diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzofo" "b/data_all_eng_slimpj/shuffled/split2/finalzofo" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzofo" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nNormalized B-bases (a comprehensive study of which can be found in\n\\cite{Pena1999} and references therein) are normalized totally positive bases\nthat imply optimal shape preserving properties for the representation of\ncurves described as linear combinations of control points and basis functions.\nSimilarly to the classical Bernstein polynomials\n\\[\n\\mathcal{B}_{n}=\\left\\{ \\dbinom{n}{i}u^{i}\\left( 1-u\\right) ^{n-i}%\n:u\\in\\left[ 0,1\\right] \\right\\} _{i=0}^{n}%\n\\]\nof degree $n\\in%\n\\mathbb{N}\n$ -- that in fact form the normalized B-basis of the vector space of\npolynomials\n\\[\n\\mathbb{P}_{n}=\\left\\{ 1,u,\\ldots,u^{n}:u\\in\\left[ 0,1\\right] \\right\\}\n\\]\nof degree at most $n$ on the compact interval $\\left[ 0,1\\right] $, cf.\n\\cite{Carnicer1993} -- normalized B-bases provide shape preserving properties\nlike closure for the affine transformations of the control polygon, convex\nhull, variation diminishing (which also implies convexity preserving of plane\ncontrol polygons), endpoint interpolation, monotonicity preserving, hodograph\nand length diminishing, and a recursive corner cutting algorithm (also called\nB-algorithm) that is the analogue of the de Casteljau algorithm of B\\'{e}zier\ncurves. Among all normalized totally positive bases of a given vector space of\nfunctions a normalized B-basis is the least variation diminishing and the\nshape of the generated curve more mimics its control polygon. Important curve\ndesign algorithms like evaluation, subdivision, degree elevation or knot\ninsertion are in fact corner cutting algorithms that can be treated in a\nunified way by means of B-algorithms induced by B-bases.\n\nThese advantageous properties make normalized B-bases ideal blending function system\ncandidates for curve (and surface) modeling. Using B-basis functions, our\nobjective is to provide control point based exact description for higher order\nderivatives of trigonometric and hyperbolic curves specified with coordinate\nfunctions given in traditional parametric form, i.e., in vector spaces%\n\\begin{equation}\n\\mathbb{T}_{2n}^{\\alpha}=\\operatorname{span}\\mathcal{T}_{2n}^{\\alpha\n}=\\operatorname{span}\\left\\{ \\cos\\left( ku\\right) ,\\sin\\left( ku\\right)\n:u\\in\\left[ 0,\\alpha\\right] \\right\\} _{k=0}^{n}\n\\label{truncated_Fourier_vector_space}%\n\\end{equation}\nor%\n\\begin{equation}\n\\mathbb{H}_{2n}^{\\alpha}=\\operatorname{span}\\mathcal{H}_{2n}^{\\alpha\n}=\\operatorname{span}\\left\\{ \\cosh\\left( ku\\right) ,\\sinh\\left( ku\\right)\n:u\\in\\left[ 0,\\alpha\\right] \\right\\} _{k=0}^{n},\n\\label{hyperbolic_vector_space}%\n\\end{equation}\nwhere $\\alpha$ is a fixed strictly positive shape (or design)\nparameter which is either strictly less than $\\pi$, or it is unbounded from above in\nthe trigonometric and hyperbolic cases, respectively. The obtained results will also be\nextended for the control point based exact description of the rational\ncounterpart of these curves and of higher dimensional multivariate (rational)\nsurfaces that are also specified by coordinate functions given in traditional\ntrigonometric or hyperbolic form along of each of their variables.\n\n\\begin{remark}\nFrom the point of view of control point based exact description of smooth\n(rational) trigonometric closed curves and surfaces (i.e., when $\\alpha=2\\pi\n$), articles \\cite{RothJuhasz2010}\\ and \\cite{JuhaszRoth2010} already provided\ncontrol point configurations by using the so-called cyclic basis functions\nintroduced in \\cite{RothEtAl2009}. Although this special cyclic basis of\n$\\mathbb{T}_{2n}^{2\\pi}$ fulfills some important properties (like positivity,\nnormalization, cyclic variation diminishing, cyclic symmetry, singularity free\nparametrization, efficient explicit formula for arbitrary order elevation), it\nis not totally positive, hence it is not a B-basis, since, as it was shown in\n\\cite{Pena1997}, the vector space (\\ref{truncated_Fourier_vector_space}) has\nno normalized totally positive bases when $\\alpha\\geq\\pi$. Therefore, by using\nthe B-basis of (\\ref{truncated_Fourier_vector_space}), the control point based\nexact description of arcs, patches or volume entities of higher dimensional\n(rational) trigonometric curves and multivariate surfaces given in traditional\nparametric form remained, at least for us, an interesting and challenging question.\n\\end{remark}\n\nThe rest of the paper is organized as follows. Section\n\\ref{sec:special_parametrizations} briefly recalls some basic properties of\nrational B\\'{e}zier curves and points out that curves described as linear\ncombinations of control points and B-basis functions of vector spaces\n(\\ref{truncated_Fourier_vector_space}) or (\\ref{hyperbolic_vector_space}) are\nin fact special reparametrizations of specific classes of rational B\\'{e}zier\ncurves. This section also defines control point based (rational) trigonometric\nand hyperbolic curves of finite order, briefly reviews some of their\n(geometric) properties like order elevation and asymptotic behavior and at the\nsame time also describes their subdivision algorithm which, to the best of our\nknowledge, were either totally not detailed or not described with full\ngenerality for these type of curves in the literature. Based on multivariate\ntensor products of trigonometric and hyperbolic curves, Section\n\\ref{sec:multivariate_surfaces} defines higher dimensional multivariate\n(rational) trigonometric and hyperbolic surfaces. Section\n\\ref{sec:basis_transformations} provides efficient and parallely implementable\nrecursive formulae for those base changes that transform the normalized B-bases of vector\nspaces (\\ref{truncated_Fourier_vector_space}) and\n(\\ref{hyperbolic_vector_space}) to their corresponding canonical (traditional)\nbases, respectively. Using these transformations, theorems and algorithms of\nSection \\ref{sec:exact_description} provide control point configurations for\nthe exact description of large classes of higher dimensional (rational)\ntrigonometric or hyperbolic curves and multivariate (hybrid) surfaces. All\nexamples included in this section emphasize the applicability and usefulness\nof the proposed curve and surface modeling tools. Finally, Section\n\\ref{sec:final_remarks} closes the paper with our final remarks.