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+{"text":"\\section{Introduction}\n\nThis paper is concerned with the problem of supervised classification, a topic of interest in both statistics and machine learning.\nHastie {\\it et al} \\citeyear{has:tib:fri00} gives a description of various classification methods. We outline our\nproblem as follows. We have a collection of training data $\\{(x_i,y_i),i=1,\\dots,n\\}$. The values in the\ncollection $\\hbox{\\boldmath$x$} = \\{x_1,\\dots,x_n\\}$ are often called features and can be conveniently\nthought of as covariates.\nWe denote the class labels as $\\hbox{\\boldmath$y$}=\\{y_1,\\dots,y_n\\}$, where each $y_i$ takes one of the\nvalues $1,2,\\dots,G$. Given a collection of incomplete\/unlabelled\ntest data $\\{(x_i,y_i),i=n+1,\\dots,n+m\\}$, the problem amounts to predicting the class labels\nfor $\\hbox{\\boldmath$y$}^*=\\{y_{n+1},\\dots,y_{n+m}\\}$ with corresponding feature vectors\n$\\hbox{\\boldmath$x$}^*=\\{x_{n+1},\\dots,x_{n+m}\\}$.\n\nPerhaps the most common approach to classification is the well-known $k-$nearest neighbours ($k-$nn) algorithm.\nThis algorithm amounts to classifying an unlabelled $y_{n+i}$ as the most common class among the\n$k$ nearest neighbours of $x_{n+i}$ in the training set $\\{(x_i,y_i),i=1,\\dots,n\\}$.\nWhile this algorithm is easy to implement, and often gives good performance, it can be\ncriticised since it does not allow any uncertainty to be associated to the test class labels, and\nto the value to $k$. Indeed the choice of $k$ is crucial to the performance of the algorithm. The value of\n$k$ is often chosen on the basis of leave-one-out cross-validation.\n\n\n\n\n\nThere has been some interest in extending the $k-$nearest neighbours algorithm to allow for\nuncertainty in the test class labelling, most notably by \\cite{hol:ada02}, \\cite{hol:ada03} and\nmore recently \\shortcite{cuc:mar08}. Each of these probabilistic variants of the $k-$nearest neighbour\nalgorithm, is based on defining a neighbourhood of each point $x_i$, consisting of the $k$ nearest neighbours\nof $x_i$. But moreover, each of these neighbouring points has equal influence in determining the missing\nclass label for $y_i$, regardless of distance from $x_i$. In this article we present a class of models,\nthe \\textit{distance nearest neighbour} model, which shares many of the advantages of these probabilistic approaches, but in contrast to these approaches, the relative influence of neighbouring points depends\non the distance from $x_i$. Formally, the distance nearest neighbour model is a discrete-valued Markov random field, and, as is typical with such models, depends on an intractable normalising constant. To overcome this problem we use the exchange algorithm of Murray {\\textit{et al.}} \\citeyear{Murray06} and illustrate that this provides a computationally efficient algorithm with very good mixing properties.\nThis contrasts with the difficulties encountered by Cucala \\textit{et al.} \\citeyear{cuc:mar08} in their implementation of the sampling scheme of M{\\o}ller \\textit{et al} \\citeyear{mol:pet06}.\n\nThis article is organised as follows. Section 2 presents a recent overview of recent probabilistic approaches\nto supervised classification. Section 3 introduces the new distance nearest neighbour model and outlines\nhow it compares and contrasts to previous probabilistic nearest neighbour approaches. We provide a computationally\nefficient framework for carrying out inference for the distance nearest neighbour model in Section 4. The performance\nof the algorithm is illustrated in Section 5 for a variety of benchmark datasets, as well as challenging\nhigh-dimensional datasets. Finally, we present some closing remarks in Section 6.\n\n\n\n\\section{Probabilistic nearest neighbour models}\n\nHolmes and Adams \\citeyear{hol:ada03} attempted to place the $k-$nn algorithm in a probabilistic setting\ntherefore allowing for uncertainty in the test class labelling. In their approach the full-conditional\ndistribution for a training label is written as\n\\[ \\pi(y_i|\\hbox{\\boldmath$y$}_{-i},\\hbox{\\boldmath$x$},\\beta,k) \\propto \\exp\\left( \\beta\\sum_{j\\sim^k i} I(y_i=y_j)\/k \\right), \\]\nwhere the summation is over the $k$ nearest neighbours of $x_i$ and where $I(y_i=y_j)$ is an indicator\nfunction taking the value $1$ if $y_i=y_j$ and $0$ otherwise. The notation, $j\\sim^k i$ means that\n$x_j$ is one of the $k$ nearest neighbours of $x_i$.\nHowever, as pointed out in \\shortcite{cuc:mar08}, there is a difficulty with this formulation, namely that\nthere will almost never be a joint probability for $\\hbox{\\boldmath$y$}$ corresponding to this collection of\nfull-conditionals. The reason is simply because the $k-$nn neighbourhood system is usually asymmetric.\nIf $x_i$ is one of the $k$ nearest neighbours of $x_j$, then it does not necessarily follow that\n$x_j$ is one of the $k$ nearest neighbours of $x_i$.\n\nCucala \\textit{et al.} \\citeyear{cuc:mar08} corrected the issue surrounding the asymmetry of the $k$-nn\nneighbourhood system. In their probabilistic $k$-nn ($pk$-nn) model, the full-conditional for class label $y_i$ appears as\n\\begin{equation}\n \\pi(y_i|\\hbox{\\boldmath$y$}_{-i},\\hbox{\\boldmath$x$},\\beta,k) \\propto \\exp\\left( \\beta\/k\n    \\left\\{\\sum_{j\\sim^k i} I(y_i=y_j) + \\sum_{i\\sim^k j} I(y_i=y_j) \\right\\} \\right),\n\\label{eqn:cuc}\n\\end{equation}\nand this gives rise to the joint distribution\n\\[ \\pi(\\hbox{\\boldmath$y$}|\\hbox{\\boldmath$x$},\\beta,k) \\propto \\exp\\left( \\beta\/k \\sum_{i=1}^n \\sum_{j\\sim^k i} I(y_i=y_j) \\right). \\]\nTherefore under this model, following~(\\ref{eqn:cuc}), mutual neighbours are given double weight, with respect to non-mutual neighbours and for this reason the model could be seen, perhaps, as an ad-hoc solution to this problem.\n\nIt is important to also note that both Holmes and Adams \\citeyear{hol:ada02} and Cucala \\textit{et al.} \\citeyear{cuc:mar08}\nallow the value of $k$ to be a variable. Therefore the neighbourhood size can vary. Holmes and Adams \\citeyear{hol:ada02}\nargue that allowing $k$ to vary has a certain type of smoothing effect.\n\n\\section{Distance nearest neighbours}\n\\label{sec:dnn}\n\nMotivated by the work of Holmes and Adams \\citeyear{hol:ada02} and Cucala \\textit{et al.} \\citeyear{cuc:mar08} our\ninterest focuses on modelling the distribution of the training data as a Markov random field. Similar to these\napproaches, we consider a Markov random field based approach, but in contrast our approach\nexplicitly models and depends on the distances between points in the training set. Specifically,\nwe define the full-conditional distribution of the class label $y_i$ as\n\\[ \\pi(y_i|\\hbox{\\boldmath$y$}_{-i},\\hbox{\\boldmath$x$},\\beta,\\sigma) \\propto \\exp\\left( \\beta\\sum_{j=1, j\\neq i}^n\n  w_j^iI(y_j=y_i)  \\right). \\]\nPositive values of the Markov random field parameter $\\beta$ encourage aggregation of the\nclass label. When $\\beta=0$, the class labels are uncorrelated.\nIn contrast to the $pk$-nn model, here the neighbourhood set of $x_i$ is constructed to be\n\\[ \\hbox{\\boldmath$x$}\\setminus\\{x_i\\} = \\{x_1,\\dots,x_{i-1},x_{i+1},\\dots,x_n\\} \\]\nand is therefore of maximal size. We consider three possible models depending on how\nthe collection of weights $\\{w_i^j\\}$ for $j=1,\\dots,i-1,i+1,\\dots,n$ are defined.\n\n\\begin{enumerate}\n\\item $d-$nn$_1$:\n\\[ w_i^j \\propto \\exp\\left\\{ -\\frac{d(x_i,x_j)^2}{2\\sigma^2}\\right\\},\\; \\mbox{for}\\; j=1,\\dots,i-1,i+1,\\dots,n,  \\]\nwhere $d$ is a distance measure such as Euclidean.\n\\item $d-$nn$_2$:\n\\[ w_i^j \\propto \\epsilon + (1-\\epsilon) I(d(x_i,x_j)<\\sigma),\\; \\mbox{for}\\; j=1,\\dots,i-1,i+1,\\dots,n,  \\]\nagain where, $I$ is an indicator function taking value $1$, if $d(x_i,x_j)<\\sigma$ and $0$, otherwise.\nFurther, $\\epsilon\\in (0,1)$ is defined as a constant, and is set to a value close to $0$. (Throughout\nthis paper we assign the value $\\epsilon=10^{-10}$.)\nA non-zero value of $\\epsilon$ guarantees that if there are no features within a distance $\\sigma$ of $x_i$ then the class of $y_i$ is modelled using the marginal proportions of the class labels.\n\\item $d-$nn$_3$:\n\\[ w_i^j \\propto \\exp\\left\\{ -d(x_i,x_j)\\sigma \\right\\},\\; \\mbox{for}\\; j=1,\\dots,i-1,i+1,\\dots,n.  \\]\n\\end{enumerate}\n\nClearly the neighbour system for both models is symmetric, and so the Hammersley-Clifford theorem guarantees that the joint distribution of the class\nlabels is a Markov random field. This joint distribution is written as\n\\begin{equation}\n \\pi(\\hbox{\\boldmath$y$}|\\hbox{\\boldmath$x$},\\beta,\\sigma) = \\frac{q(\\hbox{\\boldmath$y$}|\\beta,\\sigma,\\hbox{\\boldmath$x$})}{z(\\beta,\\sigma)} =\n \\frac{\\exp\\left( \\beta\\sum_i \\sum_{j=1, j\\neq i}^n w_j^iI(y_j=y_i) \\right) }{z(\\beta,\\sigma)}.\n \\label{eqn:joint}\n\\end{equation}\n\nAs usual, the normalising constant of such a Markov random field is difficult to evaluate in all but\ntrivial cases. It appears as\n\\begin{equation}\n z(\\beta,\\sigma) = \\sum_{y_1}\\dots\\sum_{y_n} \\exp\\left( \\beta\\sum_i\n  \\sum_{j=1, j\\neq i}^n w_j^iI(y_j=y_i)  \\right).\n\\label{eqn:nc}\n\\end{equation}\nSome comments:\n\\begin{enumerate}\n \\item The $k-$nn algorithm and its probabilistic variants always contain neighbourhoods of size $k$, regardless\nof how far each of neighbouring points are from the center point, $x_i$. Moreover, each neighbouring point $x_j$\nhas equal influence, regardless of distance from $x_i$. It could therefore be argued that these algorithms are\nnot sensitive to outliers. By contrast the distance nearest neighbour models deal with outlying points in a more\nrobust manner, since if a point $x_j$ lies further away from other neighbours of $x_i$, then it will have a relatively\nsmaller weight, and consequently less influence in determining the likely class label of $y_i$.\n\n\\item The formulation of distance nearest neighbour models includes every training point in the neighbourhood set, but the value of $\\sigma$\ndetermines the relative influence of points in the neighbourhood set. For the $d-$nn$_1$ model, small values of $\\sigma$ imply that only those points with small distance from the centre point will be influential, while for large values of $\\sigma$, points in the neighbourhood set are more uniformly weighted. Similarly, for the $d-$nn$_2$ model, points within a $\\sigma$ radius of the center point\nare weighted equally, while those outside a $\\sigma$ radius of the center point will have relatively\nlittle weight, when $\\epsilon$ is very close to $0$.\nBy contrast, for the $d-$nn$_3$ model, large values of the parameter\n$\\sigma$ imply that points close to the centre point will be influential.\n\\item For the $d-$nn$_2$ model, if there are no features in the training set within a distance $\\sigma$\nof $x_i$, then\n\\[ \\pi(y_i=j|\\hbox{\\boldmath$y$}_{-i},\\hbox{\\boldmath$x$},\\beta,\\sigma) \\propto \\exp\\left( \\beta p_j^i \\right),\\; \\mbox{for}\\; j=1,\\dots,G,  \\]\nwhere $p_j^i$ denotes the proportion of class labels $j$ in the set $\\hbox{\\boldmath$y$}\\setminus\\{y_i\\}$.\nThe parameter $\\beta$ determines the dependence on the class proportions.  A large value of $\\beta$ typically predicts the class label to be the class with the largest proportion, whereas a small value of $\\beta$ results in a prediction which is almost uniform over all possible classes.\nConversely,\nif there any feature vectors within a radius $\\sigma$ of $x_i$, then the class labels for these features\nwill most influence the class label of $y_i$.\n\\item As $\\beta\\rightarrow\\infty$, the most frequently occurring training label in the neighbourhood\nof a test point will be chosen with increasing large probability. The $\\beta$ parameter can be thought of, in a\nsense, as a tempering parameter. In the limit as $\\beta\\rightarrow\\infty$, the modal class label in the neighbourhood\nset has probability $1$.\n\n\\end{enumerate}\n\nThere has been work on extending the $k-$nearest neighbours algorithm to weight neighbours within the neighbourhood of size $k$.  For example, \\cite{dud75} weighted neighbours using the distance in a linear manner while standardizing weights to lie in $[0,1]$.\n\nA model similar to the $d-$nn$_1$ model appeared in \\cite{zhu:gha02}, but it does not contain\nthe $\\beta$ Markov random field parameter to control the level of aggregation in the spatial\nfield. Moreover, the authors outline some MCMC approaches, but note that inference for this model is\nchallenging. The aim of this paper is to illustrate how this model may be generalised and to\nillustrate an efficient algorithm to sample from this model. We now address the latter issue.\n\n\n\\section{Implementing the distance-nearest neighbours algorithm}\n\nThroughout we consider a Bayesian treatment of this problem. The posterior distribution of test\nlabels and Markov random field parameters can be expressed as\n\\begin{displaymath}\n \\pi(\\hbox{\\boldmath$y$}^*,\\beta,\\sigma|\\hbox{\\boldmath$y$},\\hbox{\\boldmath$x$},\\hbox{\\boldmath$x$}^*) \\propto \\pi(\\hbox{\\boldmath$y$},\\hbox{\\boldmath$y$}^*|\\beta,\\sigma,\\hbox{\\boldmath$x$},\\hbox{\\boldmath$x$}^*)\\pi(\\beta)\\pi(\\sigma),\n\\end{displaymath}\nwhere $\\pi(\\beta)$ and $\\pi(\\sigma)$ are prior distributions for $\\beta$ and $\\sigma$, respectively.\nNote, however that the first term on the right hand side above depends on the intractable normalising\nconstant (\\ref{eqn:nc}). In fact, the number of test labels is often much greater than the number of\ntraining labels, and so the resulting normalising constant for the distribution $\\pi(\\hbox{\\boldmath$y$},\\hbox{\\boldmath$y$}^*|\\beta,\\sigma,\\hbox{\\boldmath$x$},\\hbox{\\boldmath$x$}^*)$\ninvolves a summation over $G^{n+m}$ terms, where as before $n,m$ and $G$ are the number of test data points, training\ndata points and class labels, respectively. A more pragmatic alternative is to consider the posterior distribution of the unknown parameters for\nthe training class labels,\n\\begin{displaymath}\n \\pi( \\beta,\\sigma|\\hbox{\\boldmath$x$},\\hbox{\\boldmath$y$}) \\propto \\pi(\\hbox{\\boldmath$y$}|\\beta,\\sigma,\\hbox{\\boldmath$x$}) \\pi(\\beta) \\pi(\\sigma),\n\\end{displaymath}\nwhere now the normalising constant depends on $G^n$ terms. Test class labels can then be predicted by averaging\nover the posterior distribution of the training data,\n\\begin{displaymath}\n \\pi( y_{n+i}|x_{n+i},\\hbox{\\boldmath$x$},\\hbox{\\boldmath$y$}) = \\int \\pi(y_{n+i}|x_{n+i},\\hbox{\\boldmath$x$},\\hbox{\\boldmath$y$},\\beta,\\sigma) \\pi( \\beta,\\sigma|\\hbox{\\boldmath$x$},\\hbox{\\boldmath$y$}) d\\beta d\\sigma.\n\\end{displaymath}\nObviously, this assumes that the test class labels, $\\hbox{\\boldmath$y$}^*$ are mutually independent, given the training data, which\nwill typically be an unreasonable assumption.\nThe training class labels are modelled as being mutually independent. Clearly, this is not ideal\nfrom the Bayesian perspective.\nNevertheless, it should reduce the computational complexity of the\nproblem dramatically.\n\nIn practice, we can estimate the predictive probability of $y_{n+i}$ as an ergodic average\n\\begin{displaymath}\n \\pi( y_{n+i}|x_{n+i},\\hbox{\\boldmath$x$},\\hbox{\\boldmath$y$}) \\approx \\frac{1}{J}\\sum_{j=1}^J \\pi(y_{n+i}|x_{n+i},\\hbox{\\boldmath$x$},\\hbox{\\boldmath$y$},\\beta^{(j)},\\sigma^{(j)}),\n\\end{displaymath}\nwhere $\\beta^{(j)},\\sigma^{(j)}$ are samples from the posterior distribution $\\pi( \\beta,\\sigma|\\hbox{\\boldmath$x$},\\hbox{\\boldmath$y$})$.\n\n\\subsection{Pseudolikelihood estimation}\n\nA standard approach to approximate the distribution of a Markov random field is to use a pseudolikelihood\napproximation, first proposed in \\cite{bes74}. This approximation consists of a product of easily normalised\nfull-conditional distributions. For our model, we can write a pseudolikelihood approximation as\n\\begin{displaymath}\n \\pi(\\hbox{\\boldmath$y$}|\\hbox{\\boldmath$x$},\\beta,\\sigma) \\approx \\prod_{i=1}^n \\pi(y_i|\\hbox{\\boldmath$y$}_{-i},\\hbox{\\boldmath$x$},\\beta,\\sigma) =\n\\prod_{i=1}^n \\frac{\\exp\\left( \\beta\\sum_{j=1, j\\neq i}^n w_j^iI(y_j=y_i) \\right) }\n{\\sum_{k=1}^G \\exp\\left( \\beta\\sum_{j=1, j\\neq i}^n w_j^iI(y_j=k) \\right) }.\n\\end{displaymath}\nThis approximation yields a fast approximation to the posterior distribution, however it does ignore dependencies\nbeyond first order.\n\n\\subsection{The exchange algorithm}\n\nThe main computational burden is sampling from the posterior distribution\n\\begin{eqnarray*}\n \\pi( \\beta,\\sigma | \\hbox{\\boldmath$x$},\\hbox{\\boldmath$y$} ) &\\propto& \\pi(\\hbox{\\boldmath$y$}|\\beta,\\sigma,\\hbox{\\boldmath$x$}) \\pi(\\beta) \\pi(\\sigma) \\\\\n &=& \\frac{q(\\hbox{\\boldmath$y$}|\\beta,\\sigma,\\hbox{\\boldmath$x$})}{z(\\beta,\\sigma)} \\pi(\\beta) \\pi(\\sigma).\n\\end{eqnarray*}\nA naive implementation of a Metropolis-Hastings algorithm proposing to move from $(\\beta,\\sigma)$ to\n$(\\beta',\\sigma')$ would require calculation of the following ratio at each sweep of the algorithm\n\\begin{equation}\n \\frac{q(\\hbox{\\boldmath$y$}|\\beta',\\sigma',\\hbox{\\boldmath$x$})\\pi(\\beta')\\pi(\\sigma')}{q(\\hbox{\\boldmath$y$}|\\beta,\\sigma,\\hbox{\\boldmath$x$})\\pi(\\beta)\\pi(\\sigma)}\n \\times \\frac{z(\\beta,\\sigma)}{z(\\beta',\\sigma')}.\n\\label{eqn:MHratio}\n\\end{equation}\nThe intractability of the normalising constants, $z(\\beta,\\sigma)$ and $z(\\beta',\\sigma')$, makes this\nalgorithm unworkable. There has been work which has tackled the problem of sampling from such complicated\ndistributions, for example, \\shortcite{mol:pet06}. The algorithm presented in this paper overcomes the\nproblem of sampling from a distribution with intractable normalising constant, to a large extent. However\nthe algorithm can result in an MCMC chain with poor mixing among the parameters. The algorithm in\n\\shortcite{mol:pet06} has been extended and improved in \\cite{Murray06}.\n\nThe algorithm samples from an augmented distribution\n\\begin{equation}\n \\pi(\\beta',\\sigma',\\hbox{\\boldmath$y$}',\\sigma,\\beta|\\hbox{\\boldmath$y$},\\hbox{\\boldmath$x$}) \\propto \\pi(\\hbox{\\boldmath$y$}|\\beta,\\sigma,\\hbox{\\boldmath$x$})\\pi(\\beta)\\pi(\\sigma)\n h(\\beta',\\sigma'|\\beta,\\sigma) \\pi(\\hbox{\\boldmath$y$}'|\\beta',\\sigma',\\hbox{\\boldmath$x$}),\n\\label{eqn:exchange}\n\\end{equation}\nwhere $\\pi(\\hbox{\\boldmath$y$}'|\\beta',\\sigma',\\hbox{\\boldmath$x$})$ is the same distance nearest-neighbour distribution as the\ntraining data $\\hbox{\\boldmath$y$}$. The distribution $h(\\beta',\\sigma'|\\beta,\\sigma)$ is any arbitrary distribution\nfor the augmented variables $(\\beta',\\sigma')$ which might depend on the variables $(\\beta,\\sigma)$, for\nexample, a random walk distribution centred at $(\\beta,\\sigma)$. It is clear that the marginal distribution\nof (\\ref{eqn:exchange}) for variables $\\sigma$ and $\\beta$ is the posterior distribution of interest.\n\nThe algorithm can be written in the following concise way:\n\\begin{algorithm}\n\\step{1.} Gibbs update of $(\\beta',\\sigma',\\hbox{\\boldmath$y$}')$:\\\\\n(i) Draw $(\\beta',\\sigma') \\sim h(\\cdot,\\cdot|\\beta,\\sigma)$.\\\\\n(ii) Draw $\\hbox{\\boldmath$y$}' \\sim \\pi(\\cdot|\\beta',\\sigma',\\hbox{\\boldmath$x$})$.\n\\step{2.} Propose to move from $(\\beta,\\sigma,\\hbox{\\boldmath$y$}),(\\beta',\\sigma',\\hbox{\\boldmath$y$}')$ to $(\\beta',\\sigma',\\hbox{\\boldmath$y$}),(\\beta,\\sigma,\\hbox{\\boldmath$y$}')$.\n(Exchange move) with probability\n\\begin{displaymath}\n \\min\\left( 1, \\frac{q(\\hbox{\\boldmath$y$}'|\\beta,\\sigma,\\hbox{\\boldmath$x$})\\pi(\\beta')\\pi(\\sigma')h(\\beta,\\sigma|\\beta',\\sigma')q(\\hbox{\\boldmath$y$}|\\beta',\\sigma',\\hbox{\\boldmath$x$})}\n{q(\\hbox{\\boldmath$y$}|\\beta,\\sigma,\\hbox{\\boldmath$x$})\\pi(\\beta)\\pi(\\sigma)h(\\beta',\\sigma'|\\beta,\\sigma)q(\\hbox{\\boldmath$y$}'|\\beta',\\sigma',\\hbox{\\boldmath$x$})}\n\\times \\frac{z(\\beta,\\sigma)z(\\beta',\\sigma')}{z(\\beta,\\sigma)z(\\beta',\\sigma')} \\right).\n\\end{displaymath}\n\\end{algorithm}\n\nNotice in Step 2, that all intractable normalising constants cancel above and below the fraction.\nThe difficult step of the algorithm in the context of the $d-$nn model is Step 1 (ii), since this requires a draw from $\\pi(\\hbox{\\boldmath$y$}'|\\beta',\\sigma',\\hbox{\\boldmath$x$})$.\nPerfect sampling \\cite{pro:wil96} is often possible for Markov random field models, however a pragmatic alternative\nis to sample from $\\pi(\\cdot|\\beta',\\sigma',\\hbox{\\boldmath$x$})$ by standard MCMC methods, for example, Gibbs sampling, and take\na realisation from a long run of the chain as an approximate draw from the distribution.\nNote that this is the approach that Cucala \\textit{et al.} \\citeyear{cuc:mar08}\ntake. They argue that perfect sampling is possible for the $pk-$nn algorithm\nfor the case where there are two classes, but that the time to coalescence can be prohibitively large. They note that\nperfect sampling for more than two classes is not yet available.\n\nNote that this algorithm has some similarities with Approximate Bayesian Computation (ABC)\nmethods \\cite{sis:fan:tan07} in the sense that ABC algorithms also rely on drawing exact values from\nanalytically intractable distributions. By contrast however, ABC algorithms rely on comparing summary statistics of the auxiliary\ndata to summary statistics of the observed data.\nFinally, note that the Metropolis-Hastings ratio in step $2$ above, after re-arranging some terms, and assuming that\n$h(\\beta,\\sigma|\\beta',\\sigma')$ is symmetric can be written as\n\\[\n \\frac{q(\\hbox{\\boldmath$y$}|\\beta',\\sigma',\\hbox{\\boldmath$x$})\\pi(\\beta')\\pi(\\sigma')q(\\hbox{\\boldmath$y$}'|\\beta,\\sigma,\\hbox{\\boldmath$x$})}\n{q(\\hbox{\\boldmath$y$}|\\beta,\\sigma,\\hbox{\\boldmath$x$})\\pi(\\beta)\\pi(\\sigma)q(\\hbox{\\boldmath$y$}'|\\beta',\\sigma',\\hbox{\\boldmath$x$})}.\n\\]\nComparing this to~(\\ref{eqn:MHratio}), we see that the ratio of normalising constants, $z(\\beta,\\sigma)\/z(\\beta',\\sigma')$,\nis replaced by $q(\\hbox{\\boldmath$y$}'|\\beta,\\sigma,\\hbox{\\boldmath$x$})\/q(\\hbox{\\boldmath$y$}'|\\beta',\\sigma',\\hbox{\\boldmath$x$})$, which itself can be interpreted as an importance\nsampling estimate of $z(\\beta,\\sigma)\/z(\\beta',\\sigma')$, since\n\\[ {\\mathbf{E}}_{\\hbox{\\boldmath$y$}'|\\beta',\\sigma'} \\Big[\\frac{q(\\hbox{\\boldmath$y$}'|\\beta,\\sigma,\\hbox{\\boldmath$x$})}{q(\\hbox{\\boldmath$y$}'|\\beta',\\sigma',\\hbox{\\boldmath$x$})}\\Big] =\n \\int \\frac{q(\\hbox{\\boldmath$y$}'|\\beta,\\sigma,\\hbox{\\boldmath$x$})}{q(\\hbox{\\boldmath$y$}'|\\beta',\\sigma',\\hbox{\\boldmath$x$})} \\frac{q(\\hbox{\\boldmath$y$}'|\\beta',\\sigma',\\hbox{\\boldmath$x$})}{z(\\beta',\\sigma')}\\;d\\hbox{\\boldmath$y$}'\n = \\frac{z(\\beta,\\sigma)}{z(\\beta',\\sigma')}.\n\\]\n\n\n\\section{Results}\n\nThe performance of our algorithm is illustrated in a variety of settings. We begin by testing\nthe algorithm on a collection of benchmark datasets and follow this by exploring two real\ndatasets with high-dimensional feature vectors. Matlab computer code and all of the datasets\n(test and training) used in this paper can be found at \\texttt{mathsci.ucd.ie\/$\\sim$nial\/dnn\/}.\n\n\\subsection{Benchmark datasets}\n\nIn this section we present results for our model and in each case we compare results with the $k-$nn algorithm\nfor well known benchmark datasets.\nA summary description of each dataset is presented in Table~\\ref{tab:bench}.\n\n\\begin{table}[htp]\n\\begin{center}\n\\begin{tabular}{l|ccc}\n       & $G$ & $F$ & $N$ \\\\\n\\hline\nPima   & $2$  & $8$ & $532$ \\\\\nForensic glass & $4$  & $9$ & $214$ \\\\\nIris   & $3$   & $4$  & $150$  \\\\\nCrabs  & $4$  & $5$ & $200$ \\\\\nWine   & $3$   & $13$  & $178$ \\\\\nOlive  & $3$   & $9$  & $572$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Summary of the benchmark datasets: $G,F,N$ correspond to the number of classes, the number\nof features and the overall number of observations, respectively.}\n\\label{tab:bench}\n\\end{table}\n\nIn all situations, the training dataset was approximately $25\\%$ of the size of the overall dataset,\nthereby presenting a challenging scenario for the various algorithms. Note that the sizes of each dataset\nranges from quite small in the case of the iris dataset, to reasonably large in the case of the forensic\ndataset. In all examples, the data was standardised to give transformed features with zero mean and unit variance.\nIn the Bayesian model, non-informative $N(0,50^2)$ and $U(0,100)$ priors were chosen for $\\beta$ and $\\sigma$,\nrespectively. Each $d-$nn algorithm was run for $20,000$ iterations, with the first $10,000$ serving as\nburn-in iterations. The auxiliary chain within the exchange algorithm was run for $1,000$ iterations.\nThe $k-$nn algorithm was computed for values of $k$ from $1$ to half the number of features in the training\nset. In terms of computational run time, the $d-$nn algorithms took, depending on the size of the dataset, between\n$1$ to $12$ hours to run using Matlab code on a $2$GHz desktop machine.\n\nA summary of misclassification error rates is presented in Table~\\ref{tab:bench_res} for various benchmark datasets. In almost all of the situations $d-$nn$_1$ and $d-$nn$_3$ performs at least as well as $k-$nn and often considerably better. In general, $d-$nn$_1$ and $d-$nn$_3$ performed better than $d-$nn$_2$.\nA possible explanation for this may be due to the\ncut-off nature of the weight function in the $d-$nn$_2$ model, since if a point $x_i$ has no neighbours inside a ball of\nradius $\\sigma$, then $w_i^j$ is uniform over the entire test set, and consequently there is no effect of distance. By contrast,\nboth the $d-$nn$_1$ and $d-$nn$_3$ models, have weight functions which depend on distance, and smoothly converge to a uniform\ndistribution as $\\sigma\\rightarrow \\infty$ and $\\sigma\\rightarrow 0$, respectively. \n\n\n\\begin{table}[htp]\n\\begin{center}\n\\begin{tabular}{l|cccc}\n  & $k-$nn & $d-$nn$_1$ & $d-$nn$_{2}$ & $d-$nn$_{3}$ \\\\\n\\hline\nPima   & $30\\%$  & $29\\%$ & $32\\%$ & $30\\%$\\\\\nForensic glass & $35\\%$  & $33\\%$ & $39\\%$ & $31\\%$ \\\\\nIris   & $6\\%$   & $5\\%$  & $5\\%$ & $6\\%$ \\\\\nCrabs  & $16\\%$  & $16\\%$ & $23\\%$ & $16\\%$ \\\\\nWine   & $6\\%$   & $4\\%$  & $6\\%$ &  $4\\%$    \\\\\nOlive  & $1\\%$   & $3\\%$  &$4\\%$ & $2\\%$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Misclassification error rates for various benchmark dataset. The value of $k$ in the $k-$nn algorithm\nwas chosen as the value that minimises the leave-one-out cross-validation error rate. (In the case of a tie,\nthe smallest value of $k$ was selected.)}\n\\label{tab:bench_res}\n\\end{table}\n\n\n\\subsection{Classification with large feature sets: food authenticity}\n\nHere we consider two datasets concerned with food authentication. The first example involves\nsamples of Greek olive oil from $3$ different regions, and the second example involves\nsamples of $5$ different types of meat. In both situations each sample was analysed using\nnear infra-red spectroscopy giving rise to $1050$ reflectance values for wavelengths in the\nrange $400-2098$nm. These $1050$ reflectance values serve as the feature vector for each sample.\nThe objective in both examples is to authenticate a test sample based on a training set of\ncomplete data (both reflectance values and class labels). Details of how both datasets were collected\nappear in \\shortcite{mcel:dow:fea99}, and were analysed using a model-based clustering approach in \\shortcite{dea:mur:dow06}.\n\n\\subsubsection{Classifying meat samples}\n\nHere $231$ samples of meat were collected.\nThe aim of this study was to see if these measurements could be used to classify each meat sample according to whether it is chicken, turkey, pork, beef or lamb. The data were randomly split into $60$ training samples and $171$ test samples. The respective number of samples in each class is given in the table below.\n\\begin{table}[htp]\n\\begin{center}\n\\begin{tabular}{ccc}\n& Training & Test \\\\\n\\hline\nChicken & 15 & 40 \\\\\nTurkey & 20 & 35 \\\\\nPork & 13 & 42 \\\\\nBeef & 11 & 21 \\\\\nLamb & 11 & 23 \\\\\n\\end{tabular}\n\\end{center}\n\\caption{Number of samples within each class for both the training and test datasets}\n\\end{table}\n\nAs before, non-informative normal, $N(0,50^2)$ and uniform $U(0,10)$ priors were chosen for $\\beta$ and $\\sigma$,\nrespectively. In the exchange\nalgorithm, the auxiliary chain was run for $1000$ iterations, and the overall chain ran for $20,000$ of which\nthe first $10,000$ were discarded as burn-in iterations. The overall acceptance rate for the exchange algorithm was around $25\\%$ for each of the $d-$nn models.\n\nThe misclassification error rate for leave-one-out cross-validation on the training dataset is minimised for\n$k=3$ and $k=4$. See Figure~\\ref{fig:meat_crossval} (a). At both of these values, the $k-$nn algorithm yielded a misclassification error rate of $35\\%$ and $39\\%$, respectively, for the test dataset.\nSee Figure~\\ref{fig:meat_crossval} (b). By comparison, the $d-$nn$_1$, $d-$nn$_2$  and $d-$nn$_3$ models\nachieved misclassification error rates of $29\\%$, $33\\%$ and $27\\%$,\nrespectively, for the test dataset.\nThis example further illustrates the value of the $d-$nn models.\n\n\n\\begin{figure}[htp]\n\\begin{center}\n \\hspace*{-3cm}\\includegraphics[scale=0.95]{meat_knn.pdf}\n \\caption{Meat dataset: (a) Training data: misclassification rates of leave-one-out cross-validation for $k-$nn algorithm for varying values of $k$. (b) Test data: misclassification rates for $k-$nn algorithm for varying values of $k$.}\n\\label{fig:meat_crossval}\n\\end{center}\n\\end{figure}\n\n\n\\subsubsection{Classifying Greek olive oil}\n\nThis example concerns classifying Greek oil samples, again based on infra-red spectroscopy. Here\n$65$ samples of Greek virgin olive-oil were collected.\nThe aim of this study was to see if these measurements could be used to classify each olive-oil sample to\nthe correct geographical region. Here there were $3$ possible classes (Crete (18 locations), Peloponnese\n(28 locations) and other regions (19 locations).\n\nIn our experiment the data were randomly split into a training set of $25$ observations and a test set of $40$ observations. In the training dataset the proportion of class labels was similar to that in the complete dataset.\n\nIn the Bayesian model,  non-informative $N(0,50^2)$ and $U(0,100)$ priors were chosen for\n$\\beta$ and $\\sigma$. In the exchange algorithm, the auxiliary chain was run for $1000$ iterations, and the overall chain ran for $50,000$ of which the first $20,000$ were discarded as burn-in iterations. The overall acceptance rate\nfor the exchange algorithm was around $15\\%$ for each of the Markov chains.\n\nThe $d-$nn$_1$, $d-$nn$_2$ and $d-$nn$_3$ models achieved misclassification rates\nof $20\\%$, $26\\%$ and $20\\%$, respectively.\nIn terms of comparison with the $k-$nn algorithm, leave-one-out cross-validation was minimised for\n$k=3$ for the training dataset. See Figure~\\ref{fig:olive_crossval} (a). The misclassification rates at\nthis value of $k$ was $29\\%$ for the test dataset. See Figure~\\ref{fig:olive_crossval} (b).\n\n\n\\begin{figure}[htp]\n\\begin{center}\n \\hspace*{-3cm}\\includegraphics[scale=0.95]{olive_knn.pdf}\n \\caption{Olive oil dataset: (a) Training data: misclassification rates of leave-one-out cross-validation for $k-$nn algorithm for varying values of $k$. (b) Test data: misclassification rates for $k-$nn algorithm for varying values of $k$.}\n  \\label{fig:olive_crossval}\n\\end{center}\n\\end{figure}\n\n\nIt is again encouraging that the $d-$nn algorithms yielded  improved misclassification rates by\ncomparison.\n\n\n\\section{Concluding remarks}\n\\label{sec:conc}\n\nIn terms of providing a probabilistic approach  to a Bayesian analysis of supervised learning, our work builds on that of Cucala \\textit{et al} \\citeyear{cuc:mar08} and shares many of the advantages of the approach there, providing a sound setting for Bayesian inference.  The most likely allocations for the test dataset can be evaluated and also the uncertainty that goes with them. So this makes it possible to determine regions where allocation to specific classes is uncertain. In addition, the Bayesian framework allows for an automatic approach to choosing weights for neighbours or neighbourhood sizes.\n\nThe present paper also addresses the computational difficulties related to the well-known issue of the intractable normalising constant for discrete exponential family models. While Cucala \\textit{et al} \\citeyear{cuc:mar08}\ndemonstrated that MCMC sampling is a practical  alternative to the perfect sampling scheme of M{\\o}ller \\textit{et al} \\citeyear{mol:pet06}, there remain difficulties with their implementation of the approach of \\shortcite{mol:pet06}, namely the choice of an auxiliary distribution.  To partially overcome the difficulties of a poor choice, Cucala \\textit{et al} \\citeyear{cuc:mar08} use an adaptive algorithm where the auxiliary distribution is defined by using historical values in the Monte Carlo algorithm. We use an alternative approach based on the exchange algorithm which avoids this choice or adaptation and has very good mixing properties and therefore also has\ncomputational efficiency.\n\nAn issue with the neighbourhood model of Cucala \\textit{et al} \\citeyear{cuc:mar08}, which is an Ising or Boltzmann type model, is that it is necessary to define an upper value for the association parameter $\\beta$.  This parameter value arises from the phase change of the model and which is known for a regular neighbourhood structure but has to be investigated empirically for the probabilistic neighbourhood model.  Our distance nearest neighbour models avoid this difficulty.\n\nOur approach is robust to outliers whereas the nearest neighbour approaches will always have an outlying point having neighbours and therefore classified according to assumed independent distant points which are the nearest neighbours.\n\n\\paragraph*{Acknowledgements:} Nial Friel was supported by a Science Foundation Ireland Research Frontiers\nProgram grant, 09\/RFP\/MTH2199. Tony Pettitt's research was supported by an Australian Research Council Discovery Grant.