \\textbf{Increase} iteration index $k$ by 1.\n\\State \\textbf{Generate} $L$ candidate policies $\\pi_{1},...,\\pi_{L}$ from the random policy generation mechanism $f(\\pi,\\boldsymbol{\\mu}_{k-1},\\boldsymbol{\\sigma}_{k-1})$.\n\\State \\textbf{Select} elite policies by setting the elite threshold $\\gamma_{k}=\\max\\{\\Phi_{\\hat{\\pi}_{\\lceil(1-\\rho)L\\rceil}},\\gamma_{k-1}\\}.$\n\\State \\textbf{Update} the random policy generation mechanism by \\begin{align}\n\\mu_{R,k}&=\\frac{\\sum_{i=1}^{L}e^{(k-1)\\Phi_{\\pi}}I_{\\{\\Phi_{\\pi_{i}}\\geq\\gamma_{k}\\}}\\pi_{i}(R)}{\\sum_{i=1}^{L}e^{(k-1)\\Phi_{\\pi}}I_{\\{\\Phi_{\\pi_{i}}\\geq\\gamma_{k}\\}}},&\\forall R\\in\\mathcal{R},\\label{eq:525-1}\\\\\n\\sigma_{R,k}^{2}&=\\frac{\\sum_{i=1}^{L}e^{(k-1)\\Phi_{\\pi}}I_{\\{\\Phi_{\\pi_{i}}\\geq\\gamma_{k}\\}}[\\pi_{i}(R)-\\mu_{R}]^{2}}{\\sum_{i=1}^{L}e^{(k-1)\\Phi_{\\pi}}I_{\\{\\Phi_{\\pi_{i}}\\geq\\gamma_{k}\\}}},&\\forall R\\in\\mathcal{R}. \\nonumber \\\\ \\label{eq:525-2}\\end{align}\n\\Until{$\\max_{R\\in\\mathcal{R}}\\sigma_{R,k}<\\xi.$}\n\\end{algorithmic}\n\\caption{\\label{alg:MRAS-Method-For}MRAS-based Algorithm For Adaptive Recommendation Based Optimal Spectrum Access}\n\\end{algorithm}\n\n\n\\section{\\label{sec:Motivation-Adaptive}Motivations For Adaptive Channel Recommendation}\nThe static channel recommendation mechanism is simple to implement due to a fixed value of $P_{rec}$. However, it may lead to significant congestions when the number of recommended channels is small. In the extreme case when only $R=1$ channel is recommended, calculation (\\ref{eq:SCR-1}) suggests that every user will access that channel with a probability $P_{rec}$. When the number of users $N$ is large, the expected number of users accessing this channel $NP_{rec}$ will be high. Thus heavy congestion happens and each secondary user will get a low expected throughput.\n\nA better way is to adaptively change the value of $P_{rec}$ based on the number of recommended channels. This is the key idea of our proposed algorithm.\nTo illustrate the advantage of adaptive algorithms, let us first consider a simple heuristic  adaptive algorithm. In this algorithm, we choose the branching probability such that the expected number of secondary users choosing a single recommended channel is one. To achieve this, we need to set $P_{rec}$ as in Lemma 2.\n\\begin{lem}\\label{lemma12s}\nIf we choose the branching probability $P_{rec}=\\frac{R}{N}$, then the expected number of secondary users choosing\nany one of the $R$ recommended channels is one.\n\\end{lem}\n\nPlease refer to the Appendix for the detailed proof of Lemma \\ref{lemma12s}.\n\n\nWithout going through detailed analysis, it is straightforward to show the benefit for such adaptive approach through simple numerical examples. Let us consider a network with $M=10$ channels and $N=5$ secondary users. For each channel $m$, the initial channel state probability vector is $\\boldsymbol{p}_{m}(0)=(0,1)$\nand the transition matrix is \\[\n\\Gamma=\\left[\\begin{array}{cc}\n1-0.01\\epsilon & 0.01\\epsilon\\\\\n0.01\\epsilon & 1-0.01\\epsilon\\end{array}\\right],\\]\nwhere $\\epsilon$ is called the dynamic factor. A larger value of $\\epsilon$ implies that the channels are more dynamic over time.\nWe are interested in the time average system throughput\n\\[U=\\frac{\\sum_{t=1}^{T}\\sum_{n=1}^{N}u_{n}(t)}{T},\\]\nwhere $u_{n}(t)$ is the throughput of user $n$ at time slot $t$. In the simulation, we set the total number of time slots $T=2000$.\n\nWe implement the following three channel access schemes:\n\\begin{itemize}\n\\item Random access scheme: each secondary user selects a channel randomly.\n\\item Static channel recommendation scheme as in \\cite{key-2} with the \\emph{optimal} constant branching\nprobability $P_{rec}=0.7$.\n\\item Heuristic adaptive channel recommendation scheme with the variable branching probability\n$P_{rec}=\\frac{R}{N}$.\n\\end{itemize}\n\nFigure \\ref{fig:Comparison-of-three} shows that the heuristic adaptive channel recommendation scheme outperforms the static channel recommendation scheme, which in turn outperforms the\nrandom access scheme. Moreover, the heuristic adaptive scheme is more robust to the dynamic channel environment, as it decreases slower than the static scheme when $\\epsilon$ increases.\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[scale=0.5]{Recommendation_System_Comparisons.eps}\n\\end{center}\n\\caption{\\label{fig:Comparison-of-three}Comparison of three channel access\nschemes}\n\n\\end{figure}\n\n\nWe can imagine that an optimal adaptive scheme (by setting the right $P_{rec}(t)$ over time) can further increase the network performance. However, computing the optimal branching probability in closed-form is very difficult. In the rest of the paper, we will focus on characterizing the structures of the optimal spectrum access strategy and designing an efficient algorithm to achieve the optimum.\n\n\\section{\\label{sec:Related-Work}Related Work}\n\n\nThe spectrum access by multiple secondary users can be either \\emph{uncoordinated} or \\emph{coordinated}.\nFor the uncoordinated case, multiple secondary users compete with other for the resource. Huang \\emph{et al.} in \\cite{ key-28 } designed two auction mechanisms to allocate the interference budget among selfish users. Southwell and Huang in \\cite{key-88} studied the largest and smallest convergence time to an equilibrium when secondary users access multiple channels in a distributed fashion.\n Liu \\emph{et al.} in  \\cite{key-16} modeled the interactions among spatially separated users as congestion games with resource reuse.  Li and Han in \\cite{key-29} applied the graphic game theory to address the spectrum access problem with limited range of mutual interference. Anandkumar \\emph{et al.} in \\cite{key-25} proposed a learning-based approach for competitive spectrum access with incomplete spectrum information. Law \\emph{et al.} in \\cite{key-15} showed that uncoordinated spectrum access may lead to poor system performance.\n\nFor the coordinated spectrum access, Zhao \\emph{et al.} in  \\cite{key-26} proposed a dynamic group formation algorithm to distribute secondary users' transmissions across multiple channels. Shu and Krunz proposed a multi-level spectrum opportunity framework in \\cite{key-27}. The above papers assumed that each secondary user knows the entire channel occupancy information. We consider the case where  each secondary user only has a limited view of the system, and improve each other's information by recommendation.\n\n\n\\rev{Our algorithm design is partially inspired by the recommendation systems in the electronic commerce industry, where analytical methods such as collaborative filtering \\cite{key-10} and\nmulti-armed bandit process modeling \\cite{key-14} are useful. However, we cannot directly apply the existing methods to analyze cognitive radio networks due to the unique congestion effect in our model.}\n\n\\section{\\label{sec:Numerical-Results}Simulation Results}\n\nIn this section, we investigate the proposed adaptive channel recommendation scheme by simulations. The results show that the adaptive channel recommendation scheme\nnot only achieves a higher performance over the static channel recommendation\nscheme and random access scheme, but also is more robust to the dynamic\nchange of the channel environments.\n\n\n\n\\subsection{Simulation Setup}\n\nWe first consider a cognitive radio network consisting of multiple independent\nand stochastically homogeneous primary channels.  