\n\n\\section{Special parametrizations of a class of rational B\\'{e}zier\ncurves\\label{sec:special_parametrizations}}\n\nUsing Bernstein polynomials, a rational B\\'{e}zier curve of even degree $2n$\ncan be described as%\n\\begin{equation}\n\\mathbf{r}_{2n}\\left( v\\right) =\\frac{%\n{\\displaystyle\\sum\\limits_{i=0}^{2n}}\nw_{i}\\mathbf{d}_{i}B_{i}^{2n}\\left( v\\right) }{%\n{\\displaystyle\\sum\\limits_{j=0}^{2n}}\nw_{j}B_{j}^{2n}\\left( v\\right) },~v\\in\\left[ 0,1\\right]\n,\\label{rational_Bezier_curve}%\n\\end{equation}\nwhere $\\left[ \\mathbf{d}_{i}\\right] _{i=0}^{2n}\\in\\mathcal{M}_{1,2n+1}%\n\\left(\n\\mathbb{R}\n^{\\delta}\\right) $ is a user defined control polygon ($\\delta\\geq2$), while\n$\\left[ w_{i}\\right] _{i=0}^{2n}\\in\\mathcal{M}_{1,2n+1}\\left(\n\\mathbb{R}\n_{+}\\right) $ is also a user specified non-negative weight vector of rank $1$\n(i.e., $\\sum_{i=0}^{2n}w_{i}\\neq0$).\n\nFor any fixed ratio $v\\in\\left[ 0,1\\right] $, the recursive relations%\n\\begin{equation}\n\\left\\{\n\\begin{array}\n[c]{l}%\nw_{i}^{r}\\left( v\\right) =\\left( 1-v\\right) w_{i}^{r-1}\\left( v\\right)\n+vw_{i+1}^{r-1}\\left( v\\right) ,\\\\\n\\\\\n\\mathbf{d}_{i}^{r}\\left( v\\right) =\\left( 1-v\\right) \\dfrac{w_{i}%\n^{r-1}\\left( v\\right) }{w_{i}^{r}\\left( v\\right) }\\mathbf{d}_{i}%\n^{r-1}\\left( v\\right) +v\\dfrac{w_{i+1}^{r-1}\\left( v\\right) }{w_{i}%\n^{r}\\left( v\\right) }\\mathbf{d}_{i+1}^{r-1}\\left( v\\right) ,~r=1,2,\\ldots\n,2n,~i=0,1,\\ldots,2n-r\n\\end{array}\n\\right. \\label{rational_de_Casteljau}%\n\\end{equation}\nwith initial conditions%\n\\[\nw_{i}^{0}\\left( v\\right) \\equiv w_{i},~\\mathbf{d}_{i}^{0}\\left( v\\right)\n\\equiv\\mathbf{d}_{i},~i=0,1,\\ldots,2n\n\\]\ndefine the B-algorithm (or rational de Casteljau algorithm) of the curve\n(\\ref{rational_Bezier_curve}) (cf. \\cite{Farin2002}).\n\nWe will produce control point exact based description of trigonometric and\nhyperbolic curves, therefore we need proper bases for vector spaces\n(\\ref{truncated_Fourier_vector_space}) and (\\ref{hyperbolic_vector_space}) of\ntrigonometric and hyperbolic polynomials of order at most $n$ (or of degree at\nmost $2n$), respectively. In what follows, $\\overline{\\mathcal{T}}%\n_{2n}^{\\alpha}$ and $\\overline{\\mathcal{H}}_{2n}^{\\alpha}$ denote the normalized B-bases\nof vector spaces $\\mathbb{T}_{2n}^{\\alpha}$ and $\\mathbb{H}_{2n}^{\\alpha}$, respectively.\n\n\\subsection{Trigonometric curves and their rational\ncounterpart\\label{sec:trigonometric_curves}}\n\nLet $\\alpha\\in\\left( 0,\\pi\\right) $ be an arbitrarily fixed parameter and\nconsider the linearly reparametrized version of the B-basis\n\\begin{equation}\n\\overline{\\mathcal{T}}_{2n}^{\\alpha}=\\left\\{ T_{2n,i}^{\\alpha}\\left(\nu\\right) :u\\in\\left[ 0,\\alpha\\right] \\right\\} _{i=0}^{2n}=\\left\\{\nt_{2n,i}^{\\alpha}\\sin^{2n-i}\\left( \\frac{\\alpha-u}{2}\\right) \\sin^{i}\\left(\n\\frac{u}{2}\\right) :u\\in\\left[ 0,\\alpha\\right] \\right\\} _{i=0}^{2n}\n\\label{Sanchez_basis}%\n\\end{equation}\nof order $n$ (degree $2n$) specified in \\cite{Sanchez1998}, where the\nnon-negative normalizing coefficients%\n\\[\nt_{2n,i}^{\\alpha}=\\frac{1}{\\sin^{2n}\\left( \\frac{\\alpha}{2}\\right) }%\n\\sum_{r=0}^{\\left\\lfloor \\frac{i}{2}\\right\\rfloor }\\binom{n}{i-r}\\binom\n{i-r}{r}\\left( 2\\cos\\left( \\frac{\\alpha}{2}\\right) \\right) ^{i-2r}%\n,~i=0,1,\\ldots,2n\n\\]\nfulfill the symmetry property%\n\\begin{equation}\nt_{2n,i}^{\\alpha}=t_{2n,2n-i}^{\\alpha},~i=0,1,\\ldots,n\\text{.}\n\\label{symmetry_of_Sanchez_Reyes_constants}%\n\\end{equation}\n\n\n\\begin{definition}\n[Trigonometric curves]A trigonometric curve of order $n$ (degree $2n$) can be\ndescribed as the convex combination%\n\\begin{equation}\n\\mathbf{t}_{n}^{\\alpha}\\left( u\\right) =\\sum_{i=0}^{2n}\\mathbf{d}%\n_{i}T_{2n,i}^{\\alpha}\\left( u\\right) ,~u\\in\\left[ 0,\\alpha\\right] ,\n\\label{trigonometric_curve}%\n\\end{equation}\nwhere $\\left[ \\mathbf{d}_{i}\\right] _{i=0}^{2n}\\in\\mathcal{M}_{1,2n+1}%\n\\left(\n\\mathbb{R}\n^{\\delta}\\right) $ defines its control polygon.\n\\end{definition}\n\nAs stated in Remark \\ref{rem:trigonometric_reparametrization} curves of type\n(\\ref{trigonometric_curve}) can also be obtained as a special trigonometric\nreparametrization of a class of rational B\\'{e}zier curves of even degree.