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{}\nAxisymmetric magnetic activity on the Sun and sun-like stars increases the frequencies of the modes of acoustic oscillation. However, it is unclear how a corotating patch of activity  affects the oscillations, since such a perturbation is unsteady in the frame of the observer. In this paper we qualitatively describe the asteroseismic signature of a large active region in the power spectrum of the dipole ($\\ell=1$) and quadrupole ($\\ell=2$)  p modes.\nFirst we calculate the frequencies and the relative amplitudes of the azimuthal modes of oscillation in a frame that corotates with the active region, using first-order perturbation theory. For the sake of simplicity, the influence of the active region is approximated by a near-surface increase in sound speed.  In the corotating frame the perturbations due to (differential) rotation and the active region completely lift the ($2\\ell+1$)-fold azimuthal  degeneracy of the frequency spectrum of modes with harmonic degree $\\ell$. \nThen we transform to an inertial frame to obtain the observed power spectrum. In the frame of the observer, the unsteady nature of the perturbation leads to the appearance of $(2\\ell+1)^2$ peaks in the power spectrum of a multiplet. These peaks blend into each other to form asymmetric line profiles. In the limit of a small active region (angular diameter less than $30^\\circ$), we approximate the power spectrum of a multiplet in terms of $2\\times(2\\ell+1)$ peaks, whose amplitudes and frequencies depend on the latitude of the active region and the inclination angle of the star's rotation axis.\nIn order to check the results and to explore the nonlinear regime, we perform numerical simulations using the 3D time-domain pseudo-spectral linear pulsation code GLASS. \nFor small sound-speed perturbations, we find a good agreement between the simulations and linear theory. Larger perturbation amplitudes will induce mode mixing and lead to additional complex changes in the predicted power spectrum.\n{  However linear perturbation theory provides useful guidance to search for the observational signature of large individual active regions in stellar oscillation power spectra.}\n\n\\tiny\n \\fontsize{8}{11}\\helveticabold { \\section{Keywords:} asteroseismology, stars: oscillations, stars: activity, stars: rotation, starspots} \n\\end{abstract}\n\n\n\n\\section{Introduction}\n\n\nSurface magnetic activity shifts the frequencies of the global modes of acoustic oscillation during solar and stellar activity cycles  \\citep[e.g.,][]{Palle1989,Garcia2010,Santos2016,Kiefer2017}. Asteroseismology can in turn inform us about the strength and the latitude distribution of a band of magnetic activity on the stellar surface  \\citep{Gizon2002,Chaplin2007a,Gizon2016}. However, the signature of a single large active region in stellar p-mode oscillation power spectra has not been discussed so far in detail.  A complication inherent to this problem comes from the fact that the perturbation associated with a rotating active region is not steady in the observer's frame. \nYet, this problem is relevant, since large starspots are detected in photometric variability \\citep[{{e.g.}},][]{Mosser2009a} and using Doppler or Zeeman-Doppler imaging  \\citep[{{e.g.}},][]{Strassmeier2009}. \n\n\n\n \n\n\nWe build on preliminary work by \\citet{gizon1995, gizon1998} who considered the effect of a single sunspot in corotation on the low-degree modes of solar oscillation. \nThe problem of unsteady perturbations has been considered in the past in different contexts. The interaction with acoustic modes with an inclined magnetic field with respect to the stellar rotation axis was first studied by \\citet{Kurtz1982} to explain the oscillation power spectra of roAp stars \\citep[see also][]{Kurtz2008}.\nIn the oblique pulsator model the effect of the magnetic field dominates over rotation, and the pulsation axis is aligned with the magnetic axis of the star. Only modes of oscillations symmetric with respect to the magnetic field axis are excited.\nThe oblique pulsator model was extended by \\citet{Dziembowski1985} to include the first-order effects of the Coriolis and Lorentz forces, and then  by \\citet{Shibahashi1993}  to account for the distortion in the eigenfunctions.\nIn parallel \\citet{Dziembowski1984} and \\citet{Gough1984}  discussed the combined influence of rotation and of an inclined magnetic field in corotation  on multiplets of solar acoustic oscillations. They explicitly mentioned that each multiplet consist of $(2\\ell+1)^2$ components in the power spectrum \\citep[this was already hinted at by][]{Dicke1982}.\n\nWe focus on stars with a level of activity higher than the Sun, which may have active regions with larger surface coverage, and therefore better chances for detection. Following the same approach of  \\citet{goode1992}, we investigate the linear changes  induced in a $n\\ell$-multiplet by an unsteady perturbation, that mimics an active region (AR) rotating with the star. In particular we study the power spectra of the dipole and quadrupole multiplets. For the active region we consider a simple two-parameter model, where near-surface sound-speed perturbations have a given amplitude and surface coverage.\n\nAs a complement to the linear analysis we also explore the nonlinear regime of the active-region perturbation by means of 3D numerical simulations, by studying the combined effect of rotation and mode mixing on the observed power spectra using the wave propagation code GLASS \\citep[see][]{Hanasoge2007,Papini2015}. \n{ We note that the non-linear regime was studied in the context of strong magnetic fields in roAp stars by, e.g., \\citet{Cunha2000},  \\citet{Bigot2002}, \\citet{Saio2004}, and  \\citet{Cunha2006}.}\n\n\\section{Methods}\n\n\\subsection{Signature of an active region in the oscillation power spectrum: linear theory}\n\\label{cha:rotation:sec:lineartheory}\n\n\\subsubsection{Linear problem in the corotating frame}\n\nThe normal modes of oscillation of a spherically symmetric non-rotating star are identified by three integer numbers: the radial order $n$, the angular degree $\\ell$, and the azimuthal order $m$, with $|m|\\le \\ell$.\nIn the absence of attenuation the degenerate mode frequencies, $\\omega_{n\\ell}^{(0)}$, are real and the displacement eigenvectors ${\\mbox{\\boldmath${\\xi}$} }^{(0)}_{n\\ell m}(\\vet r) \\exp(-{{\\mathrm{i}}} \\omega_{n\\ell}^{(0)} t)$ solve\nthe linearized equation of motion\n\\begin{gather}\n\\label{eq:H0xi0}\n {\\mathcal{L}}^{(0)} [ {\\mbox{\\boldmath${\\xi}$} }_{n\\ell m}^{(0)} ]  = \\omega_{n\\ell}^{(0)2} {\\mbox{\\boldmath${\\xi}$} }_{n\\ell m}^{(0)} ,\n\\end{gather}\nwhere ${\\mathcal{L}}^{(0)}$ is a linear spatial differential operator \\citep[see, {{e.g.}}, ][]{Unno1979}. \n{ Hereafter superscripts ''(0)'' denote quantities associated with the non-rotating stellar model.}\nIn spherical polar coordinates $\\vet r=(r,\\theta,\\phi)$, the displacement eigenfunctions can be written as\n\\begin{gather}\n\\label{eq:xi_0}\n {\\mbox{\\boldmath${\\xi}$} }_{n\\ell m}^{(0)}(\\vet r)= \n      \\left [\\xi_{r,n\\ell}(r) {\\mbox{\\boldmath${e}$} }_r +\n      \\xi_{h,n\\ell}(r) \\left ( {\\mbox{\\boldmath${e}$} }_\\theta \\PD{ }{\\theta} + \n             \\frac{1}{\\sin\\theta}{\\mbox{\\boldmath${e}$} }_\\phi\\PD{ }{\\phi} \\right )\\right] Y_l^m(\\theta,\\phi) ,\n\\end{gather}\nwhere $Y_l^m(\\theta,\\phi)$ are spherical harmonics and $\\xi_{r,n\\ell}$ and $\\xi_{h,n\\ell}$ are functions of radius only that can be calculated numerically for a given stellar model \\citep[see, e.g.,][]{Aerts2010}.\n\nWe now consider two perturbations, the first perturbation due to rotation \\citep[e.g.,][]{Hansen1977} and the second arising from the presence of an active region that rotates with the star \\citep[see, e.g.,][for the effect of a sunspot on high-degree modes]{Schunker2013}.   The latter perturbation is unsteady in any inertial frame of reference. \nHere we aim to study how these two effects may affect the fine structure of the modes within a fixed multiplet $(n,\\ell)$.\nThe two perturbations taken together completely remove the $(2\\ell+1)$-fold degeneracy in $m$.\n\\begin{figure}\n \\includegraphics[width=\\columnwidth]{fig1.jpg}\n \\caption[reference frames]{Reference frames and angles of the problem. Arrows show the polar axes of the coordinate systems ${{\\mathcal{R}}}$, ${{\\mathcal{R}}}_\\beta$, and $\\widehat{{\\mathcal{R}}}_\\beta$. The frame ${{\\mathcal{R}}}$ is the inertial frame of the observer.\n The rotation axis of the star is inclined by an angle $i$ with respect to the line of sight. \n Both frames ${{\\mathcal{R}}}_\\beta$ and $\\widehat{{\\mathcal{R}}}_\\beta$ corotate with the active region (shaded area) at a constant angular velocity $\\Omega_\\beta$. The polar axis of ${{\\mathcal{R}}}_\\beta$ is aligned with the rotation axis of the star. In ${{\\mathcal{R}}}_\\beta$ the active region has a colatitude $\\beta$ and in $\\widehat{{\\mathcal{R}}}_\\beta$ it is at the pole. \n }\n \\label{fig:referenceframes}\n\\end{figure}\n\n\nProvided that there is only one active region, it is much more convenient to first tackle the problem in a reference frame that is corotating with the active region, where both perturbations are steady \\citep[e.g.,][]{Dziembowski1984,goode1992}. \n In Fig. \\ref{fig:referenceframes} we define three frames of reference, ${{\\mathcal{R}}}$, ${{\\mathcal{R}}}_\\beta$, and $\\widehat{{{\\mathcal{R}}}}_\\beta$, all three with the same origin at the center of the star.\nFrame ${{\\mathcal{R}}}$ is an inertial frame of reference, with polar axis $\\vet z$ pointing towards the observer. \nWe denote by $\\beta$ the colatitude of the active region in frame ${{\\mathcal{R}}}$. The other two frames are both corotating with the active region at the angular velocity $\\Omega_\\beta$ about the rotation axis of the star. \nThe polar axis of ${{\\mathcal{R}}}_\\beta$ is directed along the stellar rotation axis and is inclined by an angle $i$ with respect to $\\vet z$. The polar axis of the frame $\\widehat{{\\mathcal{R}}}_\\beta$ is inclined by the angle $\\beta$ with respect to the rotation axis. In ${\\mathcal{\\widehat{R}}}_\\beta$ the center of the active region is at the north pole. \nProvided that the starspot has bno proper motion, the angular velocity $\\Omega_\\beta$ is equal to the surface rotational angular velocity of the star at  colatitude $\\beta$.\nWe call $\\vet r=(r,\\theta,\\phi)$, $\\vet r_\\beta=(r,\\theta_\\beta,\\phi_\\beta)$, and $\\widehat\\vet r_\\beta=(r,\\widehat\\theta_\\beta,\\widehat\\phi_\\beta)$ the spherical-polar coordinates associated with ${{\\mathcal{R}}}$, ${{\\mathcal{R}}}_\\beta$, and ${\\mathcal{\\widehat{R}}}_\\beta$ respectively. \n\n\nWe consider the effects of rotation and the active region on the acoustic oscillations as small perturbations. In the frame ${{\\mathcal{R}}}_\\beta$ each mode is identified by the index $M$, $-\\ell \\leq M \\leq \\ell$, and we expand the displacement eigenvectors and eigenfunctions as \n\\begin{equation}\n\\label{eq:xi_pert}\n{\\mbox{\\boldmath${\\xi}$} }_{n\\ell M} (\\vet r_\\beta) = \\sum_{m=-\\ell}^{\\ell} A^{M}_{m}  {\\mbox{\\boldmath${\\xi}$} }^{(0)}_{n\\ell m} (\\vet r_\\beta) + \\delta{\\mbox{\\boldmath${\\xi}$} }_{n\\ell M} (\\vet r_\\beta) +\\cdots \n\\end{equation}\nand \n\\begin{equation}\n\\label{eq:freq_pert}\n\\omega_{n\\ell M} = \\omega^{(0)}_{n\\ell} + \\delta \\omega_{n\\ell M} + \\cdots ,\n\\end{equation}\nwhere $\\delta{\\mbox{\\boldmath${\\xi}$} }_{n\\ell M}$ is orthogonal to each unperturbed eigenvector ${\\mbox{\\boldmath${\\xi}$} }^{(0)}_{n\\ell m}$ with the same $\\ell$ and $n$ \\citep[{{e.g.}}][]{1990gough}, \nand { the coefficients $A^{M}_{m}$ are (real) amplitudes}.\nWe write the wave operator as\n\\begin{equation}\n{\\mathcal{L}} = {\\mathcal{L}}^{(0)} + \\delta {\\mathcal{L}}  + \\cdots, \n\\end{equation}\nwith\n\\begin{equation}\n\\delta {\\mathcal{L}}  =   {\\mathcal{L}}_\\Omega +  {\\mathcal{L}}_{\\rm AR},\n\\end{equation}\nwhere ${\\mathcal{L}}_{\\Omega}$ accounts for  the effects of rotation and ${\\mathcal{L}}_{\\rm AR}$  for the effects of the active region.\nTo first order, the linearized equation of motion reduces to \n\\begin{equation}\n  \\sum_{m=-\\ell}^\\ell A_{m}^M \\left ({\\mathcal{L}}_\\Omega+{\\mathcal{L}}_\\RM{AR} \\right )[{\\mbox{\\boldmath${\\xi}$} }^{(0)}_{n\\ell m}]  + {\\mathcal{L}}^{(0)} [ \\delta{\\mbox{\\boldmath${\\xi}$} }_{n\\ell M} ] = \n 2 \\omega^{(0)}_{n\\ell}  \\delta \\omega_{n\\ell M}  \\sum_{m=-\\ell}^\\ell  A^{M}_{m}  {\\mbox{\\boldmath${\\xi}$} }^{(0)}_{n\\ell m}  +     \\omega_{n\\ell}^{(0)2} \\delta{\\mbox{\\boldmath${\\xi}$} }_{n\\ell M}.\n\\label{eq:HMxiM}\n\\end{equation}\nWe define the inner product between two vectors ${\\mbox{\\boldmath${\\xi}$} }(\\vet r_\\beta)$ and ${\\mbox{\\boldmath${\\eta}$} }(\\vet r_\\beta)$ on the Hilbert space of displacement vectors as\n\\begin{gather}\n \\inner{{\\mbox{\\boldmath${\\xi}$} }}{{\\mbox{\\boldmath${\\eta}$} }} =\\int_V {\\mbox{\\boldmath${\\xi}$} }^* \\cdot {\\mbox{\\boldmath${\\eta}$} }\\,\\rho\\mathrm{d} V  ,\n\\end{gather}\nwhere $^*$ denotes the complex conjugate and $V$ is the stellar volume.\nThe unpertubed eigenmodes are normalized such that\n$$\\inner{{\\mbox{\\boldmath${\\xi}$} }^{(0)}_{n\\ell m'}}{{\\mbox{\\boldmath${\\xi}$} }^{(0)}_{n\\ell m}}=\\delta_{m'm}.$$\nWe take the inner product of Eq. (\\ref{eq:HMxiM}) with ${\\mbox{\\boldmath${\\xi}$} }^{(0)}_{n\\ell m'}$ to obtain\n\\begin{equation}\n  \\sum_{m=-\\ell}^\\ell A_{m}^M \\inner{{\\mbox{\\boldmath${\\xi}$} }^{(0)}_{n\\ell m'}}{ ({\\mathcal{L}}_\\Omega+{\\mathcal{L}}_{AR})[{\\mbox{\\boldmath${\\xi}$} }^{(0)}_{n\\ell m} ] }  \n+ \\inner{ {\\mbox{\\boldmath${\\xi}$} }^{(0)}_{n\\ell m'} }{  {\\mathcal{L}}^{(0)} [ \\delta{\\mbox{\\boldmath${\\xi}$} }_{n\\ell M} ] }  \n=2 \\omega^{(0)}_{n\\ell}  \\delta \\omega_{n\\ell M}   A_{m'}^M .   \n\\label{eq:inner1}\n\\end{equation}\nBecause ${\\mathcal{L}}^{(0)}$ is Hermitian symmetric and $\\langle  {\\mbox{\\boldmath${\\xi}$} }^{(0)}_{n\\ell m'} ,\\delta{\\mbox{\\boldmath${\\xi}$} }_{n\\ell M} \\rangle =0$, the second term on the left-hand side of the above equation vanishes\n\\begin{align*}\n\\inner { {\\mbox{\\boldmath${\\xi}$} }^{(0)}_{n\\ell m'} } \n\t   {{\\mathcal{L}}^{(0)} [ \\delta{\\mbox{\\boldmath${\\xi}$} }_{n\\ell M} ] } \n        = \\inner {{\\mathcal{L}}^{(0)} [{\\mbox{\\boldmath${\\xi}$} }^{(0)}_{n\\ell m'}] } \n       {   \\delta{\\mbox{\\boldmath${\\xi}$} }_{n\\ell M}  } =0 .\n\\end{align*}\nIntroducing the perturbation matrix elements\n\\begin{align}\nO_{m' m} = O^{\\Omega}_{m' m} + O^{\\rm AR}_{m' m},\n\\end{align}\nwhere\n\\begin{align}\n& O^{\\Omega}_{m' m} =   \\frac{1}{2\\omega^{(0)}_{n\\ell}} \n\\inner {{\\mbox{\\boldmath${\\xi}$} }^{(0)}_{n\\ell m'} }{ {\\mathcal{L}}_\\Omega [ {\\mbox{\\boldmath${\\xi}$} }^{(0)}_{n\\ell m} ]} \\\\\n& O^{\\rm AR}_{m' m}  =  \\frac{1}{2\\omega^{(0)}_{n\\ell}} \n\\inner {{\\mbox{\\boldmath${\\xi}$} }^{(0)}_{n\\ell m'} }{ {\\mathcal{L}}_{\\rm AR} [ {\\mbox{\\boldmath${\\xi}$} }^{(0)}_{n\\ell m} ] } ,\n\\end{align}\nEquation (\\ref{eq:inner1}) becomes\n\\begin{align}\n    \\sum_{m=-\\ell}^\\ell O_{m'm}  A^{M}_{m}      = \n   \\delta \\omega_{M}   A^M_{m'}.   \n\\end{align}\nTo simplify the notation we dropped the indices $n\\ell$ on $\\delta\\omega_M$.\nIn matrix form,\n\\begin{align}\n\\label{eq:eigen_problem}\n    \\vet O \\vec{A}^{M} = \n   \\delta \\omega_{M}  \\;  \\vec{A}^{M},\n\\end{align}\nwhere\n$\n\\vec{A}^M= [A_{-\\ell}^M \\; A_{-\\ell+1}^M \\;\\dots \\; A_{\\ell}^M]^T\n$\nis the vector of amplitudes. \nTo find $\\vec{A}^M$ and $\\delta\\omega_M $  we have to solve the above eigenvalue problem, Eq. (\\ref{eq:eigen_problem}).\n\n\nThe rotation perturbation matrix $\\vet O^{\\Omega}$ is diagonal in the frame ${{\\mathcal{R}}}_\\beta$.\nThe active region perturbation matrix $\\vet O^{\\mathrm{AR}}$ is not diagonal in ${{\\mathcal{R}}}_\\beta$, \n but it can be obtained in  terms of \n the diagonal perturbation matrix ${\\vet{\\widehat{O}}}^{\\mathrm{AR}}$ expressed in the frame ${\\mathcal{\\widehat{R}}}_\\beta$, \n\\begin{gather}\n\\label{eq:OtildeAR_toOAR}\n\\vet O^\\mathrm{AR}={\\vet{R}}^{(\\ell)}{\\vet{\\widehat{O}}}^\\mathrm{AR} \n({{\\vet{R}}^{(\\ell)}})^{-1}. \n\\end{gather}\nwhere the matrix ${\\vet{R}}^{(\\ell)}$ performs \na clockwise rotation of $\\beta$ about the $y$ axis that transforms the frame ${\\mathcal{\\widehat{R}}}_\\beta$ into the frame ${{\\mathcal{R}}}_\\beta$.\nMore explicitly, the elements of the rotation matrix are given by\n\\begin{gather}\n\\label{eq:rotylmabg}\nR_{m\\,m'}^{(\\ell)}= r^{(\\ell)}_{m\\,m'}(-\\beta) = r^{(\\ell)}_{m'm}(\\beta),\n\\end{gather}\nwhere  $r^{(\\ell)}_{m\\,m'}(\\beta)$ is given by \\citet{1959messiah}.\n \n \n \n\\label{sec:corot}\n\n\\subsubsection{Frequency splittings due to rotation}\n\n\nIn the corotating frame ${{\\mathcal{R}}}_\\beta$ the rotation perturbation matrix is diagonal:\n\\begin{align}\n O^\\Omega_{m'm} = \\delta_{m'm}\\delta\\omega_m^\\Omega \n    \\label{eq:rot_splitting}\n\\end{align}\nwith \n\\begin{equation}\n\\delta\\omega_m^\\Omega =m \\int_V K_{n\\ell m} (r,\\theta) \\left[ \\Omega (r,\\theta)-\\Omega_\\beta \\right] \\mathrm{d}{V} \n- m\\Omega_\\beta C_{n\\ell} +  \\eta Q_{2 \\ell m}\\,  \\omega_{n\\ell}^{(0)},\n    \\label{eq:rot_splitting2}\n\\end{equation}\nwhere $\\Omega(r,\\theta)$ is the internal angular velocity in an inertial frame.\nBy construction, the angular velocity $\\Omega_\\beta$ of the frame ${{\\mathcal{R}}}_\\beta$ is $\\Omega_\\beta = \\Omega( R,\\beta)$, where $R$ is the radius of the star.\nThe first and second terms on the right-hand side of Eq. (\\ref{eq:rot_splitting2}) describe the effect of differential rotation, where the functions $K_{n\\ell m} (r,\\theta)$ are the rotational sensitivity kernels \\citep{Hansen1977} and $C_{n\\ell}$ are the Ledoux constants \\citep{1951ledoux} that account for the effect of the Coriolis force. \nThe last term describes the quadrupole distortion of the star due to the centrifugal forces \\citep[e.g.][]{Saio1981,Aerts2010} and is proportional to the ratio of the centrifugal to the gravitational forces at the surface $\\eta=\\Omega^2 R^3\/(G M)$, where $M$ is the mass of the star and $G$ is the universal constant of gravity.\nThe term $Q_{2\\ell m}$ accounts for the quadrupolar component of the centrifugal distorsion \n\\begin{gather}\n Q_{2 \\ell m}\\simeq \\frac{2\/3 \\int_{-1}^{1} P_2(x) \\left[ P_\\ell^{|m|}(x)\\right]^2 \\mathrm{d}{x}}\n {\\int_{-1}^{1}  \\left[ P_\\ell^{|m|}(x)\\right]^2 \\mathrm{d}{x}} \n = \\frac{2\\ell(\\ell+1) - 6 m^2 }{3(2\\ell+3) (2\\ell-1)},\n\\end{gather}\nwhere $P_\\ell^m(x)$ are the associated Legendre functions and $P_2$ is the Legendre polynomial of second order.\nThe centrifugal term is very small in the case of slow rotators like the Sun \\citep{1992dziembowski}, however it increases rapidly with rotation and it is not negligible anymore for stars rotating a few times faster than the Sun \\citep[e.g.][]{2004gizon}.\n\n\n\n\\subsubsection{Frequency splittings due to the active region}\n\nIn this section we parametrize the effects  of the active region on the oscillation frequencies in the corotating frame ${\\mathcal{\\widehat{R}}}_\\beta$, where the active region is at the north pole.\n\nModeling the complex influence of surface magnetic fields on acoustic oscillations is challenging \\citep[{{e.g.}},][]{Gizon2010,Schunker2013}.\nHere we choose to drastically simplify the physics and to focus on the geometrical aspects of the problem. \n\nAssuming that the area of the active region covers a polar cap with $0\\le \\widehat \\theta_\\beta \\le \\alpha$ (see Fig.\n\\ref{fig:referenceframes}) and that the structure of the active region is separable in $r$ and $\\widehat\\theta_\\beta$, we can parametrize the perturbation matrix as follows:\n\\begin{gather}\n\\label{eq:omat_mag_epsilon}\n \\widehat O^{\\mathrm{AR}}_{m''m'} = \\delta_{m''m'} \\delta\\omega_{m'}^{\\rm AR} = \\delta_{m''m'} \\omega_{n\\ell}^{(0)} \\,\\varepsilon_{n\\ell}\\, G_\\ell^{m''}(\\alpha) \\, ,\n\\end{gather}\nwhere\n\\begin{gather}\n\\label{eq:glm}\n  G_\\ell^{m''}(\\alpha) = \\frac{(2\\ell+1)(\\ell-|m''|)!}{2(\\ell+|m''|)!} \n      \\int_{\\cos\\alpha}^1 \\left [P_\\ell^{|m''|}(x)\\right ]^2 \\mathrm{d} x\n\\end{gather}\nis a geometrical weight factor that accounts for the surface coverage of the active region.\nFor small values of $\\alpha$, $G_\\ell^{m''}(\\alpha)$ decreases fast as $|m''|$ increases (Fig. \\ref{fig:glmalpha}). \n\\begin{figure}\n \\includegraphics[width=\\columnwidth]{fig2.jpg}\n \\caption{ Left panel: Geometrical weight factor $G_\\ell^{m''}(\\alpha)$ defined by Eq.~(\\ref{eq:glm}) as a function of angular degree $\\ell$ for $m=0$, $\\pm1$, $\\pm2$, at fixed $\\alpha=8^\\circ$. Right panel: $G_\\ell^{m''}(\\alpha)$ as a function of  $\\alpha$ for $\\ell=2$.\n{ The red curves show the parabolic approximations for $G_\\ell^0(\\alpha)$ in the limit of small $\\alpha$ (see Sec. \\ref{sec:small_ar}, Eq. \\ref{eq:Gl0_approx}).} \n }\n \\label{fig:glmalpha}\n\\end{figure}\nSince the value of $G_\\ell^{m''}(\\alpha)$ does not depend on the sign of $m''$, the eigenvalues of $\\vet O^{\\rm AR}$ are degenerate in $|m|$ \\citep[see also, {{e.g.}},][]{Kurtz1986}. \n\nThe parameter $\\varepsilon_{n\\ell}$ is a measure of the relative magnitude of the active region perturbation and contains all the physics. \nFrom studies of local helioseismology \\citep[{{e.g.}}][]{Gizon2009, Moradi2010, Schunker2013}, it is known that the net effect of an active region is to increase the frequencies of acoustic modes, i.e. waves propagate faster in magnetic regions and thus $\\varepsilon_{n\\ell}$ is positive.\nSince the active region introduces a perturbation that is strongly localized near the surface, the value of $\\varepsilon_{n\\ell}$ increases with radial order $n$. \nA proper calculation of the value of $\\varepsilon_{n\\ell}$ goes beyond the scope of the present study.\nInstead, we parametrize the active region perturbation as an increase in sound speed near the surface.\nFollowing \\citet[][]{Papini2015}, we write\n\\begin{align}\n\\label{eq:epsilon_nldef}\n \\varepsilon_{n\\ell}= & \\int_\\mathrm{AR} \\Delta c^2(r) \\left( \\frac{1}{r} \\PD{\\left(r^2\\xi_{r,n\\ell}(r)\\right)}{r} - \\ell(\\ell+1)\\xi_{h,n\\ell} (r)\\right)^2 \\rho_0(r) \\mathrm{d}{r} \\nonumber\\\\\n &\\times \\left ( 2 \\omega_{n\\ell}^{(0)2} \\int_V ||{\\mbox{\\boldmath${\\xi}$} }^{(0)}_{n\\ell m}||^2\\rho_0(r)\\mathrm{d}{V}\n \\right)^{-1} ,\n\\end{align}\nwhere $\\rho_0(r)$ is the density of the unperturbed stellar background and $\\Delta c^2(r)$ is the radial change in the squared sound speed.\nHere the integral is over the radial extent of the active region.  In Section~\\ref{sec:AR_parameters} we will further specify the effective sound-speed perturbation.\n\n\nUsing Eq.~(\\ref{eq:OtildeAR_toOAR}), the perturbation matrix elements  in frame ${{\\mathcal{R}}}_\\beta$ are\n\\begin{gather}\nO^{\\mathrm{AR}}_{m'm} = \\omega_{n\\ell}^{(0)} \\varepsilon_{n\\ell} \n\\sum_{m''=-\\ell}^\\ell G_\\ell^{m''}(\\alpha) r^{(\\ell)}_{m'' m'}(\\beta) r^{(\\ell)}_{m ''\\, m}(\\beta).\n\\end{gather}\nWe note that for $\\ell\\leq 3 $ and $\\alpha \\lesssim 30^\\circ$, we have $G_\\ell^{0}(\\alpha) \\gg  G_\\ell^{m''}(\\alpha)$, where $|m''| \\neq 0$,\nand therefore the dominant eigenvalue is \n$\nO_{00}^\\mathrm{AR}~=~\\omega_{n\\ell}^{(0)}\\varepsilon_{n\\ell} G_\\ell^0(\\alpha)\n$, see Eq. (\\ref{eq:omat_mag_epsilon}).\n\n\n\n\\subsection{Numerical setup for the linear problem}\n\n\nWe now introduce the internal rotation model and the active region parameters, used to illustrate the theory.\nWe consider a star with a rotation period of $8$~days, about one third the rotation period of the Sun. \n{ This choice of rotation period ensures that the azimuthal modes in a multiplet are well separated in frequency space.}\nFor the internal rotation profile, we take\n\\begin{gather}\n  {\\Omega(r,\\theta)}\/{2\\pi} = \n  \\left\\{ \n  \\begin{array}{ll}\n  (1447 - 183 \\cos^2\\theta - 253 \\cos^4\\theta)\n  \\text{  nHz } &  r>0.7R_\\odot , \\\\\n 1447\\text{  nHz } &  r< 0.7 R_\\odot ,\n \\end{array}\n \\right.\n \\label{eq.rotprofile}\n\\end{gather}\nwhich is a scaled model of solar differential rotation as in \\citet{2004gizon}. The centrifugal term has a significant effect. It shifts the $m=0$ mode and introduces an asymmetry in the shifts for positive and negative azimuthal orders $m$, with a maximum frequency shift of more than $100~{\\mbox{$\\mathrm{nHz}$}}$ in the case of a multiplet near $3$~mHz.\nTherefore this term must be included when performing the analysis.\n\n\n\\label{sec:AR_parameters}\n\nFrom observations of $p$-mode frequency changes during the solar cycle, \\citet{Libbrecht1990} showed that the (positive) frequency shifts are almost independent of $\\ell$ and increase with frequency, thus indicating that the effects of magnetic activity on acoustic oscillations are confined to the surface.\nAssuming that the perturbation covers two pressure scale heights below the photosphere and setting $\\Delta c^2\/c^2 \\simeq 10\\%$ there, Eq. (\\ref{eq:epsilon_nldef}) gives $\\varepsilon_{n\\ell}\\simeq 0.003$  at frequencies near $3$~mHz.\nThe surface coverage of a stellar active region, as inferred from Doppler imaging, ranges from a percent up to $10\\%$ \\citep{Strassmeier2009}. Here we consider two different surface coverages of either $4\\%$ or  $7\\%$, corresponding to $\\cos\\alpha=0.92$ or $0.86$ ($\\alpha\\simeq23^\\circ$ and $30^\\circ $).\nFinally, we consider an active region at a colatitude of either $\\beta=20^\\circ$ (near the pole) or $80^\\circ$ (near the equator).\n\nIn the following section we focus on two multiplets with $(\\ell, n)=(1, 18)$ and $(2,18)$. For each of these modes we solve the eigenvalue problem (\\ref{eq:eigen_problem}) by means of Jacobi's method \\citep{numrecipes}. \nFor the calculation of the unperturbed eigenmodes $(\\omega_{n\\ell}^{(0)},{\\mbox{\\boldmath${\\xi}$} }_{n\\ell m}^{(0)})$ we use the ADIPLS software package \\citep{Christensen-Dalsgaard2008} and Solar Model S  as the reference structure model \\citep{JCD1996}.\nThe unperturbed frequencies $\\omega_{n\\ell}^{(0)}\/2\\pi$ of the dipole and quadrupole modes are $2695.40~\\mu{\\mbox{$\\mathrm{Hz}$}}$ and $2756.95~\\mu{\\mbox{$\\mathrm{Hz}$}}$ respectively.\n{ We note that the choice of reference solar model is unimportant for the present study.}\n\n\n\\subsection{Power spectrum in the observer's frame: $(2\\ell+1)^2$ peaks}\n\\label{sec:linear_PS}\n\nGiven particular values for $\\alpha$, $\\beta$, and $\\varepsilon_{n\\ell}$ the eigenvalue problem (\\ref{eq:eigen_problem}) is fully specified and can be solved.\nIn this section we use the solutions (\\ref{eq:xi_pert}) and (\\ref{eq:freq_pert}) to build a synthetic power spectrum in the observer's frame, in order to relate the results to observations.\n\nWe need to find an expression that connects the eigenmodes to the observed intensity fluctuations.\nFor the sake of simplicity, we assume that the variation $I(\\theta_\\beta,\\phi_\\beta,t)$ induced by the acoustic oscillations in the emergent photospheric intensity is proportional to the Eulerian pressure perturbation $p$ of the acoustic wavefield \\citep[{{e.g.}}][]{Toutain1993}, as measured  at the stellar surface $r=R$. \nThe pressure perturbation $p$ is related to the wavefield  displacement ${\\mbox{\\boldmath${\\xi}$} }$ through the linearized adiabatic equation\n\\begin{gather*}\np = -\\rho_0 c_0^2 \\nabla \\cdot {\\mbox{\\boldmath${\\xi}$} } - {\\mbox{\\boldmath${\\xi}$} }\\cdot \\nabla P_0,\n\\end{gather*}\nwhere $c_0$ and $P_0$ are respectively the sound speed and the pressure of the unperturbed stellar model.\nThe pressure perturbation $p_{n\\ell M}(r,\\theta_\\beta,\\phi_\\beta)$ of the mode $M$ is\n\\begin{equation}\np_{n\\ell M} = -\\rho_0 c_0^2 \\nabla \\cdot {\\mbox{\\boldmath${\\xi}$} }_{n\\ell M} - {\\mbox{\\boldmath${\\xi}$} }_{n\\ell M}\\cdot \\nabla P_0,\n\\end{equation}\nwhere ${\\mbox{\\boldmath${\\xi}$} }_{n\\ell M}$ is given by Eq. (\\ref{eq:xi_pert}).\nTo  leading order, we have\n\\begin{gather}\np_{n\\ell M}({\\vet r_\\beta})= \\sum_{m=-\\ell}^\\ell A_m^M p_{n\\ell m}^{(0)}(\\vet r_\\beta)\n\\end{gather}\nwith \n\\begin{gather*}\np_m^{(0)}(R,\\theta_\\beta,\\phi_\\beta) = \n -\\rho_0 c_0^2 \\nabla \\cdot {\\mbox{\\boldmath${\\xi}$} }_m^{(0)} - {\\mbox{\\boldmath${\\xi}$} }_m^{(0)}\\cdot \\nabla P_0 ,\n\\end{gather*}\nwhere we dropped the subscripts $n\\ell$.\nAcoustic oscillations in stars are stochastically excited and damped by turbulent convection, therefore $I(\\theta_\\beta,\\phi_\\beta,t)$ is a realization of a random process. \nSince  the perturbation is steady in the frame ${{\\mathcal{R}}}_\\beta$, this random process is stationary in that frame.\nAn expression for the intensity fluctuations $I(\\theta_\\beta,\\phi_\\beta,\\omega)$ in  Fourier space with the required  statistical properties is\n\\begin{align}\nI(\\theta_\\beta,\\phi_\\beta,\\omega) \n& \\propto \\sum_{M=-\\ell}^\\ell p_M(R,\\theta_\\beta,\\phi_\\beta)\nL_M^{1\/2}(\\omega) {{\\mathcal{N}}}_M(\\omega) \\nonumber \\\\\n & \\propto \\sum_{M=-\\ell}^\\ell \\sum_{m=-\\ell}^\\ell A_m^M {Y_\\ell^m}(\\theta_\\beta,\\phi_\\beta)\nL_M^{1\/2}(\\omega)\n    {{\\mathcal{N}}}_M(\\omega) ,\n\\label{eq:Iomega_corot}\n\\end{align}\nwhere the ${{\\mathcal{N}}}_M(\\omega)$ are independent complex Gaussian random variables, with zero mean and unit variance:\n\\begin{gather}\n\\label{eq:covNMNMp}\n  E \\left [ {{{\\mathcal{N}}}_{M'}}^*(\\omega'){{\\mathcal{N}}}_M(\\omega)\\right ] = \\delta_{M'M} \\delta_{\\omega'\\omega}.\n\\end{gather}\nIn Equation (\\ref{eq:Iomega_corot}) and (\\ref{eq:covNMNMp}) we only consider the positive-frequency part of the spectrum ($\\omega$ and $\\omega'>0$) since $I(\\theta_\\beta,\\phi_\\beta,t)$ is real. The negative-frequency part is related to the positive part by $I(\\theta_\\beta,\\phi_\\beta,-\\omega)= I^*(\\theta_\\beta,\\phi_\\beta,\\omega)$.\nThe function $L_M(\\omega)$ is a Lorentzian  \n\\begin{gather}\nL_M(\\omega)=\\left [ 1+\\left (\\frac{\\omega-\\omega_M}{\\Gamma\/2}  \\right )^2 \\right ]^{-1},\n\\end{gather}\nappropriate for describing the power spectrum of an exponentially damped oscillator  with full width at half maximum (FWHM) $\\Gamma$ \\citep[see ,e.g.,][]{Anderson1990}.\n\nWe transform Eq. (\\ref{eq:Iomega_corot}) back to the time domain by inverse Fourier transformation to obtain $I(\\theta_\\beta,\\phi_\\beta,t)$.\nThe intensity $I(\\theta,\\phi,t)$ as seen by the observer in the inertial frame ${{\\mathcal{R}}}$ is obtained by applying a passive rotation of Euler angles $(0,-i,\\Omega_\\beta t)$ to\nexpress ${Y_\\ell^m}(\\theta_\\beta,\\phi_\\beta)$ in terms of $\\theta$ and $\\phi$ \\citep{1959messiah}:   \n\\begin{gather}\nY_\\ell^{m}(\\theta_\\beta,\\phi_\\beta) = \ne^{-\\mathrm{i} m\\Omega_\\beta t}\n\\sum_{m'=-\\ell}^{\\ell} Y_\\ell^{m'}(\\theta,\\phi) r_{m'm}^{(\\ell)}(-i) .\n\\end{gather}\nIn the frequency domain, the intensity fluctuations become\n\\begin{equation}\nI (\\theta,\\phi,\\omega)  = \\sum_{M=-\\ell}^\\ell \\sum_{m=-\\ell}^\\ell\\sum_{m'=-\\ell}^\\ell \nA_m^M Y_\\ell^{m'}(\\theta,\\phi) r_{m\\,m'}^{(\\ell)}(i) \\;  L_M^{1\/2}(\\omega - m\\Omega_\\beta)\n    {{\\mathcal{N}}}_M(\\omega- m\\Omega_\\beta) ,\n\\label{eq:I_thetaphiomega}\n\\end{equation}\nwhere we used the property $r_{m'm}^{(\\ell)}(-i)=r_{m\\,m'}^{(\\ell)}(i)$.\nSince $L_M(\\omega - m\\Omega_\\beta)$ peaks at frequency $\\omega=\\omega_M + m\\Omega_\\beta$, the intensity spectrum observed in the inertial frame has $(2\\ell+1)^2$ peaks, corresponding to all combinations of $m$ and $M$.\n\nTo obtain the full-disk integrated intensity fluctuations  $I_\\text{obs}(\\omega)$ we perform an integration over the visible disk of the star:\n\\begin{gather*}\n I_\\text{obs}(\\omega) = \n\\int_0^{2\\pi} \\mathrm{d}\\phi \\int_0^{\\pi\/2} \\mathrm{d}\\theta \\;\n I(\\theta,\\phi,\\omega)  W(\\theta)\\cos{\\theta}\\sin\\theta \\,,\n\\end{gather*}\nwhere $W(\\theta)$ is the limb-darkening function.\nThe components with $m'\\ne 0$ vanish upon integration over $\\phi$, thus \n\\begin{gather}\n\\label{eq:Iobs_final}\nI_\\text{obs} (\\omega) \n =  \\sum_{M=-\\ell}^\\ell \\sum_{m=-\\ell}^\\ell\nB_m^M L_M^{1\/2}(\\omega-m\\Omega_\\beta)\n    {{\\mathcal{N}}}_M(\\omega- m\\Omega_\\beta) ,\n\\end{gather}\nwith\n$B_m^M =  A_m^M \\,V_\\ell \\; r_{m0}^{(\\ell)}(i)$ and\n\\begin{gather}\n\\label{eq:V_l}\nV_\\ell=\n\\int_0^{2\\pi} \\mathrm{d}\\phi \\int_0^{\\pi\/2} \\mathrm{d}\\theta\\; \n Y_\\ell^0(\\theta,\\phi)  W(\\theta)\\cos{\\theta}\\sin\\theta .\n \\end{gather}\nThe matrix elements $r_{m0}^{(\\ell)}(i)$ are written explicitly in terms of the associated Legendre polynomials \\citep{1959messiah}:\n\\begin{gather}\n\\label{eq:r0lm}\n\n {r}_{m0}^{(\\ell)}(i) =\n (-1)^m \\sqrt{\\frac{(\\ell-m)!}{(\\ell+m)!}} P_\\ell^m(\\cos i).\n\\end{gather}\n\n\n\n\\begin{figure*}\n\\includegraphics[width=20cm]{fig3a.jpg}\n\\includegraphics[width=20cm]{fig3b.jpg}\n \\caption{ Left side of the diagram: Perturbations to the mode eigenfrequencies in the frame that is corotating with the active region,  for the multiplets with $(\\ell,n)=(1,18)$ (top) and with $(\\ell,n)=(2,18)$ (bottom). The star is Sun-like and rotates with a scaled solar differential rotation profile (rotation period is approximately $8~\\text{days}$). The active region perturbation is specified by  $\\varepsilon_{n\\ell}=0.003$, $\\alpha=23^\\circ$, and $\\beta=80^\\circ$. The rotational frequency of the active region at colatitude $\\beta$ is $\\Omega_\\beta\/2\\pi$ is $1.504~\\mu{\\mbox{$\\mathrm{Hz}$}}$.\nThe frequency of the mode $M=0$  is the most shifted. Right side of the diagram: The  $(2\\ell+1)^2$ peaks of the power spectrum as seen in the observer's frame, for an inclination angle $i=80^\\circ$. For each $m$, the $M$-components are identified with different colors: red for $M=0$, black for $M=1$, blue for $M=-1$, orange for $M=2$, and pink for $M=-2$.  \n The peaks with the same colors are statistically correlated to each other (according to  Eq.~\\ref{eq:PS_correlation}).