The data rate of each channel is normalized to be $1$ Mbps. In order to take\nthe impact of primary user's long run behavior into account, we consider the following two types of channel state transition matrices: \\begin{eqnarray}\n\\mbox{Type 1: } \\Gamma^{1} & = & \\left[\\begin{array}{cc}\n1-0.005\\epsilon & 0.005\\epsilon\\\\\n0.025\\epsilon & 1-0.025\\epsilon\\end{array}\\right],\\label{T1}\\\\\n\\mbox{Type 2: } \\Gamma^{2} & = & \\left[\\begin{array}{cc}\n1-0.01\\epsilon & 0.01\\epsilon\\\\\n0.01\\epsilon & 1-0.01\\epsilon\\end{array}\\right],\\label{T2}\\end{eqnarray}\nwhere $\\epsilon$ is the dynamic factor. Recall that a larger $\\epsilon$ means that the channels are more dynamic over time. Using (\\ref{eq:sd-2}), we know that channel models $\\Gamma^{1}$ and $\\Gamma^{2}$ have the stationary channel idle probabilities of ${1}\/{6}$\nand ${1}\/{2}$, respectively. In other words, the primary activity level is much higher with the Type 1 channel than with the Type 2 channel.\n\nWe initialize the parameters of MRAS algorithm as follows. We set $\\mu_{R}=0.5$ and $\\sigma_{R}=0.5$ for the Gaussian distribution, which has 68.2\\% support over the feasible region $(0,1)$. We found that the performance of the MRAS algorithm is insensitive to the elite ratio $\\rho$ when $\\rho\\leq0.3$. We thus choose  $\\rho=0.1$.\n\nWhen using the MRAS-based algorithm, we need to determine how many (feasible) candidate policies to generate in each iteration.  Figure \\ref{fig:MRAS-algorithm-with}\nshows the convergence of MRAS algorithm with $100$, $300$, and $500$ candidate policies per iteration, respectively. We have two observations. First,\nthe number of iterations to achieve convergence reduces as the number\nof candidate policies increases. Second, the convergence speed is insignificant when the number changes from $300$ to $500$. We thus choose\n$L=500$ for the experiments in the sequel.\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[scale=0.45]{Number_of_candidate_policy.eps}\n\\caption{\\label{fig:MRAS-algorithm-with}The convergence of MRAS-based algorithm with different number\nof candidate policies per iteration}\n\\end{center}\n\\end{figure}\n\n\n\\subsection{Simulation Results}\nWe implement the adaptive channel recommendation\nscheme with  $M=10$ channels and $N=5$ secondary users. We also benchmark the adaptive channel recommendation\nscheme with the static channel recommendation scheme in \\cite{key-2} and the random access scheme as the benchmark. We choose the dynamic factor $\\epsilon$ within a wide range to investigate the\nrobustness of the schemes to the channel dynamics. The results are shown in Figures \\ref{fig:Type-1-Transition1} -- \\ref{fig:Performance_Gain2}. From these figures, we see that\n\\begin{itemize}\n\\item \\emph{Superior performance of adaptive channel recommendation scheme (Figures \\ref{fig:Type-1-Transition1} and \\ref{fig:Type-2-Transition1})}: the adaptive channel recommendation scheme performs better than the random\naccess scheme and static channel recommendation scheme. Typically,\nit offers 5\\%$\\scriptsize{\\sim}$18\\% performance gain over the static\nchannel recommendation scheme.\n\\item  \\emph{Impact of channel dynamics (Figures \\ref{fig:Type-1-Transition1} and \\ref{fig:Type-2-Transition1})}: the performances of both adaptive and static channel recommendation schemes degrade as the dynamic factor\n$\\epsilon$ increases. The reason is that both two schemes rely on\nthe recommendation information from previous time slots to make decisions.\nWhen channel states change rapidly, the value of recommendation information\ndiminishes. However, the adaptive channel recommendation is much more robust to the dynamic channel environment changing (See Figure \\ref{fig:Performance_Gain2}). This\nis because the optimal adaptive policy takes the channel dynamics into account while the static one does not.\n\\item \\emph{Impact of channel idleness level (Figures \\ref{fig:Performance_Gain} and \\ref{fig:Performance_Gain2})}: Figure \\ref{fig:Performance_Gain} shows the performance gain of  the adaptive channel recommendation scheme over the random access scheme under two different types of transition matrix scenarios. We see that the performance gain decreases with the idle probability of the channel. This shows that the information of channel recommendations can enhance the spectrum access more efficiently when the primary activity level increases (i.e., when the channel idle probability is low). Interestingly,  Figure \\ref{fig:Performance_Gain2} shows that the performance gain of the adaptive channel recommendation scheme over the static channel recommendation scheme trends to increase with the channel idleness probability. This illustrates that the adaptive channel recommendation scheme can better utilize the channel opportunities given the information of channel recommendations.\n\\end{itemize}\n\n\n\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[scale=0.45]{10Channels_5Users_1.eps}\n\\caption{\\label{fig:Type-1-Transition1}System throughput with $M=10$ channels and $N=5$ users under the Type 1 channel state transition matrix}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[scale=0.45]{10Channels_5Users_2.eps}\n\\caption{\\label{fig:Type-2-Transition1}System throughput with $M=10$ channels and $N=5$ users  under the Type 2 channel state transition matrix}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[scale=0.45]{Performance_Gain.eps}\n\\caption{\\label{fig:Performance_Gain}Performance gain over random access scheme. The Type 1 and Type 2 channels have the stationary channel idle probabilities of ${1}\/{6}$ and ${1}\/{2}$, respectively.}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[scale=0.45]{Performance_Gain_Static.eps}\n\\caption{\\label{fig:Performance_Gain2}Performance gain over static channel recommendation scheme. The Type 1 and Type 2 channels have the stationary channel idle probabilities of ${1}\/{6}$ and ${1}\/{2}$, respectively.}\n\\end{center}\n\\end{figure}\n\n\n\n\\subsection{Comparison of MRAS algorithm and Q-Learning}\n\nTo benchmark the performance of the spectrum access\npolicy based on the MRAS algorithm, we compare it\nwith the policy obtained by Q-learning algorithm \\cite{key-22}.\n\nSince the Q-learning can only be used over the discrete action space, we\nfirst discretize the action space $\\mathcal{P}$ into\na finite discrete action space $\\mathcal{\\hat{P}}=\\{0.1,...,1.0\\}$.\nThe Q-learning then defines a Q-value representing the estimated quality\nof a state-action combination as $\nQ:\\mathcal{R}\\times\\mathcal{\\hat{P}}_{rec}\\rightarrow\\mathbb{R}.$\nGiven a new reward $U(R(t),P_{rec}(t))$ is received, we can update the Q-value to be \\begin{multline*}\nQ(R(t),P_{rec}(t))= (1-\\alpha)Q(R(t),P_{rec}(t)),\\\\\n +\\alpha[U(R(t),P_{rec}(t))+\\max_{P_{rec}\\in\\mathcal{\\hat{P}}}Q(R(t+1),P_{rec})],\n \\end{multline*}\nwhere $0<\\alpha<1$ is the smoothing factor. Given a system\nstate $R$, the probability of choosing an action $P_{rec}$ is $\nP_{r}(P_{rec}(t)=P_{rec}|R(t)=R)=\\frac{e^{\\tau Q(R,P_{rec})}}{\\sum_{P_{rec}^{'}\\in\\mathcal{\\hat{P}}}e^{\\tau Q(R,P_{rec})}},$\nwhere $\\tau>0$ is the temperature.\n\nAfter the Q-learning converges, we obtain the corresponding spectrum\naccess policy $\\pi_{Q}$ over the discretized action space $\\mathcal{\\hat{P}}$.\nNote that $\\pi_{Q}$ is a sub-optimal policy for the adaptive channel\nrecommendation MDP over the continuous action space $\\mathcal{P}$.\n\nWe compare the Q-learning based policy with our MRAS-based optimal policy when there are $M=10$ channels and\n$N=5$ users, and show the simulation results in Figures \\ref{fig:Type-1-transition3} and \\ref{fig:Type-2-transition3}.\nFrom these figures, we see that the MRAS-based\nalgorithm outperforms Q-learning up to $10\\%$, which demonstrates the effectiveness of our proposed algorithm.