\n\n\\begin{remark}\n[Trigonometric reparametrization]\\label{rem:trigonometric_reparametrization}%\nUsing the function%\n\\begin{equation}\n\\left\\{\n\\begin{array}\n[c]{l}%\nv:\\left[ 0,\\alpha\\right] \\rightarrow\\left[ 0,1\\right] ,\\\\\n\\\\\nv\\left( u\\right) =\\dfrac{1}{2}+\\dfrac{\\tan\\left( \\frac{u}{2}-\\frac{\\alpha\n}{4}\\right) }{2\\tan\\left( \\frac{\\alpha}{4}\\right) }=\\dfrac{\\sin\\left(\n\\frac{u}{2}\\right) }{2\\cos\\left( \\frac{\\alpha}{4}-\\frac{u}{2}\\right)\n\\sin\\left( \\frac{\\alpha}{4}\\right) }%\n\\end{array}\n\\right. \\label{trigonometric_reparametrization}%\n\\end{equation}\nand weights%\n\\begin{equation}\nw_{i}=\\frac{t_{2n,i}^{\\alpha}}{\\binom{2n}{i}},~i=0,1,\\ldots,2n,\n\\label{trigonometric_weights}%\n\\end{equation}\none can reparametrize the rational B\\'{e}zier curve\n(\\ref{rational_Bezier_curve}) into the trigonometric form\n(\\ref{trigonometric_curve}). Indeed, one has that%\n\\begin{align*}\nw_{i}B_{i}^{2n}\\left( v\\left( u\\right) \\right) & =\\frac{t_{2n,i}%\n^{\\alpha}}{\\binom{2n}{i}}\\binom{2n}{i}v^{i}\\left( u\\right) \\left(\n1-v\\left( u\\right) \\right) ^{2n-i}\\\\\n& =t_{2n,i}^{\\alpha}\\cdot\\dfrac{\\sin^{i}\\left( \\frac{u}{2}\\right) }%\n{2^{i}\\cos^{i}\\left( \\frac{\\alpha}{4}-\\frac{u}{2}\\right) \\sin^{i}\\left(\n\\frac{\\alpha}{4}\\right) }\\cdot\\frac{\\sin^{2n-i}\\left( \\frac{\\alpha-u}%\n{2}\\right) }{2^{2n-i}\\cos^{2n-i}\\left( \\frac{\\alpha}{4}-\\frac{u}{2}\\right)\n\\sin^{2n-i}\\left( \\frac{\\alpha}{4}\\right) }\\\\\n& =\\frac{1}{2^{2n}\\cos^{2n}\\left( \\frac{\\alpha}{4}-\\frac{u}{2}\\right)\n\\sin^{2n}\\left( \\frac{\\alpha}{4}\\right) }\\cdot T_{2n,i}^{\\alpha}\\left(\nu\\right)\n\\end{align*}\nfor all $i=0,1,\\ldots,2n$ and $u\\in\\left[ 0,\\alpha\\right] $, therefore%\n\\[\n\\mathbf{r}_{2n}\\left( v\\left( u\\right) \\right) =\\frac{%\n{\\displaystyle\\sum\\limits_{i=0}^{2n}}\nw_{i}\\mathbf{d}_{i}B_{i}^{2n}\\left( v\\left( u\\right) \\right) }{%\n{\\displaystyle\\sum\\limits_{j=0}^{2n}}\nw_{j}B_{j}^{2n}\\left( v\\left( u\\right) \\right) }=\\frac{%\n{\\displaystyle\\sum\\limits_{i=0}^{2n}}\n\\mathbf{d}_{i}T_{2n,i}^{\\alpha}\\left( u\\right) }{%\n{\\displaystyle\\sum\\limits_{j=0}^{2n}}\nT_{2n,j}^{\\alpha}\\left( u\\right) }=%\n{\\displaystyle\\sum\\limits_{i=0}^{2n}}\n\\mathbf{d}_{i}T_{2n,i}^{\\alpha}\\left( u\\right) =\\mathbf{t}_{n}^{\\alpha\n}\\left( u\\right) ,~\\forall u\\in\\left[ 0,\\alpha\\right] ,\n\\]\nsince the function system (\\ref{Sanchez_basis}) is normalized, i.e.,\n$\\sum_{j=0}^{2n}T_{2n,j}^{\\alpha}\\left( u\\right) \\equiv1$, $\\forall\nu\\in\\left[ 0,\\alpha\\right] $. Basis functions (\\ref{Sanchez_basis}) and the\nreparametrization function (\\ref{trigonometric_reparametrization}) were\nrepeatedly applied in articles \\cite{Sanchez1990}, \\cite{Sanchez1997} and\n\\cite{Sanchez1998}, however the (inverse) transformation between bases\n$\\overline{\\mathcal{T}}_{2n}^{\\alpha}$ and $\\mathcal{T}_{2n}^{\\alpha}$ were\ncalculated only up to second order in \\cite[p. 916]{Sanchez1998} with the aid\nof a computer algebra system, moreover subdivision of such curves was\ndetailed only for very special control point configurations in\n\\cite{Sanchez1990}.\n\\end{remark}\n\n\\begin{remark}\n[B-algorithm of trigonometric curves]\\label{rem:trigonometric_subdivision}Due\nto Remark \\ref{rem:trigonometric_reparametrization}, the subdivision algorithm\nof trigonometric curves of type (\\ref{trigonometric_curve}) is a simple\ncorollary of the rational de Casteljau algorithm (\\ref{rational_de_Casteljau}%\n). One has to apply the parameter transformation\n(\\ref{trigonometric_reparametrization}) and initial weights\n(\\ref{trigonometric_weights}) in recursive formulae\n(\\ref{rational_de_Casteljau}). Fig. \\ref{fig:subdivsion}(a) shows the steps of\nthis special variant of the classical rational corner cutting algorithm in\ncase of a third order trigonometric curve.\n\\end{remark}\n\n\n\\begin{figure}\n[!h]\n\\begin{center}\n\\includegraphics[\nheight=3.243in,\nwidth=6.2388in\n]%\n{trigonometric_and_hyperbolic_subdivision_u_pi_per_4_alpha_pi_per_2.pdf}%\n\\caption{Consider the fixed control polygon $\\left[ \\mathbf{d}_{i}\\right]\n_{i=0}^{6}$ and the shape parameter $\\alpha=\\frac{\\pi}{2}$ that generate third\norder trigonometric and hyperbolic curves of the type\n(\\ref{trigonometric_curve}) and (\\ref{hyperbolic_curve}), respectively. Cases\n(\\emph{a}) and (\\emph{b}) illustrate the subdivision of these curve at the\ncommon parameter value $u=\\frac{\\pi}{4}$. (The parameter value $\\left.\nv\\left( u\\right) \\right\\vert _{u=\\frac{\\pi}{4}}=\\frac{1}{2}$ is generated by\nreparametrization functions (\\ref{trigonometric_reparametrization}) and\n(\\ref{hyperbolic_reparametrization}), respectively.)}%\n\\label{fig:subdivsion}%\n\\end{center}\n\\end{figure}\n\\qquad\n\nThe order elevation of trigonometric curves of type\n(\\ref{trigonometric_weights}) was also considered in \\cite[Section\n4.2]{Sanchez1998}. This method will be one of the main auxiliary tools used by\nthe present paper, therefore we briefly recall this process, by using our notations.\n\n\\begin{remark}\n[Order elevation of trigonometric curves]%\n\\label{rem:trigonometric_order_elevation}Multiplying the curve\n(\\ref{trigonometric_curve}) with the first order constant function%\n\\[\n1\\equiv T_{2,0}^{\\alpha}\\left( u\\right) +T_{2,1}^{\\alpha}\\left( u\\right)\n+T_{2,2}^{\\alpha}\\left( u\\right) ,~\\forall u\\in\\left[ 0,\\alpha\\right]\n\\]\nand applying the product rule%\n\\[\nT_{2n,i}^{\\alpha}\\left( u\\right) T_{2m,j}^{\\alpha}\\left( u\\right)\n=\\frac{t_{2n,i}^{\\alpha}t_{2m,j}^{\\alpha}}{t_{2\\left( n+m\\right)\n,i+j}^{\\alpha}}T_{2\\left( n+m\\right) ,i+j}^{\\alpha}\\left( u\\right)\n,~\\forall u\\in\\left[ 0,\\alpha\\right] ,\n\\]\none obtains the trigonometric curve%\n\\[\n\\mathbf{t}_{n+1}^{\\alpha}\\left( u\\right) =\\sum_{r=0}^{2\\left( n+1\\right)\n}\\mathbf{e}_{r}T_{2\\left( n+1\\right) ,r}^{\\alpha}\\left( u\\right)\n,~u\\in\\left[ 0,\\alpha\\right]\n\\]\nof order $n+1$ such that%\n\\[\n\\mathbf{t}_{n+1}^{\\alpha}\\left( u\\right) =\\mathbf{t}_{n}^{\\alpha}\\left(\nu\\right) ,~\\forall u\\in\\left[ 