\n }\n \\label{fig:degenerate2obs}\n\\end{figure*}\n\n\n\nA realization of the power spectrum is given by   \n\\begin{align}\nP(\\omega) &= |I_\\text{obs}(\\omega)|^2 \\nonumber\\\\\n &=  \\left |\\sum_{M=-\\ell}^\\ell \\sum_{m=-\\ell}^\\ell\n     B_m^M L_M^{1\/2}(\\omega-m\\Omega_\\beta)\n    {{\\mathcal{N}}}_M(\\omega- m\\Omega_\\beta) \\right |^2,\n\\label{eq:powerspectrum_realization}\n\\end{align}\nwhich depends on $2\\ell+1$ independent realizations of complex Gaussian random variables. The expectation value of $ P(\\omega)$ is\n\\begin{gather}\n\\label{eq:powerspectrum}\n\\mc{P}(\\omega) = E\\left [P(\\omega)\\right ]=  \\sum_{m=-\\ell}^\\ell \\sum_{M=-\\ell}^\\ell\n P_m^M L_M(\\omega - m\\Omega_\\beta).\n\\end{gather}\nAs mentioned earlier, the power spectrum displays $(2\\ell+1)^2$ Lorentzian peaks, centered at frequencies \n\\begin{gather}\n\\label{eq:observed_frequencies}\n \\omega_m^M := \\omega_M + m \\Omega_\\beta = \\omega_{n\\ell}^{(0)}+\\delta\\omega_M + m \\Omega_\\beta  ,\n\\end{gather}\n with peak power\n\\begin{gather}\n\\label{eq:observed_amplitude}\n P_m^M := ({B_m^M})^2 = \\frac{(\\ell-|m|)!}{(\\ell+|m|)!} \\left [V_\\ell A_m^M P_\\ell^{|m|}(\\cos i) \\right ]^2.\n\\end{gather}\nExample power spectra for a dipole and a quadrupole multiplet are shown in Fig. \\ref{fig:degenerate2obs}.\nThe frequencies and amplitudes of the $(2\\ell+1)^2$ peaks are obtained from Eqs.  (\\ref{eq:observed_frequencies}) and (\\ref{eq:observed_amplitude}),  using the limb-darkening function quoted by \\citet{Pierce2000} to calculate $V_\\ell$.\n\nFigure \\ref{fig:degenerate2obs} also displays the different contributions to the frequency splittings due to rotation and to the active region perturbation in the corotating frame ${{\\mathcal{R}}}_\\beta$.\nFor both multiplets, the $M=0$ peaks are shifted by the largest amount, they are the most affected by the AR perturbation and in the frame ${\\mathcal{\\widehat{R}}}_\\beta$  \\citep[see also][]{Papini2015}.\nThis feature, which arises from geometrical considerations only,  is preserved in the spectrum as seen in the observer's frame, where  the $M=0$ peaks are clearly visible.\nWith increasing AR surface coverage the frequency shifts of the $M\\neq 0$ modes increase and the peaks get less clustered.\n\n\\section{Results}\n\n\\subsection{Dipole and quadrupole power spectra}\nIn this section we describe the changes  imprinted by a large active region in the spectrum of two multiplets with $(\\ell,n)=(1,18)$ and $(2,18)$, \nfor a star rotating with a period of $8~\\text{days}$.\n\n\\begin{figure}\n\\begin{center}\n \\includegraphics[width=0.9\\columnwidth]{fig4.jpg}\n \\end{center}\n \\caption{Oscillation power spectra for the $(\\ell,n)= (2,18)$ multiplet observed at two inclination angles $i=30^\\circ$ (panels a and c) and $80^\\circ$ (panels b and d),\n for a star with a rotation period of $8~\\text{days}$ and for an active region with $\\varepsilon_{n\\ell}=0.003$, $\\beta=20^\\circ$ (panels a and b) or $80^\\circ$ (panels c and d), and for a surface coverage with $\\alpha=23^\\circ$. {The power spectra are normalized with respect to $V_2$ (Eq. \\ref{eq:V_l}).}  The vertical line segments show the theoretical frequencies and amplitudes for $M=\\pm1,\\pm 2$ modes (black) and the $M=0$ mode (red). \n The envelopes of the power spectra (solid black curves) are obtained by summing over Lorentzians with widths of $1~\\mu{\\mbox{$\\mathrm{Hz}$}}$. The dashed black curve shows the envelope of the pure rotational power spectrum, which includes the centrifugal distortion (Eq. \\ref{eq:rot_splitting}).\n }\n \\label{fig:l2n18spectrum}\n\\end{figure}\nFigure \\ref{fig:l2n18spectrum} shows the results for the quadrupole multiplet, for four combinations of the values of $\\alpha$ and $\\beta$ selected in Sect.~\\ref{sec:AR_parameters}: the observed power spectra are plotted for two angles of observation, $i=30^\\circ\\text{ and}~80^\\circ$, and are \nnormalized with respect to $V_2$, {{i.e.}}~with respect to the $m=0$ peak of the pure rotational spectrum seen with $i=0$.\nThe corresponding theoretical Lorentzian envelope (solid line) has been calculated \nfrom Eq. (\\ref{eq:powerspectrum}), by setting a value for the FWHM of $\\Gamma\/2\\pi=1~\\mu{\\mbox{$\\mathrm{Hz}$}}$, typical for this multiplet in the Sun \\citep[see, {{e.g.}},][]{Chaplin2005}.\nDue to the finite lifetime of the modes of oscillation, it is clear that is not possible to resolve all the $(2\\ell+1)^2$ peaks, and an observer would identify not many more than $(2\\ell+1)$ peaks in a multiplet.\nIn the cases shown here, it is possible to identify from 5 to 6 peaks for $i=80^\\circ$, the additional peak coming from the uppermost shifted $m=2,M=0$ peak. \nWe note that the Lorentzian envelope displays an asymmetric profile.\nBecause of their large shifts in frequency, the $M=0$ peaks blend with peaks from different $m$-quintuplets. Blending increases with activity level. Figure \\ref{fig:l2n18spectrum}a shows a case for which the $(M,m)=(0,0)$ and  $(M,m)=(1,1)$ peaks have close frequencies and comparable amplitudes, they contribute equally to a single peak in the power spectrum.\n \nThe  envelope of the power spectrum  is very sensitive to the latitudinal position of the AR and to the inclination angle: in Fig. \\ref{fig:l2n18spectrum}c the power spectrum is near the standard rotationally-split spectrum, while the same configuration observed from a different inclination angle (Fig. \\ref{fig:l2n18spectrum}d) shows a more asymmetric profile with additional peaks. \n{This is better seen in Fig. \\ref{fig:contours}, which shows contours of the acoustic power as a function of inclination angle, for both the $\\ell=1$ and $\\ell=2$ multiplets, for the same active region parameters as in Fig. \\ref{fig:l2n18spectrum}. For an active region at high latitude (middle panels of Fig. \\ref{fig:contours}), the central peak  shows a significant shift and overlaps with the $m=1$ peak. \nFor a low-latitude active region (bottom panels) the envelope of the power spectrum displays more than $2\\ell+1$ peaks. A distinct feature is the presence of  two peaks  instead of one when observing an $\\ell=2$ multiplet at zero inclination angle.}\n{The sensitivity of the spectrum to the colatitude of the AR, shown in Fig \\ref{fig:contours_beta}, is due in part to the variation with $\\beta$ of the non-diagonal elements of the rotation matrix ${{\\vet{R}}}^{(\\ell)}$. }\n\nAn observed power spectrum is, of course, much more difficult to interpret than its expectation value. The power spectrum in Fig.~\\ref{fig:noise_1} includes realization noise due to the stochastic nature of stellar oscillations and to additional shot noise. At each frequency the observed power is a realization of an exponential distribution (a chi-squared with two degrees of freedom) with standard deviation and mean equal to the expectation value of the power spectrum. Realization noise considerably degrades the spectrum, however in some cases it is still possible to distinguish between the pure rotational spectrum and a spectrum with the active region.\n\n\n\\begin{figure}\n\\includegraphics[width=0.97\\columnwidth]{fig5.jpg\n \\caption{\n Expectation value of power spectra of oscillation,  as functions of inclination angle $i$.\n The left panels are for the dipole multiplet $\\ell=1,n=18$, and the right panels for the quadrupole multiplet $\\ell=2,n=18$.\n The top panels are for the pure-rotation case, the middle panels are for an active region at co-latitude $\\beta=20^\\circ$, and the bottom panels for  $\\beta=80^\\circ$.\nThe active region parameters are  $\\varepsilon_{n\\ell}=0.003$, $\\alpha=23^\\circ$, and the stellar rotation period is $8$~days.\n }\n \\label{fig:contours}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=1.0\\columnwidth]{fig6.jpg\n \\caption{Expectation value of power spectra of oscillation, as functions of active-region colatitude $\\beta$, at fixed inclination angle $i=80^\\circ$. The other physical parameters are the same as in Fig.~\\ref{fig:contours}.\n }\n \\label{fig:contours_beta}\n\\end{figure}\n\n\n\\begin{figure}\n\\includegraphics[width=1.0\\columnwidth]{fig7.jpg\n \\caption{A realization of the power spectrum of an $\\ell=2$ multiplet (gray) and its expectation value (black solid curve, also shown in  Fig.~\\ref{fig:l2n18spectrum}d). The observation duration is 6 months and the signal-to-noise ratio is 50.  The dashed black curve shows the expectation value of the pure rotational spectrum.\n }\n \\label{fig:noise_1}\n\\end{figure}\n\n\n\n\\subsection{Asymptotics}\n\\subsubsection{Limit of small latitudinal differential rotation}\n\\label{sec:small_diff_rot}\nThe examples shown so far suggest that the power spectrum of a multiplet may be approximated by the sum of $2(2\\ell+1)$ peaks, for two reasons.\nFirst, in the corotating frame the splitting due to rotation is small compared to the shift induced by the AR perturbation (see, {{e.g.}}, Fig. \\ref{fig:degenerate2obs}).\nSecond, the AR perturbation induces a shift that is largest for the $M=0$ mode. Therefore, for each $m$, all the $M\\neq0$ peaks are clustered near the pure rotational frequencies and appear as a single peak, while the $M=0$ peaks are well separated. \n\nHere we wish to find an approximation for the power spectrum of a multiplet in terms of $2(2\\ell+1)$ Lorentzians only. \nIn the following we assume the rotation perturbation to be small compared to the active region perturbation, and seek an approximate solution to the eigenvalue problem (\\ref{eq:eigen_problem}). \n\nIt is convenient to solve the eigenvalue problem in the frame where the dominant perturbation is diagonal, {{i.e.}}, the frame ${\\mathcal{\\widehat{R}}}_\\beta$ of the active region. The following results are similar to Sec. 19.5 of \\citet[][]{Unno1979}. \nWe rewrite the elements of the full matrix $\\widehat{\\vet O}$ in ${\\mathcal{\\widehat{R}}}_\\beta$ as\n\\begin{gather}\n\\widehat{O}_{m''m'} = \\delta_{m''m'}\\delta\\omega_{m'}^{\\rm AR} +\n\\sum_{m=-\\ell}^{\\ell}\nr_{m''m}^{(\\ell)} (\\beta) r_{m'm}^{(\\ell)} (\\beta)\n\\delta\\omega_{m}^\\Omega ,\n\\end{gather}\nwhere the term with the sum in the right hand side correspond to $\\widehat O_{m''m'}^\\Omega $.\nWe look for solutions of the form\n\\begin{gather}\n\\delta\\omega_M = \\delta\\omega_{M}^{\\rm AR} + \\delta\\omega_M^{(1)}\n\\end{gather}\nand \n\\begin{gather}\n\\widehat\\boldsymbol{A}{}^M = \\widehat\\boldsymbol{A}{}^{M,\\rm AR} + \\widehat\\boldsymbol{A}{}^{M,(1)},\n\\end{gather}\nwhere $\\left (\\delta\\omega_M^{(1)},\\widehat\\boldsymbol{A}{}^{M,(1)}\\right )$ are the perturbations to the (partially degenerate) eigenvalues $\\delta\\omega_{M}^\\mathrm{AR}$ and eigenvectors $\\widehat\\boldsymbol{A}{}^{M,\\rm AR}$ of $\\widehat\\vet O{}^\\mathrm{AR}$, with $\\widehat{A}_m^{M,\\mathrm{AR}}=\\delta_{mM}$. \nThe eigenvector perturbation is orthogonal to the reference eigenspace  \\citep[see, {{e.g.}},][p. 687]{1959messiah}, \n\\begin{equation}\n\\widehat{\\boldsymbol{A}}^{M,(1)} \\cdot \\widehat{\\boldsymbol{A}}^{\\pm M,\\mathrm{AR}}=0,\n\\label{eq:orthoA'A}\n\\end{equation}\nwhich gives \n$\n\\widehat A_{\\pm M}^{M,(1)} = 0.\n$\nThe eigenvectors  $\\boldsymbol{A}^M$ in the frame ${{\\mathcal{R}}}_\\beta$ are given by\n\\begin{gather}\n\\boldsymbol{A}^M = {\\vet{R}}^{(\\ell)} \\widehat\\boldsymbol{A}{}^M \n\\end{gather}\nvia the rotation matrix ${\\vet{R}}^{(\\ell)}$ (see Eq.~\\ref{eq:rotylmabg}).\n\nTo first order , the eigenvalue problem (Eq.~\\ref{eq:eigen_problem})  becomes\n\\begin{gather}\n \\left ( \\widehat{\\vet O}^\\Omega - \\delta\\omega_M^{(1)} \\vet I \\right ) \\widehat\\boldsymbol{A}{}^{M,\\mathrm{AR}} + \\left ( \\widehat\\vet O^\\text{AR} -\\delta\\omega_{M}^\\mathrm{AR} \\vet I \\right )\\widehat\\boldsymbol{A}{}^{M,(1)} = 0,\n\\label{eq:eigenproblem_small}\n\\end{gather}\nwhere $\\vet I$ is the identity matrix. \nTo calculate $\\widehat\\boldsymbol{A}{}^{M,(1)}$ and $\\delta\\omega_M^{(1)}$ we multiply the above equation on the left by the transpose of $\\widehat\\boldsymbol{A}{}^{m',\\text{AR}}$ to obtain:\n\\begin{gather}\n\\widehat O_{m'M}^\\Omega  - \\delta\\omega_M^{(1)} \\delta_{m'M} + \n\\left (\\delta\\omega_{m'}^\\mathrm{AR} -\\delta\\omega_{M}^\\mathrm{AR} \\right ) \\widehat A_{m'}^{M,(1)}  = 0  ,  \\quad   -\\ell \\le m' \\le \\ell.\n\\label{eq:eigenproblem_small_component}\n\\end{gather}\nWe find the perturbed eigenvalues by setting $m'=M$ in the above equation\n\\begin{equation}\n\\delta\\omega_M^{(1)} =  \\sum_{m=-\\ell}^{\\ell}\n[r_{Mm}^{(\\ell)} (\\beta) ]^2 \n\\delta\\omega_{m}^\\Omega .\n\\end{equation}\nThe non-zero elements of $\\boldsymbol{A}^{M,(1)}$ are obtained from the $m'\\neq \\pm M$ components of Eq. (\\ref{eq:eigenproblem_small_component}):\n\\begin{gather}\n\\widehat A_{m'}^{M,(1)} = \\frac{\\widehat O_{m'M}^\\Omega}{\\delta\\omega_{M}^\\mathrm{AR}-\\delta\\omega_{m'}^\\mathrm{AR}} \\quad \\text{ for } m'\\neq \\pm M .\n\\end{gather}\nThe explicit expressions for the eigenvalues and eigenvectors of the combined perturbations are\n\\begin{align}\n\\label{eq:shifts_small_rotation}\n\\delta\\omega_M = & \\,\\delta\\omega_{M}^\\mathrm{AR} + \\sum_{m=-\\ell}^{\\ell}\n[r_{Mm}^{(\\ell)} (\\beta) ]^2 \n\\delta\\omega_{m}^\\Omega \n\\end{align}\nand\n\\begin{gather}\n\\label{eq:amplitudes_small_rotation}\n\\widehat A_{m'}^M = \\left \\{\n\\begin{aligned}\n&1 & \\text{for } m'&=M ,\\\\\n&0 & \\text{for } m'&=-M ,\\\\\n& \\ffrac{\\sum_{m=-\\ell}^{\\ell}\nr_{m'm}^{(\\ell)} (\\beta) r_{Mm}^{(\\ell)} (\\beta)\n\\delta\\omega_{m}^\\Omega}\n{\\delta\\omega_{M}^\\mathrm{AR}-\\delta\\omega_{m'}^\\mathrm{AM}}\n& \\text{for } m' &\\neq M .\n\\end{aligned}\n\\right .\n\\end{gather}\nThe explicit expression for the amplitudes $A_m^M$ in the frame ${{\\mathcal{R}}}_\\beta$ is then\n\\begin{align}\nA_m^M  = &\\sum_{m'=-\\ell}^\\ell  \\widehat A_{m'}^M r_{m'm}^{(\\ell)}(\\beta) \\nonumber\\\\\n= &r_{Mm}^{(\\ell)}(\\beta) \n + \\sum_{m'=-\\ell,m'\\neq \\pm M}^{\\ell}\n\\frac{r_{m'm}^{(\\ell)}(\\beta)}{\\delta\\omega_{M}^\\mathrm{AR}-\\delta\\omega_{m'}^\\mathrm{AR}}\n\\sum_{m''=-\\ell}^{\\ell}\nr_{m'm''}^{(\\ell)} (\\beta) r_{Mm''}^{(\\ell)} (\\beta)\n\\delta\\omega_{m''}^\\Omega .\n\\label{eq:amplitudes_asymptotics}\n\\end{align}\n\n\\subsubsection{Neglecting latitudinal differential rotation}\n\nIf we neglect differential rotation then Eq. (\\ref{eq:rot_splitting2}) reduces to\n\\begin{equation} \n\\delta\\omega_m^\\Omega = -m\\Omega_\\beta C_{n\\ell} +  \\eta Q_{2\\ell m} \\, \\omega_{n\\ell}^{(0)}.\n\\label{eq:domega_nodiff_rot}\n\\end{equation}\nThen, by using the identities \\citep{Unno1979,1990gough}\n\\begin{equation}\n\t\\sum_{m''=-\\ell}^\\ell \n    \\left [r_{m'm''}^{(\\ell)}(\\beta) \\right ]^2 m'' = m' \\cos\\beta \n\\end{equation}\nand\n\\begin{equation}\n  \\sum_{m''=-\\ell}^\\ell \\left [r_{m'm''}^{(\\ell)}(\\beta) \\right ]^2\n    Q_{2 \\ell m''}=P_2(\\cos\\beta) Q_{2 \\ell m'},\n\\end{equation}\nthe frequency shifts (Eq.~\\ref{eq:shifts_small_rotation}) simplify to\n\\begin{align}\n\\delta\\omega_M = \\delta\\omega_{M}^\\mathrm{AR} \n- M C_{n\\ell} \\Omega_\\beta \\cos\\beta \n+ \\eta P_2(\\cos\\beta) Q_{2 \\ell M} \\, \\omega_{n\\ell}^{(0)} .\n \\label{eq:lin_dom_no_diff}\n\\end{align}\nNote that for moderately fast rotating stars the Coriolis term  in the above equation is much smaller than centrifugal distortion term \\citep[e.g.,][]{2004gizon}.\n\nDue to the clustering of the $M\\neq0$ peaks, the power spectrum in the observer's frame can be approximately modeled by $2(2\\ell+1)$ Lorentzians, some of which may overlap. Half of these correspond to the peaks with $M=0$. The remaining $2\\ell+1$ (approximate) Lorentzians are obtained by summing over the $M\\neq0$ peaks; their mean frequency shifts are given by a power weighted average of the $M\\neq0$ frequency shifts $\\delta\\omega_M^m = \\delta\\omega_M +m\\Omega_\\beta$. We denote by $\\langle \\delta\\omega\\rangle_m $ this average:\n\\begin{align}\n{\\langle \\delta\\omega\\rangle}_m = \n(\\Sigma_m )^{-1}\n\\left( \\sum_{1\\leq |M| \\leq \\ell} ({A_m^M})^2 \\delta\\omega_M  \\right)\n+m\\Omega_\\beta,\n\\label{eq:linear_freq_shift_Mneq0}\n\\end{align}\nwith \n\\begin{align}\n\\Sigma_m = \\sum_{1\\leq |M| \\leq \\ell} {(A_m^M)}^2.\n\\end{align}\nThe corresponding averaged power amplitudes $\\langle P\\rangle_m $ are (see Eq.~\\ref{eq:observed_amplitude})\n\\begin{gather}\n \\langle P \\rangle_m = \\frac{(\\ell-|m|)!}{(\\ell+|m|)!} \\left [V_\\ell P_\\ell^{|m|}(\\cos i) \\right ]^2 \n \\Sigma_m.\n \\label{eq:linear_amplitude_shift_Mneq0}\n\\end{gather}\n\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig8.jpg}\n\\caption{Acoustic power spectrum for the same case as in Fig. \\ref{fig:l2n18spectrum}d, (black solid line) and the power spectrum resulting from neglecting differential rotation and by assuming a small rotation perturbation with respect to the AR perturbation (black dashed line), as calculated from Eqs. (\\ref{eq:amplitudes_asymptotics}, \\ref{eq:domega_nodiff_rot}, \\ref{eq:lin_dom_no_diff}).\nVertical red lines denote the $M=0$ peaks, and black lines the averaged $M\\neq 0$ peaks from Eqs. (\\ref{eq:linear_freq_shift_Mneq0}, \\ref{eq:linear_amplitude_shift_Mneq0}). }\n\\label{fig:spectrum_small_diff_rot}\n\\end{figure}\n\n\n\\begin{figure}[t]\n\\includegraphics[width=\\columnwidth]{fig9.jpg\n\\caption{\nBlack lines: Averaged frequency shifts $\\langle \\delta\\omega\\rangle_m$ vs. stellar rotation rate, as given by Eq. (\\ref{eq:linear_freq_shift_Mneq0}) and calculated using Eq. (\\ref{eq:eigen_problem}) (solid lines) and using the approximations of Eqs. (\\ref{eq:amplitudes_asymptotics}, \\ref{eq:domega_nodiff_rot}, \\ref{eq:lin_dom_no_diff}) (dashed lines).\nRed lines: Approximate frequency shifts $\\delta\\omega_{M=0}+m\\Omega_\\beta$ of the $M=~0$ peaks as given by Eq. (\\ref{eq:lin_dom_no_diff}) (dashed lines) and first-order exact shifts (solid lines).\n}\n\\label{fig:linear_qnl_AR}\n\\end{figure}\n\nFigure \\ref{fig:spectrum_small_diff_rot} shows how good is the $2(2\\ell+1)$-Lorentzian model  in reproducing the expected power spectrum for the case of Fig.\\ref{fig:noise_1}. The vertical red lines are for the $(2\\ell+1)$  peaks with $M=0$,  while the vertical black lines refer to the $(2\\ell+1)$ peaks with power  $\\langle P\\rangle_m $ and frequency shifts $\\langle \\delta\\omega\\rangle_m $. The envelope of power of the $2(2\\ell+1)$-model compares well with the envelope  obtained by summing over all the $(2\\ell+1)^2$ peaks.\n\nFigure \\ref{fig:linear_qnl_AR} shows the frequency shifts $\\langle \\delta\\omega\\rangle_m $ (black lines) and the shifts of the $M=0$ peaks (red lines) calculated from the  solution of Eq. (\\ref{eq:eigen_problem}) (solid lines) and from the approximate solutions (dashed lines), as a function of stellar rotation rate and for an active region with a surface coverage of $4\\%~(\\alpha=23^\\circ)$ and $\\beta = 80^\\circ$.\nThe linear approximation successfully returns the frequency shifts of the $2(2\\ell+1)$ Lorentzians, even for moderately high rotation rates.\n\n\n\n\n\\subsubsection{Limit of small-size active region ($\\alpha \\lesssim 15^\\circ$)}\n\\label{sec:small_ar}\n\n\\begin{figure*}[t]\n\\includegraphics[width=0.5\\columnwidth]{fig10a.jpg}\n\\includegraphics[width=0.5\\columnwidth]{fig10b.jpg\n\\caption{\nLeft: Contour plot of the absolute difference \n$|\\overline{\\delta\\omega}_{M=0} - \\delta\\omega_{M=0} |$ \nbetween the approximate shifts $\\overline{\\delta\\omega}_{M=0}$ (see Eq.~\\ref{eq:small_ar_shiftsM0}) and the first-order shifts ${\\delta\\omega}_{M=0}$, as a function of stellar rotation rate and surface coverage of the active region. \nRight: Contour plot of the maximum of the absolute difference\n$| \\langle \\overline{\\delta\\omega}\\rangle_m - \\langle \\delta\\omega\\rangle_m | $ over all values of $m$, as a function of stellar rotation rate and surface coverage of the active region.\n}\n\\label{fig:alphaomega_contour}\n\\end{figure*}\n\nThe above formulae for the frequency shifts simplify further when the active region has a small surface area. For small values of $\\alpha$, the integral in Eq.~(\\ref{eq:glm}) can be approximated,\n\\begin{align}\n\\int_{\\cos\\alpha}^1 \\left [P_\\ell^{|m''|}(\\mu)\\right ]^2 \\mathrm{d}{\\mu}  & \n\\underset{\\alpha \\rightarrow 0}{\\sim} \n\\left [P_\\ell^{|m''|}(0)\\right ]^2\\frac{\\alpha^2}{2} + {{\\mathcal{O}}}(\\alpha^4)  = \\delta_{m''0} \\frac{\\alpha^2}{2} + {{\\mathcal{O}}}(\\alpha^4),\n\\end{align}\nsuch that $G_\\ell^{m''}(\\alpha)$ becomes\n\\begin{gather}\n\\label{eq:Gl0_approx}\nG_\\ell^{m''}(\\alpha) = \\delta_{m''0}\\frac{\\alpha^2}{4} (2\\ell+1) + {{\\mathcal{O}}}(\\alpha^4).\n\\end{gather}\nAs shown in Fig.~\\ref{fig:glmalpha}, the above approximation is very good for $\\alpha \\lesssim 15^\\circ$ and $\\ell \\le 2$. \nUp to order $\\alpha^2$, the active region induces a shift only in the frequency of the $M=0$ mode\n\\begin{gather}\n\\delta\\omega_M^{\\rm AR} = \\delta_{M0}\\, \\omega_{n\\ell}^{(0)} \\varepsilon_{n\\ell} \\frac{\\alpha^2}{4} (2\\ell+1) + {{\\mathcal{O}}}(\\alpha^4).\n\\end{gather}\nPerturbed frequency shifts and amplitudes are obtained by following the calculation described in Sect.~\\ref{sec:small_diff_rot}, but taking into account the fact that the eigenfrequencies $\\delta\\omega_M^{\\rm AR}$ are now degenerate for $M\\neq0$.\nThe frequencies in the observer's frame can then be approximated by\n\\begin{gather}\n\\omega_m^M = \\omega_{n\\ell}^{(0)} +\\overline{\\delta\\omega}_M +m\\Omega_\\beta ,\n\\end{gather}\nwhere \n\\begin{align}\n\\overline{\\delta\\omega}_M = & \\delta_{M0} \\, \\omega_{n\\ell}^{(0)} \\varepsilon_{n\\ell} \\frac{\\alpha^2}{4} (2\\ell+1) + (1-\\delta_{M0})\\eta P_2(\\cos\\beta) Q_{2\\ell M}\\, \\omega_{n\\ell}^{(0)}\n\\,.\n\\label{eq:small_ar_shiftsM0}\n\\end{align}\nThe peak amplitudes in the power spectrum, $P_m^M$, are given by Eq.~(\\ref{eq:observed_amplitude}), where $A_m^M$ is replaced by \n\\begin{gather}\nA_m^M = \\sum_{m'=-\\ell}^\\ell \\overline{A}_{m'}^M \\; r_{m'm}^{(\\ell)}(\\beta)\n\\label{eq:linear_ampl_shifts_smallAR}\n  \\end{gather}\nwith\n\\begin{align}\n\\overline{A}_{m'}^M = \\delta_{Mm'} + (\\delta_{M0} - \\delta_{m'0})\n\\frac{\\eta\\omega_{n\\ell}^{(0)}}\n{\\delta\\omega_0^{\\rm AR}} \\sum_{m''=-\\ell}^\\ell \nr_{m'm''}^{(\\ell)}(\\beta) r_{Mm''}^{(\\ell)}(\\beta) Q_{2\\ell m''}.\n\\end{align}\n\n Figure~\\ref{fig:alphaomega_contour} shows the frequency error in $\\mu$Hz introduced by the small-$\\alpha$ approximation.\nDark blue shades indicate the regions in parameter space ($\\alpha$ and $\\Omega_\\beta$) where the approximation is very good.\nFor an active region with a  surface coverage below $2\\%$, i.e. $\\alpha<15^\\circ$, the frequency shift $\\overline{\\delta\\omega}_{M=0}$ is within $\\sim 0.1\\; \\mu$Hz of ${\\delta\\omega}_{M=0}$, even for fast rotation rates. \n\n\n\n\n\\subsection{Correlations in frequency space}\n\\label{sec:theory_correlations}\nDue to the fact that  the active region perturbation is unsteady in the observer's frame, the intensity fluctuations (Eq.~\\ref{eq:Iobs_final}) are not statistically independent in frequency space. \nGiven two frequencies $\\omega$ and $\\omega'$, the intensity covariance  is \n\\begin{align}\n \\text{Cov} & \\left [ {I_\\text{obs}}(\\omega), \\, I_\\text{obs}(\\omega')\\right ] \\nonumber\\\\\n&\n= E\\left [ {I^*_\\text{obs}}(\\omega) I_\\text{obs}(\\omega')\\right ] \\nonumber \\\\ \n& =\\sum_{M,m} \\sum_{M',m'}  B_m^M  B_{m'}^{M'} L^{1\/2}_M(\\omega-m\\Omega_\\beta)  L^{1\/2}_{M'}(\\omega'-m'\\Omega_\\beta)\nE[{{\\mathcal{N}}}^*_M(\\omega- m\\Omega_\\beta) {{\\mathcal{N}}}_{M'}(\\omega'- m'\\Omega_\\beta)]\n\\nonumber\\\\\n& =\\sum_{M=-\\ell}^{\\ell} \\sum_{m=-\\ell}^{\\ell} \\sum_{m'=-\\ell}^{\\ell} B_m^M B_{m'}^M L_M(\\omega-m\\Omega_\\beta)\\delta_{m',m+(\\omega'-\\omega)\/\\Omega_\\beta}   \\,  .\n\\label{eq:I_correlation}\n\\end{align}\nTo simplify the analysis we assumed that $k=(\\omega'-\\omega)\/\\Omega_\\beta$ is an integer (this is not a weakness of the theory though). The above expression vanishes unless   $| k | \\le 2 \\ell $.  The quantities $I_\\text{obs} (\\omega)$ and $I_\\text{obs} (\\omega')$ are correlated for frequency separations $\\Delta\\omega = \\omega'-\\omega=(m'-m)\\Omega_\\beta$. \n\n\nThe power spectrum  is also correlated for  \n$\\Delta \\omega =(m-m')\\Omega_\\beta$. Using the formulae given in appendix C of \\citet{fournier2014}, we find\n\\begin{align}\n \\text{Cov}\\left [ P(\\omega),P(\\omega') \\right ] \n &= \n \\text{Cov}\\left [ I^*_\\text{obs}(\\omega)I_\\text{obs}(\\omega) , \\, I^*_\\text{obs}(\\omega')I_\\text{obs}(\\omega') \\right ]  \\nonumber \\\\\n & = E \\left [ I^*_\\text{obs}(\\omega)I_\\text{obs}(\\omega')  \\right ]  E \\left [ I_\\text{obs}(\\omega)I^*_\\text{obs}(\\omega')  \\right ]  \\nonumber \\\\\n&= \\left | \\text{Cov} \\left [ I_\\text{obs}(\\omega), \\, I_\\text{obs}(\\omega') \\right ] \\right |^2  ,\n\\label{eq:PS_correlation}\n\\end{align}\nwhere the intensity covariance is given above.\nWe note that the values of the intensity and power spectrum covariances  (Eqs.~\\ref{eq:I_correlation} and~\\ref{eq:PS_correlation}) depend on the parameters of the model. However the existence of a  correlation is a general feature, which arises from the fact that the active region rotates with the star.   \nRemarkably, the expectation value  of the power spectrum $\\mc{P}(\\omega)$ (see Eq.~\\ref{eq:powerspectrum})  is as if all the terms in Eq.~(\\ref{eq:Iobs_final}) were statistically independent.\n\n\n\n\n\\section{Discussion}\n\\subsection{Nonlinear frequency shifts and amplitudes from numerical simulations}\n\\label{cha:rotation:sec:simulation}\n\nIn order to compare with the perturbation theory, we study the nonlinear regime by means of numerical simulations.\nWe use the GLASS wave propagation code, with the same numerical setup employed by  \\citet{Papini2015}.\n\n\\subsubsection{Rotation in post-processing}\nRunning different numerical simulations for different values of $\\beta$ and different perturbation amplitudes is computationally expensive. \nInstead we performed simulations for a 3D polar perturbation to the sound speed and for a star with no rotation, that is equivalent to solve the numerical problem in the reference frame ${\\mathcal{\\widehat{R}}}_\\beta$.\nWe introduced the effect of rotation later in processing the output.\nThis approach has the advantage that, for a given amplitude of the AR perturbation, we only need to run one simulation in order to calculate the power spectrum for any given value of $\\beta$ and rotation period.\nHowever we can only reproduce solid body rotation, and it is not possible to include the effects of the centrifugal distortion and of the Coriolis force,\ntherefore in each $n\\ell$-multiplet we expect to find only $(2\\ell+1)(\\ell+1)$ peaks.\nNonetheless the results are useful for exploring the nonlinear regime.\nWe note that, as a consequence of neglecting these rotational effects, $M$ can be identified with the azimuthal degree of the spherical harmonics $Y_\\ell^M (\\widehat\\theta_\\beta,\\widehat\\phi_\\beta) $ in the frame ${\\mathcal{\\widehat{R}}}_\\beta$ (see Sect.~\\ref{sec:corot}).\n\\subsubsection{Sound-speed perturbation}\nAs was done earlier, we approximate the perturbation introduced by the starspot by a local increase in sound speed.\nWe consider separable perturbations in the square sound speed of the form\n\\begin{equation}\n\\label{eq:deltacs2spot}\n {\\Delta c^2(r,\\widehat\\theta_\\beta)} = \\mathrm{\\epsilon} \\, c_0^2(r)  \\, f(r) \\, g(\\widehat\\theta_\\beta), \n \\end{equation}\n where $\\epsilon>0$ is a positive amplitude, $f$ is a radial profile, and $g$ a latitudinal profile.\nExplicitly, $g = 1\/2 + \\cos(\\kappa \\theta)\/2$ is a raised cosine for $0\\le \\kappa\\theta<\\pi$ and is zero otherwise, where $\\pi\/(2\\kappa)= 0.65$~rad~$=37.5^\\circ$. The function $f=\\exp(-|r-r_c|^2\/2\\sigma^2) [1\/2 + \\cos(|r-r_c|\/\\sigma)\/2]$  is a Gaussian function centered at radius $r_c=0.9985~R_\\odot$ with dispersion $\\sigma=0.004~R_\\odot$, multiplied by a raised cosine. \nThis functional form is the same as the one described by \\citet{Papini2015}.\nThe perturbation is thus placed along the polar axis at a depth of $4$~Mm, with a surface coverage of $12\\%$.  \n\n\\subsection{Synthetic power spectrum}\n\n\\label{cha:rotation:sec:simulationpowerspectrum}\nAs in Sect.~\\ref{sec:linear_PS}, we assume that the intensity fluctuations are proportional to the Eulerian pressure perturbation measured at $r_0=R+200~\\mathrm{km}$ above the surface \\citep[see ][]{Papini2015}. \nTo calculate the expectation value of the observed power spectrum, we performed a first set of simulations for which all the modes were excited with the same phase at the initial time.\n\n\nThe approximation that the changes in the eigenfunction of a mode $M$ are limited to the same angular degree $\\ell$ (Eq. \\ref{eq:xi_pert}) does not hold in the nonlinear case: the horizontal shape of an eigenfunction is a combination of spherical harmonics of different $\\ell$ values. \nHowever, since the perturbation is axisymmetric (see Eq.~\\ref{eq:deltacs2spot}), \nthe eigenfunction for a mode $M$ is a combination of spherical harmonics  $Y_\\ell^M(\\widehat\\theta_\\beta,\\widehat\\phi_\\beta)$ of different angular degree $\\ell$ and same $M$.\nTherefore the intensity fluctuations  $I_{M}(\\widehat\\theta_\\beta,\\widehat\\phi_\\beta,t)$ due to all the modes with the same $M$ take the form\n\\begin{gather}\n\\label{eq:I_Mglass}\n I_{M}(\\widehat\\theta_\\beta,\\widehat\\phi_\\beta,t) \\propto \\sum_{\\ell=|M|}^{\\ell_\\text{max}}\\Re \\left \\{ \n {p}_{\\ell M}(r_0,t)  \n Y_\\ell^M(\\widehat\\theta_\\beta,\\widehat\\phi_\\beta) \\right \\},\n\\end{gather}\nwhere ${p}_{\\ell M}(r_0,t)$ are the coefficients of the spherical harmonic decomposition of the pressure wavefield $p(r_0,\\widehat\\theta_\\beta,\\widehat\\phi_\\beta,t)$  returned by the GLASS code  in the frame $\\widehat{{\\mathcal{R}}}_\\beta$. The index $\\ell_\\mathrm{max} $ is set by the spectral resolution of the spherical harmonic transform.\n\n\nTo obtain the full-disk integrated intensity in the observer's frame \nwe express each $Y_\\ell^M(\\widehat\\theta_\\beta,\\widehat\\phi_\\beta)$ in terms of the spherical harmonics in the frame ${{\\mathcal{R}}}$ \n\\begin{gather}\nY_\\ell^{M}(\\widehat\\theta_\\beta,\\widehat\\phi_\\beta) = \n\\sum_{m=-\\ell}^{\\ell} \nr_{mM}^{(\\ell)}(\\beta) e^{-\\mathrm{i} m\\Omega_\\beta t} \n\\sum_{m'=-\\ell}^{\\ell}\nY_\\ell^{m'}(\\theta,\\phi) \\; r_{m' m}^{(\\ell)}(-i) \n  .\n\\end{gather}\nby means of two consecutive rotations of the Euler angles $(0,-i,\\Omega_\\beta t)$ and $(0,\\beta,0)$. Combining this equation with  Eq.~(\\ref{eq:I_Mglass}) and integrating over the visible disk, we obtain the full-disk integrated intensity of each $mM$-component:\n\\begin{gather}\n\\label{eq:integrated_intensity_sim}\nI_{mM}(t)= \n\\sum_{\\ell=\\max\\{|m|,|M|\\}}^{\\ell_{\\max}} V_\\ell\\; r^{(\\ell)}_{m0}(i) \\, r^{(\\ell)}_{mM}(\\beta) \\, \n\\Re \\left \\{{p}_{\\ell M}(r_0,t)  e^{-im\\Omega_\\beta t} \\right \\}.\n\\end{gather}\nWe then perform a Fourier transform to calculate the intensity $I_{mM}(\\omega)$ in the frequency domain. Finally, we derive the expression for the expectation value of the  power spectrum\n\\begin{gather}\n\\label{eq:powerspectrum_sim}\n\\mc{P}(\\omega)=  \\sum_{m=-\\ell_{\\max}}^{\\ell_{\\max}}\\sum_{M=-\\ell_{\\max}}^{\\ell_{\\max}} \n\\left |I_{mM}(\\omega)\\right |^2,\n\\end{gather}\nthat is analogous to Eq. (\\ref{eq:powerspectrum}), but for the entire wavefield. \n\n\n\\begin{figure*}\n \\includegraphics[width=0.5\\columnwidth]{fig11a.jpg}\n \\includegraphics[width=0.5\\columnwidth]{fig11b.jpg}\n \\caption{Numerical simulations of oscillation power spectra for two quadrupole multiplets with $n=12$ (left panels) and $n=18$ (right panels), for an inclination angle $i=80^\\circ$ and a \nstellar  rotation period of $8~\\text{days}$ (solid body rotation).\nThe active region has a colatitude  $\\beta=80^\\circ$ and different perturbation amplitudes of $\\epsilon=0.1$ (top), $0.2$ (middle), and $0.3$ (bottom), with $r_c=0.9985~R_\\odot$, $\\sigma = 0.004~R_\\odot$, and $\\kappa=2.4$ (see Eq. \\ref{eq:deltacs2spot}).\n The black curves show the power spectra computed with  the GLASS code, the red dot-dashed curves indicate the power spectra computed using  linear perturbation theory. Vertical lines show the  peaks from linear theory, in red for $M=0$ and in black $M\\neq0$.\n The blue curves in the right panels display the contribution of the $M=0$ peaks to the simulated power spectrum.\n The $n=12$ peaks  have a FWHM of $\\Gamma\/2\\pi\\simeq0.2~\\mu{\\mbox{$\\mathrm{Hz}$}}$, while for $n=18$ the peaks have   $\\Gamma\/2\\pi\\simeq1~\\mu{\\mbox{$\\mathrm{Hz}$}}$.\n }\n \\label{fig:l2spectrum_sim}\n\\end{figure*}\n\n\nFor the nonlinear study we chose three perturbation amplitudes   $\\mathrm{\\epsilon}=0.1$, $0.2$, and $0.3$ which, for the multiplet $(\\ell,n)=(2,18)$, correspond to $\\epsilon_{n\\ell}\\simeq 0.005$, $0.010$, and $0.015$, {{i.e.}}, roughly twice to six times the value used in the linear analysis.\nThe simulations run for $80~\\text{days}$ (stellar time), in order to reach an accuracy of $\\sim 0.14~\\mu{\\mbox{$\\mathrm{Hz}$}}$ in the frequency domain.\nThe wavefield computed by GLASS includes some numerical damping that increases with frequency with an exponential dependence.\nWe took advantage of this damping and  selected two $\\ell=2$ multiplets: one with $n=18$ and a FWHM comparable to the solar value, the other with $n=12$ and a FWHM small enough to resolve all the $mM$ peaks in the multiplet.  \nModel~S is convectively stabilized \\citep{Papini2014}, which implies that the unperturbed frequencies of the quadrupole modes are $1970.50$~$\\mu$Hz for $n=12$ and $2783.62$~$\\mu$Hz for $n=18$.\nFigure \\ref{fig:l2spectrum_sim} shows the observed power spectra of the two selected multiplets in the case   $\\beta=80^\\circ$ and $i=80^\\circ$, normalized with respect to the highest $M\\neq0$ peak.\nIn the case $\\epsilon=0.1$, the simulated power spectrum (black curve) of the $(\\ell,n)=(2,12)$ multiplet (top left panel) is well reproduced by the power spectrum computed with linear theory (red dot-dashed curves), except for the peaks corresponding to $M=0$ (vertical red curves show the $M=0$ peaks from linear theory), which are less shifted in frequency and have smaller amplitudes than predicted. \nFor the $(\\ell,n)=(2,18)$ multiplet (top right panel) the nonlinear effects are less visible, due to the overlapping  Lorentzian profiles. A blue curve, displaying the contribution to the power spectrum of the $M=0$ peaks, shows that also for this multiplet the $M=0$ component of the power spectrum deviates from the linear behaviour, both in frequency and amplitude. \n This plot also shows an example of the combined action of mode mixing and mode visibility, which almost suppresses the $m=0$, $M=0$ peak located at a frequency shift of $\\sim 3.1~\\mu{\\mbox{$\\mathrm{Hz}$}}$.