\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[scale=0.45]{MRAS_and_Qlearning_1.eps}\n\\caption{\\label{fig:Type-1-transition3}Comparison of MRAS-based algorithm and Q-learning with Type 1 channel state transition matrix}\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[scale=0.45]{MRAS_and_Qlearning_2.eps}\n\\caption{\\label{fig:Type-2-transition3}Comparison of MRAS-based algorithm and Q-learning with Type 2 channel state transition matrix}\n\\end{center}\n\\end{figure}\n\n\n\\subsection{Heuristic Heterogenous Channel Recommendation}\n\\rev{\nWe now evaluate the proposed heuristic heterogeneous channel recommendation mechanism in Section \\ref{sec:Adaptive-Channel-RecommendationII} with a network consisting of $M=10$ channels and $N=5$ users. We implement the heuristic heterogeneous channel recommendation mechanism in both homogeneous and heterogenous homogeneous environments.\n\\subsubsection{Homogeneous Channel Environment}\nWe first study how the heuristic heterogeneous channel recommendation mechanism performs in the homogeneous channel environment (which is a special case of the heterogeneous environment) in both types of $\\Gamma^{1}$ and $\\Gamma^{2}$ homogeneous channel environments, and simulate the optimal homogeneous channel recommendation (Algorithm \\ref{alg:MRAS-Method-For}) as a benchmark. . The data rate of each channel is normalized to be $1$ Mbps. The results are shown in Figures \\ref{fig:Type-1-transition3II} and \\ref{fig:Type-2-transition3II}. Comparing to the optimal channel access policy, the performance loss of the heuristic heterogeneous channel recommendation in the Type $1$ and Type $2$ channel environments are at most $12\\%$ and $5\\%$, respectively. This shows the efficiency of the heuristic heterogeneous channel recommendation in homogeneous channel environments.\n\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[scale=0.45]{HeurHomo1.eps}\n\\caption{\\label{fig:Type-1-transition3II}Comparison of heuristic heterogenous channel recommendation and optimal homogeneous channel recommendation in Type 1 homogeneous channel environment.}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[scale=0.45]{HeurHomo2.eps}\n\\caption{\\label{fig:Type-2-transition3II}Comparison of heuristic heterogenous channel recommendation and optimal homogeneous channel recommendation in Type 2 homogeneous channel environment.}\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[scale=0.45]{HeurHeter1.eps}\n\\caption{\\label{fig:Type-1-transition3III}Comparison of heuristic heterogenous channel recommendation, optimal homogeneous channel recommendation and optimal homogeneous channel recommendation in the first kind of heterogenous channel environment.}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[scale=0.45]{HeurHeter2.eps}\n\\caption{\\label{fig:Type-2-transition3III}Comparison of heuristic heterogenous channel recommendation, optimal homogeneous channel recommendation and optimal homogeneous channel recommendation in the second kind of heterogenous channel environment.}\n\\end{center}\n\\end{figure}\n\n\\subsubsection{Heterogeneous Channel Environment}\nWe next implement the heuristic heterogeneous channel recommendation mechanism in heterogenous channel environments. The data rates of $M=10$ channels are $\\{B_{1}=0.2,B_{2}=0.6,B_{3}=0.8,B_{4}=1,B_{5}=2,B_{6}=4,B_{7}=6,B_{8}=8,B_{9}=10,B_{10}=20\\}$ Mbps. We consider two kinds of stochastic channel state changing environments:\n\\begin{align}&\\{\\Gamma_{1}=\\Gamma^{2},\\Gamma_{2}=\\Gamma^{2},\\Gamma_{3}=\\Gamma^{2},\\Gamma_{4}=\\Gamma^{2},\\Gamma_{5}=\\Gamma^{2},\\nonumber \\\\\n&~\\Gamma_{6}=\\Gamma^{1},\\Gamma_{7}=\\Gamma^{1},\\Gamma_{8}=\\Gamma^{1},\\Gamma_{9}=\\Gamma^{1},\\Gamma_{10}=\\Gamma^{1}\\},\\end{align}\nand\\begin{align}\n&\\{\\Gamma_{1}=\\Gamma^{1},\\Gamma_{2}=\\Gamma^{1},\\Gamma_{3}=\\Gamma^{1},\\Gamma_{4}=\\Gamma^{1},\\Gamma_{5}=\\Gamma^{1},\\nonumber \\\\ &~\\Gamma_{6}=\\Gamma^{2},\\Gamma_{7}=\\Gamma^{2},\\Gamma_{8}=\\Gamma^{2},\\Gamma_{9}=\\Gamma^{2},\\Gamma_{10}=\\Gamma^{2}\\}.\\end{align} Here subscript denotes channel index, and superscript denote channel type index. For the first kind of channel environment, a channel with low data rate tends to have a low primary transmission occupancy. While for the second kind, a channel with high data rate tends to have a high idleness probability. We also implement static channel recommendation, the optimal homogeneous channel recommendation (Algorithm \\ref{alg:MRAS-Method-For}) and optimal heterogeneous channel recommendation (obtained by adapting the MRAS algorithm to optimize the heterogeneous channel MDP, not shown in this paper) as benchmarks. The results are depicted in Figures \\ref{fig:Type-1-transition3III} and \\ref{fig:Type-2-transition3III}. From these figures, we see that:\n\\begin{itemize}\n\\item For the first kind of channel environment, the heuristic heterogeneous channel recommendation achieves up-to\n$40\\%$ and $100\\%$ performance improvement over the optimal homogeneous channel\nrecommendation and the static channel recommendation, respectively. Comparing with\nthe optimal heterogeneous channel recommendation, the performance\nloss of the heuristic heterogeneous channel recommendation is at most\n$35\\%$. Note that the number of decision variables in the optimal heterogeneous\nchannel recommendation is $M2^{M}=10240$, while the number of decision variables\nin the heuristic heterogeneous channel recommendation is only $2M=20$.\nThe convergence of the heuristic heterogeneous channel recommendation\nhence is much faster than the optimal heterogeneous channel recommendation.\n\\item For the second kind of channel environment, the heuristic heterogeneous\nchannel recommendation achieves up-to $70\\%$ and $100\\%$ performance improvement\nover the optimal homogeneous channel recommendation and static channel recommendation, respectively. The performance\nloss is at most\n$20\\%$ comparing with the the optimal heterogeneous channel recommendation. Comparing with Figure \\ref{fig:Type-1-transition3III}, we see that  the heuristic heterogeneous channel recommendation\nperforms better if more channel opportunities are available for the\nsecondary users.\n\\end{itemize}\n} \n\\section{\\label{sec:Static-Channel-Recommendation1}Introduction To Channel Recommendation}\nIn this section, we first give a review of the static channel recommendation scheme in in  \\cite{key-2} and then discuss the motivation for adaptive channel recommendation.\n\\subsection{\\label{sec:Static-Channel-Recommendation}Review of Static Channel Recommendation}\nThe key idea of the static channel recommendation scheme is that secondary users inform each other about the available channels they have just accessed. More specifically, each secondary user executes the following four stages synchronously during each time slot (See Figure \\ref{fig:Structure-of-Each}):\n\\begin{itemize}\n\\item \\emph{Spectrum sensing:} sense one of the channels based on channel selection result made at the end of the previous time slot.\n\\item \\emph{Channel Contention:} if the channel sensing result is idle, compete for the channel with the backoff mechanism described in Section \\ref{sec:System-Model}.\n\\item \\emph{Data transmission:}  transmit data packets if the user successfully grabs the channel.\n\\item \\emph{Channel recommendation and selection:}\n\\begin{itemize}\n\\item \\emph{Announce recommendation:} if the user has successfully accessed an idle channel, broadcast this channel ID to all other secondary users.\n\\item \\emph{Collect recommendation:} collect recommendations\nfrom other secondary users and store them in a buffer. Typically,\nthe correlation of channel availabilities between two slots diminishes as the time difference increases.\nTherefore, each secondary user will only keep the recommendations received from the most recent $W$ slots and discard the out-of-date information. The user's own successful transmission history within $W$ recent time slots is also stored in the buffer. $W$ is a system design parameter and will be further discussed later.