0,\\alpha\\right] ,\n\\]\nwhere%\n\\begin{equation}\n\\left\\{\n\\begin{array}\n[c]{lcl}%\n\\mathbf{e}_{0} & = & \\mathbf{d}_{0}\\dfrac{t_{2n,0}^{\\alpha}t_{2,0}^{\\alpha}%\n}{t_{2\\left( n+1\\right) ,0}^{\\alpha}}=\\mathbf{d}_{0}\\\\\n& & \\\\\n\\mathbf{e}_{1} & = & \\mathbf{d}_{0}\\dfrac{t_{2n,0}^{\\alpha}t_{2,1}^{\\alpha}%\n}{t_{2\\left( n+1\\right) ,1}^{\\alpha}}+\\mathbf{d}_{1}\\dfrac{t_{2n,1}^{\\alpha\n}t_{2,0}^{\\alpha}}{t_{2\\left( n+1\\right) ,1}^{\\alpha}},\\\\\n& & \\\\\n\\mathbf{e}_{r} & = & \\mathbf{d}_{r-2}\\dfrac{t_{2n,r-2}^{\\alpha}t_{2,2}%\n^{\\alpha}}{t_{2\\left( n+1\\right) ,r}^{\\alpha}}+\\mathbf{d}_{r-1}%\n\\dfrac{t_{2n,r-1}^{\\alpha}t_{2,1}^{\\alpha}}{t_{2\\left( n+1\\right)\n,r}^{\\alpha}}+\\mathbf{d}_{r}\\dfrac{t_{2n,r}^{\\alpha}t_{2,0}^{\\alpha}%\n}{t_{2\\left( n+1\\right) ,r}^{\\alpha}},~r=2,3,\\ldots,2n,\\\\\n& & \\\\\n\\mathbf{e}_{2n+1} & = & \\mathbf{d}_{2n-1}\\dfrac{t_{2n,2n-1}^{\\alpha}%\nt_{2,2}^{\\alpha}}{t_{2\\left( n+1\\right) ,2n+1}^{\\alpha}}+\\mathbf{d}%\n_{2n}\\dfrac{t_{2n,2n}^{\\alpha}t_{2,1}^{\\alpha}}{t_{2\\left( n+1\\right)\n,2n+1}^{\\alpha}},\\\\\n& & \\\\\n\\mathbf{e}_{2\\left( n+1\\right) } & = & \\mathbf{d}_{2n}\\dfrac{t_{2n,2n}%\n^{\\alpha}t_{2,2}^{\\alpha}}{t_{2\\left( n+1\\right) ,2\\left( n+1\\right)\n}^{\\alpha}}=\\mathbf{d}_{2n}.\n\\end{array}\n\\right. \\label{trigonometric_order_elevation}%\n\\end{equation}\n\n\\end{remark}\n\nDue to normality of functions systems $\\overline{\\mathcal{T}}_{2}^{\\alpha}$,\n$\\overline{\\mathcal{T}}_{2n}^{\\alpha}$ and $\\overline{\\mathcal{T}}_{2\\left(\nn+1\\right) }^{\\alpha}$, one has the simple equality%\n\\[\n1^{n+1}=\\sum_{r=0}^{2\\left( n+1\\right) }T_{2\\left( n+1\\right) ,r}^{\\alpha\n}\\left( u\\right) =\\left( \\sum_{i=0}^{2}T_{2,i}^{\\alpha}\\left( u\\right)\n\\right) \\left( \\sum_{j=0}^{2n}T_{2n,j}^{\\alpha}\\left( u\\right) \\right)\n=1\\cdot1^{n},~\\forall u\\in\\left[ 0,\\alpha\\right]\n\\]\nfrom which follows that%\n\\begin{align*}\n1 & =\\frac{t_{2n,0}^{\\alpha}t_{2,0}^{\\alpha}}{t_{2\\left( n+1\\right)\n,0}^{\\alpha}}=\\frac{t_{2n,2n}^{\\alpha}t_{2,2}^{\\alpha}}{t_{2\\left(\nn+1\\right) ,2\\left( n+1\\right) }^{\\alpha}},\\\\\n1 & =\\frac{t_{2n,0}^{\\alpha}t_{2,1}^{\\alpha}}{t_{2\\left( n+1\\right)\n,1}^{\\alpha}}+\\frac{t_{2n,1}^{\\alpha}t_{2,0}^{\\alpha}}{t_{2\\left( n+1\\right)\n,1}^{\\alpha}}=\\frac{t_{2n,2n-1}^{\\alpha}t_{2,2}^{\\alpha}}{t_{2\\left(\nn+1\\right) ,2n+1}^{\\alpha}}+\\frac{t_{2n,2n}^{\\alpha}t_{2,1}^{\\alpha}%\n}{t_{2\\left( n+1\\right) ,2n+1}^{\\alpha}},\\\\\n1 & =\\frac{t_{2n,r-2}^{\\alpha}t_{2,2}^{\\alpha}}{t_{2\\left( n+1\\right)\n,r}^{\\alpha}}+\\frac{t_{2n,r-1}^{\\alpha}t_{2,1}^{\\alpha}}{t_{2\\left(\nn+1\\right) ,r}^{\\alpha}}+\\frac{t_{2n,r}^{\\alpha}t_{2,0}^{\\alpha}}{t_{2\\left(\nn+1\\right) ,r}^{\\alpha}},~r=2,3,\\ldots,2n,\n\\end{align*}\ni.e., all combinations that appear in the order elevation process\n(\\ref{trigonometric_order_elevation}) are convex. This observation implies\nthat the order elevated control polygon is closer to the shape of the curve\nthan its original one. Therefore, repeatedly increasing the order of the\ntrigonometric curve (\\ref{trigonometric_curve}) from $n$ to $n+z$ ($z\\geq1$),\nwe obtain a sequence of control polygons that converges to the curve generated\nby the starting control polygon. This geometric property is illustrated in\nFig. \\ref{fig:trigonometric_order_elevation} and it will be essential in case\nof control point based exact description of higher dimensional rational\ntrigonometric curves and multivariate surfaces given in traditional parametric form.%\n\n\\begin{figure}\n[!h]\n\\begin{center}\n\\includegraphics[\nheight=3.0268in,\nwidth=2.9551in\n]%\n{trigonometric_order_elevation_alpha_pi_per_2.pdf}%\n\\caption{A plane third order trigonometric curve ($\\alpha=\\frac{\\pi}{2}$) with\nits original and order elevated control polygons which form a sequence that\nconverges to the curve generated by the original control polygon.}%\n\\label{fig:trigonometric_order_elevation}%\n\\end{center}\n\\end{figure}\n\n\n\\begin{remark}\n[Asymptotic behavior]\\label{rem:trigonometric_asymptotic_behavior}As proved in\n\\cite[Proposition 2.1, p. 249]{JuhaszRoth2014}, the basis $\\overline\n{\\mathcal{T}}_{2n}^{\\alpha}$ degenerates to the classical Bernstein polynomial\nbasis $\\mathcal{B}_{2n}$ defined over the unit compact interval as the shape\nparameter $\\alpha$ tends to $0$ from above. In this case the trigonometric\ncurve (\\ref{trigonometric_curve}) becomes a classical B\\'{e}zier curve of\ndegree $2n$, while the subdivision algorithm presented in Remark\n\\ref{rem:trigonometric_subdivision} degenerates to the classical non-rational\nde Casteljau algorithm. Fig. \\ref{fig:effect_of_trigonometric_shape_parameter}\nillustrates the effect of the shape parameter $\\alpha\\in\\left( 0,\\pi\\right)\n$ on the image of a third order trigonometric curve.\n\\end{remark}\n\n\n\\begin{figure}\n[!h]\n\\begin{center}\n\\includegraphics[\nheight=3.0398in,\nwidth=3.0952in\n]%\n{effect_of_trigonometric_shape_parameter.pdf}%\n\\caption{Effect of the design parameter $\\alpha\\in\\left( 0,\\pi\\right) $ on\nthe shape of a third order trigonometric curve. In the limiting case\n$\\alpha\\rightarrow0$ the curve becomes a classical B\\'{e}zier curve of degree\n$6$ (as it is expected, in this special case, the B-algorithm of the\ntrigonometric curve degenerates to the classical corner cutting de Casteljau\nalgorithm, i.e., each subdivision point is determined by the same ratio along\nthe edges of the control polygon.)