\n\n\nWith increasing $\\epsilon$ (middle and bottom panels) the interaction of the wavefield  with the active region becomes strongly nonlinear, and for a perturbation with $\\epsilon=0.3$ results in a massive distortion of the power spectrum with respect to that  one predicted by linear theory. \nHere two different behaviors are evident: in the $(\\ell,n)=(2,12)$ multiplet the $m=0, M=0$  peaks are almost suppressed, while  \nthe peak with $m=0, M=0$ in the $(\\ell,n)=(2,18)$ multiplet, that was suppressed in the case $\\epsilon=0.1$, increases in amplitude as $\\epsilon$ increases (middle and bottom right panels).\nMoreover the peaks with $M=1$ start to deviate from the linear prediction.\nThis is in agreement with what found in the non-rotating case by \\citet{Papini2015}, who also showed that second-order perturbations would correct most of the differences.\n\n\\subsection{Correlations in synthetic power spectra}\n\n\n\\begin{figure}\n  \\includegraphics[width=\\columnwidth]{fig12.jpg}\n \\caption{ Autocorrelation of the intensity spectrum as defined by  Eq. (\\ref{eq:autocorrelation_PS_I}) for a starspot with $\\epsilon=0.1$ (red curve).\n  A stellar rotation period $2\\pi\/\\Omega_\\beta = 8$~days, an inclination angle $i=80^\\circ$, and a starspot colatitude $\\beta=80^\\circ$ were chosen for the post processing. Vertical dotted lines denote frequency separations $\\Delta\\omega=j\\Omega_\\beta$ where $j=1,2,3, 4$.  For comparison, the black curve is the case with no starspot.\n  }\n  \\label{fig:correlation_glass}\n \\end{figure}\n \nIn Sect.~\\ref{sec:theory_correlations} we showed that in presence of an active region rotating with the star, the power spectrum of a multiplet is correlated at frequency separations that are multiple of the rotational frequency of the active region. However, for small perturbations this correlation is too weak to be observed. Here we ask whether such a correlation can be measured in presence of a perturbation of moderate amplitude. \nFor that purpose, we ran a second set of simulations, in which the acoustic waves were excited by applying a random forcing function at  $150~\\text{km}$ below the surface at each time step, as described by \\citet{Hanasoge2007}.\nThe duration of the simulation is 80 days (stellar time).\nIn the frequency domain, the observed intensity is \n\\begin{equation}\nI_\\mathrm{obs} (\\omega) = \\sum_{m=-\\ell_{\\max}}^{\\ell_{\\max}}\\sum_{M=-\\ell_{\\max}}^{\\ell_{\\max}} I_{mM}(\\omega) ,\n\\end{equation}\nwhere $I_{mM}(\\omega)$ is obtained by Fourier transformation of  Eq.~(\\ref{eq:integrated_intensity_sim}) using the numerical realizations of $p_{\\ell M}(r_0, t)$.\n\n\nThe autocorrelation of  the intensity spectrum is\n\\begin{gather}\n C_I(\\Delta\\omega) = \n \\frac{\\sum_{n=15}^{25}  \\int_{[\\omega_{n\\ell}]}  \n I_\\text{obs}^*(\\omega) I_\\text{obs}(\\omega+\\Delta\\omega)\\mathrm{d}{\\omega} }\n {\\sum_{n=15}^{25} \\int_{[\\omega_{n\\ell}]}  \n |I_\\text{obs}(\\omega)|^2 \\mathrm{d}{\\omega} } ,\n \\label{eq:autocorrelation_PS_I} \n\\end{gather}\nwhere $[\\omega_{n\\ell}]$ denotes an appropriate frequency interval of size $\\sim20~\\mu$Hz containing the multiplet $n\\ell$ but excluding the nearby $l=0$ mode. \nThe average is performed over all $\\ell=2$ multiplets with $n$ ranging from 15 to 25.\n\nFigure \\ref{fig:correlation_glass} shows the real part of $C_I$  for the special case of an inclination angle  $i=80^\\circ$ and a starspot at colatitude  $\\beta=80^\\circ$ with $\\epsilon=0.1$.\nA correlation is clearly visible at frequency separations $\\Delta\\omega=2\\Omega_\\beta$ and  $4\\Omega_\\beta$.\nThis suggests that the frequency-domain  autocorrelation function could be used as a diagnostic tool to identify \nunsteady perturbations in the time series of stellar oscillation. \nWe note that the imaginary part of $C_I$ contains no visible signal above the noise level.\n\n\n\n\n\n\\subsection{Towards a physical model for mode interaction with an active region}\n\\label{SectPhys}\n\n{ \nIn this paper we replaced the  active region  by a localized  increase in sound-speed near the stellar surface.\nWe focused on the geometrical aspects of the problem rather than on the physics.\nOne may ask, however,  what would be the difference in the obtained results if we had instead considered a realistic model for the magnetic active region.\nAlthough we will not answer this question here, it is worth listing some of the steps involved.\n\n\n\nA typical solar active region consists of a pair of sunspots surrounded by plage with strong vertical field. \nLocal helioseismology of the visible disk and the far side indicates that p modes are strongly scattered by both sunspots and extended plage \\citep[see, e.g.,][and references therein]{Gizon2009}.\nSome studies of the interaction of high-degree p~modes with magnetostatic sunspots  \\citep[e.g.,][]{Moradi2010,Cameron2011} have been carried out using MHD wave propagation codes \\citep{Cameron2008,Felipe2016}. \nOther studies are based on  numerical magneto-convective simulations \\citep[e.g.,][]{Rempel2009,DeGrave2014}. \nThe main conclusion of these simulations is that the interaction takes place in the top few hundred kilometres below the surface, where the direct effects of the magnetic field and the indirect effects due to changes in thermodynamic structure with respect to the reference atmosphere (e.g. the Wilson depression) are large.\nWave simulations indicate that the outgoing p modes are phase shifted with respect to the incoming p modes in such a way that the effective wave speed is increased, as observed. \nThe physical interaction involves the conversion of p modes  into fast and slow magnetoacoustic modes in the sunspot  \\citet[e.g.][]{Cameron2008,Khomenko2006}. A fraction of the incoming p-mode  energy is tunnelled downward in the form of slow MHD waves, leading to absorption  \\citep{Braun1995,ZhaoH2016}. See also, e.g.,  \\citet{Saio2004} and  \\citet{Cunha2006} for mode conversion calculations in roAp stars.\n\n\n\nUsing 2D ray tracing, \\citet{Liang2013} showed that high-degree helioseismic waveforms can be reproduced by increasing the effective wave speed by 10\\%  in the sunspot.\n This provides some justification for the values that we have used in the present paper, although the extension to low-degree p modes has not been studied.\nClearly, much additional work will be needed to determine the correct active-region perturbation amplitude from first principles. Until then, a simple calibration can be obtained from the observational study by \\citet{Santos2016} who estimated empirically the contribution of sunspots to the low-degree p-mode frequency shifts associated with the solar cycle. By  combining our Eq. (\\ref{eq:omat_mag_epsilon}) with Eq. (3) from \\citet{Santos2016}, we find  $\\varepsilon_{n\\ell}=-\\Delta\\delta_\\mathrm{ch}\/I_\\ell$, where $\\Delta\\delta_\\mathrm{ch}$ is the integral phase difference introduced in the mode eigenfunction by a sunspot and $I_\\ell$ is related to mode inertia. Using  the value $\\Delta\\delta_\\mathrm{ch}=-0.44$  estimated by \\citet{Santos2016}, we have  $\\varepsilon_{n\\ell} \\sim 0.05$ for a quadrupole mode, {{i.e.}}\\  a larger value than proposed by \\citet{Liang2013} and used in the present paper. This may suggest that for large active regions the spectra calculations may have to be carried out in the nonlinear regime. However, only realistic numerical modelling would help settle this question. The full problem would also have to include multiple scattering by collections of flux tubes in plage  \\citep[see, e.g.,][]{Hanson2015}. \n}\n\n\n\\section{Conclusions}\n\n\n\nIn this paper we investigated the changes in global acoustic oscillations caused by a localised sound-speed perturbation on a rotating star mimicking  a large active region, using both linear perturbation theory and 3D numerical simulations. In an inertial frame, the active region perturbation is unsteady.\n\nWe find that the power spectra of low-degree modes have a complex structure. The combined effects of the active region and differential rotation cause each $n\\ell$-multiplet to appear as $(2\\ell+1)^2$ peaks, each with a different amplitude. \nMost of the peaks are clustered near the classical rotationally-split frequencies, and only $2\\ell+1$ peaks (the $M=0$ peaks, which correspond to the axisymmetric mode in the reference frame of the AR) are shifted to higher frequencies. This leads to an apparent asymmetry in the line profiles. However, due to the finite lifetime of acoustic oscillations, most of the peaks cannot be resolved.\n{ For solar-type stars, the results are not very sensitive to the choice of latitudinal differential rotation profile.}\n\nThe structure of the power spectra  is sensitive to the latitudinal position of the active region and to the inclination angle $i$ of the stellar rotation axis. The latter plays a major role in determining the relative visibility of the individual peaks.\nWe find that the envelope of the power spectrum becomes more complex as the latitude of the active region decreases. \nIn practice, it would be very difficult to perform a fit of the $(2\\ell+1)^2$ peaks in a multiplet, due to peak blending and noise. However, by neglecting differential rotation it is possible to derive a simplified formula that approximates the power spectrum of a multiplet to a sum of only $2(2\\ell+1)$ Lorentzian profiles. For small-area  active regions, the formula further simplifies and directly links the frequencies of the peaks in the power spectrum  to the active region parameters. Such formula may find applications in the analysis of real asteroseismic observations.\n\n\nNumerical simulations were performed to explore the nonlinear regime of the perturbation. We find that the $M=0$ peaks deviate from the linear behavior for active-region perturbation amplitudes $\\varepsilon_{n\\ell} \\gtrsim 0.005$. Depending on  each particular case, the amplitude of these peaks is either reduced or enhanced compared to first-order linear theory, due to mixing  with modes with other values of $\\ell$ and $m$ \\citep{Papini2015}. \n\n\nWe found that there are correlations in the power spectrum at frequency separations that are multiples of the active-region rotation rate. In the linear regime the correlation signal is too weak to be observed. However the numerical simulations show that for active-region perturbations of moderate amplitude, such a correlation might be detectable, provided that  the frequency intervals are carefully selected  to increase the signal to noise ratio.\n\nWe note that perturbation theory can easily be extended to compute the effect of multiple active regions provided that latitudinal differential rotation is small. The  treatment of  several active regions rotating at different rotation rates  would require a different setup, since there is no frame in which these perturbations are steady. A numerical approach would be preferable in such a case.\n\n\n{ The work presented in this paper uses simplified physics, but  it should provide useful guidance to identify the seismic signature of a large active region in the power spectrum of stellar oscillations. \nGiven that the values of $\\epsilon_{n\\ell}$ are uncertain, we believe that it is worth searching  for low-degree multiplets consisting of $2(2\\ell+1)$ components in available asteroseismic observations.}\nIdeal targets are stars that are known to have high-quality oscillation power spectra \\citep[high SNR, narrow line profiles, clear rotational splitting, see e.g.][]{Nielsen2014} \nand  show evidence for long-lived starspots \\citep[e.g.][]{Nielsen2013, Nielsen2018}. \nThe catalogue of potential targets, currently limited to CoRoT and {\\it Kepler},  will increase fast with  TESS \\citep{TESS2015} and  PLATO \\citep{plato2014}.\n\n\n\n\\section*{Conflict of Interest Statement}\n\nThe authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.\n\n\\section*{Author Contributions}\n\nL.G. provided the basic theory and E.P. the numerical applications. Both authors contributed to the analysis of the results and to the writing of the article.\n\n\n\\section*{Acknowledgments}\n\nThis paper is a contribution to PLATO PSM activity `Seismic diagnostics of stellar activity'. We acknowledge financial support from the German Aerospace Center DLR and from the Max Planck Society.\nWe thank Shravan Hanasoge for making the GLASS code available. \nComputational resources were provided by the German Data Center for SDO.\nThis work appeared in part in the PhD thesis of \\citet{Papini-thesis}. \n\n\\bibliographystyle{frontiersinSCNS_ENG_HUMS} \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nModern $\\gamma$-ray~  telescopes operating at energies above the MeV window, provide\nevent-resolved observational data. Each event (after the reconstruction process)\nis typically described by a tuple (i.e. an ordered list of elements) storing sky \ncoordinates, arrival time, and energy. \nDetection of discrete sources (either point-like or extended) is\nperformed using various methods. Given the discrete topological nature of $\\gamma$-ray~\nimages, methods based on cluster search, like the {\\it  { Minimum} Spanning Three}\n(MST) \\citep{MSTI,MSTII} have successfully been used. One of the main advantages of topometric\nmethods, { compared to methods using the spatial binning},  is to minimize \nthe impact of the poor energy-dependent Point Spread Function (PSF), typical of\n$\\gamma$-ray~ telescopes, preserving the spatial information of each event. Moreover,\nthese methods are able to detect sources compounded by a small amount of events,\nbut they need to be fine tuned to take into account properly the background. The\nproblem of background rejection is the most penalizing feature of topometric\nmethods, for this reason in this paper, for the first time, we present a \nmethod based on the DBSCAN~ algorithm \\citep{Ester96DBScan}. \nThe DBSCAN~ is a topometric algorithm used to\ncluster spatial data that are affected by background noise. Compared to other\ntopometric methods, it has the advantage to embed inside the algorithm itself the\ndiscrimination between signal (cluster) and background (noise), according to the\nlocal density of events within a typical scanning brush { i.e. within \na given scanning area}.\n\n{ The aim of the present paper is to show the potentialities of the method,\nand  its statistical characterization when applied to astrophysical $\\gamma$-ray~ data.} \nWe apply this method to the detection of point-like sources in the {\\it Fermi}-LAT~ data. \nWe explore a large volume of the $\\gamma$-ray DBSCAN~ parameter space, by means  of simulated \ndata, and we provide a statistical characterization the $\\gamma$-ray DBSCAN, finding signatures \nthat differentiate  purely random fields, from fields with sources. We define a \nsignificance level for the detected clusters, and we successfully test this significance \nwith our simulated data. We apply the method to real {\\it Fermi}-LAT~ $\\gamma$-ray~ data, and we \nfind an excellent agreement with the results  obtained with simulated data.\n\n\nIn a companion paper \\citep{dbscan_inprep}, we will apply the method\nto the\t {\\it Fermi}-LAT~ sky, {investigating specific issues related to the {\\it Fermi}-LAT~ response \nfunctions}, showing the potentiality for the discovery of new sources,\nin particular of small clusters located at high galactic latitude, or   \nclusters on the galactic plane, affected by a strong background.\n\nThe paper is organized as follows. In Sec. \\ref{sec:dbscan} we describe the logic of\nthe DBSCAN~ method, and we present the algorithm implemented to analyse $\\gamma$-ray~\ndata, the $\\gamma$-ray DBSCAN.\nIn Sec. \\ref{sec:caveat} we discuss some caveats regarding the\napplication of the $\\gamma$-ray DBSCAN~ algorithm to $\\gamma$-ray~ data. \nIn Sec. \\ref{sec:stats_sim} we study the statistical properties of the\n$\\gamma$-ray DBSCAN~ detection, using a simulated test field with only noise,\nand five simulated test fields with noise plus point-like sources.\nIn Sec. \\ref{sec:det_sim}  we evaluate the detection  performance of the method \nin terms of positional accuracy, cluster reconstruction,  and rejection of spurious clusters.\nIn Sec. \\ref{sec:signif_sim} we investigate the significance of the clusters,\nand describe our algorithmic implementation.\nIn Section \\ref{sec:realdata} we finally use our method with real {\\it Fermi}-LAT~ data,\ninvestigating the detection performance, and comparing the $\\gamma$-ray DBSCAN~ clusters \nsignificance, to that returned by the Maximum Likelihood method with\nstandard {\\it Fermi}-LAT~ software \\footnote{http:\/\/fermi.gsfc.nasa.gov\/ssc\/data\/analysis\/scitools\/overview.html}.\nIn Section \\ref{sec:conclusions}, we present our conclusions, and we discuss\nfuture developments and applications.\n\n\n\n\\begin{figure*}\n\\centering\n\\begin{tabular}{l} \n\t\\includegraphics[width=18cm]{sim_detection_plot.pdf}\\\\\n\\end{tabular}\n\\caption{ Photon map for the {\\it sky} test field 1, \nwith the result of the $\\gamma$-ray DBSCAN~ detection for $K = 5$ and $\\varepsilon=0.17$ deg.\nThe blue crosses refer to the simulated sources, the green boxes  \nto  51 detected {\\it true} clusters, and the red boxes to the 2 {\\it fake} ones. \nThe black dots represent the background events, the remaining colors indicate\ncluster events.}\n\\label{fig:sim_detection}\n\\end{figure*}\n\n\n\\section{The $\\gamma$-ray~ DBSCAN~ algorithm}\n\\label{sec:dbscan} \nThe DBSCAN~ \\citep{Ester96DBScan} is a topometric algorithm used  to cluster\nspatial data that are affected by background noise.  Some modifications have\nbeen developed to adapt the original DBSCAN~ algorithm to our study. Our\nalgorithm is mainly built upon the following criteria: \n\\begin{enumerate}\n\\item{} given a list of photons $D$, where each element $p_i$ is  a  tuple storing  \npositional sky coordinates, let $\\rho(p_k,p_l)$ be the angular distance between \ntwo photons $p_k$ and $p_l$.\n\\item{} We iterate over the full photon list $D$. A seed cluster $C_m^*$ is built \nwhen a minimum number of photons $K+1$ is enclosed within a circle of radius $\\varepsilon$~\ncentered on $p_i$ \n\\item{} For each photon  $p_l \\in C^*_m$,  we build the photon list $C_m^+$ \nby collecting all the photons $p_k$ respecting the condition:  \n$\\rho(p_l,p_k)<\\varepsilon$, and $p_k \\notin C^*_m$. \n\\item For each photon  $p_j \\in C_m^+$, if the number of photons enclosed within a circle of  \nradius $\\varepsilon$~ centered on $p_j$ is $\\leq K$ and $p_j \\notin C^*_m$, then $p_j$ will be attached to the \n{ final photon list of the cluster   without a recursive search for further neighbours}, \nthese points are defined  {\\it density-reachable}.\n\\item For each photon  $p_j \\in C_m^+$, if the number of photons enclosed within a circle of \nradius $\\varepsilon$~ centered on $p_j$ is $>K$ and $p_j \\notin C^*_m$, $p_j$ is attached to the $C_m^*$, and  \nand step { 3} is repeated recursively.\n\\item{} { When both conditions at step 4 and 5 are false, the cluster $C_m$ is built by joining\nthe {\\it density-reachable} events to those in the $C^*_m$ and in the  $C_m^+$  lists.}\n{ \\item{} The process starts again from step 1 searching for new clusters, skipping all the\nevents already flagged as {\\it noise} or {\\it clusters}, until all the\nevents in $D$ are flagged as {\\it cluster}, or {\\it noise}, or\n{\\it density-reachable}, events. }\n\\item{} At the end of the process the full photon list will be partitioned as \nfollows: \n\\begin{eqnarray}\nD &=& D_{cls} \\cup D_{noise}= (p_i \\in \\cup_m C_m) \\cup (p_i \\notin  \\cup_m C_m) \\\\\n\\emptyset&=&D_{cls} \\cap D_{noise} \\nonumber\n\\end{eqnarray}\n\\end{enumerate}\nIn this way high densely populated areas are classified as clusters (sources),\nconversely  low densely populated areas  are classified as noise (background). \nThe recursive call  of step 3, is not implemented in the original\nDBSCAN~ algorithm, and { represents} a novelty. This new feature, allows to reconstruct\nclusters with a size significantly larger than the $\\varepsilon$~ radius, making rare the  \npossibility to fragment a single clusters in small satellite clusters. Moreover, allows\nthe possibility to reconstruct extended { structures}, in particular extended sources, or\nfilamentary structures in the background.\n\n\n\n\n\n\nAfter the clustering process, each photon in $D$ will be described by a tuple, storing:\n{ \nthe photon position (both in galactic  and celestial  coordinates),\nthe photon class type ({\\it noise} or {\\it cluster}),\nand the ID of the cluster the photon belongs to. \nEach cluster $C_m$,  will be described by a tuple storing \nthe position of the centroid  with his positional error,   the ellipse  \nof the cluster containment, the cluster effective radius ($r_eff$),\nand number of photons in the cluster ($N_p$).\nThe ellipse of the cluster containment, is defined by major and \nminor semi-axis ($\\sigma_x$  and $\\sigma_y$, respectively), and the inclination angle\n($\\sigma_{alpha}$) of the major semi-axis w.r.t. the latitudinal coordinate. ($b$ or $DEC$). \nTo evaluate the ellipse axis we use the Principal Component Analysis method (PCA) \\citep{Jolliffe1986}. This method uses the eigenvalue decomposition of the \ncovariance matrix of the  the two position arrays $\\mathbf x$, and $\\mathbf y$. \nBy definition, the square root of the first eigenvalue will correspond \nto $\\sigma_x$, and the second to $\\sigma_y$.\nThe axes represent the two orthogonal directions of maximum variance of the cluster.\nThe effective radius is defined as $r_{eff}=\\sqrt{\\sigma_x^2 + \\sigma_y^2}$.\nTo find the centroid of the cluster and its uncertainty, we use a weighted average of the \nposition of each photon in $C_m$, as follows:\n\\begin{itemize}\n\\item{} we define the first order centroid ($C_{ave}$) as the average of the \nposition of each cluster photon: $C_{ave}=(<\\mathbf x>,<\\mathbf y>)$.\n\\item{} We define the weight array, according to the distance between\n$p_k \\in C_m$ and $C_{ave}$: $w_k=1\/\\rho(p_k,C_{ave})$.\n\\item {} The cluster centroid $C_{ctr}$ will result from average of the position  of\neach cluster point weighted by $w_k$.\n\\item {} The centroid position uncertainty ($pos_{err}$) is determined by propagating the  \nerror on the weighted average of $C_{ctr}$.  \nWe have numerically verified that  $pos_{err}$ corresponds to a $\\approx 95\\%$ \npositional uncertainty. \n\n\\end{itemize}\n\n\n\\begin{figure}[!h]\n\\includegraphics[width=9cm]{sim_det_zooming.pdf}\n\\caption{\nA close-up of two {\\it true} clusters reported in Fig. \\ref{fig:sim_detection}. \nThe  ellipses correspond to the ellipse of  the cluster containment. \nThe purple and orange points represent the cluster points, the black dots represent the background\nevents, the blue crosses the position of the simulated sources, and green boxes the position of \nthe clusters centroid\n}\n\\label{fig:closeup}\n\\end{figure}\n\n\n\n\n\n\\section{Caveat on the application to $\\gamma$-ray~ data }\n\\label{sec:caveat}\nThe application of clustering methods, such as the $\\gamma$-ray DBSCAN, leads to deal\nwith practical difficulties, related mostly to the instrument PSF, and\nto gradient and\/or structures in the background. In order to deal with these \nissues, without biasing the detection results, it's recommended to apply some\ncriteria that we discuss in the following.\n\n \n As first, we comment on the PSF impact. The PSF imposes a limit on the capability\nof an instrument to resolve sources separated by a distance smaller then the PSF\nsize. Sources with sizes below the PSF { } classified as point-like, { otherwise} are classified as extended. A further complication is that the PSF\noften depends on the energy; in the case of {\\it Fermi}-LAT,  the $68\\%$ containment angle of the \nreconstructed incoming photon direction, for normal incidence photons, has a typical \nsize of a couple of degrees at 100 MeV \\citep{LATCalib}, and scales down to few tenths of \ndegree above the GeV energies \n\\footnote{http:\/\/www.slac.stanford.edu\/exp\/glast\/groups\/canda\/\\\\\tlat\\_Performance.html}. \nThe size of the PSF is strongly connected to the size $\\varepsilon$~ of the \n$\\gamma$-ray DBSCAN~ scanning brush. Indeed, if $\\varepsilon$~ is much smaller than the PSF size, it \nmight occur the risk to loose clusters characterized by small $N_p$, or to fragment a \ncluster with large $N_p$ in smaller fake {\\it satellite} clusters.  \nWe stress that the formation of {\\it satellite} clusters is a very rare event, \nthanks to our recursive  DBSCAN implementation, that is explained in Sec. \\ref{sec:dbscan}. \nOn the contrary, if $\\varepsilon$~ is much larger w.r.t the PSF, it is likely to build extended \n{ clusters} contaminated by the background, or by  close sources. \\\\\n{\nA careful and self-consistent analysis of the effects of the energy dependence\nof the PSF, and in general of  issues related to the {\\it Fermi}-LAT~ response function, \nis beyond the scope of this paper, where we focus mostly on a statistical \ncharacterization  of the method.\nThese subjects will be investigated in the companion paper \\citep{dbscan_inprep}.\n}\n\nA second relevant issue, is the inhomogeneity of the background, that affects\nboth the choice of $\\varepsilon$~ and $K$. If the background is homogeneous over the\nentire field, the optimal choice of a single pair of values of $\\varepsilon$~ and $K$,\nguarantees a safe rejection of the background. Indeed, values of $\\varepsilon$~ and $K$,\nsuch that the average density of photons within $\\varepsilon$~ is significantly larger\nthe the average density of the background photons, make rare the chance to grow\na cluster from a background fluctuation. Unfortunately, the $\\gamma$-ray~ sky shows\nstrong gradients of background, in particular at low galactic latitudes. To\nsolve this issue, one could think to adapt the value of $\\varepsilon$~ and $K$ according\nto a local value of the background photon density. Since $\\varepsilon$~ has a strong\nconstraint imposed by the PSF, one should tune mostly the value of $K$. The\ndrawback is that as we increase the value of $K$ to compensate for the\nbackground, we decrease the capability to detect cluster with small $N_p$. To\n{ overcome} this difficulty, we adopt an alternative solution. We use a unique pair\nof values of $\\varepsilon$~ and $K$, for each field, where $\\varepsilon$~ is mostly constrained by\nthe PSF, and $K$, by the field average background, ad we take into account the\nbackground inhomogeneities by defining a significance level of the cluster,\naccording to the signal to noise ratio \\citep{LiMa1983}, evaluated  from the\nlocal background. This is explained in detail in Section \\ref{sec:signif_sim}. The\ncapability to reject clusters according to a low significance level, allows to\nrelax the constrain on  $\\varepsilon$~ and $K$, increasing the number of clusters \ndetected, hence increasing the detection ratio, and at the same time allows to\nreject spurious sources, due to the significance threshold. \nAnyhow, to avoid that the background is { so high}, that the fluctuations \nin the background events, can lead to densities comparable to those\nof weak sources, it's recommended to apply a cut in energy, to make\nthis possibility rare. \nIn order   to optimize the ratio between background and clusters events, in the \nfollowing we use a threshold energy of 3 GeV, that mitigates the possible bias\ndue to the background fluctuations.\n\n\n\n\n\\begin{figure*}\n\\centering\n\\begin{tabular}{ll} \n\\includegraphics[width=9cm]{r68_rand.pdf}&\n\\includegraphics[width=9cm]{r68_campo1.pdf}\\\\\n\\end{tabular}\n\\caption{{\\it Panel a:} distribution of the values of $\\log_{10} r_{eff}$ for \nthe {\\it random} field case, for  the full parameter space (black line) and fit \nby means  of Gaussian distribution (blue line). {\\it Panel b:} \nthe same as in the top panel, for the case of $K=3$ and $\\varepsilon=0.3$ deg. \n{\\it Panel c:}  distribution of $\\log_{10} r_{eff} $ for the case of\nthe {\\it sky} test field 1, for {\\it fake} clusters (red solid line), \nand {\\it true} clusters (blue solid line, hatched histogram). \n{ the  dashed lines represent a Gaussian best fit.}\n}\n\\label{fig:r_68}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\centering\n\\begin{tabular}{ll} \n\\includegraphics[width=9cm]{r68_vs_Eps_stat_rand.pdf}&\n\\includegraphics[width=9cm]{r68_vs_Eps_stat.pdf}\\\\\n\\end{tabular}\n\\caption{\n{\\it: Left panel:} the $r_{eff}\/\\varepsilon$ statistical distribution as a function of $\\varepsilon$, \nfor the {\\it random} field case. The blue solid circles represent the median, and the grey shaded \narea represents  the 1-$\\sigma$ confidence level region,  for each value of $\\varepsilon$. \n{\\it Right panel:} the same as in the bottom left panel, for the case of the {\\it sky} test\nfield 1. The red solid circles represent the median of the {\\it true} clusters case,  and\nthe grey area the 1-$\\sigma$ confidence level region. The  dashed line shows the \n1-$\\sigma$ confidence level region, for the case of the {\\it fake} clusters.\n}\n\\label{fig:r_68_vs_Eps}\n\\end{figure*}\n\n\t\\begin{figure*}\n\\centering\n\\begin{tabular}{ll} \n\\includegraphics[width=9cm]{Np_poiss_vs_lognorm.pdf}&\n\\includegraphics[width=9cm]{Np.pdf}\\\\\n\\end{tabular}\n\\caption{\n{\\it Left panels:} the distribution of $N_p$ for the {\\it random} test field,\nfor the case of $K=2$, $\\varepsilon=0.20$ deg (panel a, red solid boxes). The empty blue bars\nline represent a Poissonian best fit. The panel b shows the case of\n $K=2$ $\\varepsilon=0.30$ deg (purple solid triangles). \n The panel c shows the case for the full $K$-$\\varepsilon$~ parameter space,\n the solid black line represent a  log-normal best fit.\n{\\it Right panels:} panel c shows the distribution of $r_{eff}^2$ (black solid line),\nand it's best fit by means of a log-normal distribution (red dashed line).\nPanel  d shows the $N_p$ distribution for the {\\it fake} clusters in the {\\it sky} test { field} 1 (red \nsolid circles),  and the blue empty bars a Poissonian best fit. Panel  e shows the $N_p$ distribution \nfor the {\\it fake} clusters in the {\\it sky} test { field} 1   (blue hatched histogram),  the log-normal best\nfit (red dashed line), and the Poissonian fit (solid black line.)\n}\n\\label{fig:Np}\n\\end{figure*}\n\n\\section{Statistical properties of the $\\gamma$-ray DBSCAN~ clusters}\n\\label{sec:stats_sim}\n\\subsection{The test fields}\nIn this section we study the statistical properties of the clusters,\nlooking for signatures that characterize random Poissonian fields,\nand fields with point-like sources. To accomplish this task we\ncompare results obtained for a test field with only noise ({\\it random} test\nfield), and the five test  fields with noise plus point-like sources ({\\it sky} test\nfields 1-5).\n\nAs {\\it sky } test fields we use the same fields used in the\n\\cite{MSTII}. Each of these five {\\it sky} fields covers a broad sky region, with\na galactic longitude extension of $80^\\circ<l<170^\\circ$, and a galactic latitude\nextension of $40^\\circ<b<65^\\circ$. The $\\gamma$-ray~ background has been simulated\nusing the standard {\\ttfamily gtobssim}\n\\footnote{http:\/\/fermi.gsfc.nasa.gov\/ssc\/data\/analysis\/scitools\/help\/gtobssim.\ntxt} tool, developed by the {\\it Fermi}-LAT~ collaboration, simulating both the Galactic\nand isotropic components for a 2-year long period, using a threshold energy of 3\nGeV, for a total amount of 9322 photons.  To this photon list  { we added} 70\nsimulated sources: for each source, the number of photons was chosen from a\nprobability distribution given by a power-law, with exponent 2, from a minimum\nvalue of 4 up to 40 photons, joined to a constant tail up to 240 photons. \n{ The number of the sources is similar to that reported in  the Fermi-LAT \nSecond Source Catalog  \\citep[2FGL hereafter]{2FGL}, in the same region of  the sky.} \nThe source events are spatially distributed with a bivariate Gaussian \nprobability density function (PDF) with $\\sigma^{sim}_x=\\sigma^{sim}_y = 0.2$ \ndeg., centered at the source location. \nFive simulated test fields have been generated, adding the simulated sources to\nthe diffuse background. The only difference in the five realizations is the\nsource location,  randomly chosen to have different brightness contrast between\nsources and the background.\nThe {\\it random} test field covers the same area of the {\\it sky} test fields,\nand a number of events equal to  the {\\it sky} test field-1 (background and\nsources), for a total amount of   11044 events.\n\nIn Fig. \\ref{fig:sim_detection} we show  the photon map\nfor the {\\it sky} test field 1, and the result of the $\\gamma$-ray DBSCAN~ detection\nfor $K=5$ and $\\varepsilon=0.17$ deg. \nWe detect 51 {\\it true} clusters, and only 2 {\\it fake} ones. \nA cluster is defined {\\it true}, { if the position of the simulated source  falls within \na circle centered  on the cluster centroid, with a radius equal to $2pos_{err}$. }\nWe call {\\it fake}, the remaining clusters. \nIn Fig. \\ref{fig:closeup}, we show a close-up of two\n{\\it true} clusters. The black ellipses correspond to the ellipses\nof the positional error, and the purple  and orange thick points represent the cluster\npoints, while the black thick dots represent the background.\n\n\\begin{figure*}\n\\centering\n\\begin{tabular}{ll} \n\\includegraphics[width=9cm]{Np_vs_K_stat_rand.pdf}&\n\\includegraphics[width=9cm]{Np_vs_K_stat.pdf}\\\\\n\\end{tabular}\n\\caption{\n{\\it Panel b:}  the $N_p$ statistical distribution as a function of $K$, for the {\\it  \nrandom}  field case. The blue solid circles represent the median, and the grey shaded area represents  \nthe 1-$\\sigma$ confidence level region around the median, for each value of $K$. The dashed black line represents the  \n$N_p=K+1$ law.