\n\\item \\emph{Select channel}: choose a channel to sense at the next time slot by putting more weights on the recommended channels according to a \\emph{static branching probability $P_{rec}$}. Suppose that the user has $0<R<M$ different channel recommendations in the buffer, then the probability of accessing\na channel $m$ is \\begin{equation}\nP_{m}=\\begin{cases}\n\\frac{P_{rec}}{R}, & \\mbox{if channel $m$ is recommended,}\\\\\n\\frac{1-P_{rec}}{M-R}, & \\mbox{otherwise.}\\end{cases}\\label{eq:SCR-1}\\end{equation}\n A larger value of $P_{rec}$ means that putting more weight on the recommended channels. When $R=0$ (no channel is recommended) or $M$ (all channels are recommended), the random access is used and the probability of selecting channel $m$ is $P_{m}=\\frac{1}{M}$.\n\\end{itemize}\n\\end{itemize}\n\nTo illustrate the channel selection process, let us take the network in Figure \\ref{ChannelRec} as an example. Suppose that the branching probability $P_{rec}=0.4$. Since only $R=1$ recommendation is available (i.e., channel 4), the probabilities of choosing the recommended channel 4 and any unrecommended channel are $\\frac{0.4}{1}=0.4$ and $\\frac{1-0.4}{6-1}=0.12$, respectively.\n\nNumerical studies in \\cite{key-2} showed that the static channel recommendation scheme achieves a higher performance over the traditional random channel access scheme without information exchange. However, the fixed value of $P_{rec}$ limits the performance of the static scheme, as explained next.\n\n\n\\subsection{\\label{sec:Motivation-Adaptive}Motivations For Adaptive Channel Recommendation}\nThe static channel recommendation mechanism is simple to implement due to a fixed value of $P_{rec}$. However, it may lead to significant congestions when the number of recommended channels is small. In the extreme case when only $R=1$ channel is recommended, calculation (\\ref{eq:SCR-1}) suggests that every user will access that channel with a probability $P_{rec}$. When the number of users $N$ is large, the expected number of users accessing this channel $NP_{rec}$ will be high. Thus heavy congestion happens and each secondary user will get a low expected throughput.\n\nA better way is to adaptively change the value of $P_{rec}$ based on the number of recommended channels. This is the key idea of our proposed algorithm.\nTo illustrate the advantage of adaptive algorithms, let us first consider a simple heuristic  adaptive algorithm in a homogeneous channel environment, i.e., for each channel $m$, its data rate $B_{m}=B$ and channel state changing probabilities $p_{m}=p,q_{m}=q$. In this algorithm, we choose the branching probability such that the expected number of secondary users choosing a single recommended channel is one. To achieve this, we need to set $P_{rec}$ as in Lemma \\ref{lemma12s}.\n\\begin{lem}\\label{lemma12s}\nIf we choose the branching probability $P_{rec}=\\frac{R}{N}$, then the expected number of secondary users choosing\nany one of the $R$ recommended channels is one.\n\\end{lem}\n\nDue to space limitations, we give the detailed proof of Lemma \\ref{lemma12s}  in \\cite{key-21}. Without going through detailed analysis, it is straightforward to show the benefit for such adaptive approach through simple numerical examples. Let us consider a network with $M=10$ channels and $N=5$ secondary users. For each channel $m$, the initial channel state probability vector is $\\boldsymbol{p}_{m}(0)=(0,1)$\nand the transition matrix is \\[\n\\Gamma_{m}=\\left[\\begin{array}{cc}\n1-0.01\\epsilon & 0.01\\epsilon\\\\\n0.01\\epsilon & 1-0.01\\epsilon\\end{array}\\right],\\]\nwhere $\\epsilon$ is called the dynamic factor. A larger value of $\\epsilon$ implies that the channels are more dynamic over time.\nWe are interested in time average system throughput\n$U=\\frac{\\sum_{t=1}^{T}\\sum_{n=1}^{N}u_{n}(t)}{T},$\nwhere $u_{n}(t)$ is the throughput of user $n$ at time slot $t$. In the simulation, we set the total number of time slots $T=2000$.\n\nWe implement the following three channel access schemes:\n\\begin{itemize}\n\\item Random access scheme: each secondary user selects a channel randomly.\n\\item Static channel recommendation scheme as in \\cite{key-2} with the \\emph{optimal} constant branching\nprobability $P_{rec}=0.7$.\n\\item Heuristic adaptive channel recommendation scheme with the variable branching probability\n$P_{rec}=\\frac{R}{N}$.\n\\end{itemize}\n\nFigure \\ref{fig:Comparison-of-three} shows that the heuristic adaptive channel recommendation scheme outperforms the static channel recommendation scheme, which in turn outperforms the\nrandom access scheme. Moreover, the heuristic adaptive scheme is more robust to the dynamic channel environment, as it decreases slower than the static scheme when $\\epsilon$ increases.\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[scale=0.45]{Recommendation_System_Comparisons.eps}\n\\end{center}\n\\caption{\\label{fig:Comparison-of-three}Comparison of three channel access\nschemes}\n\n\\end{figure}\n\nWe can imagine that an optimal adaptive scheme (by setting the right $P_{rec}(t)$ over time) can further increase the network performance. However, computing the optimal branching probability in closed-form is very difficult. In the rest of the paper, we will focus on characterizing the structures of the optimal spectrum access strategy and designing an efficient algorithm to achieve the optimum.\n\n\n\\section{System Model}\\label{sec:System-Model}\n\\rev{\nWe consider a cognitive radio network with $M$ parallel and stochastically heterogeneous primary\nchannels. $N$ homogeneous secondary users try to access these channels using a slotted transmission structure (see Figure \\ref{fig:Structure-of-Each}). The secondary users  can exchange information by broadcasting messages over a common control channel\\footnote{Please refer to \\cite{key-20} for the details on how to set up and maintain a reliable common control channel in cognitive radio networks.}. We assume that the secondary users are located close-by, thus they experience similar spectrum availabilities and can hear one another's broadcasting messages. To protect the primary transmissions, secondary users need to sense\nthe channel states before their data transmission.\n\nThe system model\nis described as follows:\n\\begin{itemize}\n\\item \\emph{Channel state:} For each primary channel $m$, the channel state at time slot $t$ is \\[\nS_{m}(t)=\\begin{cases}\n0, & \\mbox{if channel $m$ is occupied by}\\\\\n& \\mbox{primary transmissions,}\\\\\n1, & \\mbox{if channel $m$ is idle.}\n\\end{cases}\\]\n\n\\item \\emph{Channel state transition:} The states of different channels change according to independent Markovian processes (see Figure\n\\ref{fig:Markovian-Channel-Model}). We denote the channel state probability\nvector of channel $m$ at time $t$ as $\n\\boldsymbol{p}_{m}(t)\\triangleq(Pr\\{S_{m}(t)=0\\},Pr\\{S_{m}(t)=1\\}),$\n which follows a two-state Markov chain as $\n\\boldsymbol{p}_{m}(t)=\\boldsymbol{p}_{m}(t-1)\\Gamma_{m},\\forall t\\geq 1,$\n with the transition matrix \\[\n\\Gamma_{m}=\\left[\\begin{array}{cc}\n1-p_{m} & p_{m}\\\\\nq_{m} & 1-q_{m}\\end{array}\\right].\\]\nNote that when $p_{m}=0$ or $q_{m}=0$, the channel\nstate stays unchanged. In the rest of the paper, we will look at the more interesting and challenging cases where $0<p_{m}\\leq1$ and $0<q_{m}\\leq1$. The stationary distribution\nof the Markov chain is given as \\begin{eqnarray}\n\\lim_{t\\rightarrow\\infty}Pr\\{S_{m}(t) & = & 0\\}=\\frac{q_{m}}{p_{m}+q_{m}},\\label{eq:sd-1}\\\\\n\\lim_{t\\rightarrow\\infty}Pr\\{S_{m}(t) & = & 1\\}=\\frac{p_{m}}{p_{m}+q_{m}}.