}%\n\\label{fig:effect_of_trigonometric_shape_parameter}%\n\\end{center}\n\\end{figure}\n\n\n\\begin{definition}\n[Rational trigonometric curves]The non-negative weight vector $\\mathbf{%\n\\boldsymbol{\\omega}%\n}=\\left[ \\omega_{i}\\right] _{i=0}^{2n}$ of rank $1$ associated with the\ncontrol polygon $\\left[ \\mathbf{d}_{i}\\right] _{i=0}^{2n}\\in\\mathcal{M}%\n_{1,2n+1}\\left(\n\\mathbb{R}\n^{\\delta}\\right) $ and the normalized linearly independent rational (or\nquotient) functions%\n\\[\nR_{2n,i}^{\\alpha,\\mathbf{%\n\\boldsymbol{\\omega}%\n}}\\left( u\\right) =\\frac{\\omega_{i}T_{2n,i}^{\\alpha}\\left( u\\right) }{%\n{\\displaystyle\\sum\\limits_{j=0}^{2n}}\n\\omega_{j}T_{2n,j}^{\\alpha}\\left( u\\right) },~u\\in\\left[ 0,\\alpha\\right]\n,~i=0,1,\\ldots,2n\n\\]\ndefine the rational counterpart%\n\\begin{equation}\n\\mathbf{t}_{n}^{\\alpha,\\mathbf{%\n\\boldsymbol{\\omega}%\n}}\\left( u\\right) =\\sum_{i=0}^{2n}\\omega_{i}\\mathbf{d}_{i}R_{2n,i}%\n^{\\alpha,\\mathbf{%\n\\boldsymbol{\\omega}%\n}}\\left( u\\right) =\\frac{%\n{\\displaystyle\\sum\\limits_{i=0}^{2n}}\n\\omega_{i}\\mathbf{d}_{i}T_{2n,i}^{\\alpha}\\left( u\\right) }{%\n{\\displaystyle\\sum\\limits_{j=0}^{2n}}\n\\omega_{j}T_{2n,j}^{\\alpha}\\left( u\\right) },~u\\in\\left[ 0,\\alpha\\right]\n\\label{rational_trigonometric_curve}%\n\\end{equation}\nof the trigonometric curve (\\ref{trigonometric_curve}).\n\\end{definition}\n\n\\begin{remark}\n[Pre-image of rational trigonometric curves]The rational trigonometric curve\n(\\ref{rational_trigonometric_curve}) can also be considered as the central\nprojection of the higher dimensional curve%\n\\begin{equation}\n\\mathbf{t}_{n,\\mathcal{\\wp}}^{\\alpha,\\mathbf{%\n\\boldsymbol{\\omega}%\n}}\\left( u\\right) =\\sum_{i=0}^{2n}\\left[\n\\begin{array}\n[c]{c}%\n\\omega_{i}\\mathbf{d}_{i}\\\\\n\\omega_{i}%\n\\end{array}\n\\right] T_{2n,i}^{\\alpha}\\left( u\\right) ,~u\\in\\left[ 0,\\alpha\\right]\n\\label{trigonometric_pre_image}%\n\\end{equation}\nin the $\\delta+1$ dimensional space from the origin onto the $\\delta$\ndimensional hyperplane $x^{\\delta+1}=1$ (assuming that the coordinates of $%\n\\mathbb{R}\n^{\\delta+1}$ are denoted by $x^{1},x^{2},\\ldots,x^{\\delta+1}$). The curve\n(\\ref{trigonometric_pre_image}) is called the pre-image of the rational curve\n(\\ref{rational_trigonometric_curve}), while the vector space $%\n\\mathbb{R}\n^{\\delta+1}$ is called its pre-image space. This concept will be useful in\ncase of control point based exact description of smooth rational trigonometric\ncurves given in traditional parametric form expressed in the canonical basis\n$\\mathcal{T}_{2n}^{\\alpha}$.\n\\end{remark}\n\n\\subsection{Hyperbolic curves and their rational counterpart}\n\nIn this case, let $\\alpha>0$ be an arbitrarily fixed parameter and consider\nthe B-basis%\n\\begin{equation}\n\\overline{\\mathcal{H}}_{2n}^{\\alpha}=\\left\\{ H_{2n,i}^{\\alpha}\\left(\nu\\right) :u\\in\\left[ 0,\\alpha\\right] \\right\\} _{i=0}^{2n}=\\left\\{\nh_{2n,i}^{\\alpha}\\sinh^{2n-i}\\left( \\frac{\\alpha-u}{2}\\right) \\sinh\n^{i}\\left( \\frac{u}{2}\\right) :u\\in\\left[ 0,\\alpha\\right] \\right\\}\n_{i=0}^{2n} \\label{Wang_basis}%\n\\end{equation}\nof order $n$ (degree $2n$) of the vector space (\\ref{hyperbolic_vector_space})\nintroduced in \\cite{ShenWang2005}, where the non-negative normalizing coefficients%\n\\[\nh_{2n,i}^{\\alpha}=\\frac{1}{\\sinh^{2n}\\left( \\frac{\\alpha}{2}\\right) }%\n\\sum_{r=0}^{\\left\\lfloor \\frac{i}{2}\\right\\rfloor }\\binom{n}{i-r}\\binom\n{i-r}{r}\\left( 2\\cosh\\left( \\frac{\\alpha}{2}\\right) \\right) ^{i-2r}%\n,~i=0,1,\\ldots,2n\n\\]\nfulfill the symmetry property%\n\\begin{equation}\nh_{2n,i}^{\\alpha}=h_{2n,2n-i}^{\\alpha},~i=0,1,\\ldots,n\\text{.}%\n\\end{equation}\n\n\n\\begin{definition}\n[Hyperbolic curves]The convex combination%\n\\begin{equation}\n\\mathbf{h}_{n}^{\\alpha}\\left( u\\right) =\\sum_{i=0}^{2n}\\mathbf{d}%\n_{i}H_{2n,i}^{\\alpha}\\left( u\\right) ,~u\\in\\left[ 0,\\alpha\\right] ,\n\\label{hyperbolic_curve}%\n\\end{equation}\ndefines a hyperbolic curve of order $n$ (degree $2n$), where $\\left[\n\\mathbf{d}_{i}\\right] _{i=0}^{2n}\\in\\mathcal{M}_{1,2n+1}\\left(\n\\mathbb{R}\n^{\\delta}\\right) $ forms a control polygon.\n\\end{definition}\n\nSimilarly to Subsection \\ref{sec:trigonometric_curves} it is easy to observe\nthat curves of type (\\ref{hyperbolic_curve}) are in fact special\nreparametrizations of a class of rational B\\'{e}zier curves of even degree\n$2n$. Instead of trigonometric sine, cosine, and tangent functions one has to\napply the hyperbolic variant of these functions, i.e., instead of parameter\ntransformation (\\ref{trigonometric_reparametrization}) and weights\n(\\ref{trigonometric_weights}) one has to substitute the reparametrization\nfunction%\n\\begin{equation}\n\\left\\{\n\\begin{array}\n[c]{l}%\nv:\\left[ 0,\\alpha\\right] \\rightarrow\\left[ 0,1\\right] ,\\\\\n\\\\\nv\\left( u\\right) =\\dfrac{1}{2}+\\dfrac{\\tanh\\left( \\frac{u}{2}-\\frac{\\alpha\n}{4}\\right) }{2\\tanh\\left( \\frac{\\alpha}{4}\\right) }=\\dfrac{\\sinh\\left(\n\\frac{u}{2}\\right) }{2\\cosh\\left( \\frac{\\alpha}{4}-\\frac{u}{2}\\right)\n\\sinh\\left( \\frac{\\alpha}{4}\\right) }%\n\\end{array}\n\\right. \\label{hyperbolic_reparametrization}%\n\\end{equation}\nand weights%\n\\begin{equation}\nw_{i}=\\frac{h_{2n,i}^{\\alpha}}{\\binom{2n}{i}},~i=0,1,\\ldots,2n\n\\label{hyperbolic_weights}%\n\\end{equation}\ninto the rational B\\'{e}zier curve (\\ref{rational_Bezier_curve}), respectively.\n\nUsing observations similar to Remarks \\ref{rem:trigonometric_subdivision},\n\\ref{rem:trigonometric_order_elevation} and\n\\ref{rem:trigonometric_asymptotic_behavior}, the subdivision, order elevation\nand asymptotic behavior of hyperbolic curves of type (\\ref{hyperbolic_curve})\ncan also be formulated. With the exception of the subdivision algorithm and\nwithout the observation of the parameter transformation\n(\\ref{hyperbolic_reparametrization}) and special weight settings\n(\\ref{hyperbolic_weights}), the asymptotic behavior and the order elevation of\nhyperbolic curves were first studied in \\cite{ShenWang2005}. The steps of the\nsubdivision of a third order hyperbolic curve is presented in Fig.