\n{\\it Panel e:} same as in  panel b, for the {\\it sky} test field 1 case.\n{\\it Panel a:} number of detected clusters for the {\\it random} test field case (blue solid   \npoints), as a function of $K$, and best fit by means of a Poissonian survival  function  \n(red empty bars).\n{\\it Panel c:} number of detected cluster for the {\\it sky} test field 1 case (black solid  \npoints),  for the case of  {\\it fake} clusters, as a function of $K$, and best fit by means of a  \nPoissonian survival function (red empty bars).\n{\\it Panel d:} number of detected cluster for the {\\it sky} test field 1 case (red solid  \nboxes),  for the case of  {\\it true} clusters, as a function of $K$, and best fit by means of a  \nPoissonian survival  function (black empty boxes).\n}\n\\label{fig:Np_vs_K}\n\\end{figure*}\n\n\\subsection{Test strategy}\n\n\nWe want to investigate the statistical properties of the $\\gamma$-ray DBSCAN~\nclusters, in particular signatures that distinguish purely random fields\nfrom fields with point-like sources,  and their dependence on $K$ and $\\varepsilon$. \nTo investigate systematically a broad volume of the parameter space, \nwe use a parametric approach. We set the range of $\\varepsilon$~ in [0.1$\\div$0.50] deg. with a \nstep of 0.01 deg., and the range of $K$ in [2$\\div$15],  with a step of 1. The\ntotal amount of detection trials for each test field is 574. \nWe collect the statistics of the trials, and we investigate the distribution\nof  $r_{eff}$ and $N_p$, and their connection with $\\varepsilon$~ and $K$, respectively.\n\n\n\n\n\\subsection{Statistics of $r_{eff}$ and connection with $\\varepsilon$}\nWe start by investigating the distribution of the $\\log(r_{eff})$ values, in the\ncase of the  {\\it random}, and of the {\\it sky} test field-1. The distribution for the\ndetections collected over the full $K$-$\\varepsilon$~  parameter space (top left panel of\nFig. \\ref{fig:r_68}),  shows a symmetric shape \nwell fitted by a Gaussian distribution (log-normal w.r.t. $r_{eff}$), with\nthe mean value of $<\\log_{10}(r_{eff})>\\simeq -0.45$ (corresponding to\n$<r_{eff}>\\simeq 0.3$ deg) and a dispersion of $\\sigma_{\\log_{10}( r_{eff})}\\simeq$ 0.23.\n{\nThe log-normal distribution  provides a reasonable\ndescription of the empirical distributions also for individual pairs of ($K$,$\\varepsilon$) values.\nAn example is\ngiven in panel c of  Fig. \\ref{fig:r_68}, for the case $K=3$, $\\varepsilon=0.3$ deg.,\nwhere the best fit values are $<\\log_{10}(r_{eff})>\\simeq -0.51$, and $\\sigma_{\\log_{10}( r_{eff})}\\simeq$ 0.16.\nWe now investigate the empirical distribution of  $\\log_{10}(r_{eff})$ for \nfields with point-like sources. In the right panel of Fig. \\ref{fig:r_68},\nwe show the case of  the {\\it  sky}  test field-1. The distributions of $\\log_{10}(r_{eff})$ are \nstill described by a by a normal. In the case of {\\it fake} clusters (red dashed line), the \nbest fit values of the mean ($<\\log_{10}(r_{eff})>\\simeq -0.46$) and of the\ndispersion ($\\sigma_{\\log_{10} r_{eff}}\\simeq$ 0.24), are very similar to those found  in the\ncase of the {\\it random} test field. On the contrary, the {\\it true} cluster distribution (blue hatched histogram) \nis peaking around the value of $\\log_{10}(r_{eff})\\simeq-  0.67$ deg, corresponding \nto $r_{eff}\\simeq 0.21$. deg., very close to  the value of the \ndispersion $\\sigma^{sim}=0.20$ deg., used to simulate the sources. }\nSince the simulation parameter $\\sigma^{sim}$ reproduces the\neffect of the instrumental PSF, we observe that for non-{\\it random} fields, the\ntypical size of the reconstructed clusters is constrained by the PSF, suggesting\nthe empirical rule to set the value of $\\varepsilon$~ of the order of the PSF size.\n\n\nTo investigate more accurately the connection between $\\varepsilon$~ and the PSF, we analyse \nthe statistical properties of the quantity  $r_{eff}\/\\varepsilon$ as a function of $\\varepsilon$. \nFor each value of $\\varepsilon$, we determine the median, and the  two-sided 1-$\\sigma$ \nconfidence level (CL) interval around the median,  of the $r_{eff}\/\\varepsilon$ distributions.\nIn the left  panel of Fig. \\ref{fig:r_68_vs_Eps} we plot the $r_{eff}\/\\varepsilon$ \nmedian   (blue solid circles) and 1-$\\sigma$  CL region, as  a function of $\\varepsilon$,\nfor the {\\it random} field.  \nWe note that  the $r_{eff}\/\\varepsilon$ trend is slightly increasing with $\\varepsilon$, \nand that the 1-$\\sigma$ CL region is consistent with the case $r_{eff}\/\\varepsilon=1$,\nbut  the upper boundary shows a systematic increase, compared to the lower\nboundary, for $\\varepsilon$$\\gtrsim 0.30$ deg.\n{ The trend for the case of the {\\it true} clusters in {\\it sky} test field 1 \n(right panel Fig.\\ref{fig:r_68_vs_Eps}), shows a different behaviour. \nThe median of $r_{eff}\/\\varepsilon$ (red solid circles) is slightly decreasing with\n$\\varepsilon$, showing that, for {\\it true } clusters, $r_{eff}$ is not sensitive to the \nsize of $\\varepsilon$, being mostly constrained by the simulated PSF size. As expected,  for the case of \n{\\it fake} clusters (blue dashed line), the trend is almost identical to that \nof the clusters in the {\\it random} field.\n}\n \n\n \n\n\n\n\n\n\n\n\n\n\\subsection{Statistics of $N_p$ and connection with $K$}\nWe now investigate the statistics of the distribution of the number of \nphotons per cluster. In the case of {\\it random} fields, we expect  that\nthe number of photons in a cluster attends a Poisson distribution. Indeed,\nfor a generic two-dimensional Poisson process,  the probability to observe\na number of events ($N(S)=j$) enclosed by a surface $S$  is given by:\n\\begin{equation}\nP(N(S)=j)=\\frac{(\\lambda |S|)^j \\exp{(-\\lambda |S|)}}{j!},\n\\label{eq:Poisson}\n\\end{equation}\nwhere $\\lambda$ is the average spatial density.\nTranslating $S$ in terms of $\\varepsilon^2$, we can rewrite:\n\\begin{equation}\nP(N(\\varepsilon^2)=j)=\\frac{(\\lambda |\\varepsilon^2|)^j \\exp{(-\\lambda |\\varepsilon^2|)}}{j!},\n\\label{eq:Poisson_Eps}\n\\end{equation}\nfrom which follows that, given the value of $K$ and $\\varepsilon$, the probability\nto find a cluster  as function of $K$ and $\\varepsilon$~ will be given by\n\\begin{equation}\nP_{clus}(\\varepsilon,K)=P(N(\\varepsilon^2)>K)=1- \\sum\\limits_{j=0}^{K} \\frac{(\\lambda |\\varepsilon^2|)^j\\exp{(-\\lambda |\\varepsilon^2|)}}{j!},\n\\label{eq:P_clus}\n\\end{equation}\nnamely the Poissonian  survival function. \nAnyhow, due to  the logic of the  DBSCAN~ \nclustering process, the Poisson statistics can't be extended  from $\\varepsilon$~ \nto $r_{eff}$, for any value of $\\varepsilon$.\nIndeed, a cluster  is not a simple collection of points enclosed\nwithin a surface $S$, this holds only within the $\\varepsilon$-sized circle, namely the\n{\\it seed} of the cluster ($C^*$). If we consider the annulus defined  between $\\varepsilon$~ \nand the cluster radius  $r_{clus}$, not all the points in the annulus will be cluster\nmember, but only those that are at least density reachable. \nThis implies that we expect a deviation from the Poisson \nstatistics, when $r_{eff}$  is significantly larger than $\\varepsilon$, i.e. $\\varepsilon$ $\\gtrsim$ 0.3 deg. \n(according to the analysis presented in the previous section).  This expected deviation from the \nPoissonian statistics,  is confirmed by the plots in the left panels of Fig. \\ref{fig:Np}.\nIn panel a we show the distribution of $N_p$ for the\ncase $K=2$ and $\\varepsilon=0.20$ deg. We note that the Poisson distribution (Eq. \\ref{eq:Poisson_Eps}) \ngives a reasonable  description of the empirical distribution. On the contrary, for the case of\n$\\varepsilon=0.30$ deg. (panel b), we observe that the Poisson distribution shows larger deviations,\nin particular for $K>6$.  When we take into account the $N_p$ distribution for the full \nparameter space (panel c), we note the Possonian distribution is failing in  providing\na reasonable description of the empirical distribution, whilst a log-normal one gives a \ngood fit.\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=18cm]{param_run.png}\n\\caption{\nIsolevel maps for $D_{fake}$ (panel a),  $D_{true}$ (panel b), $D_{eff}$ (panel c), and $Q$ \n(panel d), for the {\\it sky} test field 1. The white lines show the\n$\\mathrm{isolevel}=0$, the black lines show the $\\mathrm{isolevel}=0.68$, and the blue lines \nshow the $\\mathrm{isolevel}=0.95$.\n}\n\\label{fig:param_run}\n\\end{figure*}\n\n\n\nThe log-normal trend of $N_p$ is consistent with the log-normal trend of the  \ndistribution of $r_{eff}$.  Since the number of photons in a cluster will be \napproximatively $N_p \\propto \\lambda r_{eff}^2$,\nwe can write the PDF of $N_p$:\n\n\\begin{equation}\nf(N_p)\\propto f(r_{eff}^2)\\lambda.\n\\label{eq:f_N_p}\n\\end{equation}\n\nTo evaluate the distribution of $ r_{eff}^2$ we can use the\nstandard theory of the transformation of Random Variables (RV) \\citep{papoulis}.\nIt can be easily proved that, given a RV $X$ having a log-normal distribution,\n\n\\begin{equation}\nf(X)=\\frac{1}{X\\sqrt{2\\pi\\sigma^2}}\\exp\\Big({\\frac{-\\ln(X)-\\mu}{2\\sigma^2}}\\Big), \n\\end{equation}\nthe RV $Y=X^2$, will follow a log-normal distribution given by:\n\n\\begin{equation}\nf(Y)=\\frac{1}{2Y\\sqrt{2\\pi\\sigma^2}}\\exp\\Big({\\frac{-\\ln(Y)-2\\mu}{4\\sigma^2}}\\Big). \n\\end{equation}\t\t\t\n\nIndeed, our $r_{eff}^2$ distribution, for the {\\it random} field (panel d, \nFig. \\ref{fig:Np}), is fitted by a log-normal distribution peaking at $\\simeq 0.03$ deg$^2$.\nHence, according to Eq. \\ref{eq:f_N_p} we expect that also $f(N_{p})$ will follow a log-normal\ndistribution, when $N_p$ is not ruled by a Poissonian statistics.\n\nWe verify, that the same statistical trends, describe the {\\it real} sky fields. \nThe  panels e and f in Fig. \\ref{fig:Np}, show \nthe statistical distribution of $N_p$ for the {\\it sky} test field 1 case. \nIn agreement with the analysis concerning the {\\it random} test field, \nwe see that the {\\it fake } clusters  ($\\varepsilon=0.30$ deg., panel e in \nFig. \\ref{fig:Np}), are described  by a Possonian statistic, whilst, the  {\\it true } \nclusters (panel f in Fig.  \\ref{fig:Np}), are  better described by a log-normal \ndistribution (red dashed line), compared to a Poissonian  one (solid black line). \nWe  also observe that the log-normal law,  describes reasonably \nthe empirical distribution, only for values of $N_p\\lesssim 50$, whilst shows \nsignificant deviation  in the tail, consistent with the statistics of our simulated sources    \npopulation.\n \n\n\nTo complete this statistical characterization, we investigate the distribution\nof the number of detected clusters, as a function of the threshold $K$.\nAccording to Eq. \\ref{eq:P_clus}, we expect that the number of detected cluster,\nfor a {\\it random} field, follows a Poisson survival distribution. The  plot a of \nFig. \\ref{fig:Np_vs_K} confirms our hypothesis, indeed\nthe Poisson  survival function provides a reasonable description of the \nempirical distribution. The same holds for the case of {\\it fake} clusters of\nthe {\\it sky} test field 1 (plot c Fig. \\ref{fig:Np_vs_K}). On the\ncontrary, in the case of {\\it true} clusters (panel d Fig.\n\\ref{fig:Np_vs_K}), the Poisson survival distribution is not able to reproduce\nthe observed trend, consistently with the non-poissonian statistic of the\nsimulated clusters . The panels b and e of Fig. \\ref{fig:Np_vs_K}, show the\n$1-\\sigma$ CL region for the $N_p$, as a function of $K$. We note, that both in the\ncase of {\\it random} and {\\it sky} field {\\it true} clusters, the lower boundary\nof the region is constrained by the equation $y=K+1$, that is consistent with\nthe $\\gamma$-ray DBSCAN~ logic.\nOn the contrary, the upper boundary shows a different\nbehaviour. In the case of the {\\it random} field, the upper boundary\ndeviates from the lower boundary compatibly with the fluctuations \nof the events around the $\\varepsilon$~ circle, and ranges from about 8 to about 16.\nOn the contrary, in the case of {\\it sky} field {\\it true} the upper boundary\nis constrained by the statistics of the number of events in the simulated sources,\nand rages from about 60 to 100.\n \n\n\n\n\n\n\n\\section{Testing  the detection performance with simulated $\\gamma$-ray~ data}\n\\label{sec:det_sim}\nIn this section we investigate the detection performance of the\n$\\gamma$-ray DBSCAN.\nAs first point, we study the dependency of the detection efficiency on $K$ and $\\varepsilon$, \nand their impact on the spurious ratio, and on the detection efficiency. \nThen, we investigate the capability of\nthe algorithm to reconstruct the simulated clusters, and the positional accuracy\nof the reconstructed centroids.  \nWe test the detection performance of the $\\gamma$-ray DBSCAN~ using as benchmark the \nfive {\\it sky} test fields used in the previous section, and exploring\nthe same parameter space.\n\n\n\n\\subsection{Detection efficiency and spurious ratio as a function of $K$ and \n$\\varepsilon$}\n\nTo investigate the detection performance of the $\\gamma$-ray DBSCAN,  we   run, \nfor each of the five {\\it sky} test fields, and  for each pair of values $K$,$\\varepsilon$,\na $\\gamma$-ray DBSCAN~ detection. For each detection run, we build a  {\\it cluster catalog}.\nStarting from the {\\it cluster catalog}, we build the corresponding {\\it candidate catalog}. \nThe {\\it candidate catalog} is a list of  sources built by  taking into  \naccount two possible biases, the {\\it confusion}, and the {\\it multiple association}, \nin detail:\n\\begin{itemize}\n\\item{} a cluster is defined {\\it true}, i.e. with a possible counterpart, \n{    if the position of the simulated source \n falls within a circle centered  on the cluster centroid, with a radius equal to \n $2 pos_{err}$. }\n\\item{} Two or more {\\it true} clusters are defined {\\it confused}, if they have the same counterpart\n\\item{} A {\\it true} cluster has a  {\\it multiple association}, if has more than one counterpart.\n\\end{itemize}\nWe stress that, the number of confused clusters is negligible, \n{ indeed the average number of \n{\\it confused} clusters per run is about 0.08, and no  {\\it confused} clusters \nare found for $K>4$, and that  the average number of  {\\it multiple associations}\nper run is about 0.2.\n}\n\n\n\n\nThe final {\\it candidate catalog} will count a number  of candidate sources $N_{src}$,  \neach identified by a unique $SRC_{ID}$.  The  number of spurious sources  will be \n$N_{fake}=N_{src}-N_{true}$. In order to to characterize the performance, we\ndefine the following parameters:\n\n\\begin{itemize}\n\\item{} the detection efficiency:\n\\begin{equation}\nD_{eff}= \n\\left\\{\n\\begin{array}{c l}\n\\frac{N_{true}-N_{fake}}{N_{sim}(N_p sim.>K)}, &\\mathrm{if~}  (N_{true}-N_{fake}) \\leq N_{sim}(N_p sim.>K) \\\\\n1.0                                          , &\\mathrm{if~}  (N_{true}-N_{fake})>N_{sim}(N_p sim.>K) \\end{array} \n\\right.\n\\label{eq:Deff}\n\\end{equation}\nwhere $N_{sim}(N_p sim.>K)$ is the number of simulated sources\nwith a number of simulated events larger than $K$\n\\item{} the true detection ratio  $D_{true}=N_{true}\/N_{src}$\n\\item{} the spurious detection ratio  $D_{fake}=N_{fake}\/N_{src}$\n\\item{} the overall detection quality factor ($Q$), that takes into account \nthe tradeoff between $D_{eff}$  and $D_{fake}$, defined as:\n\\begin{equation}\nQ=D_{eff}\\Big(1-\\frac{N_{fake}}{N_{src}}\\Big)\n\\label{eq:Qeff}\n\\end{equation}\n\\end{itemize}\n\n\n\n\n\n\n\n\n\n\\begin{figure*}\n\\centering\n\\begin{tabular}{ll} \n\\includegraphics[width=9cm]{position_recon_vs_Np.pdf}&\n\\includegraphics[width=9cm]{position_recon.pdf}\\\\\n\\end{tabular}\n\\caption{\n{\n{\\it Panel a:} red solid boxes show the mean positional error of the centroid, for\n{\\it true} clusters in {\\it sky} test field 1, and the standard deviation (vertical error bar), vs. $N_p$.  \nThe clusters are  binned in $N_p$, with the bin width indicated by the horizontal error bar. \nThe black solid circles represent the corresponding trend for the distance between the cluster \ncentroid and the simulated source position.\n} \n{\\it Panel b:} the distribution of the distance between the simulated source position and the \ncluster centroid, expressed in arcsec, for the case of $\\varepsilon=0.10$ deg. (black line), \n$\\varepsilon=0.15$ deg. (blue line), and $\\varepsilon=0.20$ deg. (red lines). {\\it Panel c:} the cumulative \ndistributions  corresponding to panel b.\n}\n\\label{fig:Pos_prec}\n\\end{figure*}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=9cm]{Np_sym_vs_detected.pdf}\n\\caption{\n{\n{\\it Top panel:} the average number  of photons associated to each clusters \n$N_p$, and their dispersion (vertical bar) vs. the number of photons  \nsimulated  ($N_p$ sim). The red points refer to\nthe sub parameter space $\\varepsilon=0.15$ deg., and the solid blue  circles to the\n$\\varepsilon=0.20$ deg. sub space. The solid green lines represent the\nlaw  $N_p=N_p$ sim. The dashed lines represent the law $N_p=N_p$ sim. $\\pm$ 10.\n{\\it Bottom panel:} The corresponding fractional deviation $(N_p-N_p sim.)\/N_p sim$.\n}\n}\n\\label{fig:Np_rec}\n\\end{figure}\n\n\n\n{ The $D_{eff}$ parameter shows the fraction of simulated clusters, \nabove the threshold $N_p sim=K$, detected by the method,\nnet of the {\\it fake} ones. Hence, does not provide an indication\nof the spurious contamination. For this reason we have introduced\nthe $Q$ parameter, which rescale the $D_{eff}$ according to the ratio \nbetween {\\it fake} clusters, and found clusters $N_{src}$.}\nWe remind that, according to the $D_{eff}$ definition in Eq. \\ref{eq:Deff},\nit's possible to obtain values of $D_{eff}>1.0$. Assume to have a simulated\ncluster such that, for a given $K$ and $\\varepsilon$,  the corresponding seed cluster  \nhas a size $N^*=N_p sim.=K$. In the case of no background events within\nthe circle of radius $\\varepsilon$, this cluster  will be rejected. \nIf we have one or more background events contained within the circle\t\nof radius $\\varepsilon$, i.e.  $N^*>K$, the cluster will be detected.\nFor this reason, in such a case, we report a value of $D_{eff}=1.0$. \nThe same applies to $Q$.\n\nIn Fig.  \\ref{fig:param_run} we summarize the detection runs for the \ncase of {\\it sky} test field 1, for the full parameters space with \n$K>2$. The  panel a shows the isolevel map of the {\\it fake} \nclusters detection ratio. The gradient in the isolevel map is quite \nsharp, and  roughly half of the parameter space shows no {\\it \nfake} clusters (white isolevel line). To have a better understanding of \nthe impact of {\\it fake} clusters, it's interesting to compare the \n$D_{fake}$ isolevel map  to the $D_{true}$ isolevel map (panel b \nFig. \\ref{fig:param_run}). Also in this case the map shows a \nsharp  gradient, and the region with $D_{true}>0.95$ overlaps the \n$D_{fake}=0$ region. These two maps, clearly show the region\nof the parameter space where the algorithm has the best performance,\nbut the $D_{true}$ and $D_{fake}$ ratios do not provide information on the { ratio} \nbetween the number of {\\it true} detected clusters and the number\nof simulated clusters. At this regard more information are provided\nby the  $D_{eff}$ isolevel map (panel c, Fig. \\ref{fig:param_run}). To focus on the \n\"effective\" volume of the parameter space,  we hide by a white area \nthe region where $D_{eff}<0$. \nWe note that the isolevel lines $D_{eff}=0$ \nand the isomap lines in the maximum gradient area show a positive \ncorrelation between  $K$ and $\\varepsilon$, meaning that  an increased  value of $\\varepsilon$~, \nrequires an increased value of $K$, to have better background rejection. \nTo evaluate better the trade-off \nbetween $D_{true}$ and $D_{fake}$, we plot in the panel d  \nof Fig. \\ref{fig:param_run}, the isolevel map of $Q$. This plot \nshows that the area corresponding to $Q>0.95$, is consistent with that found in the \ncase of $D_{eff}$.\nIn  Tab. 1 we report the $D_{eff}$ values obtained for all the five \n{\\it  sky} fields, for  detections with a number  of {\\it fake}  sources \n$\\leq$ 6.  We note that the average values of {\\it true} clusters ranges\nbetween 44 and 51, with the {\\it fake} ones ranging between 1 and 3, \nand an average  $D_{eff}$ between 0.96 and 1.0.\nThis is a very promising result.  \n \n\n\n\n\n\n\n\n\n\n\\subsection{Cluster reconstruction, and positional accuracy}\nThe positional accuracy of the topometric methods, is probably\nthe most important feature of this class of algorithms.\nIn Sec. \\ref{sec:dbscan}, we have described our weighting \nmethod to reconstruct the centroid of the cluster.\n\n\n\n{\nThe panel a of Fig. \\ref{fig:Pos_prec} shows by red solid boxes\nthe mean positional error of the clusters centroid and the standard \ndeviation (vertical error bar) vs. $N_p$, for the {\\it true} clusters \nof the {\\it sky} test field 1 with  $\\varepsilon \\leq 30$ deg.\nThe clusters are binned in $N_p$, with the bin width indicated by the horizontal \nerror bar. \t\nAs expected, the uncertainty on the reconstructed cluster centroid \nis  $pos_{err} \\approx \\sigma_{sim}\/\\sqrt N_p$ (solid red line).\nThe solid black circles represent the corresponding trend \nfor the separation between the simulated cluster position\nand the reconstructed cluster centroid. For $N_p \\gtrsim 30$,\nthe separation is below $2'$. \nIn  the panel b   of Fig. \\ref{fig:Pos_prec} we plot the\nhistogram of the distribution of the angular separation \nbetween the position of the simulated source, and the\nposition of the cluster centroid. \nFor the three cases of $\\varepsilon=0.10$ deg., $\\varepsilon=0.15$ deg.,  \nand  $\\varepsilon=0.20$ deg., the positional error is below the $1.5'$, for \nthe $68\\%$ of the sample. \n}\n\n\\begin{table}\n\\input{spur_Cut_det_stat_table.tex}\n\\label{tab:tab1}\n\\caption{\nSummary of the\tdetections obtained for all the five {\\it sky} fields, \nfor detections with a number of fake sources $\\leq 6$. $N_{sim}^{cut}>K$, is is the number of \nsimulated sources with a number of simulated events larger than $K$, the number separated by the \n$\/$ symbol, indicates the full number of simulated sources.\n}\n\\end{table}\n\nBesides positional accuracy, is also important to understand\nthe capability of the $\\gamma$-ray DBSCAN~ to reconstruct the simulated\ncluster in terms of number of photons. Indeed, this information\ngives an idea of the average number of background photons\ncontaminating the reconstructed cluster. In the top left panel \nof Fig. \\ref{fig:Np_rec}, we show the scatter plot\nof $N_p$ vs. the number of simulated events ($N_p$ sim.). \nThe solid points represent the average value of $N_p$,\nfor a given value of  $N_p$ sim., and the error bar, corresponds \nto the standard deviation.\nThe solid green line represents the case $N_p=N_p$ sim.,\nand the dashed upper and lower lines, represent $N_p=N_p\\pm 10$ sim.,\nrespectively. \nBoth for the cases of $\\varepsilon=0.15$ deg.,  and  $\\varepsilon=0.20$ deg.,\nthe scatter is bounded by the dashed lines, showing that the largest\nexcess in the $N_p$ is about 10 photons, independently of $N_p$ sim.\nWe note, that in the case of $\\varepsilon=0.15$ deg., the number of reconstructed\nphotons, systematically underestimates the simulated number, whilst, \nthe $\\varepsilon=0.20$ deg. case does not shows this bias.\nIt's possible to appreciate better this effect, in the  bottom left panel   \nof Fig. \\ref{fig:Np_rec}, where we show\nthe fractional reconstruction error ($N_p-N_p$ sim.)\/$N_p$ sim., vs. \n$N_p$ sim. The solid green line represent the case with 0 error, \nand the dashed lines represent the $\\pm 20\\%$ boundaries. \nThe bias on $N_p$ in the case of $\\varepsilon=0.15$ deg., shows again\nthe strong correlation between $\\varepsilon$~ and the PSF radius. When $\\varepsilon$~ is\nsmaller then the $\\sigma^{sim}$ (that in our simulations reproduces\nthe PSF effect), the number of reconstructed events $N_p$ is systematically\nsmaller than $N_p$ sim., on the contrary, when the $\\varepsilon$~ radius matches the PSF\nradius size ($\\varepsilon=0.20$ deg.), the bias disappears.\n \n\n\n\n\n\\begin{figure*}\n\\centering\n\\begin{tabular}{ll} \n\\includegraphics[width=9.0cm]{sig_distr_fake.pdf}&\n\\includegraphics[width=8.5cm]{param_run_sigcut4.png}\\\\\n\\end{tabular}\n\\caption{\n{\\it Left panel:} The distribution (blue line) of the square of the significance, for \nthe {\\it fake} clusters in the {\\it sky} test field 1, for the full $K$,$\\varepsilon$~ parameter space, \ncompared to a $\\chi^2$ distribution with one degree of freedom.\n{\\it Right panel:} the  spurious  ratio\n$D_{fake}$ for $S_{cls}>4.0$, the white line shows the isolevel $D_{fake}=0.0$. \n}\n\\label{fig:signif}\n\\end{figure*}\n\n\n\n\\section{Cluster significance, background inhomogeneities, and  rejection of spurious clusters}\n\\label{sec:signif_sim}\n\nEven though, we have identified the region of the $K$-$\\varepsilon$~ parameter space,  \nwhere the detection efficiency is larger,  and the probability to detect {\\it fake} \ncluster is lower, in the application to real data, it's mandatory to \nprovide a significance level, expressing the probability of  a \ncluster being not originated in a background fluctuation.\nWe propose a method derived   from the \\cite{LiMa1983} approach, \nbased on the evaluation of the  signal to noise (S\/N) ratio. \nA significance method based on the S\/N ratio fits well the the  \n$\\gamma$-ray DBSCAN~ implementation, because the algorithm directly provides \na partition of the photon list in {\\it cluster} and {\\it noise} \nevents. Hence, for each  cluster we can evaluate easily the S\/N \nratio, knowing the exact nature of each event. The procedure to \nevaluate the significance is summarized by the following items:\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=18cm]{MST_region_clean_3GeV_Cmap_crop.pdf}\n\\caption{Aitoff projection of the Fermi sky region. The purple boxes represent the $\\gamma$-ray DBSCAN~\n sources ($K=8,\\varepsilon=0.21$ deg.). The green crosses are the 2FGL sources with $TS>16$, the\n red one those with $TS\\leq16$. There are no {\\it fake} sources, and the $\\gamma$-ray DBSCAN~ finds all the\n sources with $TS>16$, except only one, enclosed by the red circle, and with the center positioned \n at the edge of the field.}\n \\label{fig:Fermi_Cmap}\n\\end{figure*}\n\n\\begin{enumerate}\n\\item{} for each cluster, we define an annular region,  with \nan inner radius  $r_{in}$, and an external radius  $r_{out}$.\n\\item{} $r_{in}$ is set to an initial value of $r_{in}=2 r_{eff}$, and\n is adaptively increased with a step of $r_{in}\/10$, for a\nmaximum of 10 trials, until at least the $95\\%$ of the cluster events are enclosed within $r_{in}$.\n\\item{} $r_{out}$ is set to $3 r_{in}$.\n\\item{} We count all the cluster events $N^{in}_{src}$  and all the background events   \n$N^{in}_{bkg}$, enclosed  within the circle with  radius $r_{in}$ and centered on the cluster  \ncentroid.  \n\\item{} We determine the $N^{out}_{bkg}$ background level, rescaling the number of \nbackground events in  $r_{in}<r<r_{out}$, to a circle with  radius $r_{in}$.  \n\\item{} To evaluate possible gradients in the background, we select a region enough far from the \ncluster to sample properly the background level, and enough close to the cluster, to measure \na \"local\" background level. At this regard, we define the radius $r_{out}^{ave}=(r_{out}+r_{in})\/2$, \nand we evaluate the average background  level ($N^{local}_{bkg}$)  in a circle of radius $r_{in}$, centered on each point in  $r_{out}^{ave}<r<r_{out}$.\n\\item{} If no background points are { found} in  $r_{out}^{ave}<r<r_{out}$, we set \n$N^{local}_{bkg}=N^{out}_{bkg}$.  \n\\item {} By\tcomparing\t$N^{local}_{bkg}$ to $N^{in}_{bkg}$,\twe\tevaluate the\tfraction\nof {\\it noise} already resolved by the $\\gamma$-ray DBSCAN, and we evaluate the effective background level \n$N^{eff}_{bkg}$, by correcting $N^{local}_{bkg}$ for $N^{in}_{bkg}$.\n\\item{}  we evaluate the significance according to the Likelihood Ratio Test (LRT)\nmethod proposed by \\cite{LiMa1983}:\n\n\\begin{equation}\nS_{cls}=\\sqrt{2 \\Big  ( N^{in}_{src} \\ln \\Big[ \\frac{2 N^{in}_{src}}{N^{in}_{src}+N^{eff}_{bkg}} \\Big]+\nN^{eff}_{bkg} \\ln \\Big[ \\frac{2N^{in}_{src}}{ N^{in}_{src}+N^{eff}_{bkg}} \\Big ]  \\Big  )}\n\\label{eq:signif}\n\\end{equation}\n\n\\end{enumerate}\n\n\n\n\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=9cm]{position_err_cluster_vs_pos_err_2FGL.pdf}\n\n\\caption{\n{\nThe scatter plot the positional error of the $\\gamma$-ray DBSCAN~ clusters vs. the \npositional error of the corresponding associated 2FGL$_{TS>16}$  sources. \nFor each  2FGL$_{TS>16}$  source associated to one or more $\\gamma$-ray DBSCAN~ clusters, \nwe plot the error on the position  of the reconstructed cluster centroid  and \nits standard deviation (represented by the error bar). \nThe dashed red line represents a linear best fit with a slope\nof $\\simeq$ 0.99, and an intercept of $\\simeq$ 9.53. \n}\n}\n\\label{fig:Pos_rec_RealSky}\n\\end{figure}\n\n\n\nUnder the hypothesis that a cluster is due to a background fluctuation,\nthe variable $S_{cls}^2$, is expected to follow a chi square distribution,\nwith one degree of freedom ($\\chi(1)^2$).  \nIn the left panel of Fig.  \\ref{fig:signif}, we plot the  distribution of $S_{cls}^2$,\nfor the fake clusters in the {\\it sky} test field 1  (blue histogram), \ncompared to a  $\\chi(1)^2$ distribution. \nThe empirical distribution, is well described by the expected $\\chi(1)^2$ distribution, proofing \nthat the value of $S_{cls}$, can be used  as  the \"significance\" of the detected cluster. \nA very illustrative example of the power of $S_{cls}$ in rejecting {\\it fake} clusters,  is  \ngiven by the plot in the right panel in Fig \\ref{fig:signif}, where we plot the\n$D_{fake}$ ratio isolevel map, applying the selection  $S_{cls}>4.0$. { The {\\it fake} ratio is 0 \nfor the parameter space with $\\varepsilon\\lesssim 0.25$ deg.  For 0.25 deg. $\\lesssim \\varepsilon \\lesssim$ 0.35 deg.,  \nthere are { fluctuations} showing  $D_{fake}\\lesssim 0.05$. Only for $\\varepsilon$$\\gtrsim 0.40$ deg. and \n$K\\lesssim 8$, the  {\\it fake} ratio shows a significant increase, but we stress that in this \nregion of the parameter  space, $\\varepsilon$~ is more then double of the PSF size, hence \nthis is a region of the parameter space that should not be used in the detection with real data.\n}\n\n\n\n\n\\section{Application to real {\\it Fermi}-LAT~ data }\n\\label{sec:realdata}\n\n\n\nThe last step in our investigation of the $\\gamma$-ray DBSCAN, is the application\nto real {\\it Fermi}-LAT~ $\\gamma$-ray~ data. We select the same region of the sky\nused for the simulated test { field} ( $80^\\circ<l<170^\\circ$, and $40^\\circ<b<65^\\circ$),\nand we extract all the photons with energy $E>3$ GeV. The photons\nare collected for the same time span of the 2FGL\ncatalog . We repeat the detection \ntest performed in the case of simulated data (see Sec. \\ref{sec:det_sim} and  \nSec. \\ref{sec:signif_sim}), restricting the parameter space to $2\\leq K \\leq10$,  \nand $0.10\\leq \\varepsilon \\leq0.30$ deg.\n\n\\begin{figure*}\n\\centering\n\\begin{tabular}{l} \n\\includegraphics[width=17cm]{Fermi_Sky_det_2FGL_SigCut4.png}\\\\\n\\includegraphics[width=17cm]{Fermi_Sky_det_SigCut2_2FGL_SigCut4.png}\\\\\n\\includegraphics[width=17cm]{Fermi_Sky_det_SigCut4_2FGL_SigCut4.png}\n\\end{tabular}\n\\caption{\n$D_{fake}$ (left panels) and $D_{eff}$ (right panels), for the\nreal sky detections, using the 2FGL$_{TS>16}$ catalog.\n{\\it Panels a,b:} no cut on $S_{cls}$ applied.\n{\\it Panels c,d:} $S_{cls}>2.0$\n{\\it Panels e,f:} $S_{cls}>4.0$ \n}\n\\label{fig:Fermi_Det}\n\\end{figure*}\n\n\\begin{figure*}[t]\n\\centering\n\\begin{tabular}{ll} \n\\includegraphics[width=9cm]{sig_vs_TS.pdf}&\n\\includegraphics[width=9cm]{sig_distr_fake_Fermi.pdf}\\\\\n\\end{tabular}\n\\caption{\n{\\it Left panel: }scatter plot of $S_{cls}$ vs. $\\sqrt{TS}$. For \neach source in our 2FGL$_{TS>16}$ list, associated to one or more $\\gamma$-ray DBSCAN~ cluster, we plot the  $\\sqrt{TS}$ in the 3-10 GeV band, \nv.s. the average values of $S_{cls}$ and its standard deviation\n(represented by the error bar).\n{\\it Right panel: }the distribution (blue line) of the square of the significance, for \nthe {\\it fake} clusters in the {\\it Fermi}-LAT~ real sky, for the full $K$,$\\varepsilon$~ parameter space, \ncompared to a $\\chi^2$ distribution with one degree of freedom.\n}\n\\label{fig:sig_vs_true_rate_RealSky}\n\\end{figure*}\n\n\nTo properly understand the detection performance, we need\nto take into account that the 2FGL catalog has been built using\nphotons with an energy threshold of 100 MeV, whilst we use\na value of 3 GeV. A possibility is to select sources with\na reported flux larger than zero,  \nin  the 3-10 GeV band flux column of the 2FGL.\nThis flux-based selection, is not the best way to \nstudy the detection performance of the $\\gamma$-ray DBSCAN~, indeed\nthe flux does not contain a unambiguous relation with\nthe significance of the detection, for that energy threshold. \nA more reliable  criterion is to select the sources according \nto the significance  reported in the 2FGL. The 2FGL detection \nsignificance is given by the $\\sqrt{TS}$.  The $TS$ is the test \nstatistic  defined as  $TS=2(\\log L(\\mathrm{source}) - \\log L(\\mathrm{no ~ source}) )$,\nwhere $L$ is the likelihood of the data given the model with or without \na source present at a given position on the sky \\citep{2FGL}.\nWe apply a selection according to $\\sqrt{TS}>4$, and we refer\nto the corresponding source list (counting 35 sources) as\n2FGL$_{TS>16}$. \n\n\n\n\nAn example of the application of the $\\gamma$-ray DBSCAN~ to real {\\it Fermi}-LAT~ \ndata is given in Fig. \\ref{fig:Fermi_Cmap}, where we report an Aitoff \nprojection in galactic coordinates of the analysed $\\gamma$-ray~ sky region. The red\ncrosses represent the 2FGL sources with $TS<16$ in the 3-10 GeV band, and the \ngreen crosses represents those with $TS\\geq 16$. The purple boxes\nrepresent the $\\gamma$-ray DBSCAN~ sources found for $K=8,~\\varepsilon=0.21$ deg. \nFor this choice of parameters, we find no {\\it fake} sources,  and we find all \nthe sources with with $TS>16$, except only one, enclosed by the red circle, and \npositioned at the edge of the sky region, with a galactic latitude $l=64.85$ deg.