\\label{eq:sd-2}\\end{eqnarray}\n\n\\item \\emph{Heterogeneous channel throughput:} When a secondary user transmits successfully on an idle channel $m$, it achieves a data rate of $B_{m}$. Different channels can support different data rates.\n\\item \\emph{Channel Contention:} To resolve the transmission collision when multiple secondary users access the same channel, a backoff mechanism is used (see Figure \\ref{fig:Structure-of-Each} for illustration). The contention stage of a time slot is divided into $\\lambda^{*}$ mini-slots, and each user $n$ executes the following two steps:\n\\begin{enumerate}\n\\item Count down according to a randomly and uniformly chosen integral backoff time (number of mini-slots) $\\lambda_{n}$ between $1$ and $\\lambda^{*}$.\n\\item Once the timer expires, monitor the channel and transmit RTS\/CTS messages to grab the channel if the channel is clear (i.e., no ongoing transmission). Note that if multiple users choose the same backoff mini-slot, a collision will occur with RTS\/CTS transmissions and no users can grab the channel. Once successfully grabing the channel, the user starts to transmit its data packet.\n\\end{enumerate}\nSuppose that $k_{m}$ users choose channel $m$ to access. Then\nthe probability that user $n$ (out of the $k_{m}$\nusers) successfully grabs the channel $m$ is \\begin{eqnarray}\nPr_{n} & = & Pr\\{\\min\\{\\lambda_{1},...,\\lambda_{k_{m}}\\}=\\lambda_{n}\\}\\nonumber\\\\\n &  & \\cdot \\sum_{\\lambda=1}^{\\lambda^{*}}Pr\\{\\lambda_{n}=\\lambda\\}Pr\\{\\min_{i\\neq n}\\{\\lambda_{i}\\}>\\lambda|\\lambda_{n}=\\lambda\\}\\nonumber\\\\\n & = & \\frac{1}{k_{m}}\\sum_{\\lambda=1}^{\\lambda^{*}}\\frac{1}{\\lambda^{*}}\\left(\\frac{\\lambda^{*}-\\lambda}{\\lambda^{*}}\\right)^{k_{m}-1}.\\end{eqnarray}\nFor the ease of exposition, we focus on the asymptotic case where $\\lambda^{*}$ goes to $\\infty$. This is a good approximation when the number of mini-slots $\\lambda^{*}$ for backoff is much larger than the number of users $N$ and collisions rarely occur. It simplifies the analysis as \\begin{equation} \\lim_{\\lambda^{*}\\rightarrow\\infty}\\frac{1}{\\lambda^{*}}\\sum_{\\lambda=1}^{\\lambda^{*}}(\\frac{\\lambda^{*}-\\lambda}{\\lambda^{*}})^{k_{m}-1}=1,\\end{equation} and thus the expected throughput of user $n$ is \\begin{equation}\nu_{n}(t)=\\frac{B_{m}S_{m}(t)}{k_{m}}.\\label{eq:uu}\\end{equation}\n\\end{itemize}\n}\n\n\\begin{figure}[tt]\n\\begin{center}\n\\includegraphics[scale=0.6]{Time_Slot.eps}\n\\caption{\\label{fig:Structure-of-Each}Structure of each spectrum access time\nslot}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[tt]\n\\begin{center}\n\\includegraphics[scale=0.9]{Markov_Channel_Model.eps}\n\n\\caption{\\label{fig:Markovian-Channel-Model}Two states Markovian channel model}\n\\end{center}\n\\end{figure}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn order to understand to what extent non-spherical planetary nebulae (PNe) are shaped by binary interactions, a fundamental test is the determination of the PN binary fraction. This is a notoriously difficult quantity to determine.\nThe short period binary fraction is estimated to be 15-20\\% (\\cite{Bond2000}, \\cite{Miszalski2009}). This is biased to binaries with periods shorter than about $\\sim 3$ days. \n\nThe field of view of of the Kepler satellite of 14~x~14 square degrees, 4 PNe are known so far and 2 additional possible objects. We seek to detect periodic light variability due to binarity (irradiation effects, eclipses or ellipsoidal variablity) with an amplitude smaller than detectable by  ground observations in order to quantify the extent of the bias. If the currently known short period binary fraction of 15-20\\% is accurate, we should see 1-2 binary PN in our Kepler sample.\n\n\n\\section{The PNe in the Kepler field}\n\n\\begin{figure}[!hbt]\n\\centering\n\\includegraphics[scale=0.16]{Kn61crop.png}\n\\caption{Kn 61, newly detected PN in the Kepler field. Credit: Gemini Observatory and T.~A.~Rector (University of Alaska Anchorage). http:\/\/www.gemini.edu\/node\/11656} \n\\label{kn61}\n\\end{figure}\nThere are four confirmed PNe in the Kepler field, including one (Pa 5) discovered by D.~Patchick in the framework of the Deep Sky Hunters (DSH) project (\\cite{Jacoby2010}).\nOne additional PN candidate, Kn 61 was discovered in the Kepler field by M. Kronberger using DSS false color images also in the context of the DSH search (\\cite{KronbergerIAU}), and then imaged by us in March 2011 with  a 10 min exposure at the 2.1m NOAO Kitt Peak observatory.  A Gemini Observatory image of the object is presented in Fig.~\\ref{kn61}. From its structure, that is similar to PN A 43, it is classified as a probable PN, although a spectrum of the nebula is required before considering Kn 61 as a confirmed PN. The sixth object, PaTe 1, is classified as possible PN due to its unusual morphology.\n\n\n\n\n\n\n\n\n\\begin{table}[!htb]\n\\begin{center}\n\\begin{tabular}{l c c c c}\n\\toprule\nName        &     PN status     &  Period (days) &         Amplitude (mmag)      &             Notes              \\\\\n\\midrule\nNGC 6826    &    confirmed      &     0.619     &            10                 &            Possible binary                                                     \\\\\nNGC 6742    &    confirmed      &     0.37:     &            0.2:               &      Possible periodic variability                         \\\\\nPa 5        &    confirmed      &      1.12     &            4                  &            Possible binary, see \\cite{Ostensen2010}                           \\\\\nPaTe 1      &    possible       &      0.17     &            2                  &              Possible binary                                                   \\\\\nA 61        & confirmed         &       1.47:     &   2:                         &          Variable                               \\\\\nKn 61       &   probable        &       -       &               -               &              Recently discovered, no data   \\\\\n\\bottomrule\n\n\\end{tabular}\n\\label{chart}\n\\caption{Preliminary results on the variability properties of the PNe in the Kepler field. The numbers followed by : are to be confirmed. Out of 5 possible or confirmed PNe, 4 show periodic light variations that could be attributed to binarity.}\n\\end{center}\n\n\\end{table}\n\n\n \n\n\n\\section{Early results and future work}\n\nWhile the smallest detectable amplitude for flux variability is 0.1 mag for ground-based observations, Kepler can detect light variability of about 1 mmag. In Table 1 we show that  4 of the 5 confirmed and possible PNe that have already been observed by Kepler have periodic light variations consistent with binarity (though a proper analysis is required to confirm that the perodic variability of these objects is due to binarity) with amplitudes less than the ground-based detection limit. \n\nWe will take thirty minute exposures of these six potential PNe in the Kepler field regularly throughout the year to detect potential long period flux variations. Wind and pulsations effects will be assessed with 1 minute sampling for each object over a single quarter. The advantages of this monitoring with respect to ground-based observations are a higher photometric precision (up to $1\\mu$mag for bright stars and well monitored objects) and a longer time baseline with no day, bad weather nor lunar interruptions insuring a constant coverage and a better time sampling.\n\n\n\n\nWhen the full data set is available, we will be able to impose some limits on how many short binaries may have avoided detection. Even with a small sample of PNe, we can put constraints on the actual short period binary fraction amongst the central stars of PNe.\n\n\n\n\n\n \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nYoung supernova remnants (SNRs) are widely believed to be\nthe main source of Galactic cosmic rays.