\n\\ref{fig:subdivsion}(\\emph{b}).\n\nThe rational variant of the hyperbolic curve (\\ref{hyperbolic_curve}) and its\npre-image can also be easily described.\n\n\\begin{definition}\n[Rational hyperbolic curves]Consider the non-negative weight vector\n$\\mathbf{%\n\\boldsymbol{\\omega}%\n}=\\left[ \\omega_{i}\\right] _{i=0}^{2n}$ of rank $1$ associated with the\ncontrol polygon $\\left[ \\mathbf{d}_{i}\\right] _{i=0}^{2n}\\in\\mathcal{M}%\n_{1,2n+1}\\left(\n\\mathbb{R}\n^{\\delta}\\right) $. Normalized quotient basis functions%\n\\[\nS_{2n,i}^{\\alpha,\\mathbf{%\n\\boldsymbol{\\omega}%\n}}\\left( u\\right) =\\frac{\\omega_{i}H_{2n,i}^{\\alpha}\\left( u\\right) }{%\n{\\displaystyle\\sum\\limits_{j=0}^{2n}}\n\\omega_{j}H_{2n,j}^{\\alpha}\\left( u\\right) },~u\\in\\left[ 0,\\alpha\\right]\n,~i=0,1,\\ldots,2n\n\\]\ngenerate the rational counterpart%\n\\begin{equation}\n\\mathbf{h}_{n}^{\\alpha,\\mathbf{%\n\\boldsymbol{\\omega}%\n}}\\left( u\\right) =\\sum_{i=0}^{2n}\\omega_{i}\\mathbf{d}_{i}S_{2n,i}%\n^{\\alpha,\\mathbf{%\n\\boldsymbol{\\omega}%\n}}\\left( u\\right) =\\frac{%\n{\\displaystyle\\sum\\limits_{i=0}^{2n}}\n\\omega_{i}\\mathbf{d}_{i}H_{2n,i}^{\\alpha}\\left( u\\right) }{%\n{\\displaystyle\\sum\\limits_{j=0}^{2n}}\n\\omega_{j}H_{2n,j}^{\\alpha}\\left( u\\right) },~u\\in\\left[ 0,\\alpha\\right]\n\\label{rational_hyperbolic_curve}%\n\\end{equation}\nof the hyperbolic curve (\\ref{hyperbolic_curve}), the pre-image of which is%\n\\begin{equation}\n\\mathbf{h}_{n,\\mathcal{\\wp}}^{\\alpha,\\mathbf{%\n\\boldsymbol{\\omega}%\n}}\\left( u\\right) =\\sum_{i=0}^{2n}\\left[\n\\begin{array}\n[c]{c}%\n\\omega_{i}\\mathbf{d}_{i}\\\\\n\\omega_{i}%\n\\end{array}\n\\right] H_{2n,i}^{\\alpha}\\left( u\\right) ,~u\\in\\left[ 0,\\alpha\\right] .\n\\label{hyperbolic_pre_image}%\n\\end{equation}\n\n\\end{definition}\n\n\\section{(Hybrid) (rational) trigonometric and hyperbolic multivariate\nsurfaces\\label{sec:multivariate_surfaces}}\n\nBy means of tensor products of curves of type (\\ref{trigonometric_curve}) and\n(\\ref{hyperbolic_curve}) one can introduce the following multivariate higher\ndimensional surface modeling tools. Let $\\delta\\geq2$ and $\\kappa\\geq0$\narbitrarily fixed natural numbers and consider the also fixed vector\n$\\mathbf{n}=\\left[ n_{j}\\right] _{j=1}^{\\delta}$ of orders, where $n_{j}%\n\\geq1$ for all $j=1,2,\\ldots,\\delta$.\n\n\\begin{definition}\n[Trigonometric surfaces and their rational counterpart]%\n\\label{trigonometric_surface_definitions}Let\n\\[\n\\mathbf{%\n\\boldsymbol{\\alpha}%\n=}\\left[ \\alpha_{j}\\right] _{j=1}^{\\delta}\\in\\times_{j=1}^{\\delta}\\left(\n0,\\pi\\right)\n\\]\nbe a fixed vector of shape parameters and consider the multidimensional\ncontrol grid\n\\begin{equation}\n\\left[ \\mathbf{d}_{i_{1},i_{2},\\ldots,i_{\\delta}}\\right] _{i_{1}%\n=0,i_{2}=0,\\ldots,i_{\\delta}=0}^{2n_{1},2n_{2},\\ldots,2n_{\\delta}}%\n\\in\\mathcal{M}_{2n_{1}+1,2n_{2}+1,\\ldots2n_{\\delta}+1}\\left(\n\\mathbb{R}\n^{\\delta+\\kappa}\\right) . \\label{trigonometric_control_grid}%\n\\end{equation}\nThe multivariate surface%\n\\begin{align}\n\\mathbf{t}_{\\mathbf{n}}^{\\mathbf{%\n\\boldsymbol{\\alpha}%\n}}\\left( \\mathbf{u}\\right) & =\\mathbf{t}_{n_{1},n_{2},\\ldots,n_{\\delta}%\n}^{\\alpha_{1},\\alpha_{2},\\ldots,\\alpha_{\\delta}}\\left( u_{1},u_{2}%\n,\\ldots,u_{\\delta}\\right) \\label{trigonometric_surface}\\\\\n& =\\sum_{i_{1}=0}^{2n_{1}}\\sum_{i_{2}=0}^{2n_{2}}\\cdots\\sum_{i_{\\delta}%\n=0}^{2n_{\\delta}}\\mathbf{d}_{i_{1},i_{2},\\ldots,i_{\\delta}}T_{2n_{1},i_{1}%\n}^{\\alpha_{1}}\\left( u_{1}\\right) T_{2n_{2},i_{2}}^{\\alpha_{2}}\\left(\nu_{2}\\right) \\cdot\\ldots\\cdot T_{2n_{\\delta},i_{\\delta}}^{\\alpha_{\\delta}%\n}\\left( u_{\\delta}\\right) ,~\\mathbf{u}=\\left[ u_{j}\\right] _{j=1}^{\\delta\n}\\in\\times_{j=1}^{\\delta}\\left[ 0,\\alpha_{j}\\right] \\nonumber\n\\end{align}\nis called $\\delta$-variate trigonometric surface of order $\\mathbf{n}$ taking\nvalues in $%\n\\mathbb{R}\n^{\\delta+\\kappa}$. Assigning the non-negative multidimensional weight matrix\n\\begin{equation}\n\\boldsymbol{\\Omega}=\\left[ \\omega_{i_{1},i_{2},\\ldots,i_{\\delta}}\\right] _{i_{1}%\n=0,i_{2}=0,\\ldots,i_{\\delta}=0}^{2n_{1},2n_{2},\\ldots,2n_{\\delta}}%\n\\in\\mathcal{M}_{2n_{1}+1,2n_{2}+1,\\ldots2n_{\\delta}+1}\\left(\n\\mathbb{R}\n_{+}\\right) \\label{trigonometric_weight_grid}%\n\\end{equation}\nof rank at least $1$ to the control grid (\\ref{trigonometric_control_grid}),\none obtains the $\\delta$-variate rational trigonometric surface%\n\\begin{align}\n\\mathbf{t}_{\\mathbf{n}}^{\\mathbf{%\n\\boldsymbol{\\alpha}%\n},\\boldsymbol{\\Omega}}\\left( \\mathbf{u}\\right) & =\\mathbf{t}_{n_{1},n_{2}%\n,\\ldots,n_{\\delta}}^{\\alpha_{1},\\alpha_{2},\\ldots,\\alpha_{\\delta},\\boldsymbol{\\Omega}\n}\\left( u_{1},u_{2},\\ldots,u_{\\delta}\\right)\n\\label{trigonometric_rational_surface}\\\\\n& =\\frac{%\n{\\displaystyle\\sum\\limits_{i_{1}=0}^{2n_{1}}}\n{\\displaystyle\\sum\\limits_{i_{2}=0}^{2n_{2}}}\n\\cdots%\n{\\displaystyle\\sum\\limits_{i_{\\delta}=0}^{2n_{\\delta}}}\n\\omega_{i_{1},i_{2},\\ldots,i_{\\delta}}\\mathbf{d}_{i_{1},i_{2},\\ldots\n,i_{\\delta}}T_{2n_{1},i_{1}}^{\\alpha_{1}}\\left( u_{1}\\right) T_{2n_{2}%\n,i_{2}}^{\\alpha_{2}}\\left( u_{2}\\right) \\cdot\\ldots\\cdot T_{2n_{\\delta\n},i_{\\delta}}^{\\alpha_{\\delta}}\\left( u_{\\delta}\\right) }{%\n{\\displaystyle\\sum\\limits_{j_{1}=0}^{2n_{1}}}\n{\\displaystyle\\sum\\limits_{j_{2}=0}^{2n_{2}}}\n\\cdots%\n{\\displaystyle\\sum\\limits_{j_{\\delta}=0}^{2n_{\\delta}}}\n\\omega_{j_{1},j_{2},\\ldots,j_{\\delta}}T_{2n_{1},j_{1}}^{\\alpha_{1}}\\left(\nu_{1}\\right) T_{2n_{2},j_{2}}^{\\alpha_{2}}\\left( u_{2}\\right) \\cdot\n\\ldots\\cdot