\nIn  Tab. 2 we summarize the detection performance, for  detections with a number  \nof {\\it fake}  sources $\\leq$ 4.  We note that  values of {\\it true} clusters ranges\nbetween 35 and 34, out of the 35 present in the 2FGL$_{TS>16}$. \nThe {\\it fake} ones range between { 1} and 4, and we obtain an average detection\nefficiency of  $D_{eff}= 0.94$. \n\n\nIn  Fig. \\ref{fig:Pos_rec_RealSky} we compare the localization \nperformance of the $\\gamma$-ray DBSCAN~ algorithm  with that returned by the \nlikelihood analysis implemented in the Fermi Science Tools. \nFor each source in our 2FGL$_{TS>16}$ list, \nassociated to one or more $\\gamma$-ray DBSCAN~ clusters, we plot the  the error on the position \nof the reconstructed cluster centroid  and its standard deviation (represented by the error bar) \nvs. the 95$\\%$ positional  uncertainty reported in the 2FGL. We evaluate the\n2FGL  95$\\%$ positional  uncertainty as $\\sqrt{\\sigma_{95,min}~\\sigma_{95,max}}$,\nwhere $\\sigma_{95,min}$ and $\\sigma_{95,max}$, are the semimajor and semiminor axes of the \n95$\\%$ confidence source location region, respectively.\nThe dashed red line represents a linear best fit, with a slope\nof $\\simeq$ 0.99, and an intercept of $\\simeq$ 9.53, showing that \nthe error on the position of the reconstructed cluster centroid, performed with a threshold of 3 GeV, is \nof the same order of the $95\\%$ positional uncertainty reported in the 2FGL catalog, performed \nabove 100 MeV.\n\n\nTo test the reliability of  the significance $S_{cls}$ to reject spurious sources, \n in Fig. \\ref{fig:Fermi_Det}  we plot the $D_{fake}$ and $D_{eff}$, based\non the 2FGL$_{TS>16}$ catalog. The panels a and b,  correspond to the case of\nno selection on $S_{cls}$. Both the $D_{fake}$  and the $D_{eff}$ trends are \nvery similar to the case of the simulated sky.\nIf we apply a significance cut of $S_{ls}>2.0$ (panels c,d), we observe that\nthe number of spurious ratio is $D_{fake}\\leq$ 0.05 for  almost half of the parameter space\n(region to the right of the purple line). \n The more severe cut of $S_{cls}>4.0$\n(panels d,e), removes  all the {\\it fake} clusters except two, { for \n$\\varepsilon \\lesssim$ 0.15 deg. Only for  $\\varepsilon \\gtrsim$ 0.25 deg., the $D_{fake}$ \nratio shows a significant increase, ranging from 0.05  up to $\\simeq$ 0.1.  \nIn agreement with our analysis on simulated data, the region of the parameter space \nwhere $\\varepsilon$~ is comparable to the PSF size, gives the better performance.\n} \n\nTo have a further confirmation about the robustness of our significance,\nwe plot in the right panel of Fig. \\ref{fig:sig_vs_true_rate_RealSky},\n$S_{cls}$ vs. $\\sqrt{TS}$. For each source in our 2FGL$_{TS>16}$ list, \nassociated to one or more $\\gamma$-ray DBSCAN~ cluster, we plot the  $\\sqrt{TS}$ \nin the 3-10 GeV band, \nv.s. the average value of $S_{cls}$ and its standard deviation\n(represented by the error bar). The average value of $S_{cls}$  and its\nstandard deviation  are evaluated from  the list of all the cluster associated\nto  the same  2FGL source. The solid blue boxes represent the full $K$,$\\varepsilon$~\nparameter space case, and the red solid circles represent the $\\varepsilon=0.10$ deg. case.\nThe dashed black line represents a linear best fit. \nThe slope of the linear fit is $\\simeq 0.5$. The strong correlation in\nthe scatter plots ($r\\simeq 0.98$, for both data sets), proves that  our\nsignificance implementation is consistent with the $\\sqrt{TS}$  reported\nin the 2FGL, and the slope of the linear fit suggests that $S_{cls}\\simeq 0.5 \\sqrt{TS}$. \n\n\n\n\n\n\n\n\n\n\n\n\\begin{table}\n\\input{spur_Cut_det_stat_realsky_table.tex}\n\\label{tab:tab2}\n\\caption{Summary of the detection performance for the\nreal {\\it Fermi}-LAT~  field, for detections with a number of fake sources $\\leq 4$.\n}\n\\end{table}\n\n\\section{Conclusions}\n\\label{sec:conclusions}\nFor the first time, we have used the DBSCAN~ for the detection of sources \nin $\\gamma$-ray~  astrophysical images. We have implemented a new version of\nthe  DBSCAN, the $\\gamma$-ray DBSCAN, that is optimized for the application to $\\gamma$-ray~   \nastrophysical images, with relevant background noise. Our $\\gamma$-ray DBSCAN, presents\nthe novelty of recursive call of the DBSCAN~ algorithm, that allows an\nexcellent reconstruction of the cluster, with an effective background\nrejection.\nWe have tested the algorithm with a sample of simulated $\\gamma$-ray~{\\it Fermi}-LAT~ fields,\nto give a statistical characterization of the method, and to benchmark\nthe detection performance. The results, with the simulated $\\gamma$-ray~ data,\nare summarized by the following items:\n\\begin{itemize}\n\\item{} The radius of $\\gamma$-ray DBSCAN~ scanning brush $\\varepsilon$~, has a strong \ncorrelation with the instrumental PSF radius. We find that \nthe typical size of the reconstructed {\\it true} cluster is of the\norder of the simulated  PSF size  $\\sigma^{sim}$,\n{ and that the precision of the reconstructed\ncentroid is of the order of  $\\sigma^{sim}\/\\sqrt{N_p}$. \n}\n\\item{} The number of reconstructed events $N_p$ is ruled by the \nPoissonian statistics in the {\\it random } fields, and for\nthe {\\it fake} clusters. On the contrary, for {\\it true} clusters,\nthe statistics of $N_p$, is ruled by that of the simulated sources.\n\\item{} The fractional error  on the reconstructed events number is \nof the order of $20\\%$ for  $N_p sim. \\lesssim 50$, and is negligible\nfor larger values, with best performance obtained when \n$\\varepsilon$$\\simeq \\sigma^{sim}$.\n\\item{} We have investigated the detection performance, for \na wide range of the $K$,$\\varepsilon$~ parameter space, and we have\nidentified the region with the best performance in terms\nof detection efficiency, and spurious ratio.\n\\item{} We have implemented an algorithm for the estimate \nof the Signal to Noise (S\/N) ratio, able to deal with local\nbackground inhomogeneities and nearby sources contamination, \nand we have successfully used the S\/N estimate to determine \nthe significance of the clusters, using the definition in \\cite{LiMa1983}.\n\\item{} Our cluster significance, $S_{cls}$, for random clusters,\nfollows the $\\chi(1)^2$ statistics, and can be used to\nreject spurious sources. The chance to find spurious\nsources for $S_{cls}>4$, is negligible. This means, that\nour $S_{cls}$ is a robust a reliable tool to reject spurious sources,\nand that $\\chi(1)^2$ statistics can be used to evaluate the probability of\na cluster to be spurious.\n\\end{itemize}\n\nWe have successfully applied the $\\gamma$-ray DBSCAN~ to real {\\it Fermi}-LAT~ data.\nWe have found an excellent agreement with  results from the simulated fields.\nWe tested our detection performance using as catalog, the  2FGL\nsourced with a $\\sqrt(TS)>4$ cut.\nThe results, with the real {\\it Fermi}-LAT~ $\\gamma$-ray~ data,\nare summarized by the following items:\n\\begin{itemize}\n{\n\\item{} the error on the position of the reconstructed cluster centroid, \nperformed with a threshold of 3 GeV, is of the same order of the 95$\\%$ \npositional uncertainty reported in the 2FGL,  performed above 100 MeV. \n}\n\\item{} We tested the $\\gamma$-ray DBSCAN~ significance, finding that it is\nstrongly correlated with the $TS$ provided in the 2FGL.\nThe significance cut, allows to remove safely spurious \nclusters.\n\\item{} The detection efficiency with real data is excellent,\nwe are able to find all the 35 sources with $\\sqrt(TS)>4$.\n\\item{}  When working with $\\varepsilon$~ of the order of the instrumental\nPSF size, we obtain the best performance, in terms of spurious \nrejection, and detection efficiency \n\\end{itemize}\n\nIn general, we find that the $\\gamma$-ray DBSCAN~ is a very powerful detection method\nto find clusters in $\\gamma$-ray~ images, corresponding to real sources.\nIt has the great advantage to deal self-consistently  with\ngradient in the background, providing an effective rejection of spurious\nclusters. Our implementation of the detection significance, in addition\nto the algorithm to evaluate local fluctuations in the background,\nallows to apply statistically significant selection, making even\nmore effective the rejection of spurious sources.\n\nIn a companion paper \\citep{dbscan_inprep}, we will a apply the method\nto the {\\it Fermi}-LAT~ sky, showing the potentiality for the discovery of new sources,\nin particular of small clusters located at high galactic latitude, or\ncluster on the galactic plane, affected by a strong background.\n{\nWe will also investigate how to plug the energy dependence\nof the PSF into the $\\gamma$-ray DBSCAN~ algorithm, and how to improve\nthe detection performance taking into account other \n{\\it Fermi}-LAT~ calibration properties.\n}\n\nWe  remark that, since the $\\gamma$-ray DBSCAN~ provides also density maps,\nit can potentially be used in the detection of large  scale structures \nin the galactic $\\gamma$-ray~ background, providing patterns to compare to the\ninterstellar gas distribution.  \nWe also stress, that the application of this method are not limited \nto $\\gamma$-ray~ images, but can be potentially used for any application\nrelated to the detection of spatial, and\/or spatio\/temporal clusters.\n \n\n\n\n\n\n\n\\begin{acknowledgements}\nWe are grateful to Enrico Massaro, Riccardo Camapana, and Enrico Bernieri, \nfor helpful comments, and for providing us the simulated test fields.\nWe are grateful to Gino Tosti, for helpful comments.\nWe thank the anonymous referee for providing us with constructive comments and useful suggestions.\n\\end{acknowledgements}\n\n\\bibliographystyle{aa} %\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe remarkable similarity between the evolution of the star formation\nhistory of galaxies and the emissivity of AGN from $z\\sim1$ to the\npresent \\citep[e.g., ][]{bo98,fr99,me04,sh07,si08b} suggests a common\nmechanism that regulates their growth.  Major mergers of galaxies\n\\citep{mi96} and secular evolution \\citep{ko04,hop06} can both\npotentially funnel gas to the nuclear region that can then power\nconcurrent star formation and accretion onto supermassive black holes.\nFully consistent with this scheme, young stellar populations are known\nto be prevalent within the bulges of nearby Seyfert galaxies\n\\citep[e.g.,][]{te90,go01,wh03} and luminous quasars\n\\citep[]{ja04a,let07}.  Enhanced levels of ongoing star formation are\nalso evident in nearby Seyfert 2s \\citep{gu06} and the hosts of some\nmid-infrared selected quasars \\citep{la07} although are not high\nenough to put them in the same class as the Ultraluminous Infrared\nGalaxies \\citep{st06}.  On the other hand, the lack of star formation\nbased on [OII] emission in the hosts of a significant sample of PG\nquasars \\citep{ho05} and type 1 AGN \\citep{ki06} from the Sloan\nDigital Sky Survey (SDSS) may indicate an elevated role for AGN in\nsuppressing star formation \\citep[][]{gr04,sp05,cr06,cav07,hop08a}.\nHowever, there do appear to be high levels of star formation in the\nhosts of PG quasars when considering their far-infrared emission\n\\citep{sc06,ne07}, and type 2 quasars from the SDSS\n\\citep[$0.3<z<0.8$; ][]{za03,ki06} that may hint at an underlying\nevolutionary sequence (i.e., a modification to the unified model with\na correlation between star formation and nuclear obscuration) or the\nsystematic increase in the global star formation history of galaxies\n\\citep[e.g., ][]{li95,ho06,no07,tr07,zh07}.  These seemingly\ndiscrepant results highlight the complexity in determining if a casual\nrelationship between star formation and AGN activity exists.  This is\nmost likely due to the challenges in measuring the host galaxy\nproperties under the glare of a bright AGN, widely varying selection\nmethods, and inadequate control samples.\n\nThe ability to study the host galaxies of AGN selected from a large\nparent sample of galaxies has dramatically improved in the last few\nyears.  The SDSS has generated an unprecedented sample of galaxies to\nselect large numbers of low redshift ($z<0.3$) narrow-line AGN\n\\citep{ka03b,ke06,sta06,he06,wild07} to cleanly study their host\nproperties.  Powerful AGN have been clearly shown to reside in\nbulge-dominated galaxies with young stellar populations \\citep{ka03b}.\nIn conjunction with GALEX observations, AGN have optical-ultraviolet\ncolors placing them in an intermediate region \\citep {ma07,sa07,sch07}\nbetween the red and blue galaxy populations thus stirring interest in\ntheir relation to transitional galaxies \\citep{hop07}.  Remarkably,\n\\citet{ka07} find a strong correlation between the mass accretion rate\nonto SMBHs and the presence of young stars in the bulge that is\ntypically accompanied by a star-forming outer disk that may be\nsupplying the nuclear fuel.  It is imperative to extend such studies\nto epochs closer to the peak ($z\\sim1$) of the cosmic AGN and star\nformation history since AGNs may have an impact on the gas content of\ngalaxies and their subsequent demise.\n\nX-ray emission is a unique probe of AGN activity due to its ability to\npenetrate a moderately obscuring medium ($N_H \\lesssim 10^{24}$\ncm$^{-2}$), the lack of confusion with stellar sources, and because it\nallows a more direct probe of the mass accretion rate \\citep[see][for\na review]{br05}.  Selection of narrow line AGN (i.e., obscured; type\n2) by X-rays \\citep[e.g., ][]{ba03,sz04,si05,br07} can compensate for\nthe deficiencies of optical selection based on emission line ratios at\n$z>0.3$.  X-ray observations with both $Chandra$ and XMM-$Newton$ are\nbeing exploited to cleanly study the host galaxies of X-ray selected\nAGN \\citep{gr05,na07,pi07,ga08}.  Recently, \\citet{si08a} show that the\nbulge-dominated hosts of X-ray selected AGN in the Extended $Chandra$\nDeep Field - South have rest-frame colors that are bluer with\nincreasing redshift, possibly related to the star formation history of\ngalaxies.  In a related study, the mean star formation rate based on\nbroad-band photometry of 58 X-ray selected AGN ($z\\sim0.5-1.4$) in the\n1 Msec CDF-S has been shown to be similar to IRAC-selected galaxies in\nan equivalent mass regime \\citep{ah07}.  To date, few studies using a\nsignificant sample of AGN selected from a large parent sample of\ngalaxies with optical spectroscopy have been undertaken at these\nhigher redshifts ($z>0.3$).\n\nTo do so, we utilize the rich multiwavelength observations of the\nCOSMOS field \\citep{sc07} to carry out such a study at $0.5 \\lesssim z\n\\lesssim1.0$.  The COSMOS survey is roughly a 2 square degree region\nof the sky selected to be accessible from all major observatories both\nfrom the ground (e.g., Subaru, VLT) and space (e.g., $HST$, $Spitzer$,\nXMM-$Newton$, $Chandra$).  The zCOSMOS survey \\citep{lilly07} targets\nobjects for optical spectroscopy with the VLT in two separate\nobserving programs. A bright sample ($i<22.5$) is observed with a red\ngrism to provide a wavelength coverage of $5500-9500$ {\\rm \\AA}\nideal for identifying galaxies ($L_*$) up to $z \\sim 1.2$.  A deep\nprogram, not utilized in the present study, targets faint galaxies\n($B<25$), selected to be in the redshift range $1.5\\lesssim z\\lesssim\n2.5$, using a blue grism with a wavelength coverage of\n$3600<\\lambda<6700$ {\\rm \\AA}.  Here, we select a sample of galaxies\nbased on their stellar mass with reliable spectroscopic redshifts from\nthe zCOSMOS bright program.  Those that host AGN are identified by\ntheir X-ray emission as detected by XMM-$Newton$ \\citep{ha07,cap07}.\nWe measure the strength of emission lines (i.e., [OII]$\\lambda$3727,\n[OIII]$\\lambda$5007), using our automated pipeline\n\\citep[\"platefit\\_vimos\";][]{lam08} to determine star formation rates\nfor the entire galaxy sample including those hosting AGN.  An\nadditional spectral indicator ($D_n4000$) enables us to discern the\nage of the stellar populations on longer timescales.  X-ray\nluminosities of the AGN give us a handle on their bolometric output,\nless affected by obscuration, to infer mass accretion rates and\ndetermine any trends with star formation rate and redshift.  Finally,\nwe point out that a companion paper \\citep{si08c} based on the zCOSMOS\n10k catalog expands on the current study by investigating the\nenvironmental impact on AGN activity and their host galaxy properties.\n\nThroughout this work, we assume $H_0=70$ km s$^{-1}$ Mpc$^{-1}$,\n$\\Omega_{\\Lambda}=0.75$, and $\\Omega_{\\rm{M}}=0.25$.\n\n\\section{Data and derived broad-band properties}\n\n\\subsection{Parent galaxy sample and AGN identification}\n\nWe use the zCOSMOS 10k spectroscopic 'bright' catalog to construct a\nwell-defined sample of galaxies including those hosting X-ray selected\nAGN up to $z\\sim1$.  Specifically, we identify 7543 galaxies with a\nselection magnitude $i_{ACS} \\leq 22.5$ and high quality spectra hence\nreliable redshifts up to $z=1.02$.  A galaxy is included in our sample\nif the redshift has a quality flag 2.0 or higher that amounts to a\nconfidence of $\\sim99\\%$ for the overall sample.  The redshift success\nrate is not strongly dependent on galaxy color over $0.5\\lesssim z\n\\lesssim1.0$ given the quality of the spectra and presence of strong\nfeatures (i.e., 4000 \\AA~break; Ca H+K, [OII]).  Full details on data\nacquisition, reduction, redshift measurements and quality assurance\ncan be found in \\citet{lilly07,lilly08}.\n\nWe use the catalog of X-ray sources \\citep{cap07} generated from the\nuniform XMM-$Newton$ coverage ($\\sim50$ ks depth) of the COSMOS field\n\\citep{ha07} to identify galaxies within our parent sample that harbor\nAGN.  The X-ray catalog includes 1848 point--like sources detected\nabove a given threshold using a maximum likelihood method in either\nthe soft (0.5--2 keV), hard (2--10 keV) or ultra-hard (5--10 keV)\nbands down to a limiting flux of 5$\\times 10^{-16}$, 2$\\times\n10^{-15}$ and 5$\\times 10^{-15}$ erg cm$^{-2}$ s$^{-1}$ in the\nrespective band.  The adopted threshold ($Likelihood > 10$)\ncorresponds to a probability $\\sim 4.5\\times10^{-5}$ that a source is\na background fluctuation.  An additional 26 faint XMM sources\ncoincident with diffuse emission (Finoguenov et al. in preparation)\nwere excluded.  Both the soft (0.5-2.0 keV) and hard band (2.0-10.0\nkeV) detections comprise our AGN sample in order to include the\nlow-to-moderate luminosity and\/or absorbed sources.  The higher\nsensitivity of the XMM-$Newton$ observations in the soft band enables\nus to probe lower luminosity AGN ($log~L_X\\sim43$) with $0.8\\lesssim z\n\\lesssim 1.0$ not sufficiently represented in the hard-band catalog.\n\nOptical and near-infrared counterparts to the XMM-$Newton$ X-ray\nsources are found using a maximum-likelihood method \\citep{br07}.\nMost X-ray sources (84\\%) have reliable counterparts while the\nremaining are uncertain due to the presence of multiple objects within\nthe X-ray error box that have similar probababilities of being the\ntrue counterpart.  We use the $Chandra$ observations (Elvis et al. in\npreparation; Civano et al. in preparation) that cover the central 1\nsquare degree to further refine our identifications.  We elect to\ninclude counterparts with lower confidence given that $\\sim50\\%$ of\nthem are likely to be correct based on the overlap between\nXMM-$Newton$ and $Chandra$ identifications (M. Brusa, private\ncommunication).  All results presented here are confirmed based on the\nslightly smaller catalog of highly reliable counterparts.  \n\nThe zCOSMOS 'Bright' program provides optical spectra for 357 of the\n1093 optical counterparts to X-ray sources having $i<22.5$.  Reliable\nspectroscopic redshifts are available for 90\\% of which 164 have\nredshifts between $0<z<1.02$.  Higher redshift AGN up to $z\\sim4$ have\nbeen identified by the zCOSMOS 'Bright' program but are not the focus\nof this study.  Since a high fraction of the zCOSMOS galaxies with\nassociated X-ray emission have detections in the hard 2-10 keV band\n(84\\%), we do not expect the inclusion of soft-selected sources to\ninduce any significant bias.  A subset (54\\%) of the X-ray sources has\nbeen designated as 'Compulsory' targets when designing the VIMOS masks\nwhich essentially lessens their randomness and results in a 71.9\\%\nsampling rate compared to the current rate of 29.8\\% for galaxies.  We\nset priority to these sources since they were not included in the\nspectroscopic followup program using Magellan \\citep{trump07}.  Since\nthe number of AGNs is not overwhelmingly large, we chose to not select\na smaller pseudo-random sample but rather incorporate correction\nfactors applied to derived measurements where applicable.\n\n\\begin{figure}\n\n\\epsscale{1.2}\n\n\\plotone{f1.eps}\n\\caption{X-ray luminosity (0.5-10 keV) versus redshift for 152 AGN\n(large symbols).  Filled circles are AGN primarily characterized by\nnarrow emission lines while open symbols denote those with at least one\nbroad optical emission line falling within the observed spectral\nrange.  The upper limit to the broad-band X-ray luminosity for each\nzCOSMOS galaxy is shown by the small grey dots.  The solid horizontal\nline is the minimum X-ray luminosity that we enforce to measure the\nfraction of galaxies hosting AGN to ensure a significant underlying\ngalaxy population.  The dotted line shows our chosen upper limit to\navoid AGN contamination.}\n\n\\label{lx_z}\n\\end{figure}\n\nModerate-luminosity AGN ($L_X\\sim10^{43}$ erg s$^{-1}$), detected in\nmedium-to-deep X-ray observations, have been shown to be ideal\nlaboratories for the study of their host galaxies due to their low\noptical brightness which in many cases is due to their nuclear\nobscuration \\citep[e.g.,][]{to06,main07}.  Our AGNs predominately have\nluminosities below that of typical QSOs ($L_X>10^{44}$ erg s$^{-1}$)\nbut are the dominant contributor to the Cosmic X-ray Background\n\\citep{gi07}.  In specific cases, we further select AGN based on their\nX-ray luminosity ($42.0 < log~L_{0.5-10.0~{\\rm keV}}<43.7$), as done\nin \\citet{si08a}, to isolate a sample for which we can cleanly\ndetermine their host properties (i.e., stellar masses, rest-frame\n$U-V$, $D_n4000$).  \\citet{ga08} have demonstrated that the optical\nemission from a nearly equivalent AGN sample in COSMOS is primarily\nattributed to their host galaxies; AGN dominated galaxies are\npreferentially at $z>0.8$ due to the fact that their X-ray\nluminosities are typically above our cutoff ($L_X\\sim10^{43.7}$ erg\ns$^{-1}$).  We conclude that the derived properties of the host\ngalaxies of AGN in zCOSMOS are likely to be reliable.  Where feasible,\nwe include higher luminosity AGN (QSOs; 16 with $L_{0.5-10~{\\rm\nkeV}}>10^{44}$ erg s$^{-1}$), since we can measure their stellar\nproperties (i.e., [OII] strength) even under the glare of an\nintrinsically, bright AGN (see Section~\\ref{sfr_measure} for details).\nFor these cases, we refrain from analyses based on their host\nproperties such as mass-weighted SFR or rest-frame color.  One caveat\nof our luminosity-selected sample is a potential bias towards specific\naccretion modes such as the \"Seyfert mode\" \\citep{hop06} or specific\nevolutionary stages in their fueling such as pre- or post-mergers.\nFortunately, we can test the later since merger-driven models\n\\citep{hop08a} predict vastly different properties (i.e., SFRs,\nmorphology) of the hosts of AGNs before and after the coalescence of\nmassive disk galaxies.\n\nWe list in Table~\\ref{sample} the statistics of the parent galaxy\nsample and those hosting X-ray selected AGN.  The redshift and X-ray\nluminosity distribution of the full AGN sample (152) is shown in\nFigure~\\ref{lx_z} up to $z=1.02$.  Symbols denote their optical\nspectral properties with 82\\% lacking broad ($FWHM>1000$ km s$^{-1}$)\nemission lines and most characterized as narrow emission-line\ngalaxies.  As further detailed in subsequent sections, we isolate\nvarious subsamples based on redshift, galaxy mass, and AGN luminosity\nto optimize our analyses.  For instance, as described above, we\nenforce an upper limit to the X-ray luminosity\n($log~L^{max}_{X}=43.7$) of the AGN when measuring a quantity with\nrespect to its host galaxy.  Also, we slightly increase our minimum\nX-ray luminosity ($log~L^{min}_{X}$=42.48) when determining the\nfraction of galaxies hosting AGN to ensure that a significant parent\nsample of zCOSMOS galaxies are capable of detecting each AGN to avoid\neffects based on limited statistics\\footnote{Specifically, an\noverabundance of AGN at $z\\sim0.35$ falling at the flux limit of the\n$XMM$-Newton observations, seen in Figure~\\ref{lx_z}, have an adverse\neffect on our determination of the fraction of galaxies hosting an\nAGN, due to the limited sample of zCOSMOS galaxies capable of\ndetecting AGN with $L_X\\sim10^{42}$ erg s$^{-1}$, that disappears when\nimplementing a slightly higher selection on luminosity.}.\n\n\\subsection{Stellar mass measurements}\n\nStellar masses, including rest-frame absolute magnitudes (AB system;\n$M_U$, $M_V$), are derived from fitting stellar population synthesis\nmodels from the library of \\citet{bc03} to both the broad-band optical\n\\citep[CFHT: $u$, $i$, $K_s$; Subaru: $B$, $V$, $g$, $r$, $i$,\n$z$;][]{capak07} and near-infrared \\citep[$Spitzer$\/IRAC: $3.6 \\mu $,\n$4.5 \\mu $;][]{sand07} photometry using a chi-square minimization for\neach galaxy.  The measurement of stellar mass (M$_{*}$) includes (1)\nthe assumption of a Chabrier initial mass function, (2) a star\nformation history with both a constant rate and an additional\nexponentially declining component covering a range of time scales\n($0.1 < \\tau < 30$ Gyr), (3) extinction ($0<A_V<3$) following\n\\citet{ca00}, and (4) solar metallicities.  Further details on mass\nmeasurements can be found in \\citet{bo08} and \\citet{me08}.  For\ncompatibility with the star formation rate calibration of \\citet{mo06}\nimplemented in subsequent sections, we convert all masses to an\nequivalent based upon a Salpeter IMF by applying a multiplicative\nfactor of 1.7 \\citep{po07}.  We show the mass distribution for the\nparent galaxy population, including those harboring X-ray selected\nAGN\\footnote{It is worth mentioning that there may be a potential\nproblem that galaxies with even moderate-luminosity AGN may have\ninaccurate mass estimates; a bluer continuum will essentially reduce\nthe stellar age and hence lower the mass measurements since the\nderived mass-to-light ratio depends strongly on the spectrum.}  with\n$log~L_{0.5-8.0~{\\rm keV}}>42.0$, as both a function of redshift\n(Figure~\\ref{selection}) and rest-frame color ($U-V$;\nFigure~\\ref{color_mass}).  All masses are expressed in solar units\n(M$_{\\sun}$) throughout this work.\n\n\\begin{figure}\n\\epsscale{1.2}\n\n\\plotone{f2.eps}\n\\caption{Stellar-mass versus redshift for 7543 zCOSMOS galaxies (small\ngrey circles). The box marks the mass-selected subsample of galaxies\n($log~M>10.6$; $0.48<z<1.02$) for which (specific) star formation\nrates are determined using [OII]$\\lambda$3727.  Galaxies hosting X-ray\nselected AGN are further marked by a larger black circle (filled:\n$42.0<log~L_{0.5-10~{\\rm keV}}<43.7$; open: $log~L_{0.5-10~{\\rm\nkeV}}>43.7$).}\n\n\\label{selection}\n\\end{figure}\n\n\\begin{figure*}\n\\includegraphics[angle=90,scale=0.75]{f3.eps}\n\n\\caption{Rest-frame color $U-V$ versus stellar mass split into three\nredshift intervals for galaxies and AGN with symbols described in\nFigure~\\ref{selection}.  The vertical line marks our imposed mass\nlimit.}\n\n\\label{color_mass}\n\\end{figure*}\n\nWe determine a minimum mass threshold that all galaxies must satisfy\nup to $z\\sim 1.0$.  The mass limit is set in order to ensure a fairly\ncomplete representation of both blue and red galaxies at all redshifts\nconsidered.  In Figure~\\ref{color_mass}, it is clearly evident that\nthe mass limit of $log~M=10.6$ is essentially imposed by the red\ngalaxy population at $z\\gtrsim0.8$ (rightmost panel).  The lack of red\ngalaxies below this limit is due to our initial selection on apparent\nmagnitude.  \\citet{me08} estimate based on a series of mock catalogs\nfrom the Millennium simulation that the zCOSMOS 'Bright sample' is\nessentially complete for galaxies with $log~M\\approx10.6$ at $z=0.8$\nwhile the completeness drops to $\\sim50\\%$ at $z=1$.  In total, we\nhave a sample of 2540 galaxies ($0.1<z<1.02$) above this mass limit of\nwhich 105 host X-ray selected AGN (see Table~\\ref{sample} for\nstatistics regarding subsamples employed herein.).\n\n\n\n\\section{[OII] as a star-formation rate indicator and the AGN contribution}\n\\label{sfr_measure}\n\nOur primary aim is to use the emission line [OII]$\\lambda$3727 to\nmeasure the SFRs of a well-defined sample of galaxies including those\nhosting AGN, as commonly used for star-forming galaxies at these\nredshifts \\citep[e.g., ][]{ke04,co07}.  The availability of [OII]\nwithin the spectral window of the zCOSMOS 'bright' program requires\nour sample of galaxies to fall within redshift range $z=0.48-1.02$.\nThe [OII] line is one of a suite of spectral features (e.g.,\nH$\\alpha$$\\lambda$6563, H$\\beta$$\\lambda$4861, [OIII]$\\lambda$5007)\nmeasured by an automated routine \"platefit\\_vimos\" \\citep{lam08} that\nsimultaneously fits all lines with a gaussian function to determine\nline flux and equivalent widths for the entire zCOSMOS 10k sample.\nEmission line fluxes are corrected for slit loss based on a comparison\nof their spectroscopic and photometric ($i_{ACS}$) magnitudes.  Line\nmeasurements with a significance less than 1.15$\\sigma$ are quoted\nhere as upper limits.  \n\nAn assessment of the AGN contribution to the [OII] emission line is\nrequired even though the production of this low ionization line has\nbeen shown to be relatively weak. \\citet{cr02} postulate, based on a\ncomparison of the equivalent width of [OII] in the composite 2dF QSO\nspectrum to that in a composite 'normal' galaxy spectrum, that [OII]\nis mainly produced by star-formation in the host galaxies of AGN over\na wide range of absolute magnitude.  The [OII] strength has been\nobserved to be $\\sim10-30\\%$ of the [OIII]$\\lambda$5007 emission line\nflux \\citep{fe86,ho05,ki06}.  \\citet{ho05} has explored the\nfeasibility of using [OII] as an indicator of ongoing star formation\nfor a sample of PG quasars that in many cases provided only an upper\nlimit.  Upon further investigation using emission line ratio\ndiagnostics, \\citet{ki06} conclude based on a sample of $\\sim3600$\ntype 1 AGN ($z<0.3$) selected from the SDSS that [OII] emission is\nmainly attributed to AGN photoionization and does not have a strong\nHII component.    \n\nThe aforementioned results do not rule out the potential effectiveness\nof using [OII] as a SFR indicator for different samples of AGNs such\nthose having higher luminosities, significant intrinsic obscuration or\nbeing at higher redshifts.  For example, the stellar populations of\n'strong' type 2 AGN from the SDSS have young ages similar to\nlate-type galaxies \\citep{ka03b} based on the age indicator $D_n4000$.\nWe illustrate that this is likely related to the presence of ongoing\nstar formation.  In Figure~\\ref{emlines}, we show the [OII]\/[OIII]\nratio for type 1 (panel $a$) and type 2\\footnote{The luminosity\nselection of type 2 AGNs from the SDSS is equivalent to the\ndefinition of 'strong' AGN ($L>10^7$ L$_{\\sun}$) given in\n\\citet{ka03b}.  This luminosity selection, based on\nextinction-corrected values, guarantees that the sample is dominated\nby Seyfert galaxies comparable to the luminosities of the zCOSMOS\nAGN.}  (panel $b$;$L_{\\rm [OIII]}>10^{40.58}$ erg s$^{-1}$) AGNs from\nthe SDSS having $z<0.3$.  The data for 25000 type 2 AGNs is taken\nfrom the high level products available from MPA based on the SDSS DR4\nrelease; we require a $S\/N>3$ detection for [OII], H$\\beta$, [OIII]\nand H$\\alpha$.  First of all, it is worth highlighting that the\nbest-fit linear relation to the type 1 AGN (panel $a$) illustrates a\nscenario where [OII] emission scales with [OIII] ($log~L_{\\rm\n[OII]}\\propto 0.36\\times L_{\\rm [OIII]}$) but not with a one-to-one\ncorrespondence.  Even without correcting for extinction, we see that\nthere is evidence (Fig.~\\ref{emlines}$b$) for elevated [OII]\/[OIII]\nfor type 2 AGNs as seen by the flatter slope of the best-fit relation\n(slanted dashed line; $log~L_{\\rm [OII]}\\propto 0.58\\times L_{\\rm\n[OIII]}$) compared to that of the type 1 AGNs.  The change in slope\nappears to signify that the strength of an additional component to\n[OII] (i.e., star formation) increases with AGN luminosity possibly\nrelated to the results of \\citet{ka07} where a decline in stellar age\nis seen for AGNs with higher mass accretion rates.  Finally, we see\nconclusively that the AGN host galaxies in zCOSMOS, while having\nhigher [OIII] luminosities, have elevated [OII]\/[OIII] ratios compared\nto type 1 AGNs (panel $a$), and exhibit a similar slope to the type 2\nAGNs in the SDSS and slightly further enhanced [OII] emission.\n\n\\begin{figure*} \n\n\\includegraphics[angle=90,scale=0.75]{f4.eps} \n\n\\caption{Emission-line properties of AGN.  ($a$) Observed (i.e., no\nextinction correction) line ratio [OII]\/[OIII] versus [OIII]\nluminosity for zCOSMOS galaxies ($log~M>10.6$; small green points) and\nthose hosting AGN shown by larger red circles.  For comparison,\nemission line properties of type 1 AGN from the SDSS \\citep[$z<0.3$;\n][]{ki06} are marked by the small black dots that are further\ncharacterized by the best fit linear relation (solid line) and mean\nratio $<[OII\/[OIII]>=0.27$ (open white square).  The dashed horizontal\nline denotes our single assumption for the value of a purely AGN\ndominated [OII]\/[OIII] ratio (see text for further details).  ($b$)\nSame as panel $a$ but with type 1 AGNs replaced by type 2 AGNs from\n\\citet{ka03b}.  The slanted lines are the best fit linear relation to\ntype 1 (solid) and type 2 (dashed) AGNs.  ($c$) Histogram of the\nextinction-corrected [OII]\/[OIII] distribution for the type 1\/2 AGN\n(short dash\/long dash) from the SDSS and our zCOSMOS sample (solid)\nall having $log~L_{\\rm [OIII]}>41.3$.