\n\\citet{koyama1995} discovered synchrotron X-rays\nfrom shells of SN~1006,\nwhich was the first observational clue of \ncosmic-ray electrons being accelerated up to the TeV energy range.\nLater, several young SNRs were found to have synchrotron X-ray shells\n\\citep[c.f.,][]{koyama1997,slane1999,bamba2000,vink2003}.\nAnother piece of evidence for cosmic-ray acceleration in SNRs \nwas obtained from gamma-ray observations.\nVery-high-energy (VHE) and GeV gamma-rays have been detected \nfrom several SNRs,\nalthough, it is still unclear whether their origin\nis hadronic or leptonic\n\\citep[e.g.,][]{aharonian2004,aharonian2007,\nabdo2010b,abdo2010c,abdo2011}.\n\nDespite these various pieces of evidence\nin favor of acceleration,\nit is still unclear how the acceleration process\nevolves in SNRs,\nin particular how the accelerated particles cool down and\nhow they escape from SNRs.\nOne reason for this lack of understanding is that\npast observational studies have concentrated on\nindividual sources alone.\nIn this paper, we investigate, for the first time,\nthe time evolution of SNR synchrotron X-rays,\nusing all available data in literature for sources\nin which this emission component is bright.\nIn section~\\ref{sec:sample},\nwe describe the evolution of synchrotron X-ray luminosity\nin our sample.\nAn interpretation and a simple model to support it\nare in section~\\ref{sec:model}.\nFinally, section~\\ref{sec:discuss} is devoted to\ndiscussion of the results.\n\n\\section{Sample selection and Results}\n\\label{sec:sample}\n\nWe searched the literature for reports of synchrotron X-ray emission\nfrom SNRs and found a total of 14 such sources.\nHowever, we could not use all of these for the current study,\nfor various reasons.\nKepler \\citep{bamba2005,reynolds2007}\nand G330.2+1.0 \\citep{park2009} had to be removed from our sample,\nbecause of the difficulty of estimating\nthe total synchrotron X-ray luminosity,\ncaused by a low detection significance or\nby contamination from thermal X-rays.\nThus our sample consists of the 12 SNRs\nlisted in Table~\\ref{tab:sample}.\nWe used the latest value of the 2--10~keV synchrotron X-ray unabsorbed flux\nfrom the references listed in the table.\nWhen the reported flux was not absorption-corrected or\nwas for a different energy band,\nwe normalized it unabsorbed flux in the 2--10~keV\nusing the best-fit model in the references.\nSynchrotron X-ray emission may have a spectral roll-off\nin the 2--10~keV band.\nWhen significant roll-off was reported in the references,\nwe took such a component into account while computing the flux.\nSome SNRs (Cas~A, Tycho, SN~1006)\nhave bright thermal X-ray emission,\nwhich makes it difficult to estimate the synchrotron X-ray flux.\nIn these cases,\nwe inferred the synchrotron X-ray flux\nbased on wide-band information and\/or detailed spectroscopy \nto isolate synchrotron X-rays from thermal component,\nas we believe such analysis can provide the most reliable \nresults currently available.\nTwo of the sample sources, W28 and G156.7+5.7, are largely extended but\nwere only partly observed with enough exposure in X-rays.\nAs the X-ray luminosities can be deduced only for \nthe observed regions\n(5\\% for W28 and 26\\% for G156.7+5.7, respectively),\npossible contributions from the unobserved regions\nare included in the upper error bars.\nWhen we estimated these contributions,\nwe assumed that the surface brightness of the unobserved region\nin each source is identical to that of the observed one.\nThe distance uncertainty is also included into the errors\non the luminosity.\nWe used a 10\\% distance uncertainty \nfor cases where \nno uncertainty was available in the literature.\nIn summary, we took into account the following uncertainties;\nstatistical errors given in literatures, systematic distance uncertainty,\nand the limited coverage of the SNR extension.\n\nIn order to study the time evolution, we need to know\nthe SNR ages.\nHowever, we have only four SNRs whose ages are historically known.\nAnother parameter, the ionization time scale of heated plasma,\nis also difficult to determine\nsince several SNRs show no thermal emission.\nWe thus use the physical radius of each SNR, $R_s$, as the age indicator.\n$R_s$ is taken from \\citet{green2009}.\nThe distance uncertainty is included in the error computation.\nSome SNRs have distorted shapes,\nas shown in Tab.~\\ref{tab:sample}.\nThis is also included in the errors on the radii.\nAll the derived parameters are shown in Tab.~\\ref{tab:sample}.\nThe radius of a SNR depends on the density of the interstellar medium,\nbut rather insensitively.\nFor remnants in the Sedov phase, $R_s\\propto n_0{}^{-1\/5}$\n(where $n_0$ is the upstream number density).\nThe radius changes by only a factor of $\\sim$2 \neven if the upstream density changes by 2~orders of magnitude.\nThe radius also has very weak dependency on the explosion energy $E_0$,\n$R_s\\propto E_0^{1\/5}$.\nOn the other hand, $R_s\\propto t_{\\rm age}{}^{2\/5}$\n(where $t_{\\rm age}$ is the age of the remnant).\nThus the radius is a good age indicator.\n\nFigure~\\ref{fig:r_Lx} shows the synchrotron X-ray luminosity\nas a function of the radius.\nOne can see that\nthe luminosity is in the order of $10^{34}$--$10^{36}$~ergs~s$^{-1}$\nwhen the SNRs are smaller than $R_s\\sim$5~pc,\nwhich corresponds to an age of a few hundred years,\nwhereas it decreses to $10^{32}$--$10^{35}$~ergs~s$^{-1}$\nbeyond $R_s\\sim$5~pc.\nit appears that the luminosity drops off rapidly\nat $R_s\\sim$5~pc,\nalthough the scatter is rather large.\nNote that all the SNRs with $R_s<5$~pc display\nsynchrotron X-rays brighter than $10^{34}$~ergs~s$^{-1}$ \\citep{green2009},\nwhereas most of larger SNRs do not have significant synchrotron X-rays.\nThis fact makes the difference of luminosities larger\nbetween the two regions.\nThe drop off in non-thermal X-ray luminosity reaches two or three\norders of magnitude,\nwhich is much larger than the errors on the individual luminosities.\nWe have carried out a similar analysis for the thermal X-ray luminosity\nfor SNRs in the Large Magellanic Cloud \\citep{williams1999}\nand the radio luminosity of Galactic SNRs \\citep{green2009,case1998}.\nHowever, no drop off like that observed for synchrotron X-rays\nhas been found.\n\n\n\\section{Evolution of Synchrotron X-rays}\n\\label{sec:model}\n\nHere we consider the cause of the decrease in synchrotron X-ray emission\nidentified in the previous section.\nThe $\\nu F_\\nu$-spectrum of synchrotron X-ray radiation \nhas a peak around the roll-off frequency, $\\nu_{\\rm roll}$,\nabove which the flux rapidly decays towards higher frequencies.\nAs SNRs evolve, $\\nu_{\\rm roll}$ is known to decrease\n\\citep[e.g.,][]{bamba2005}.\nThe observed rapid decay in synchrotron X-rays around \n$R_s\\sim10~{\\rm pc}$ could be caused by\n$\\nu_{\\rm roll}$ passing through the X-ray band\nto lower energies.\nIn this section, we discuss this possibility in detail.\nWe introduce essential argument first,\nand show a simple model to support it later.\n\nWe assume that the electron acceleration is energy-loss-limited,\nin which the maximum energy of electrons, $E_e{}^{\\max}$,\nis determined from \nthe balance of the synchrotron loss and acceleration:\n\\begin{equation}\nE_e{}^{max} = \\frac{24}{\\xi^{1\/2}}\\left(\\frac{v_s}{10^8~{\\rm cm~s^{-1}}}\\right)\n\\left(\\frac{B_d}{10~\\mu{\\rm G}}\\right)^{-1\/2}\\ \\ {\\rm (TeV)}\\ \\ ,\n\\label{eq:Emax_e}\n\\end{equation}\nwhere $\\xi$ is the gyro-factor.\nIn deriving Eq.~(\\ref{eq:Emax_e}), we equate the acceleration time,\n$t_{\\rm acc}(E)=20\\xi cE\/3ev_s{}^2B_d$ with the synchrotron cooling time, \n$t_{\\rm syn}(E)=125~{\\rm yr}(E\/10~{\\rm TeV})^{-1}(B_d\/100~\\mu{\\rm G})^{-2}$,\nwhere $v_s$ and $B_d$ are the shock velocity and the downstream\nmagnetic field, respectively.