T_{2n_{\\delta},j_{\\delta}}^{\\alpha_{\\delta}}\\left( u_{\\delta\n}\\right) }\\nonumber\n\\end{align}\nof the same order, which is the central projection of the pre-image%\n\\begin{align}\n\\mathbf{t}_{\\mathbf{n},\\wp}^{\\mathbf{%\n\\boldsymbol{\\alpha}%\n},\\boldsymbol{\\Omega}}\\left( \\mathbf{u}\\right) & =\\mathbf{t}_{n_{1},n_{2}%\n,\\ldots,n_{\\delta},\\wp}^{\\alpha_{1},\\alpha_{2},\\ldots,\\alpha_{\\delta},\\boldsymbol{\\Omega}\n}\\left( u_{1},u_{2},\\ldots,u_{\\delta}\\right) \\label{trigonometric_preimage}%\n\\\\\n& =\\sum_{i_{1}=0}^{2n_{1}}\\sum_{i_{2}=0}^{2n_{2}}\\cdots\\sum_{i_{\\delta}%\n=0}^{2n_{\\delta}}\\left[\n\\begin{array}\n[c]{c}%\n\\omega_{i_{1},i_{2},\\ldots,i_{\\delta}}\\mathbf{d}_{i_{1},i_{2},\\ldots\n,i_{\\delta}}\\\\\n\\omega_{i_{1},i_{2},\\ldots,i_{\\delta}}%\n\\end{array}\n\\right] T_{2n_{1},i_{1}}^{\\alpha_{1}}\\left( u_{1}\\right) T_{2n_{2},i_{2}%\n}^{\\alpha_{2}}\\left( u_{2}\\right) \\cdot\\ldots\\cdot T_{2n_{\\delta},i_{\\delta\n}}^{\\alpha_{\\delta}}\\left( u_{\\delta}\\right) \\nonumber\n\\end{align}\nin $%\n\\mathbb{R}\n^{\\delta+\\kappa+1}$ from its origin onto the $\\delta+\\kappa$ dimensional\nhyperplane $x^{\\delta+k+1}=1$ (provided that the coordinates of $%\n\\mathbb{R}\n^{\\delta+\\kappa+1}$ are labeled by $x^{1},x^{2},\\ldots,x^{\\delta+k+1}$).\n\\end{definition}\n\n\\begin{remark}\n[$3$-dimensional $2$-variate trigonometric surfaces]The simplest variant of\nmultivariate surfaces introduced in Definition\n\\ref{trigonometric_surface_definitions} corresponds to $\\delta=2$ and\n$\\kappa=1$, when the $2$-variate trigonometric surface\n(\\ref{trigonometric_surface}) is a $3$-dimensional traditional tensor product\nsurface of curves of the type (\\ref{trigonometric_curve}). In this special\ncase, the grid (\\ref{trigonometric_control_grid}) of control points\\ and the\nmultidimensional weight matrix (\\ref{trigonometric_weight_grid}) degenerate to\na traditional control net and rectangular weight matrix, respectively.\n\\end{remark}\n\n\\begin{remark}\n[$3$-dimensional trigonometric volumes]Using settings $\\delta=3$ and\n$\\kappa=0$, Definition \\ref{trigonometric_surface_definitions} describes\n$3$-dimensional volumes (solids) by means of $3$-variate tensor product of\ncurves of the type (\\ref{trigonometric_curve}).\n\\end{remark}\n\n\\begin{definition}\n[Hyperbolic surfaces and their rational counterpart]%\n\\label{trigonometric_surface_definitions copy(1)}Let\n\\[\n\\mathbf{%\n\\boldsymbol{\\alpha}%\n=}\\left[ \\alpha_{j}\\right] _{j=1}^{\\delta}\\in\\times_{j=1}^{\\delta}\\left(\n0,+\\infty\\right)\n\\]\nbe a fixed vector of shape parameters and consider the non-negative\nmultidimensional weight matrix\n\\[\n\\boldsymbol{\\Omega}=\\left[ \\omega_{i_{1},i_{2},\\ldots,i_{\\delta}}\\right] _{i_{1}%\n=0,i_{2}=0,\\ldots,i_{\\delta}=0}^{2n_{1},2n_{2},\\ldots,2n_{\\delta}}%\n\\in\\mathcal{M}_{2n_{1}+1,2n_{2}+1,\\ldots2n_{\\delta}+1}\\left(\n\\mathbb{R}\n_{+}\\right)\n\\]\n(of rank at least $1$) associated with the control grid\n\\[\n\\left[ \\mathbf{d}_{i_{1},i_{2},\\ldots,i_{\\delta}}\\right] _{i_{1}%\n=0,i_{2}=0,\\ldots,i_{\\delta}=0}^{2n_{1},2n_{2},\\ldots,2n_{\\delta}}%\n\\in\\mathcal{M}_{2n_{1}+1,2n_{2}+1,\\ldots2n_{\\delta}+1}\\left(\n\\mathbb{R}\n^{\\delta+\\kappa}\\right) .\n\\]\nThe multivariate hyperbolic surface of order $\\mathbf{n}$, its rational\ncounterpart, and the pre-image of the rational variant are%\n\\begin{align}\n\\mathbf{h}_{\\mathbf{n}}^{\\mathbf{%\n\\boldsymbol{\\alpha}%\n}}\\left( \\mathbf{u}\\right) & =\\mathbf{h}_{n_{1},n_{2},\\ldots,n_{\\delta}%\n}^{\\alpha_{1},\\alpha_{2},\\ldots,\\alpha_{\\delta}}\\left( u_{1},u_{2}%\n,\\ldots,u_{\\delta}\\right) \\label{hyperbolic_surface}\\\\\n& =\\sum_{i_{1}=0}^{2n_{1}}\\sum_{i_{2}=0}^{2n_{2}}\\cdots\\sum_{i_{\\delta}%\n=0}^{2n_{\\delta}}\\mathbf{d}_{i_{1},i_{2},\\ldots,i_{\\delta}}H_{2n_{1},i_{1}%\n}^{\\alpha_{1}}\\left( u_{1}\\right) H_{2n_{2},i_{2}}^{\\alpha_{2}}\\left(\nu_{2}\\right) \\cdot\\ldots\\cdot H_{2n_{\\delta},i_{\\delta}}^{\\alpha_{\\delta}%\n}\\left( u_{\\delta}\\right) ,\\nonumber\\\\\n\\mathbf{u} & =\\left[ u_{j}\\right] _{j=1}^{\\delta}\\in\\times_{j=1}^{\\delta\n}\\left[ 0,\\alpha_{j}\\right] ,\\nonumber\\\\\n& \\nonumber\\\\\n\\mathbf{h}_{\\mathbf{n}}^{\\mathbf{%\n\\boldsymbol{\\alpha}%\n},\\boldsymbol{\\Omega}}\\left( \\mathbf{u}\\right) & =\\mathbf{h}_{n_{1},n_{2}%\n,\\ldots,n_{\\delta}}^{\\alpha_{1},\\alpha_{2},\\ldots,\\alpha_{\\delta},\\boldsymbol{\\Omega}\n}\\left( u_{1},u_{2},\\ldots,u_{\\delta}\\right)\n\\label{rational_hyperbolic_surface}\\\\\n& =\\frac{%\n{\\displaystyle\\sum\\limits_{i_{1}=0}^{2n_{1}}}\n{\\displaystyle\\sum\\limits_{i_{2}=0}^{2n_{2}}}\n\\cdots%\n{\\displaystyle\\sum\\limits_{i_{\\delta}=0}^{2n_{\\delta}}}\n\\omega_{i_{1},i_{2},\\ldots,i_{\\delta}}\\mathbf{d}_{i_{1},i_{2},\\ldots\n,i_{\\delta}}H_{2n_{1},i_{1}}^{\\alpha_{1}}\\left( u_{1}\\right) H_{2n_{2}%\n,i_{2}}^{\\alpha_{2}}\\left( u_{2}\\right) \\cdot\\ldots\\cdot H_{2n_{\\delta\n},i_{\\delta}}^{\\alpha_{\\delta}}\\left( u_{\\delta}\\right) }{%\n{\\displaystyle\\sum\\limits_{j_{1}=0}^{2n_{1}}}\n{\\displaystyle\\sum\\limits_{j_{2}=0}^{2n_{2}}}\n\\cdots%\n{\\displaystyle\\sum\\limits_{j_{\\delta}=0}^{2n_{\\delta}}}\n\\omega_{j_{1},j_{2},\\ldots,j_{\\delta}}H_{2n_{1},j_{1}}^{\\alpha_{1}}\\left(\nu_{1}\\right) H_{2n_{2},j_{2}}^{\\alpha_{2}}\\left( u_{2}\\right) \\cdot\n\\ldots\\cdot H_{2n_{\\delta},j_{\\delta}}^{\\alpha_{\\delta}}\\left( u_{\\delta\n}\\right) }\\nonumber\n\\end{align}\nand%\n\\begin{align*}\n\\mathbf{h}_{\\mathbf{n},\\wp}^{\\mathbf{%\n\\boldsymbol{\\alpha}%\n},\\boldsymbol{\\Omega}}\\left( \\mathbf{u}\\right) & =\\mathbf{h}_{n_{1},n_{2}%\n,\\ldots,n_{\\delta},\\wp}^{\\alpha_{1},\\alpha_{2},\\ldots,\\alpha_{\\delta},\\boldsymbol{\\Omega}\n}\\left( u_{1},u_{2},\\ldots,u_{\\delta}\\right) \\\\\n& =\\sum_{i_{1}=0}^{2n_{1}}\\sum_{i_{2}=0}^{2n_{2}}\\cdots\\sum_{i_{\\delta}%\n=0}^{2n_{\\delta}}\\left[\n\\begin{array}\n[c]{c}%\n\\omega_{i_{1},i_{2},\\ldots,i_{\\delta}}\\mathbf{d}_{i_{1},i_{2},\\ldots\n,i_{\\delta}}\\\\\n\\omega_{i_{1},i_{2},\\ldots,i_{\\delta}}%\n\\end{array}\n\\right] H_{2n_{1},i_{1}}^{\\alpha_{1}}\\left( u_{1}\\right) H_{2n_{2},i_{2}%\n}^{\\alpha_{2}}\\left( u_{2}\\right) \\cdot\\ldots\\cdot H_{2n_{\\delta},i_{\\delta\n}}^{\\alpha_{\\delta}}\\left( u_{\\delta}\\right) ,\n\\end{align*}\nrespectively.