}\n\n\\label{emlines} \n\\end{figure*} \n\nWhen correcting for internal extinction based on the Balmer\ndecrements, we find that the type 2 AGNs have a significantly higher\ndistribution of [OII]\/[OIII] than the type 1s (Fig.~\\ref{emlines}$c$).\nThis further suggests that star formation is more prevalant in the\ntype 2 AGN population compared to those with an observable broad line\nregion; such a hypothesis has been put forth by \\citet{ki06} to\nexplain the [OII] emission ($<[OII]\/[OIII]>=-0.12$) evident in the\nmore luminous, type 2 QSOs \\citep{za03}.  To further justify our use\nof [OII], we demonstrate in Figure~\\ref{type2_sdss} that a correlation\nbetween total [OII] emission (HII+AGN) and the spectral index\n$D_n4000$ exists and is in agreement with significant levels of\nongoing star formation seen in the hosts of type 2 AGN \\citep{gu06,yan06}.\n\n\\begin{figure}\n\\epsscale{1.0}\n\\plotone{f5.eps}\n\n\\caption{Relation between $D_n4000$ and (a) [OII] luminosity and (b)\n[OII]\/[OIII] ratio for 4609 luminous($L_{\\rm [OIII}> 10^7$\nL$_{\\sun}$), type 2 AGN from the SDSS.  In panel $b$, the median\n[OII]\/[OIII] ratio is shown (large red circles) in bins of $D_n4000$\nhaving a width of 0.1.}\n\\label{type2_sdss}\n\\end{figure}\n\nFor our purpose, we use the higher ionization line [OIII]$\\lambda5007$\nwhen present in our spectra, and an empirical relation between [OII]\nand [OIII] to statistically remove the component of the detected [OII]\nline that can be attributed to the AGN.  This entails an assumption\nthat the [OIII] line is purely of AGN origin \\citep{ka03b} that may\nlead us to underestimate SFRs due to a significant stellar\ncontribution but only in very few cases since the [OIII] luminosities\nof Seyfert galaxies in SDSS, and also zCOSMOS AGN hosts (see\nFigure~\\ref{emlines}$a$), are substantially higher than that of\ntypical HII galaxies \\citep{ke06}.  \\citet{ki06} find that the median\nvalue of the observed [OII]\/[OIII] ratio is 0.27 with significant\ndispersion (see Fig.~\\ref{emlines}$a$).  We chose to use a slightly\nlower ratio ([OII]\/[OIII]=0.21) that is the mean value for type 1 AGN\nin the SDSS sample (M. Kim private communication) with\n$log~L_{[OIII]}> 41.5$, a luminosity regime similar to our zCOSMOS AGN\nsample.  This higher luminosity cut guarantees that the ratio best\nreflects that produced by the AGN, while effectively minimizing a\ncontribution from HII regions.  This results in slightly higher SFRs\nfor our AGN sample than if we chose to use the mean of the entire type\n1 SDSS sample but our final results in this study are consistent using\neither value.  We make no attempt to use a luminosity-dependent\ncorrection, as inferred by the SDSS type 1 AGN, since a simple linear\nrelation may not be evident at higher luminosities\n($log~L_{[OIII]}>42$).  Also, there is no need to correct this\nrelation for extinction given the low levels of dust attenuation\n\\citep[$A_V\\approx0.2$ mag;][]{ki06} observed in these type 1 AGN.  We\nstrongly note, given the high dispersion in the [OII]\/[OIII] ratio for\ntype 1 SDSS AGN, that this is only a statistical correction applied to\nthe population as a whole to infer the global or mean properties of\nthe sample; the individual measurements of [OII] strength associated\nwith star formation in a particular galaxy are expected to be\ninaccurate.  We further note that even with this correction the more\nluminous AGN tend to have higher SFRs as evident by the best-fit\nrelation for type 2 AGNs in the SDSS (Fig~\\ref{emlines}$b$;\n$log~L_{\\rm [OII]}\\propto 0.37\\times L_{\\rm [OIII]}$).  In addition,\nwe assess the impact of the dispersion in the [OII]\/[OIII] ratio for\ntype 1 SDSS AGNs by assuming a normal distribution of the line ratio\n($<log [OII]\/[OIII]>=-0.69$; $\\sigma=0.25$) and perform many\niterations in our determination the distribution of SFR.\n\nWe consider extinction due to dust as an important factor in\ndetermining the component of the observed [OII] emission attributed to\nAGN photoionization given that our sample is predominantly composed of\ntype 2 AGN (see Fig.~\\ref{lx_z}).  An extinction-corrected [OIII]\nluminosity is found for each zCOSMOS AGN with detected [OIII] line\nemission using $A_V=0.8$.  Since most of our sample does not have\nH$\\alpha$ and H$\\beta$ within our observed spectral window, we chose\nto implement a level of extinction\\footnote{This amount of optical\nattenuation is similar to that of X-ray selected AGN; we find a median\n$A_V$ of 0.98 based on the Balmer decrements of 20 AGN in our sample\nwith significant $H\\alpha$ and $H\\beta$ line measurements (Mainieri et\nal. in preparation).} based on the mean Balmer decrement of type 2\nSeyferts in SDSS \\citep{ke06}.  This same level of attenuation is then\nreapplied to the inferred AGN component to [OII], based on the\nempirical relation mentioned above, and subtracted from the observed\n(i.e. no dust correction) [OII] emission line luminosity to provide an\nestimate free of any AGN contribution.  We are confident that this\nmethod is applicable to the zCOSMOS sample given that the [OII]\nluminosities of AGN hosts are all systematically higher\n($\\sim3-4\\times$) than that expected from gas photoionized by an AGN\nas indicated by the SDSS type 1 AGN (Fig.~\\ref{emlines}$c$) of similar\nluminosities ($log~L_{\\rm [OIII]}>41.3$).  Also, we highlight that the\nenhancement of [OII] relative to [OIII] for our zCOSMOS AGN is not\npurely induced by our extinction corrections as evident in observed\nrelation (i.e., no extinction correction; Fig.~\\ref{emlines}$a,b$).\n\n\\begin{figure} \n\\epsscale{1.15}\n\\plotone{f6a.eps}\n\\epsscale{1.05}\n\\plotone{f6b.eps} \n\n\n\\caption{($a$) Extinction-corrected [OIII]$\\lambda$5007 luminosity\nversus X-ray luminosity (2-10 keV) for AGN.  A clear correlation is\nevident as shown by the best-fit linear relation to these data (solid\nline) and local Seyferts \\citep[dotted line; ][]{pa06}.  ($b$)\nFraction of the [OII] line luminosity attributed to an AGN and split\nby the method of determining the AGN contribution.  The number\ndistribution of galaxies with detected [OIII] emission, used to\nestimate the amount of [OII] due to AGN activity, is given by the\nsolid histogram.  Galaxies without [OIII] detections are indirectly\nassessed by their hard X-ray luminosity (dashed histogram).  There is\nno evidence for a systematic offset between both methods.}\n\n\\label{o3_lx} \n\\end{figure} \n\n\nFor AGN host galaxies at $z\\gtrsim0.8$, we need to modify our method\nsince [OIII] is no longer within our observed spectral window.\nFortunately, we can utilize the strong correlation between hard X-ray\n(2-10 keV) and [OIII] luminosity \\citep{he05,pa06} to estimate the AGN\ncontribution for these cases.  It is important to highlight that our\nAGN are X-ray selected by the 0.5-10 keV band thus limiting the\ninclusion of more heavily absorbed type 2 AGN such as the\nCompton-thick population \\citep{fi08}, for which this correlation is\nless evident if the X-ray luminosities are not corrected for X-ray\nabsorption \\citep{he05} as is the case here.  In\nFigure~\\ref{o3_lx}$a$, we plot the [OIII] line luminosity, corrected\nfor extinction as done in the previous paragraph, as a function of the\nrest-frame 2-10 keV luminosity for AGN in the zCOSMOS sample with\nredshifts over a larger baseline ($0.2<z<1.0$).  We use these lower\nredshift AGN in order to provide higher statistics to measure the\nbest-fit linear relation between these two quantities.  We determine\nthe best fit $L_X-L_{[OIII]}$ relation (Equation~\\ref{lx-o3}) for our\nsample by implementing a bivariate linear regression (EM) algorithm\nwith ASURV \\citep[Survival Analysis for Astronomy package Rev. 1.2;\n][]{la92} that considers limits as well as detections.\n\n\\begin{equation}\nlog~L_{[OIII]}=(0.729\\pm0.101)~log~L_{2-10~keV}+(10.307\\pm4.381)\n\\label{lx-o3}\n\\end{equation}\n\n\\noindent This relation (Fig.~\\ref{o3_lx}$a$), as shown by the solid\nline, is similar to previous studies based on a sample of low redshift\nAGN \\citep[e.g.,][]{pa06} shown by the dotted line.  We find a nearly\nequivalent slope although a higher normalization (i.e., larger values\nof $L_{{\\rm [OIII]}}$ for a given $L_{2-10~{\\rm keV}}$).  The strong\nlinear relation (Pearson correlation coefficient $r=0.63$) strengthens\nour assumption that most of the [OIII] line luminosity is due to AGN\nphotoionization.  Any significant contribution from stars to the\n[OIII] luminosity would increase our derived star formation rates in\nAGN hosts.  In Figure~\\ref{o3_lx}$b$, we demonstrate that there are no\nsystematic offsets by introducing this indirect probe of the [OIII]\nline luminosity; both methods essentially remove a similar fraction\n($\\lesssim 60\\%$) of the [OII] luminosity attributed to AGN\nphotoionization.\n\nWe determine SFRs using the empirical calibration of \\citet{mo06}.\nThis relation considers interdependent factors (i.e., dust extinction,\nmetallicity, and ionization) to provide a SFR tracer that is\napplicable to samples for which these quantities are inadequately\nknown.  To derive the following empirical relation, we fit a linear\nrelation to the bright end ($log~L(B)>9.5$) of the dependence of SFR\non absolute $B$-band magnitude \\citep[$M_B$; see Figure 19 of\n][]{mo06}:\n\n\\begin{equation}\nlog~SFR({\\rm [OII]})=log~L_{{\\rm [OII]}}-41-0.195 \\times M_{B}-3.434\n\\label{eq:sfr}\n\\end{equation}\n\n\\noindent The AGN component to the emission line luminosity\n$L_{[OII]}$ is removed as described above and no correction for dust\nextinction is applied that is in essence an assumption that star\nformation is external to a dusty NLR and circumvents the potential\nproblem that extinction of the nuclear region may differ from that of\nHII regions \\citep[see][]{gu06}.  We do not attempt to adjust (i.e.,\nlower) the upper limits on nondetections by considering the\ncontribution of an AGN.  It is worth noting that an additional caveat\nof our method is that there is an underlying assumption that the\nmetallicity and extinction of galaxies with or without AGN are\nsimilar.\n\n\\section{Physical properties of AGN host galaxies}\n\n\\subsection{Stellar masses}\n\\label{mass}\n\n\\begin{figure}\n\\epsscale{1.1}\n\\plotone{f7.eps}\n\\caption{Stellar mass distribution of galaxies hosting AGN in the\nredshift range $0.1<z<0.5$.  Fifty-two galaxies ($log~M_*>9.5$) with\nX-ray selected AGN ($42<Log~L_{0.5-10~{\\rm keV}}<43.7$) are shown by\nthe solid histogram.  The observed parent distribution of 3356\ngalaxies (dotted histogram) is renormalized to match the AGN\ndistribution.  The data points represent the AGN fraction\n($42.48<Log~L_{0.5-10~{\\rm keV}}<43.7$; scale on the right hand\nvertical axis) in fixed mass intervals (shown by the horizontal bars\ncentered on each data point) with $1\\sigma$ error bars.}\n\n\\label{mass_frac}\n\\end{figure}\n\nWe measure the fraction of galaxies that host X-ray selected AGN as a\nfunction of their stellar mass to determine the characteristic mass\nabove which AGN activity is most prevalent.  First of all, it is\napparent in Figure~\\ref{selection} that galaxies hosting X-ray\nselected AGN are preferentially massive ($log~M\\gtrsim10.5$) as\nevident over the full redshift range and in agreement with most\nstudies to date \\citep[e.g.,][]{ka03b,bu08}.  For this exercise, we\nconsider the redshift interval $0.1<z<0.5$ for which we have a high\ndegree of completeness for galaxies over a wide mass baseline ($log~M\n\\gtrsim 9.5$).  We follow the technique discussed in $\\S$~3.1 of\n\\citet{le07} to determine the AGN fraction of our parent population of\ngalaxies that accounts for the spatially varying sensitivity limits of\nthe {\\it XMM} observations of the COSMOS field \\citep[see Fig.~17\nof][]{cap07}.  The necessity of this approach is demonstrated in\nFigure~\\ref{lx_z}, which shows the limiting X-ray luminosity as a\nfunction of redshift for the entire galaxy sample.  Even though the\nsensitivity of the $XMM$ coverage is remarkably uniform, some\ndispersion is present as shown by the relatively narrow distribution\nof the X-ray upper limits at each redshift.  To properly account for\nthe luminosity-redshift relation, we determine the contribution of\neach AGN separately to the total fraction.  The AGN fraction ($f$; see\nequation{~\\ref{fraction} below) is determined by summing over the full\nsample of AGN ($N$) with $N_{\\rm gal,i}$ representing the number of\ngalaxies in which we could have detected an AGN with X-ray luminosity\n$L^{i}_{\\rm X}$.  The different sampling rates, based soley on slit\nplacement, of the galaxies ($S_{gal}$) and AGN (i.e., X-ray\nsources\\footnote{In many cases, X-ray sources have redshifts based on\na random placement of a slit for which the sampling reflects the\noverall galaxy population ($S_x=S_{gal}$)}, $S_x$) are taken into\naccount in equation~\\ref{fraction}.  We estimate the associated\n1$\\sigma$ error (equation~\\ref{fraction_error}) using binomial\nstatistics where $N^{eff}_{agn}$ is the number of AGN that would be\ndetected if all galaxies have the same limiting X-ray sensitivity and\nthe sample of AGN was randomly selected.  Here, we only consider AGN\nwith $42.48<log~L_{0.5-10 {\\rm keV}}<43.7$.  As previously mentioned,\nthe lower limit ensures that we have a statistically significant\nsample of parent galaxies ($\\gtrsim700$) that could host each AGN\nwhile the upper limit restricts the sample to low-to-moderate\nluminosities thus securing the accuracy of their host-galaxy masses.\nWe refer the reader to \\citet{si08a} for further details and results\nemploying this method.\n\n\\begin{equation}\nf=\\sum_{i=1}^{N}\\frac{1\/S_x}{N_{\\rm gal,i}\/S_{gal}}\n\\label{fraction}\n\\end{equation}\n\n\\begin{equation}\n\\sigma^2=N^{eff}_{agn}\\times(N_{gal}-N^{eff}_{agn})\/N^3_{gal}\n\\label{fraction_error}\n\\end{equation}\n\nIn Figure~\\ref{mass_frac}, we plot the fraction of galaxies that host\nAGN and the number distribution (Table~\\ref{sample}; Sample A) of both\ngalaxies and those with AGN over a slightly wider luminosity interval\n($42.0<log~L_{0.5-10 {\\rm keV}}<43.7$).  The absolute fraction is low\n($\\sim1-4\\%$) due to our restrictions on X-ray luminosity.\n\\citet{ka03b} measure a fraction $\\sim5\\times$ higher although they\ninclude in their sample AGN $\\sim80\\times$ fainter.  The difference in\nthese fractions is mainly due to the steep faint-end slope of the\nluminosity function at low redshifts \\citep{sh06}.  Here, we are\nmainly interested in the relative change of the AGN fraction with\ngalaxy mass.  Clearly, we see that the fraction of galaxies that host\nAGN monotonically increases with stellar mass similar to results based\non obscured \\citep{ka03b} and radio-loud \\citep{be05} AGN in the SDSS.\n\n\\subsection{Star formation rates}\n\nIn Figure~\\ref{sfr1}, we show the SFRs for 1820 zCOSMOS galaxies\n(Table~\\ref{sample}: Sample B\\footnote{The sample is slightly smaller\nthan that given in Table~\\ref{sample}; four AGNs have [OII] falling\noutside the observed spectral window that can vary slightly from\nslit-to-slit.}) including 43 of them that host moderate-luminosity AGN\n($42.0<log~L_{0.5-10.0~{\\rm keV}}<43.7$).  The results are presented\nbefore (panels a and b) and after (panels c and d) the correction for\nthe AGN contribution to the [OII] luminosity based on a single-value\n(0.21) of the observed, median [OII]\/[OIII] ratio of type 1 AGNs in\nthe SDSS.  First of all, we see that 88\\% of AGNs have detectable\n[OII] emission\\footnote{The fraction of AGN with detected [OII]\nemission is 68\\% ($S\/N>2$) and 61\\% ($S\/N>3$) depending on the given\nline significance.  We have confirmed our results based on the lower\nS\/N ratio (1.15) with those based on these smaller, high confidence\nsamples of AGN.} (panel $a$).  It is apparent by comparing the SFR\ndistribution for AGN hosts in panels $b$ and $d$ that a significant\nshift in the distribution occurs when removing the AGN contribution to\nthe [OII] emission line.  Even so, we find that the SFRs of AGN host\ngalaxies are almost exclusively between 1-100 M$_\\sun$ yr$^{-1}$\n(Fig.~\\ref{sfr1}$c$,$d$), a range consistent with analytic models of\nAGN hosts with star-forming disks \\citep{bal08} and supportive of\nconstraints on the cosmic IR background \\citep{bal07}.\n\n\\begin{figure*}\n\\epsscale{0.9}\n\\plotone{f8.eps}\n\n\\caption{Star formation rates ($M_{\\sun}$ yr$^{-1}$) versus stellar\nmass for galaxies ($0.48<z<1.02$; $log~M_*>10.6$) before (a) and after\n(c) the removal of the AGN contribution.  Measurements are shown by\neither a small black (emission-line galaxy) or large red (AGN;\n$42.0<log~L_{0.5-10~{\\rm keV}}<43.7$) circle.  Upper limits are shown\nas an arrow with their color descriptive of the type as given above.\n($b$, $d$) For ease of comparison, the distributions, including those\nwith upper limits, are shown by either a solid (AGN hosts) or dashed\n(all galaxies) histogram with the full galaxy sample scaled down to\nmatch the AGNs.}\n\n\\label{sfr1}\n\\end{figure*}\n\nTo quantitatively test whether the SFR distribution of AGN hosts\ndiffers from that of the underlying galaxy population, we have\ndetermined the SFR distribution, including those with upper limits,\nwhile also considering the dispersion in the AGN contribution to the\nobserved [OII] emission-line luminosity.  We assume a log normal\ndistribution of the [OII]\/[OIII] ratio with parameters given in\nSection~\\ref{sfr_measure}.  Based on one thousand iterations, we\ndetermine the mean SFR distribution and perform statistical tests on\neach individual iteration compared to the parent galaxy population.\nThe results of this exercise are shown in Figure~\\ref{sfr2} for two\nAGN samples of differing luminosities.  First, we consider AGN, as\ndone above, that fall within the luminosity interval $42<log~L_X<43.7$\nfor which we have high confidence in our well-matched, mass-selected\nparent sample of galaxies.  Based on this limited sample of AGNs, we\ncannot significantly discriminate between the two distributions\n(Fig.~\\ref{sfr2}$a$) given the results of the individual K-S tests\n(panel $b$).  Although, we do find that the median SFRs shown in panel\n$c$ are for the most part higher than that of the full galaxy sample\n(4.7 M$_\\sun$ yr$^{-1}$) in each iteration.\n\nWe are able to use a less restrictive sample that includes higher\nluminosity AGN ($log~L_X>43.7$) thus improving our statistics although\nat the expense of maintaining reliable stellar masses across our\nsample.  In Figure~\\ref{sfr2}$d-f$, we present the equivalent analysis\nfor AGNs with $log~L_X>42$.  Based on this larger sample, we see that\nthe SFR distribution of AGN hosts is shifted to higher values than the\noverall population (panel $d$).  The K-S tests on the individual\ndistributions now reject the null hypothesis (i.e., distributions are\nequivalent) at the $\\sim99\\%$ level ($>2.5\\sigma$) in essentially all\niterations (panel $e$).  Moreover, the median SFRs are all\nsystematically higher ($SFR\\sim9.5$ M$_\\sun$ yr$^{-1}$) than that of\nthe full galaxy sample.  We further point out that the SFR\ndistribution of AGN hosts while being elevated in comparison to the\nunderlying massive galaxy population is essentially equivalent to\nthose forming stars (i.e., emission-line galaxies).  Therefore, we\nconclude that a plentiful gas reservoir is a necessary ingredient for\nthe fueling of AGN as indicated by the presence of significant star\nformation with rates reaching up to $\\sim100$ M$_\\sun$ yr$^{-1}$.\n\n\\begin{figure*}\n\n\\includegraphics[angle=90,scale=0.75]{f9.eps}\n\n\\caption{Star formation rates of AGN hosts with dispersion in the\napplied correction for the AGN contribution.  The two rows are given\nfor AGN spanning a different luminosity range: ($a-c$)\n$42<log~L_X<43.7$, ($d-f$) $log~L_X>42$.  The left panels ($a$, $d$),\nequivalent to those in Figure~\\ref{sfr1}, show the mean distribution\nof SFR, including upper limits, based on 1000 iterations.  The middle\npanels ($b$, $e$) give the probability distribution based on a KS\ntests that the two samples are equivalent.  The right panels ($c$,\n$d$) display the distribution of the median SFR for each iteration\nwith the median SFR (4.73 $M_{\\sun}$ yr$^{-1}$) of all galaxies\ndepicted by the vertical dashed line.}\n\n\\label{sfr2}\n\\end{figure*}\n\n\nWe investigate how star formation in galaxies hosting AGN is evolving\ncompared to that of the parent galaxy population.  In\nFigure~\\ref{sfr_evol}, we show SFR as a function of redshift for\ngalaxies with $log~M>10.6$ and those hosting AGN (Table~\\ref{sample};\nSample B).  A systematic shift of the distribution of zCOSMOS galaxies\ntowards higher SFRs with increasing redshift is clearly evident with\nthose hosting AGN exhibiting a similar behavior.  Therefore, we find\nthat the SFRs of AGN hosts are dictated by that of the underlying\ngalaxy population thus solidifying similar evidence based on the color\nevolution of AGN hosts in the E-CDF-S \\citep{si08a}.  As a\nconsequence, ongoing star formation is most likely responsible for the\nsignificant population of AGN hosts over the redshift range\n$0.5\\lesssim z \\lesssim 1.4$ having blue rest-frame colors\n\\citep{sa04,bo07,na07} with evidence for such a trend remaining in\nplace for quasar hosts at higher redshifts $1.8<z<2.8$ \\citep{ja04b}.\n\nIt is illuminating to compare the SFRs from zCOSMOS to AGN at lower\nredshifts.  For consistency with zCOSMOS, we measure SFRs of the type\n2 AGN from SDSS \\citep[$z<0.3$;][]{ka03b} using the [OII]$\\lambda3727$\nand [OIII]$\\lambda5007$ emission line fluxes from \\citet{br04} to\ndetermine an AGN-corrected [OII] luminosity.  We further select only\nthose galaxies having stellar masses above our zCOSMOS limit\n($log~M_*>10.6$) based on the conversion factor (Kroupa to Salpeter\nIMF) given in \\citet{br04}.  Here, we use the calibration of\n\\citet{ke04} to derive SFRs\\footnote{We have confirmed that the\n\\citet{mo06} relation gives the same results.} in order to have\nconsistency with the SFRs measured for the type 1 SDSS sample\n\\citep{ki06}.  Initially, we demonstrate that the ongoing SFRs shown\nin Figure~\\ref{sfr_evol} are in agreement with the findings of\n\\citet{ka03b} based on $D_n(4000)$: weak AGNs ($L_{[OIII]}< 10^7\nL_{\\sun}$; blue points) have low SFRs ($\\sim$0.2 M$_\\sun$ yr$^{-1}$)\nwhile strong AGN ($L_{[OIII]}> 10^7 L_{\\sun}$; small red points) are\nmore actively forming stars ($\\sim$1 M$_\\sun$ yr$^{-1}$).  For ease of\nvisualization, we show the best-fit relation $log~SFR \\propto\nlog(1+z)$ for only the zCOSMOS galaxies with significant [OII]\nemission (upper limits are not considered; black curve) and those\nhosting AGN (red curve).  Remarkably, we further find that the SDSS\nAGN may be the low redshift analogs of the AGNs in the zCOSMOS survey\ngiven their close proximity to an extrapolation of the evolution of\nzCOSMOS galaxies.  In addition, the low-to-moderate levels of star\nformation ($\\approx 0.5-3$ M$_\\sun$ yr$^{-1}$) in type 1 SDSS AGN\n\\citep[shown roughly by the green triangle in\nFig.~\\ref{sfr_evol};][]{ki06} are in agreement with the type 2 AGN\nfrom SDSS and the aforementioned passive evolutionary scenario.  In\nlight of these results from the SDSS that effectively extend our\nredshift baseline, we reiterate our conclusion that the SFRs of AGN\nhost galaxies are reflective of the overall star formation history of\ngalaxies and provide no indication of the suppression or truncation\ndue to a mechanism related to the AGN itself.\n\n\\begin{figure*}\n\\epsscale{0.8}\n\\plotone{f10.eps}\n\n\\caption{Cosmic evolution of star formation.  At $z>0.48$, we show the\nSFR-$z$ distribution for all zCOSMOS galaxies with $log~M>10.6$ (small\nblack circles and crosses) and those hosting AGN with $log ~L_X>42$\n(73; large red circles and arrows).  The best-fit linear relation for\nthe emission-line objects is shown for both zCOSMOS populations\n(black: galaxies; red: AGN hosts) with an extrapolation to lower\nredshifts (dashed lines).  For comparison, we plot SFRs of AGN hosts\nfrom the SDSS with an equivalent selection on stellar mass; obscured\nAGN (type 2) from the sample of \\citet{ka03b} are shown with strong\nAGN ($log~L_{OIII}>40.5$) in red and those of lower luminosity in\nblue.  A large green triangle marks the mean value of the SFR for SDSS\ntype 1 AGN \\citep{ki06}.}\n\n\\label{sfr_evol}\n\\end{figure*}\n\n\\subsection{Stellar ages}\n\nTo complement our study of the ongoing SFRs of AGN hosts, we use the\nspectral index $D_n(4000)$ \\citep{ba99}, determined by our\n\"platefit\\_vimos\" routine for each galaxy in the zCOSMOS sample to\ninfer the age of the overall stellar population on longer timescales\n($>0.1$ Gyr).  This index is the ratio of the average flux density\n$F_{\\lambda}$ in the continuum bands 3850-3950 {\\rm \\AA} and\n4000-4100 {\\rm \\AA}.  This is essentially a measure of the strength\nof the 4000-{\\rm \\AA} break with galaxies having experienced a\nrecent episode of star formation exhibiting a smaller index due to the\npresence of young stars.  In Figure~\\ref{age}, we show the values of\n$D_n(4000)$ for galaxies with respect to their specific SFR. We\nimplement two mass limits ($log~M>10.6$: panels $a$, $b$;\n$log~M>11.1$: panels $c$, $d$) in order to check for consistency when\nthe underlying $D_n(4000)$ distribution of galaxies is significantly\ndifferent: the higher mass cut results in a distribution dominated by\nmore evolved galaxies as evident by its peak at $D_n(4000)\\sim1.8$.\nInitially, it is apparent (Fig.~\\ref{age}$a$) that there is a good\ncorrespondence between $D_n(4000)$ and specific SFR for all galaxies\nincluding those with AGNs.  We then use the relation given in\n\\citet{ka03a} that assumes an instantaneous burst model of\nsolar-metallicity to infer stellar age from the value of $D_N(4000)$\nas given by the scale bar in Figure~\\ref{age}a; this relation provides\nus with a rough assessment of the actual ages since many of the\nzCOSMOS galaxies including those with AGN may have had a more sedative\nexistence in their recent past.  As a result, we find that most\ngalaxies with AGN contain a young stellar component since 70\\% have\n$D_n(4000)<1.6$ ($age \\lesssim 2~ Gyr$; Fig~\\ref{age}a, c).\nFurthermore, we measure the fraction of galaxies hosting an AGN as a\nfunction of $D_n(4000)$ and demonstrate (Fig.~\\ref{age}$b$, $d$) that\na galaxy is more likely to have an accreting SMBH if there is a\nsufficient supply of gas, fully consistent with the mass measurements\nof HI \\citep{ho08} and CO \\citep{sco03} in AGN hosts, given the higher\nrate of occurance in galaxies with younger ages.  These results\neffectively extend such studies based on $D_n(4000)$ at low redshift\n\\citep{ka03b} up to $z\\sim1$.\n\n\\begin{figure*}\n\\epsscale{1.0}\n\n\\plotone{f11.eps} \n\n\\caption{($a$) Specific SFR versus the stellar age indicator\nD$_n$(4000).  Symbols are equivalent to those in Figure~\\ref{sfr1}a.\nAn approximate age scale is given based on Figure 2 of\n\\citet{ka03a}. ($b$) Number distribution and the fraction of galaxies\nhosting an AGN as a function of $D_n(4000)$.  (c, d) The equivalent\nplots are shown for a higher mass cut as given.  AGN activity is\nprimarily associated with young stellar populations over a broad range\nin host galaxy mass.}\n\n\\label{age} \n\\end{figure*}\n\n\nThere is a noteable discrepancy between our claim for a higher level\nof AGN activity for star-forming galaxies and the location of AGN\nhosts on the color-magnitude diagram.  Both X-ray selected\n\\citep{na07,si08a} and optically-selected AGN \\citep{ma07} exhibit a\nlow AGN fraction along the red sequence, enhanced activity in the\nintermediate region (i.e., \"green valley\") between the blue and red\ngalaxy populations, and a dropoff towards very blue galaxies.  On the\ncontrary, the mass-selected AGN sample of \\citet{ka03b} clearly have\nstellar properties very similar to late-type galaxies (see their\nFig. 14) and, thus a low fraction of AGN within the 'blue cloud'\nreported by the aforementioned studies is surprising.  Interestingly,\n\\citet{le08} find that the fraction of the stacked X-ray signal of\nlate-type galaxies attributed to AGN emission continuously rises with\nSFR.\n\nTo investigate this further, we measure the fraction of galaxies\nhosting AGN as a function of their rest-frame optical color ($U-V$)\nfor both a luminosity and mass-selected sample\n(Figure~\\ref{color_mag_mass}$a$, $b$).  Since we are not constrained\nby the use of [OII] for this exercise, we have extended the redshift\nbaseline $0.1<z<1.02$ to improve the statistics.  In the first panel,\nwe find that the fraction of galaxies with X-ray selected AGN is\nstrongly peaked in the \"green valley\" for the luminosity-selected\nsample (panel a) thus agreeing with the results of \\citet{si08a}.  In\ncontrast, the mass-selected sample, behaves similarly for colors\n$U-V>1.4$, but does not have a decline towards bluer colors $U-V<1.4$.\nThis is easily understood since blue galaxies are known to have lower\nmass-to-light ratios and the AGN fraction rises substantially with\nhost galaxy mass as demonstrated in Section~\\ref{mass}.  We conclude\nthat blue, star forming galaxies do have enhanced levels of AGN\nactivity and there is no evidence for a significant delay in the\nemergence of nuclear activity with respect to the onset of star\nformation.  This result is consistent with the recent findings of\n\\citet{si08a}, based the morphology of the host galaxy, that blue\n($U-V<0.7$), bulge-dominated ($n_{sersic}>2.5$) galaxies have the\nhighest incidence (21.3\\%) of AGN activity, and the blue rest-frame\ncolors of low redshift ($z<0.2$) quasars irrespective of their\nmorphology \\citep{ja04a}.  However, there remains the possibility that\nthe lower fraction of red galaxies hosting AGN is due to some form of\nself-regulating feedback that effectively reduces the AGN luminosity\nbelow our flux limit.  Nonetheless, these results lend no support for\na simple model prescription that attributes the truncation of star\nformation or the color evolution of galaxies to AGN feedback and bring\ninto question whether studies based on the optical emission line\ndiagostics of AGN activity miss a significant number of those residing\nin star-forming galaxies \\citep[see][for a discussion on this\ntopic]{sch07}.\n\n\\begin{figure*}\n\\epsscale{1.0}\n\n\\plotone{f12.eps}\n\n\\caption{Rest-frame color distribution $U-V$ of galaxies\n($0.1<z<1.02$) and the fraction hosting AGN.  We show the distribution\nfor 58 galaxies (solid histogram) selected by optical luminosity\n($M_V<-21.5$; panel a) and 65 galaxies above a fixed mass limit\n($log~M>10.6$; panel b) that host AGN.  For comparison, the\ndistribution for the parent galaxy population (dashed histogram) is\nnormalized to match the AGNs in each panel.  The decline in the AGN\nfraction towards the bluest colors (panel a) appears to be due to the\ninclusion of galaxies with low mass-to-light ratios that are not\npresent in the mass-selected sample (panel b).}\n\n\\label{color_mag_mass}\n\\end{figure*}\n\n\\section{AGN-star formation connection and co-evolution}\n\nHaving established an association between AGN activity and star\nformation, we are motivated to determine how closely these phenomena\nare related on a case-by-case basis that can potentially signify an\nunderlying causal connection.  The existence of such a relationship\nmay be realized given the recent findings that AGN accretion power as\nprobed by [OIII] luminosity is higher for galaxies with younger\nstellar populations \\citep{ka07}.  Ideally, we would like to\ninvestigate such a relation using a quantity more closely associated\nwith the accretion process, namely X-ray emission, known to originate\ncloser to the black hole and an indicator of the instantaneous star\nformation rate.\n\nTo do so, we have plot in Figure~\\ref{agn_gal_relations}a the SFRs of\nour AGN sample, as measured by [OII] strength, versus hard X-ray\nluminosity (2-10 keV) and inferred mass accretion rate while\nimplementing a bolometric correction as given in \\citet{ma04} and an\naccretion efficiency of 0.1.  We find that a weak correlation exists\nbased upon a Pearson correlation coefficient of 0.17 and a linear fit\nthat has a shallow slope with significant dispersion in its value\n($0.28\\pm0.22$).  We conclude that underlying complexities such as the\nefficiency of transferring gas to nuclear region over kiloparsec\nscales, and varying duty cycles for star formation and accretion may\ncontribute to the large dispersion in these relations.\n\n\\begin{figure}\n\n\\plotone{f13.eps}\n\n\\caption{AGN-galaxy relations: ($a$) SFR versus hard X-ray luminosity\nand mass accretion rate including the best fit relation.  Absorbed AGN\nwith detected star formation are further highlighted by an open box.\n($b$) Ratio of mass accretion to SFR versus redshift.  The horizontal\ndashed line marks the median ratio.  Measurements are shown by\na solid circle in both panels while limits either upper ($a$) or lower\n($b$) are given by an arrow.}\n\n\\label{agn_gal_relations} \n\n\\end{figure}\n\nThere have been some recent claims that an obscured phase, coupled\nwith enhanced star formation, may represent an early stage in the\nsubsequent evolution of AGN\n\\citep[e.g.,][]{pa04,al05,ki06,hop08a,po08}.  To test this scenario,\nwe have marked those AGN in Figure~\\ref{agn_gal_relations}a that have\nexcessive X-ray absorption ($N_{\\rm H} \\gtrsim 10^{22}$ cm$^{-2}$)\nbased on their hardness ratio [$HR=(H-S)\/H+S)> -0.2$] determined by\nthe X-ray counts in the soft (S: 0.5-2.0 keV) and hard (H: 2-10 keV)\nbands assuming a powerlaw spectrum with a photon index of 1.9 at an\neffective redshift of $z\\sim0.7$.  We find that the absorbed sources\nspan the same range of SFR as the unabsorbed sources.  A KS test give\na probability of 65\\% that both absorbed and unabsorbed AGN could be\ndrawn from the same parent population.  These results are not\ndependent on the chosen division in hardness ratio.  Therefore, we\nconclude that an amendment, such as the aforementioned evolutionary\nscenario, to the unification model \\citep{an93} for these\nmoderate-luminosity AGNs is not supported by our findings.  