\nIn this case, we derive \n\\begin{equation}\nh\\nu_{\\rm roll}\\sim0.4~{\\rm keV}~\n\\xi^{-1}(v_s\/10^8~{\\rm cm}~{\\rm s}^{-1})^2 ~~,\n\\label{eq:rolloff}\n\\end{equation}\nwhich is independent of $B_d$ \\citep[e.g.,][]{aharonian1999,yamazaki2006}.\nThe roll-off frequency and the shock velocity have been measured\nin several SNRs\n\\citep[Cas~A, SN~1006;][]{reynolds1999,patnaude2009,bamba2008,ghavamian2002}.\nFor these SNRs, eq.(\\ref{eq:rolloff}) is consistent\nwith the observational values\nto within 1 order of magnitude\nassuming $\\xi = 1$.\nAlso note that the effect of particle escape from the shock region\n\\citep[e.g.,][]{reynolds1998,ohira2010} is not considered in this paper.\nThis effect might be important for older SNRs\n\\citep{ohira2011b}.\nHowever, our present model reproduces the observed trend well,\nsuggesting that particle escape is not yet significant for these SNRs\n(see also the last paragraph in section~4).\n\nThe X-ray spectrum of synchrotron radiation is well \napproximated analytically.\nIt is mainly determined by shock dynamics in the SNR\nif the synchrotron X-ray-emitting electrons with energy $E$ satisfy\n$t_{\\rm syn}(E)<t_{\\rm age}$, where $t_{\\rm age}$ is the age of the SNR.\nIn this case, the energy spectrum of electrons is generally given by\n\\begin{equation}\nN_e(E) = AE^{-p}(1+E\/E_b)^{-1}\n\\exp[-(E\/E_{e}{}^{max})^2]\\ \\ ,\n\\label{eq:N_e}\n\\end{equation}\nwhere the break energy, $E_b$,  is determined by \n$t_{\\rm syn}(E_b)=t_{\\rm age}$.\nDuring the acceleration, electrons with $E>E_b$ suffer energy loss via\nsynchrotron cooling, which causes a steepening of the\nenergy spectrum \\citep[e.g.,][]{longair94}.\nWe also note that the shape of the cutoff is analytically obtained in\n\\citet{zirakashvilli2007}.\nGiven the electron distribution, we calculate the approximate\nformula of the X-ray luminosity, $L_\\nu$.\nThe characteristic frequency, $\\nu_{\\rm syn}(E)$, of the synchrotron radiation\nemitted by electrons with energy $E$ is given by\n\\begin{equation}\nh\\nu_{\\rm syn}(E)\\sim\n0.12~{\\rm keV}(B_d\/10~\\mu{\\rm G})(E\/10~{\\rm TeV})^2~~.\n\\label{eq:character}\n\\end{equation}\nThen, as long as $\\nu<\\nu_b=\\nu_{\\rm syn}(E_b)$,\nwe can apply the standard formula,\n$L_\\nu\\propto AB_d^{(p+1)\/2}\\nu^{-(p-1)\/2}$\n\\citep[e.g.,][]{longair94}.\nOn the other hand, if $\\nu_b<\\nu$, the spectral slope\nsteepens ($L_\\nu\\propto\\nu^{-p\/2}$) due to the \nsteepening of the electron distribution, and we derive\n\\begin{eqnarray}\nL_\\nu &\\propto& AB_d{}^{(p+1)\/2}\\nu_b{}^{-(p-1)\/2} (\\nu\/\\nu_b)^{-p\/2}\n\\exp(-\\sqrt{\\nu\/\\nu_{\\rm roll}})\n\\nonumber \\\\\n&\\propto&\nAB_d{}^{(p-2)\/2}\\nu^{-p\/2} \\exp(-\\sqrt{\\nu\/\\nu_{\\rm roll}})\n~~,\n\\label{eq:Luminoosity}\n\\end{eqnarray}\nwhere we assume that $L_\\nu$ is continuous at $\\nu = \\nu_b$,\nand we again use the result of \\citet{zirakashvilli2007}\nfor the cutoff shape.\nCalculating $\\nu_b$ as\n\\begin{equation}\nh\\nu_b \\sim 0.19~{\\rm keV}\n(B_d\/10~\\mu{\\rm G})^{-3}(t_{\\rm age}\/10^4{\\rm yr})^{-2}~~,\n\\end{equation}\nwe find that throughout the evolution of the SNR,\nthe X-ray band (2--10~keV) always lies above $\\nu_b$ because\nfor young SNRs ($t_{\\rm age}\\lesssim10^3$~yr),\nthe magnetic field may be amplified to $B_d\\gg10~\\mu$G\n\\citep[e.g.,][]{bamba2003,bamba2005,vink2003}.\nEquation.~(\\ref{eq:Luminoosity}) is thus a good approximation\nof the X-ray luminosity for any arbitrary epoch.\nIn particular, when $p\\approx2$, the X-ray luminosity\nis insensitive to the magnetic field,\ndepending instead mainly on $\\nu_{\\rm roll}$.\nAs the SNR ages, the shock velocity $v_s$ decreases and\n$\\nu_{\\rm roll}$ becomes smaller. \nEquation~(\\ref{eq:rolloff}) tells us that\n$\\nu_{\\rm roll}$ is below the X-ray band (2--10~keV)\nwhen $v_s\\lesssim10^8$cm~s$^{-1}$, \nso that the X-ray luminosity drops off.\n\n\nIn order to demonstrate the above argument,\nwe construct a simple model\nto calculate the synchrotron X-ray flux.\nIn our model, a simple shock dynamics scenario is considered.\nWe assume that the forward shock velocity of SNRs $v_s$ \nis a function of the age of SNR $t_{age}$ as follows:\n\\begin{eqnarray}\nv_s &=& \\left\\{\n\\begin{array}{ll}\nv_i & (t_{age}<t_1;\\ {\\rm Free\\ expansion\\ phase}), \\\\\nv_i \\left(\\frac{t_{age}}{t_1}\\right)^{-3\/5} & (t_1 < t_{age} < t_2;\\\n{\\rm Sedov\\ phase}), \\\\\nv_i\\left(\\frac{t_2}{t_1}\\right)^{-3\/5}\\left(\\frac{t_{age}}{t_2}\\right)^{-2\/3}\n& (t_2<t_{age};\\ {\\rm Radiative\\ phase}), \\\\\n\\label{eq:velocity}\n\\end{array}\n\\right.\\\\\nt_1 &=& 2.1\\times 10^{2}(E_{51}\/n_0)^{1\/3}v_{i,9}{}^{-5\/3}\\ {\\rm yr},\\\\\nt_2 &=& 4\\times 10^{4}E_{51}{}^{4\/17}n_0{}^{-9\/17}\\ {\\rm yr},\n\\end{eqnarray}\nwhere $v_i = v_{i,9}\\times 10^9~{\\rm cm~s^{-1}}$,\n$E_{\\rm ej} = E_{51}\\times 10^{51}$~erg,\nand $n_0$ are the initial velocity,\ninitial energy of ejecta,\nand the upstream number density, respectively\n\\citep[e.g.,][]{blondin1998,yamazaki2006}.\nThe model assumes that\nthe SNR forward shock propagates into a homogeneous medium.\nThis is, however, not always in the case,\nsince the SNRs could be located inside of superbubbles\nmade by pre-explosion stellar winds.\nMore detailed models\nfor core-collapsed type SNRs, such as RCW~86\n\\citep[e.g.,][]{vink1997},\nshould be applied.\nBut it is left for future work.\n\nAs illustrated above, the X-ray luminosity has little dependence on the\nmagnetic field. Nevertheless, we  model the evolution of the \ndownstream magnetic field, $B_d$, as follows.\nUsually, the amplified magnetic field $B_{\\rm amp}$\nis simply assumed to scale with $v_s$  \\citep{volk2005,vink2008},\nbecause magnetic field evolution remains ill-understood.\nHere we adopt an assumption similar to \\citet{volk2005},\nleading to a dependence of $B_d\\propto v_s$. \nLet the energy density of the amplified magnetic field \n$U_B = B_{\\rm amp}{}^2\/8\\pi$ be proportional to the thermal energy density $U_{th}$,\n\\[\n\\frac{B_{\\rm amp}{}^2}{8\\pi} = \\epsilon_BU_{th} = \\frac{3}{2}\\epsilon_Brn_0kT_d\\ \\ ,\n\\]\nwhere $\\epsilon_B$, $r$, and $kT_d$ are\nthe energy-partition parameter, a compression ratio, \nand a downstream temperature\nobtained by the Rankine-Hugoniot equation\n$kT_d = [(r-1)\/{r^2}]m_pv_s{}^2$, respectively.\nThen, we obtain\n$B_{\\rm amp} = [12\\pi m_p(r-1)\\epsilon_B n_0\/r]^{1\/2} v_s$.\nThe downstream magnetic field is thus given by\n$B_d={\\max}\\{B_{\\rm amp},rB_{\\rm ISM}\\}$,\nwhere $B_{\\rm ISM}=10~\\mu$G is the field strength in the interstellar matter.\n\n\n\nFor given  $t_{\\rm age}$, \nwe calculate $v_s$ according to Eq.~(\\ref{eq:velocity}),\nthe shock radius $R_s$ by the integration of \n$v_s$, and the luminosity of synchrotron X-rays numerically\nby using the value of $B_d$ and the electron distribution \ngiven by Eqs.~(\\ref{eq:Emax_e}) and (\\ref{eq:N_e}).\nWe adopt $E_{51} = v_{i,9} = 1$, $p=2.0$, $r=4$, $\\xi=1$,\nand $\\epsilon_B = 0.01$ \nas  fiducial parameters.\nNote that for $\\epsilon_B\\sim0.01$, the evolution of the\namplified magnetic field in young\nSNRs is approximately reproduced \\citep{volk2005}.\nThis is also consistent with previous observational implications for SNRs\n\\citep{ghavamian2002,bamba2003,parizot2006}.\nOne finds that with the above assumption for the magnetic field,\nthe cooling time of synchrotron X-ray-emitting electrons\nis always smaller than the SNR age, $t_{\\rm age}$,\nso that the X-ray luminosity is well approximated\nby Eq.~(\\ref{eq:Luminoosity}).\nAlso note that the value of $\\xi$ is near unity,\nimplying that acceleration is near the Bohm limit.\nWe assume that the normalization factor of the electron distribution\n$A$ is constant with time.