Although,\nthere remains substantive evidence that enhanced star formation may be\nassociated with nuclear obscuration for the more luminous QSOs\n\\citep[e.g.,][]{pa04,la07,za08} possibly due to different physical\nmechanisms for triggering mass accretion on SMBHs.\n\nIt is of much interest to determine the relative growth rate of a\ngalaxy and its central SMBH as a function of redshift in light of the\nwell-established local relation\n$M_{BH}\/M_{bulge}\\approx1.5\\times10^{-3}$ \\citep[e.g.,][]{mc02,ha04}.\nWe plot in Figure~\\ref{agn_gal_relations}b the ratio of mass accretion\nrate onto a SMBH to the SFR as a function of redshift.  The median\nvalue ($1.9\\times10^{-2}$) is roughly an order-of-magnitude higher\nthan the local ratio $M_{BH}\/M_{bulge}$.  This difference is likely\ndue to the varying timescales between SMBH accretion and star\nformation $\\Delta t_{BH}\/\\Delta t_{gal}\\approx M_{BH}\/M_{bulge} \\times\nSFR\/dM_{accr} dt^{-1}\\approx 0.1$.  We note that our SFR measurements\nmost likely include a significant disk component thus a more rigorous\nassessment is required and beyond the scope of this work.\nNonetheless, by assuming that star formation occurs over rougly a\ndynamical timescale $\\Delta t_{gal}\\approx10^9$ yr, the duty cycle for\nAGN activity ($\\approx10^8$ yr) is consistent with the\nluminosity-dependent model predictions of \\citet{hop08b} for SMBHs\nwith $M_{BH}\\sim10^8$ M$_{\\sun}$ and accreting above an Eddington\nratio of 0.01, both within a physical regime spanned by our sample\nassuming these AGNs have already settled on a SMBH - bulge relation at\ntheir respective redshifts.  Alternatively, if we assume a timescale\nfor galaxy growth to be the inverse of their sSFRs as shown in\nFigure~\\ref{age}, the AGN lifetimes can reach up to $\\sim10^{10}$ yr\nthat fully illustrates that these AGNs and their host galaxies are\nclose to being fully matured although have SMBH growth times at least\nan order-of-magnitude less than those in the SDSS \\citep{he04}.\nFurthermore, we find that there is no dependence of the ratio\n($dM_{accr} dt^{-1}\/SFR$) on redshift thus supporting a\nco-evolutionary scenario where both the average SFR and mass accretion\nrates onto SMBHs are rapidly declining with equivalent rates from\n$z\\sim1$ to the present possibly due to diminishing fuel supplies.\n\nThese comparisons suggest that beyond the local universe the $M_{BH} -\nM_{bulge}$ relation should display higher intrinsic dispersion given\nthe large spread over two orders of magnitude ($10^{-3}-10^{-1}$) in\nthe relation between accretion rate and SFR\n(Fig.~\\ref{agn_gal_relations}$b$).  \\citet{ro06} point out that AGN\nsamples at high redshift may have larger intrinsic scatter in the\nvelocity dispersion of their hosts due to their immature dynamical\nstate. The observational situation is so far unclear: limited AGN\nsamples at higher redshifts \\citep{wo06,wo08} show preferentially\nhigher black hole masses relative to that expected by the local $M -\n\\sigma$ relation, while other studies of low redshift AGNs, undergoing\nsubstantial rates of accretion ($L_{Bol}\/L_{Edd}>0.1$), have black\nhole masses lower than that of inactive galaxies of equivalent host\nmass \\citep{ho08} or luminosity \\citep{ki08}.  Notwithstanding the\nlarge observational biases \\citep{lau07}, it is possible that the\ncoeval growth of galaxies and their SMBHs may be intermittent with\neither stars or SMBHs growing somewhat faster or slower than each\nother, as indicated by our findings exemplified in\nFigure~\\ref{agn_gal_relations}$b$, while still resulting in a tight\nblack hole-bulge mass relation for inactive galaxies at $z=0$ as\nsuggested by \\citet{wo06}.\n\n\\section{Summary and conclusions}\n\nWe have utilized the zCOSMOS 10k catalog of galaxies with reliable\nspectroscopic redshifts to investigate the properties of those that\nhost AGN.  X-ray observations with $XMM$-Newton enable us to identify\n152 AGN, from a parent sample of 7543 zCOSMOS galaxies, that include\nthose with significant obscuration and of low optical luminosity.  The\nderived properties of the full zCOSMOS sample such as stellar mass,\nrest-frame colors, and spectral properties (i.e., emission line\nstrength, $D_n4000$) enable us to determine the prevalence of AGN\nactivity as a function of galaxies with the aforementioned\ncharacteristics.\n\nSpecifically, we measure the SFR of galaxies using the\n[OII]$\\lambda$3727 line luminosity.  We account for the contribution\nfrom the underlying AGN component most likely arising from the narrow\nline region by using the [OIII]$\\lambda$5007 luminosity and the\ntypical [OII]\/[OIII] ratio found from previous studies of AGN and\nquasars \\citep{ho05,ki06}.  The [OIII] line luminosity is measured\ndirectly from our spectra if present.  For the subsample with [OIII]\noutside our observed spectral bandpass, we infer the [OIII] strength\nfrom the hard (2--10 keV) X-ray luminosity and the well known\ncorrelation between these two quantities.\n\nOverall, we find that the two main requirements for a galaxy to host\nan actively, accreting SMBH are (1) its stellar mass and (2) a\nsignificant amount of gas content as inferred by the observed levels\nof star formation and the age of their stellar populations.  These\nfindings essentially extend those of the SDSS \\citep{ka03b} out to\n$z\\sim1$.\n\nWe particularly draw the following conclusions:\n\n\\begin{itemize}\n\n\\item We confirm with many previous studies that the fraction of\ngalaxies hosting AGN rises with increasing mass with most host\ngalaxies having $M_*>4\\times10^{10}$ $M_{\\sun}$.\n\n\\item The host galaxies of AGN have ongoing star formation with a\nbroad range of rates ($\\approx1-100$ $M_{\\sun}$ yr$^{-1}$) higher than\nthat of the overall massive ($log~M>10.6$) galaxy population and\nessentially equivalent to those forming stars (i.e., emission-line\ngalaxies).  The association of AGN activity and young stellar\npopulations is further substantiated by an observed increase in the\nfraction of galaxies harboring AGN at low values of the spectral index\n$D_n(4000)$ and blue rest-frame color $U-V$.\n\n\\item The enhancement of AGN in young, star-forming galaxies with\n$1.0<D_n(4000)<1.4$ and $0.8<U-V<1.5$ lessens the evidence for AGN\nprefering to reside in galaxies of intermediate colors and their role\nin galaxy evolution.  We demonstrate that previous studies based on\nluminosity-selected samples are misleading due to the inclusion of\ngalaxies with low mass-to-light ratios that are less likely to harbor\nAGN given their lower masses.\n\n\\item The SFRs of AGN hosts evolves with redshift in an equivalent\nmanner to the overall star-forming galaxy population.  This\nessentially brings the evidence for a co-evolution scenario between\naccretion onto SMBHs and the star-formation history of galaxies to a\ncloser physical scale (i.e., within the same galaxies).\n\n\\item A direct relationship between the consumption of gas into stars\nand that accreted onto SMBHs is weak, suggesting that additional\nphysical complexities or varying timescales may be inherent.  On\naverage, a co-evolution scenario for the overall population is clearly\nevident given the constancy of the ratio ($\\sim10^{-2}$) between mass\naccretion rate onto SMBHs and SFR with redshift possibly indicative of\ndepleting gas reservoirs from $z\\sim1$ to the present.  The\norder-of-magnitude increase in this ratio compared to the locally\nmeasured value of $M_{BH}\/M_{bulge}$, is consistent with an AGN\nlifetime substantially shorter than that of star formation.\nFurthermore, the significant dispersion in this ratio may be\nindicative of larger scatter in the BH-bulge relations at higher\nredshift.\n\n\\end{itemize}\n\nOur results put important constraints on physical models of AGNs and\ntheir evolution.  The considerable rates of star formation ($\\sim10$\nM$_\\sun$ yr$^{-1}$) in their hosts, coupled with the lack of\nstructural signs of galaxy interactions \\citep{gr05,ga08} and\ntimescales for accretion, indicate that a \"Seyfert mode\" of accretion,\ndriven by secular processes, is more likely than merger-driven models\nfor this class of moderate-luminosity AGN.  In particular, there is no\nindication of the suppression or truncation of star formation at\nlevels expected from models implementing AGN feedback.  Therefore, we\nfind no conclusive evidence that AGNs, undergoing this mode of\naccretion, are a key factor in the evolution of galaxies.\n\n\\acknowledgments\n\nWe are most grateful to the referee for providing a critical review\nthat significantly improved the paper and to Minjin Kim for allowing\nus to use his emission line measurements of SDSS AGNs and fielding\nsubsequent communications.  This work is fully based on observations\nundertaken at the European Southern Observatory (ESO) Very Large\nTelescope (VLT) under the Large Program 175.A-0839 (P.I., Simon\nLilly).\n\n{\\it Facilities:} \\facility{XMM}, \\facility{VLT:Melipal (VIMOS)}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nVarious approaches of perturbative QCD including the\nnext-to-leading order calculations (NLO QCD) have been applied \nto analyze the heavy flavour particle production in hadron \nreactions at high energies. \n\nSuch reactions are usually analyzed\nwithin the hard parton scattering model (HPSM) suggested in \n\\cite{Efremov,FF}. This model was applied to\nthe charmed meson production both in proton-proton and meson-proton \ninteractions at high energies, see for example \\cite{Bednyakov:1995}. \nThe HPSM is significantly improved by applying the QCD parton approach \n\\cite{Nasson,Kniehl2}, see details in \\cite{LKSB:09} and references\ntherein.\nUnfortunately the QCD approach including the NLO \n has some uncertainties related to the renormalization\nparameters especially at small transverse momenta $p_t$ \\cite{LKSB:09}. \n\nIn this paper we study the charmed and beauty meson production\nwithin the QGSM \\cite{kaid1} or the dual parton model (DPM) \\cite{Capella:1994}\nin $pp$ and $p{\\bar p}$ collisions at high energies based on the $1\/N$ \nexpansion in QCD \\cite{tHooft:1974,Veneziano:1974}.   \nWe show that this approach can be applied rather successfully at not very large values\nof $p_t$.\nIn addition, we investigate the open charm and beauty baryon production\nin $pp$ collisions at LHC energies and very small $p_t$ within the QGSM to find\nnew information on the sea charmed and beauty quark distributions in the proton.\nAnd at the end of the paper we use this approach to analyze the charmed meson production\nin the $p{\\bar p}$ collision at not large energies because the obtained results would be\nvery interesting for the future experiments at the GSI (Darmstadt) planned by the PANDA\nCollaboration.\n\\section{Charmed and beauty hadron production in \n$pp$ collisions at high energies }\n\\subsection{Heavy flavour meson production }\nFirst, let us analyze the $D$ and $B$ meson production in the $pp$ \ncollisions within the QGSM \nincluding the transverse motion of quarks and diquarks in\ncolliding protons \\cite{LAS}. As is known, the cylinder type\ngraphs for the $pp$ collision presented in Fig.1 make\nthe main contribution to this process \\cite{kaid1}. \nThe left diagram of Fig.1, the so-called\none-cylinder graph, corresponds to the case where two colorless\nstrings are formed between the quark\/diquark ($q\/qq$) and the\ndiquark\/quark ($qq\/q$) in colliding protons; then, after their\nbreakup, $q{\\bar q}$ pairs are created and fragmentated to a hadron,\nfor example, $D$ meson. The right diagram of Fig.1, the\nso-called multicylinder graph, corresponds to creation of the same\ntwo colorless strings and many strings between sea\nquarks\/antiquarks $q\/{\\bar q}$ and sea antiquarks\/quarks ${\\bar\nq}\/q$ in the colliding protons.\n\n\\begin{figure}[htb]\n\\vspace{9pt}\n\\includegraphics[width=0.45\\textwidth]{cylgr_2_1.eps}\n\\caption{The one-cylinder graph (left diagram) and the multi-cylinder \ngraph (right diagram) for the inclusive $p p\\rightarrow h X$ process.}\n\\label{Fig.1}\n\\end{figure}\n\nThe general form for the invariant inclusive hadron spectrum\nwithin the QGSM is \\cite{kaid2,LAS}\n\n\\begin{align}\nE\\frac{d\\sigma}{d^3{\\bf p}}\\equiv\n\\frac{2E^*}{\\pi\\sqrt{s}}\\frac{d\\sigma}{d x d p_t^2}=\n\\sum_{n=1}^\\infty \\sigma_n(s)\\phi_n(x,p_t)~, \n\\label{def:invsp}\n\\end{align}\n\nwhere $E,{\\bf p}$ are the energy and the three-momentum of the\nproduced hadron $h$ in the laboratory system (l.s.) of colliding protons, \n$E^*,s$ are the energy of $h$ and the square of the initial energy in the\nc.m.s of $pp$; $x$ and $p_t$ are the Feynman variable and the transverse\nmomentum of $h$; $\\sigma_n$ is the cross section for production of\nthe $n$-Pomeron chain (or $2n$ quark-antiquark strings) decaying\ninto hadrons, calculated within the ``eikonal approximation''\n\\cite{Ter-Mart}, the function $\\phi_n(x,p_t)$ has the following\nform \\cite{LAS}:\n\n\\begin{eqnarray}\n\\phi_n(x,p_t)=\\int_{x^+}^1 d x_1\\int_{x_-}^1 \nd x_2\\psi_n(x,p_t;x_1,x_2)~, \n\\label{def:phin}\n\\end{eqnarray}\nwhere\n\\begin{align}\n&\\psi_n(x,p_t;x_1,x_2) = \\notag \\\\\n&\\quad F_{qq}^{(n)}(x_+,p_t;x_1)F_{q_v}^{(n)}(x_-,p_t;x_2)\/F_{q_v}^{(n)}(0,p_t) \n\\notag \\\\\n&\\quad +F_{q_v}^{(n)}(x_+,p_t;x_1)F_{qq}^{(n)}(x_-,p_t;x_2)\/F_{qq}^{(n)}(0,p_t) \n\\notag \\\\\n&\\quad +2(n-1)F_{q_s}^{(n)}(x_+,p_t;x_1) \\notag \\\\\n&\\qquad\\qquad \\times F_{{\\bar q}_s}^{(n)}(x_-,p_t;x_2)\/F_{q_s}^{(n)}(0,p_t)~,\n\\label{def:psin}\n\\end{align}\nand $x_{\\pm}=(\\sqrt{x^2+x_t^2}\\pm x)\/2$, \\\\\n$x_t=2\\sqrt{(m_h^2+p_t^2)\/s}$,\n\n\\begin{align}\n&F_\\tau^{(n)}(x_\\pm,p_t;x_{1,2}) \\notag \\\\\n&\\quad =\\int d^2k_t{\\tilde f}_\\tau^{(n)}(x_{1,2},k_t){\\tilde G}_{\\tau\\rightarrow h}\n\\left(\\frac{x_\\pm}{x_{1,2}},k_t;p_t\\right)~, \n\\label{def:Ftaux} \\\\\n&F_\\tau^{(n)}(0,p_t) \\notag \\\\\n&\\quad = \\int_0^1 dx^\\prime d^2k_t {\\tilde\nf}_\\tau^{(n)}(x^\\prime,k_t) {\\tilde G}_{\\tau\\rightarrow\nh}(0,p_t)  \\notag \\\\ \n&\\quad ={\\tilde G}_{\\tau\\rightarrow h}(0,p_t)~.\n\\label{def:Ftauzero}\n\\end{align}\n\nHere $\\tau$ means the flavor of the valence (or sea) quark or\ndiquark, ${\\tilde f}_\\tau^{(n)}(x^\\prime,k_t)$ is the quark\ndistribution function depending on the longitudinal momentum\nfraction $x^\\prime$ and the transverse momentum $k_t$ in the\n$n$-Pomeron chain; ${\\tilde G}_{\\tau\\rightarrow h}(z,k_t;p_t)=\nz{\\tilde D}_{\\tau\\rightarrow h}(z,k_t;p_t)$, ${\\tilde\nD}_{\\tau\\rightarrow h}(z,k_t;p_t)$ is the fragmentation function\n(FF) of a quark (antiquark) or diquark of flavour $\\tau$ into a hadron\n$h$ ($D$ meson in our case). \nWe present the quark distributions\n and the FF in the factorized forms ${\\tilde f}_\\tau(x,k_t)=\nf_\\tau(x)g_\\tau(k_t)$, and, according to \\cite{LS:1992},\n${\\tilde G}_{\\tau\\rightarrow h}(z,k_t;p_t)=G_{\\tau\\rightarrow h}(z)\n{\\tilde g}_{\\tau\\rightarrow h}({\\tilde k}_t)$, where ${\\tilde{\\bf k}}_t=\n{\\bf p}_t-z{\\bf k}_t$.\n We take the quark distributions $f_\\tau(x)$ and the FF  $G_{\\tau\\rightarrow h}(z)$\nobtained within the QGSM from \\cite{kaid2,Shabelsky,Piskunova},\nwhereas their $k_t$ distributions are chosen in the form suggested in\n\\cite{Piskunova:1984,LS:1992} (see the details in \\cite{LKSB:09})\n\\begin{align}\n& \\qquad g_\\tau(k_t)=(B_q^2\/2\\pi)\\exp(-B_q A(k_t))~,\n\\label{def:ktdistr} \\\\\n& \\qquad {\\tilde g}_{\\tau\\rightarrow h}({\\tilde k}_t)=(B_c^2\/2\\pi)\n\\exp(-B_c A(k_t))~,\n\\label{def:ktFF}\n\\end{align}\nwhere $A(k_t)=(\\sqrt{k^2_t+m^2_D}-m_D))$ and $m^2_{Dt}=p_t^2+m_D^2$\nAfter the integration of eq.~(\\ref{def:Ftaux}) over $d^2k_t$ \nwe have, according to \\cite{LS:1992}, \n\\begin{align}\n&F_\\tau^{(n)}(x_\\pm,p_t;x_{1,2}) \\notag \\\\\n&\\quad={\\tilde f}_\\tau^{(n)}(x_{1,2})\nG_{\\tau\\rightarrow h}(z)I_n(z,p_t)~, \n\\label{def:Fnew}\n\\end{align}\nwhere $z=x_\\pm\/x_{1,2}$,\n$I_n(z,p_t)=B_z^2\/(2\\pi(1+B_zm_D))\\exp(-B_z(m_{Dt}-m_D))$, \n$B_z=B_c\/(1+n\\rho z^2)$, $\\rho=B_c\/B_q$. \nThe function $B_z$ also can be presented in the equivalent form\n$B_z=B_q\/({\\tilde\\rho}+n z^2)$, where ${\\tilde\\rho}=B_q\/B_c$. \nThe differential cross section $d\\sigma\/d p_t^2$\nfor $D$ mesons produced in $pp$ collisions is written in the  \nform \\cite{kaid2}\n\\begin{align}\n\\qquad \\frac{d\\sigma}{d p_t^2}=\\frac{\\pi}{2}\n\\sqrt{s}\\sum_{n=1}^\\infty\\sigma_n(s) \\int\\phi_n(x,p_t)\\frac{d x}{E^*}~. \n\\label{def:dsigdptsq}\n\\end{align} \nTo describe the experimental \ndata on the $p_t$-spectra in the $p_t$ region where the NLO QCD calculation has a big uncertainty\nwe chose $\\tilde{\\rho}=7$ both for the Tevatron and the LHC energies.\nWhen $B_c~<~1$ (GeV\/c)$^{-1}$, Eq.(\\ref{def:ktFF}) can be approximately presented in the form \n${\\tilde g}_{\\tau\\rightarrow h}({\\tilde k}_t)\\simeq a^2\/(a^2+{\\tilde k}^2_t)$ at\n${\\tilde k}^2_t<2m_D$, where $a=\\sqrt{2m_D\/B_c}$ \\cite{Greco:1994}. \nThis form is similar to the form for the FF of heavy quarks obtained within the perturbative \nQCD. \nNote that the function $I_n(z,p_t)$ in Eq.(\\ref{def:Fnew}) was obtained in \n\\cite{LS:1992,LAS} on the assumption of the consequent sharing of the transverse  momentum\n$p_t$ in the proton (antiproton) between $n$-Pomeron chains. It allowed us to describe rather \nsatisfactorily the experimental data on the inclusive $p_t$ spectra of charmed and beauty mesons\nproduced in $p{\\bar p}$ collisions at moderate values of the transverse momentum $p_t~<~10$ GeV\/c\n\\cite{LKSB:09}.\nAs is shown in  \\cite{LS:1992}, this version of the QGSM describes\nrather satisfactorily the experimental data on inclusive spectra of $D$\nmesons produced in $pp$ collisions at $\\sqrt{s}=27.4$ GeV.\n\nIt allows us to make \nthe predictions for inclusive $p_t$ spectra of $D^0$ and\n$B^+$ mesons produced in the $pp$ collision at LHC energies and\ncompare our results with the NLO QCD calculation for the produced charmed quarks\n\\cite{ALICE}, see  Figs.2, 3.\n\\begin{figure}[htb]\n\\vspace{9pt}\n\\includegraphics[width=0.45\\textwidth]{D0lhcB4r7_02_new.eps}\n\\caption{The inclusive spectrum for $D^0$ mesons produced in the\n$pp$ collision at the LHC energy $\\sqrt{s}=14$ TeV obtained within\nthe QGSM for charmed mesons \\cite{LKSB:09}\nand the NLO QCD for $c$ quarks\n \\cite{ALICE}.}\n\\label{Fig.2}\n\\end{figure}\n\\begin{figure}[htb]\n\\vspace{9pt}\n\\includegraphics[width=0.45\\textwidth]{B+mes_lhc13_04_09_new.eps}\n\\caption{The inclusive spectrum for $B^+$ mesons produced in the\n$pp$ collision at the LHC energy $\\sqrt{s}=14$ TeV obtained within\nthe QGSM for beauty mesons \\cite{LKSB:09}\nand the NLO QCD for single\n$b$ quarks \\cite{ALICE}.}\n\\label{Fig.3}\n\\end{figure}\nThe solid  lines in Figs.2, 3 correspond to our calculations within the\nQGSM, whereas the hatched regions show the calculations within the\nNLO QCD including uncertainties \\cite{ALICE}. A big difference\nbetween the QGSM and NLO QCD calculations at $p_t>10$\nGeV\/c for $D$ and $B$ mesons can be due to the following.\nFirst, the NLO QCD calculation \\cite{ALICE}\ndoes not include the hadronization of quarks to heavy mesons,\nwhereas the QGSM calculation includes it. Second, we do not\ninclude the contribution of gluons and their hard scatterings off\nquarks and gluons which can be sizable at large values of $p_t$.\n\\subsection{Heavy flavor baryon production}\nNow let us analyze the charmed and beauty baryon production in the $pp$\ncollision at LHC energies and very small $p_t$ within the soft QCD, e.g.,\nthe QGSM. This study can be interesting for it may allow \npredictions for future LHC experiments like TOTEM and ATLAS and \nan opportunity to find new information on the distribution\nof sea charmed ($c$) and beauty ($b$) quarks at very low $Q^2$. \nAccording to the QGSM, the distribution of $c({\\bar c})$ quarks in the $n$th Pomeron \nchain (Fig.1(right)) is, see for example \\cite{LAS} and references therein,\n\\begin{eqnarray}\n\\lefteqn{\nf_{c({\\bar c})}^{(n)}(x) = C_{c({\\bar c})}^{(n)}\\delta_{c({\\bar c})}\nx^{-\\alpha_\\psi(0)}} \\nonumber \\\\\n& {\\times}(1-x)^{\\alpha_\\rho(0)-2\\alpha_B(0)+(\\alpha_\\rho(0)-\\alpha_\\psi(0))\n+n-1}~,\\quad\n\\end{eqnarray}\nwhere $\\delta_{c({\\bar c})}$ is the weight of charmed pairs in the quark sea, \n$C_{c({\\bar c})}^{(n)}$\nis the normalization coefficient \\cite{kaid2},\n $\\alpha_\\psi(0)$ is the intercept of the $\\psi$- Regge trajectory.\nIts value can be $-2.18$ assuming that this trajectory $\\alpha_\\Psi(t)$ \nis linear and the intercept and the slope $\\alpha_\\Psi^\\prime(0)$ can be\ndetermined by drawing the trajectory through the $\\Psi$-meson mass \n$m_\\Psi=3.1$ GeV and the $\\chi$-meson mass $m_\\chi=3.554$ GeV \n\\cite{Boresk-Kaid:1983}. Assuming that the $\\psi$- Regge trajectory is \nnonlinear one can get  $\\alpha_\\psi(0)\\simeq 0$, which follows from perturbative \nQCD, as it was shown in \\cite{Kaid-Pisk:1986}. \nThe  distribution of $b({\\bar b})$ quarks in the $n$th Pomeron \nchain (Fig.1(right)) has the similar form  \n\\begin{eqnarray}\n\\lefteqn{\nf_{b({\\bar b})}^{(n)}(x) = C_{b({\\bar b})}^{(n)}\\delta_{b({\\bar b})}\nx^{-\\alpha_\\Upsilon(0)}}\n\\nonumber \\\\\n &{\\times}(1-x)^{\\alpha_\\rho(0)-2\\alpha_B(0)+(\\alpha_\\rho(0)-\n\\alpha_\\Upsilon(0))\n+n-1}~,\\quad\n\\end{eqnarray}\nwhere $\\alpha_\\rho(0)=1\/2$ is the well known intercept of the $\\rho$-trajectory; $\\alpha_B(0)\\simeq -0.5$\nis the intercept of the baryon trajectory, $\\alpha_\\Upsilon(0))$ is the intercept  of the \n$\\Upsilon$- Regge trajectory, its value also has an uncertainty. Assuming its linearity \none can get $\\alpha_\\Upsilon(0)=-8, -16$, while for nonlinear ($b{\\bar b}$) Regge trajectory\n$\\alpha_\\Upsilon(0)\\simeq 0$, see details in \\cite{Piskunova}.\nInserting these values to the form\nfor $f_{c({\\bar c})}^{n)}(x)$ and $f_{b({\\bar b})}^{n)}(x)$ we get the large sensitivity\nfor the $c$ and $b$ sea quark distributions in the $n$th Pomeron chain.\nNote that the FFs also depend on the parameters of these Regge trajectories. Therefore,\nthe knowledge of the intercepts and slopes of the heavy-meson Regge trajectories is\nvery important for the theoretical analysis of open charm and beauty production in\nhadron processes.\n \n\\begin{figure}[ht]\n   \\begin{center}\n {\\epsfig{file=lambda_c_v2.eps,width=0.85\\linewidth  }}\n \\caption[Fig.10]{The differential cross section $d\\sigma\/dx$ for the\n inclusive process $pp\\rightarrow\\Lambda_c X$ at $\\sqrt{s}=62~\\mathrm{GeV}$.\n } \n   \\end{center}\n \\end{figure}\n\nThe information on the charmonium ($c{\\bar c}$) and botomonium ($b{\\bar b}$) Regge trajectories\ncan be found from the experimental data on the charmed and beauty baryon production in $pp$ \ncollisions at high energies. For example, Fig.4 illustrates the sensitivity of the inclusive\nspectrum $d\\sigma\/dx$ of the produced charmed  baryons $\\Lambda_c$ to different values for \n$\\alpha_\\psi(0)$. \nThe solid line corresponds to $\\alpha_\\psi(0)=0$, whereas the dashed curve corresponds to \n$\\alpha_\\psi(0)=-2.18$.\nUnfortunately the experimental data presented in Fig.4 have big uncertainties; \ntherefore, one can't extract the information on the $\\alpha_\\psi(0)$ values from the \nexisting experimental data.\n\nA high sensitivity of the inclusive spectrum $d\\sigma\/dx$ of the produced beauty baryons \n$\\Lambda_b$ to different values for $\\alpha_\\Upsilon(0)$ is presented in Fig.5 (left).\n\\begin{figure}[htb]\n\\begin{center}\n\\begin{tabular}{cc}\n\\mbox{\\epsfig{file=lambda_b_v2.eps,width=0.45\\linewidth}} &\n\\mbox{\\epsfig{file=lambda_b_pt2.eps,width=0.45\\linewidth}}\n\\end{tabular}\n\\end{center}\n \\caption[Fig.11]{The differential cross section $d\\sigma\/dx$ (left) and\n$d\\sigma\/dP_t^2$ (right) for the\n inclusive process $pp\\rightarrow\\Lambda_b X$ at $\\sqrt{s}=4~\\mathrm{TeV}$.\n}\n\\end{figure}\nThe $p_t$-inclusive spectrum of $\\Lambda_b$ has much lower sensitivity to this quantity,\naccording to the results presented in Fig.5 (right).\nActually, our results presented in Fig.5 could be considered as some predictions for \nfuture experiments at LHC.\n\\section{Charmed and beauty hadron production in \n$p{\\bar p}$ collisions}\n\\subsection{Heavy flavour meson production at high energies }\nThe production of heavy\nmesons like $D$ and $B$ mesons in proton-antiproton collisions at\nhigh energies is usually analyzed within the different schemes of\nQCD. To study these processes within the QGSM we have to include\nat least one additional graph corresponding to the creation of\nthree chains between quarks in the initial proton and antiquarks\nin the colliding antiproton, as is illustrated in Fig.6(c). \n\nThe diagrams in Fig.6(a,b) are similar to the\none-cylinder and multicylinder diagrams for the $pp$ collision in\nFig.1 with a following difference. In the $p{\\bar p}$\ncollision two colorless strings between quark\/diquark ($q\/qq$) in\nthe initial proton and antiquark\/antidiquark (${\\bar q}\/{\\bar\nqq}$) are created. Many quark-antiquark ($q-{\\bar q}$) strings for\n$p{\\bar p}$ collision (Fig.6b) are the\nsame as for the $pp$ collision (Fig.1, right diagram).\nTherefore, the invariant inclusive spectrum of hadrons produced in\nthe $p{\\bar p}$ collision calculated within the QGSM has the\nfollowing form:\n\n\\begin{align}\nE\\frac{d\\sigma^{p{\\bar p}}}{d^3{\\bf p}} & =\n\\sigma_1(s)[\n(1-\\omega)\\phi^{p{\\bar p}}_1(x,p_t)+\\omega{\\tilde\\phi}(x,p_t)]\n\\notag \\\\\n&\\quad +\\sum_{n=2}^\\infty \\sigma_n(s)\\phi^{p{\\bar p}}_n(x,p_t)\n\\label{def:sppbarp}\n\\end{align}\n\n\\begin{figure}[htb]\n\\vspace{9pt}\n\\includegraphics[width=0.45\\textwidth]{cyl1_3chains_new.eps}\n\\caption{The one-cylinder graph (a) and\n the multi-cylinder graph (b), and the three-chains graph \n(c) for the $p{\\bar p}\\rightarrow h X$ inclusive process.}\n\\label{Fig.6}\n\\end{figure}\n\nwhere $1-\\omega$ is the probability of contribution of the cut\none-cylinder (one-Pomeron exchange) and cut multicylinder\n(multiPomeron exchanges) graphs (the left and right\ndiagrams in Fig.6), whereas $\\omega$ is the probability\nof the contribution of the three-chain diagram \nto the inclusive spectrum. The value of $\\omega$ can be estimated\nas the ratio of the $p{\\bar p}$ annihilation cross section\n$\\sigma_{p{\\bar p}}^{ann}$ to the total $p{\\bar p}$ cross section\n$\\sigma_{p{\\bar p}}^{tot}$. The cross section $\\sigma_{p{\\bar\np}}^{tot}$ is well known in the wide range of the initial energies\nto the Tevatron energy, whereas experimental data on\n$\\sigma_{p{\\bar p}}^{ann}$ are available only for the antiproton\ninitial energy up to $10$ GeV, see \\cite{Uzh02} and references\ntherein. However, some theoretical predictions, for example\n\\cite{GosNus80,BZK:1988}, show that asymptotically\n$\\sigma_{p{\\bar p}}^{ann}$ goes to about $2-4$ mb. It corresponds\nto $\\omega\\simeq \\sigma_{p{\\bar p}}^{ann}\/\\sigma_{p{\\bar\np}}^{tot}~<~ 0.1$ at the Tevatron energy. Note that in addition\nto the graph of Fig.6c there can be diagrams consisting of these three chains \nand multicilynder chains between sea quarks and antiquarks. However, as our estimations \nshow, their contribution to the inclusive spectrum is much smaller than the contribution\nfrom the three-chain graph (Fig.6c). Therefore, we neglect \nit. \n\nThe form for the function $\\phi^{p{\\bar p}}_n(x,p_t)$ is similar\nto $\\phi_n(x,p_t)$ entering into (3) by replacing\n$F_{q_v}^{(n)}(x_-,p_t;x_2)$, $F_{q_v}^{(n)}(0,p_t)$ to $F_{\\bar\n{qq}}^{(n)}(x_-,p_t;x_2)$, $F_{\\bar {qq}}^{(n)}(0,p_t)$\nrespectively, and replacing $F_{qq}^{(n)}(x_-,p_t;x_2)$,\n$F_{qq}^{(n)}(0,p_t)$ to $F_{{\\bar q}}^{(n)}(x_-,p_t;x_2)$ and\n$F_{{\\bar q}}^{(n)}(0,p_t)$ respectively. The additional term\n${\\tilde\\phi}(x,p_t)$ in (\\ref{def:sppbarp}) has the\nfollowing form\n\n\\begin{eqnarray}\n{\\tilde\\phi}(x,p_t)=3{\\tilde F}_{q_v}(x_+,p_t){\\tilde F}_{{\\bar\nq}_v}(x_-,p_t)\/{\\tilde F}_q(0,p_t)~, \\label{def:tphi}\n\\end{eqnarray}\nwhere ${\\tilde F}_{q_v({\\bar\nq}_v)}(x_\\pm,p_t)=F^{(n=1)}_{q_v({\\bar q}_v)}(x_\\pm,p_t)$ and\n${\\tilde F}_q(0,p_t)=F_q^{(n=1)}(0,p_t)$.\n\nThe inclusive $p_t$ spectra of $D^0$ and $B^+$ mesons produced in\nthe $p{\\bar p}$ collision at the Tevatron energy $\\sqrt{s}=1.96$\nTeV are presented in Figs.(7,8), see \\cite{LKSB:09}.\nThe hatched regions in\nFigs.(7.8) show the calculations within the NLO\nQCD including uncertainties \\cite{CDF_Dmes}. \nNote that the our calculation showed that the contribution\nof the three-chain graph (Fg.6c) is very small at the\nTevatron energy. It is due to small values of the $p{\\bar p}$ annihilation cross \nsection at very high energies \\cite{GosNus80,BZK:1988}.   \n\n\\begin{figure}[htb]\n\\vspace{9pt}\n\\includegraphics[width=0.40\\textwidth]{D0Tev13_04_09_new.eps}\n\\caption{ The inclusive $p_t$-spectrum\nfor $D^0$ mesons produced in the $p{\\bar p}$ collision at the\nTevatron energy $\\sqrt{s}=1.96$ TeV obtained within the QGSM \n\\cite{LKSB:09} (the solid line) and within the NLO QCD\n\\cite{CDF_Dmes} (the hatched regions). \nThe experimental data are taken from \\cite{CDF1,CDF2}}\n\\label{Fig.7}\n\\end{figure}\n\\begin{figure}[htb]\n\\vspace{9pt}\n\\includegraphics[width=0.40\\textwidth]{BPTev13_04_09_01_new.eps}\n\\caption{The inclusive $p_t$-spectrum\nfor $B^+$ mesons produced in the $p{\\bar p}$ collision at the\nTevatron energy $\\sqrt{s}=1.96$ TeV obtained within the QGSM\n\\cite{LKSB:09} \n(the solid line) and within the NLO QCD\n\\cite{CDF_Dmes} (the hatched regions).\nThe experimental data \nare taken from \\cite{CDF1,CDF2}}\n\\label{Fig.8}\n\\end{figure}\n\\subsection{Charmed meson production at intermediate energies } \nLet us now use the QGSM to analyze the inclusive charmed meson production in\n${\\bar p}p$ collisions at not large energies, less than 15-20 GeV because it\nwould be interesting for future experiments at the GSI (Darmstadt) planned by the PANDA\nCollaboration. At these energies the contribution of the three-chain graph (Fig.6c)\ncan be sizable because the cross section of the ${\\bar p}p$ annihilation\nis not small \\cite{Uzh02}. Note that at energies close to the threshold of the $D$-meson \nproduction the binary process ${\\bar p}p\\rightarrow{\\bar D}D$ is dominant, according\nto \\cite{Kaid-Volkov:94}. Therefore, the total cross section of $D$ mesons produced in\n${\\bar p}p$ collisions at energies starting from the threshold is the sum of cross\nsections for the binary process $\\sigma_{{\\bar p}p\\rightarrow{\\bar D}D}$ and the inclusive reaction \n $\\sigma_{{\\bar p}p\\rightarrow D X}$. The cross section  $\\sigma_{{\\bar p}p\\rightarrow{\\bar D}D}$\nwas calculated in \\cite{Kaid-Volkov:94} within the simple Regge pole model.   \n\\begin{figure}[htb]\n\\vspace{9pt}\n\\includegraphics[width=0.45\\textwidth]{p_pbar_g2.eps}\n\\caption{ The cross section for $D({\\bar D})$ meson production in \n${\\bar p}-p$ collision as a function of the antiproton momentum $p_{lab}$ in the \nl.s. }\n\\label{Fig.9}\n\\end{figure}\nIn Fig.9 the cross sections $\\sigma_{{\\bar p}p\\rightarrow{\\bar D}D}$ (dashed curve)\nand $\\sigma_{{\\bar p}p\\rightarrow D X}$ (dash-dotted line and dotted line)\nare presented as functions of the incident momentum of the antiproton in the l.s. \nof ${\\bar p}p$. \nThe dash-dotted curve corresponds to our calculation of the\ninclusive process ${\\bar p}p\\rightarrow D({\\bar D})X$ with neglect of the graph of Fig.6c\n($\\omega=0$), and the dotted line corresponds to the similar calculation but with\nthe Fig.6c graph included; the probability for the three-chain graph (Fig.6c) \n$\\omega=\\sigma_{{\\bar p}p}^{ann}\/\\sigma_{{\\bar p}p}^{tot}$ was taken from the fit of\nthe experimental data \\cite{Uzh02}.\nThe solid line is the total yield of $D$ mesons produced in ${\\bar p}p$\ncollision.\n\\section{Conclusion}\n We have shown that the modified QGSM including the\nintrinsic longitudinal and transverse motion of quarks\n(antiquarks) and diquarks in colliding protons allowed us to\ndescribe rather satisfactorily the existing experimental data on\ninclusive spectra of heavy flavour mesons produced in $pp$ collisions \nand to make some predictions for similar spectra at LHC energies. \n\nTo verify whether these predictions can be reliable or not we apply\nthe QGSM to the analysis of charmed and beauty meson production in\nproton-antiproton collisions at Tevatron energies including graphs\nlike those in Fig.6c corresponding to annihilation of\nquarks and antiquarks in colliding $p$ and ${\\bar p}$, and\nproduction of heavy flavour mesons. \nWe got a satisfactory QGSM description ($p_t~<~10$ GeV\/c)\nof the experimental data on\n$p_t$ spectra of $D^0$ and $B^+$ mesons produced in the $p{\\bar\np}$ collisions which were obtained by the CDFII Collaboration at\nthe Tevatron \\cite{CDF_Dmes}. \n\nTo describe these spectra and make some predictions for the future LHC experiments\nin a wide region of transverse momenta one can combine the ``soft QCD'' (the QGSM)\nat small values of $p_t$ with the NLO QCD at large $p_t$.\n\nAt the Tevatron energies the contribution of the three-chain graph (Fig.6c) \nto the inclusive spectra of heavy mesons is too small, as was\nshown in \\cite{LKSB:09}; therefore, it can be neglected. However, at the\nantiproton energies about a few GeV the three-chain graph contribution is\nsizable and can  amount to 30-40 percent, as is shown in Fig.9.\nIt has also been shown that at the incident momenta of antiprotons\n$p_{lab}$ above $8-9$ GeV\/c the inclusive production of $D$ mesons in \n${\\bar p}p$ collisions should be included in addition to the binary \n${\\bar p}p\\rightarrow {\\bar D}D$ process to get the total yield of \n$D$ mesons at $p_{lab}\\leq 14-15$ GeV\/c. These results would be interesting\nfor future experiments at the GSI (Darmstadt) with the antiproton beam.\n\nWe also made some predictions for future LHC forward experiments on the beauty \nbaryon production in $pp$ collisions which can give us new information on\nthe beauty quark distribution in the proton and very interesting information on\nthe Regge trajectories of ($b{\\bar b}$) mesons.\n\n\n\\section{Acknowledgments}\n We thank P. Braun-Munzinger, W. Cassing, M. Deile, S. Dubnicka, A. Z. Dubnickova,\nA. V. Efremov, K. Eggert, D. Elia, A.Galoyan, S. B. Gerasimov, A. B. Kaidalov, Z. M. Karpova, \nB. Z. Kopeliovich, A. D. Martin, K. Peters, T. V. Lyubushkina, M. Poghosyan, K. Safarik, \nV. V. Uzhinsky and U. Wiedner for very useful discussions. \nThis work was supported in part by the RFBR grant N 08-02-01003.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}