\nThis is justified if the amount of accelerated particles is\nproportional to the product of the fluid ram pressure and \nthe SNR volume, that is,\n$A\\propto (n_0v_s{}^2)R_s{}^3$.\nIn the Sedov phase ($R_s\\propto t^{2\/5}$)\nduring which  $\\nu_{\\rm roll}$ crosses the X-ray band, \nwe find that $A$ is constant with time.\n\n\nFig.~\\ref{fig:r_Lx}  shows the results\nfor $n_0=5.0$, 1.0, and 0.1~cm$^{-3}$.\nFor the  first two cases,\nwe set $A=9.0\\times 10^{45}(n_0\/1~{\\rm cm}^{-3})$,\nwhereas $A=9.0\\times 10^{45}$\nfor $n_0=0.1$~cm$^{-3}$.\n$A$ is expressed in the cgs units.\nOne can see that our model roughly reproduces\nthe observed trend with reasonable model parameters.\nThe assumed density, $n_0 = 0.1-5.0~{\\rm cm}^{-3}$,\nis typical for the interstellar medium around SNRs,\nalthough there are discrepancies for some individual samples.\nSome SNRs are likely to be located in bubbles or very low density regions.\nFor example,\n\\citet{vink1997} showed that RCW~86 is in a super-bubble\nand expands into extremely low density materials.\nAs a result, this SNR should have a very large radius\ncompared with its age.\n$A$ is also a free parameter\nwhich may change from source to source.\nAdjusting $A$ to within 1~order of magnitude,\nit is possible to find a solution for the observed density\nin each case.\nSince our aim is simply to reproduce \nthe overall evolutionary trend in an approximate manner,\nwe ignore such fluctuations in this work.\n\n\n\\section{Discussion}\n\\label{sec:discuss}\n\nWe have reproduced the observed fast drop-off in\nsynchrotron X-ray luminosity with SNR radius using a quite simple model.\nRecently, \\citet{patnaude2010} discovered\na decline of a few percent in the synchrotron X-ray power from one source,\nCas~A,\nand reached a similar conclusion.\n\nThe total and maximum energy\nof accelerated particles can be derived from our model.\nWith the adopted values of $A$ and Eq.~(\\ref{eq:N_e}),\nwe can calculate the total energy of accelerated electrons\nwith the energy above $m_ec^2$\nto be  $10^{47}$--$10^{48}$~ergs,\nwhich is reasonable since \nthis range is much smaller than $E_{\\rm ej}$.\nNote that these estimated values could increase by 1 order of magnitude\nif we increase $\\xi$ by up to 10.\nEven if we alter the value of $p$, the results remain similar\nif we appropriately reset the normalization $A$.\nThis is expected from equation~(\\ref{eq:Luminoosity}).\nFor example, the results for $p=2.2$ are almost perfectly\nin agreement with the case of $p=2.0$ if we choose\n$A=1.7\\times 10^{46}(n_0\/1~{\\rm cm}^{-3})$\n$n_0=5.0$ and 1.0~cm$^{-3}$\nand $A=1.7\\times 10^{46}$\nfor $n_0=0.1$~cm$^{-3}$.\nIn these cases, the total energy of accelerated electrons\nwith energy above $m_ec^2$ is again\n$10^{47}$--$10^{48}$~ergs.\n\\citet{bamba2003} estimated the energy of injected accelerated electrons\nin SN~1006 for the first time.\nThis was found to be $\\sim 10^{44}$--$10^{45}$~ergs\nper small segments of filaments,\nalthough with rather large uncertainty.\nThe total energy of accelerated electrons\nis approximately $10^{46}$--$10^{48}$~ergs\nconsidering the size of these segments,\nwhich is consistent with our result.\n\\citet{bamba2005} show that the enegy injected into accelerated electrons\nis always similar in young SNRs,\nwhich is also consistent with our results.\nTheoretically, \\citet{berezhko2006} estimated the energy injected into\naccelerated protons to be $10^{50}$~ergs \nin the case of RX~J1713$-$3946.\nThis is also consistent with our results\nwhen we consider the electron and proton ratio of $10^{-4}$\nin cosmic rays.\n\n\nAt present, no direct information on the accelerated protons is \nobtained by X-ray observations.\nOne might consider whether we could estimate\nthe total energy of accelerated protons\nunder the assumption of a similar $A$ as for electrons.\nHowever, this is not possible because the injection efficiency is\ndifferent in each case.\nOn the other hand,\nthe maximum energy of accelerated protons, $E_{p}{}^{max}$,\ncan be estimated according to our  model.\nThe synchrotron and $\\pi^0$-decay loss timescales are negligible\nfor protons,\nand thus $E_{p}{}^{max}$ is determined by the condition\n$t_{\\rm acc}=t_{\\rm age}$, and we obtain\n\\begin{equation}\nE_{p}{}^{max}(t_{\\rm age}) = \\frac{4.8\\times 10^2}{\\xi}\n\\left(\\frac{v_s}{10^9 {\\rm cm~s^{-1}}}\\right)^2\n\\left(\\frac{B_d}{10~\\mu{\\rm G}}\\right)\n\\left(\\frac{t_{\\rm age}}{10^3~{\\rm yr}}\\right)\\ \\ {\\rm TeV}\\ .\n\\end{equation}\nAlthough the effect of the wave damping and\/or escape may be significant\nin older systems\n\\citep[e.g.,][]{zirakashvilli2007,ohira2010}, \nwe ignore it here for simplicity.\nFigure~\\ref{fig:Emax_evolution} shows\n$E_{p}{}^{max}$ as a function of $t_{\\rm age}$.\nIn the free expansion phase, \n$E_{p}{}^{max}$ increases in proportion to the age,\nand protons are accelerated quickly up to around the knee energy\nin $\\sim$100~yr.\nWhen the shock velocity decreases,\n$E_{p}{}^{max}$ becomes smaller.\nHence $E_{p}{}^{max}$ peaks at $t_{\\rm age}=t_1$.\nIn contrast to the case of electron acceleration,\n$E_{p}{}^{max}(t_1)$\n(that is the maximum of $E_{p}{}^{max}$) is larger\nfor higher upstream density $n_0$,\nand it changes only within a factor of $\\sim2$ \nfor $n_0$ = 0.1--5.0~cm$^{-3}$.\nSince  $B_d \\propto n_0{}^{1\/2}$ and \n$t_1 \\propto n_0{}^{-1\/3}$, one finds\n$E_{p}{}^{max}(t_1)\\propto B_dt_1\\propto n_0{}^{1\/6}$.\nProtons are accelerated up to $E_{p}{}^{max}(t_1)\\sim 10^{15-16}$~eV\nin any SNR, with little density dependence.\nThis result might be important for explaining the break at\n$\\sim 10^{15.5}$~eV in the observed cosmic-ray spectrum\nwith the shock acceleration on SNRs.\nOn the other hand, one can find \n$E_{e}{}^{max}\\propto B_d{}^{-1\/2}v_s\\propto v_s{}^{1\/2}$\nif $B_d=B_{\\rm amp}\\propto v_s$.\nHence the maximum energy of electrons scales as\n$E_{e}{}^{max}\\propto t^0$ and $\\propto t^{-3\/10}$\nfor the free expansion and Sedov phases, respectively,\nso that it depends only weakly on the SNR age.\n\nRecent {\\it Fermi} observations of middle-aged SNRs have shown the\npresence of a gamma-ray spectral break around energies of a few GeV \n\\citep[e.g.,][]{abdo2009,abdo2010}.\nSeveral interpretations\nhave been given \\citep[e.g.,][]{inoue2010,li2010,li2011,ohira2011},\n one of which is the escape\nof particles from acceleration sites. In this scenario, \naccelerated particles with energies of more than  about 10-100~GeV\nhave already escaped from the acceleration region \\citep{ohira2011}.\nIn contrast, our present model has shown that even for \n$t_{\\rm age}\\gtrsim10^4$~yr, $\\sim10$~TeV electrons still exist around the\nshock front (see Figure~\\ref{fig:Emax_evolution}). \nThis apparent discrepancy may come from the fact that\nmost SNRs observed by {\\it Fermi} are interacting with molecular clouds,\nwhereas in our sample there is only one (W~28)\nundergoing such a collision.\nHence molecular clouds may play an important role\nin dispersing cosmic rays from the SNR shock into the interstellar matter.\n\n\n\\acknowledgments\n\nWe would like to thank the anonymous referee for the fruitful comments.\nWe also thank\nS.~Wagner and G.~P{\\\" u}lhofer\nfor the analysis of individual sources.\nWe also thank P.~Gandhi for the English correction.\nThis work was supported in part by\nGrant-in-Aid for Scientific Research\nof the Japanese Ministry of Education, Culture, Sports, Science\nand Technology (MEXT) of Japan, No.~22684012 (A.~B.),\nNo.~21740184 and 21540259 (R.~Y.).\nand No.~18071001, 22244030, and 21111006 (K.~K.).\nK.K. was partly supported by the Center for the Promotion of\nIntegrated Sciences (CPIS) of\nSokendai.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}