\n\\end{definition}\n\n\\begin{remark}\n[Hybrid multivariate surfaces]Naturally, one can\\ also mix the trigonometric\nor hyperbolic type of B-basis functions in directions $\\left[ u_{j}\\right]\n_{j=1}^{\\delta}$, i.e., one can also define higher dimensional hybrid\nmultivariate (rational) surfaces.\n\\end{remark}\n\n\\section{ Basis transformations\\label{sec:basis_transformations}}\n\nWe are going to derive recursive formulae for the transformation of B-bases\n$\\overline{\\mathcal{T}}_{2n}^{\\alpha}$ and $\\overline{\\mathcal{H}}%\n_{2n}^{\\alpha}$ to the canonical bases $\\mathcal{T}_{2n}^{\\alpha}$ and\n$\\mathcal{H}_{2n}^{\\alpha}$ of the vector spaces $\\mathbb{T}_{2n}^{\\alpha}$\nand $\\mathbb{H}_{2n}^{\\alpha}$, respectively.\n\n\\subsection{The trigonometric\ncase\\label{sec:trigonometric_basis_transformation}}\n\nLet $k\\in\\left\\{ 0,1,\\ldots,n\\right\\} $ be an arbitrarily fixed natural\nnumber. Assume that the unique representations of trigonometric functions\n$\\sin\\left( ku\\right) $ and $\\cos\\left( ku\\right) $ in the basis\n(\\ref{Sanchez_basis}) of order $n$ are%\n\\begin{equation}\n\\sin\\left( ku\\right) =\\sum_{i=0}^{2n}\\lambda_{k,i}^{n}T_{2n,i}^{\\alpha\n}\\left( u\\right) ,~u\\in\\left[ 0,\\alpha\\right] \\label{sine_form}%\n\\end{equation}\nand%\n\\begin{equation}\n\\cos\\left( ku\\right) =\\sum_{i=0}^{2n}\\mu_{k,i}^{n}T_{2n,i}^{\\alpha}\\left(\nu\\right) ,~u\\in\\left[ 0,\\alpha\\right] , \\label{cosine_form}%\n\\end{equation}\nrespectively, where coefficients $\\left\\{ \\lambda_{k,i}^{n}\\right\\}\n_{i=0}^{2n}$ and $\\left\\{ \\mu_{k,i}^{n}\\right\\} _{i=0}^{2n}$ are unique real\nnumbers. The basis transformation from the first order B-basis $\\overline\n{\\mathcal{T}}_{2}^{\\alpha}$ to the first order trigonometric canonical basis\n$\\mathcal{T}_{2}^{\\alpha}$ can be expressed in the matrix form%\n\\[\n\\left[\n\\begin{array}\n[c]{c}%\n1\\\\\n\\sin\\left( u\\right) \\\\\n\\cos\\left( u\\right)\n\\end{array}\n\\right] =\\left[\n\\begin{array}\n[c]{ccc}%\n\\mu_{0,0}^{1} & \\mu_{0,1}^{1} & \\mu_{0,2}^{1}\\\\\n\\lambda_{1,0}^{1} & \\lambda_{1,1}^{1} & \\lambda_{1,2}^{1}\\\\\n\\mu_{1,0}^{1} & \\mu_{1,1}^{1} & \\mu_{1,2}^{1}%\n\\end{array}\n\\right] \\left[\n\\begin{array}\n[c]{c}%\nT_{2,0}^{\\alpha}\\left( u\\right) \\\\\nT_{2,1}^{\\alpha}\\left( u\\right) \\\\\nT_{2,2}^{\\alpha}\\left( u\\right)\n\\end{array}\n\\right] ,~\\forall u\\in\\left[ 0,\\alpha\\right] ,\n\\]\nwhere%\n\\begin{equation}\n\\left\\{\n\\begin{array}\n[c]{l}%\n\\mu_{0,0}^{1}=\\mu_{0,1}^{1}=\\mu_{0,2}^{1}=1,\\\\\n\\\\\n\\lambda_{1,0}^{1}=0,~\\lambda_{1,1}^{1}=\\tan\\left( \\frac{\\alpha}{2}\\right)\n,~\\lambda_{1,2}^{1}=\\sin\\left( \\alpha\\right) ,\\\\\n\\\\\n\\mu_{1,0}^{1}=\\mu_{1,1}^{1}=1,~\\mu_{1,2}^{1}=\\cos\\left( \\alpha\\right) .\n\\end{array}\n\\right. \\label{trigonometric_initial_conditions}%\n\\end{equation}\n\n\nUsing initial conditions (\\ref{trigonometric_initial_conditions}), our\nobjective is to derive recursive formulae for the matrix elements of the\nlinear transformation that changes the higher order B-basis $\\overline\n{\\mathcal{T}}_{2\\left( n+1\\right) }^{\\alpha}$ to the canonical trigonometric\nbasis $\\mathcal{T}_{2\\left( n+1\\right) }^{\\alpha}$.\n\nPerforming order elevation on functions (\\ref{sine_form}) and\n(\\ref{cosine_form}), one obtains that%\n\\[\n\\sin\\left( ku\\right) =\\sum_{r=0}^{2\\left( n+1\\right) }\\lambda_{k,r}%\n^{n+1}T_{2\\left( n+1\\right) ,r}^{\\alpha}\\left( u\\right)\n\\]\nand%\n\\[\n\\cos\\left( ku\\right) =\\sum_{r=0}^{2\\left( n+1\\right) }\\mu_{k,r}%\n^{n+1}T_{2\\left( n+1\\right) ,r}^{\\alpha}\\left( u\\right) ,\n\\]\nwhere%\n\\begin{align*}\n\\lambda_{k,0}^{n+1} & =\\lambda_{k,0}^{n},\\\\\n\\lambda_{k,1}^{n+1} & =\\lambda_{k,0}^{n}\\frac{t_{2n,0}^{\\alpha}%\nt_{2,1}^{\\alpha}}{t_{2\\left( n+1\\right) ,1}^{\\alpha}}+\\lambda_{k,1}^{n}%\n\\frac{t_{2n,1}^{\\alpha}t_{2,0}^{\\alpha}}{t_{2\\left( n+1\\right) ,1}^{\\alpha}%\n},\\\\\n\\lambda_{k,r}^{n+1} & =\\lambda_{k,r-2}^{n}\\frac{t_{2n,r-2}^{\\alpha}%\nt_{2,2}^{\\alpha}}{t_{2\\left( n+1\\right) ,r}^{\\alpha}}+\\lambda_{k,r-1}%\n^{n}\\frac{t_{2n,r-1}^{\\alpha}t_{2,1}^{\\alpha}}{t_{2\\left( n+1\\right)\n,r}^{\\alpha}}+\\lambda_{k,r}^{n}\\frac{t_{2n,r}^{\\alpha}t_{2,0}^{\\alpha}%\n}{t_{2\\left( n+1\\right) ,r}^{\\alpha}},~r=2,3,\\ldots,2n,\\\\\n\\lambda_{k,2n+1}^{n+1} & =\\lambda_{k,2n-1}^{n}\\frac{t_{2n,2n-1}^{\\alpha\n}t_{2,2}^{\\alpha}}{t_{2\\left( n+1\\right) ,2n+1}^{\\alpha}}+\\lambda_{k,2n}%\n^{n}\\frac{t_{2n,2n}^{\\alpha}t_{2,1}^{\\alpha}}{t_{2\\left( n+1\\right)\n,2n+1}^{\\alpha}},\\\\\n\\lambda_{k,2\\left( n+1\\right) }^{n+1} & =\\lambda_{k,2n}^{n}%\n\\end{align*}\nand%\n\\begin{align*}\n\\mu_{k,0}^{n+1} & =\\mu_{k,0}^{n},\\\\\n\\mu_{k,1}^{n+1} & =\\mu_{k,0}^{n}\\frac{t_{2n,0}^{\\alpha}t_{2,1}^{\\alpha}%\n}{t_{2\\left( n+1\\right) ,1}^{\\alpha}}+\\mu_{k,1}^{n}\\frac{t_{2n,1}^{\\alpha\n}t_{2,0}^{\\alpha}}{t_{2\\left( n+1\\right) ,1}^{\\alpha}},\\\\\n\\mu_{k,r}^{n+1} & =\\mu_{k,r-2}^{n}\\frac{t_{2n,r-2}^{\\alpha}t_{2,2}^{\\alpha}%\n}{t_{2\\left( n+1\\right) ,r}^{\\alpha}}+\\mu_{k,r-1}^{n}\\frac{t_{2n,r-1}%\n^{\\alpha}t_{2,1}^{\\alpha}}{t_{2\\left( n+1\\right) ,r}^{\\alpha}}+\\mu_{k,r}%\n^{n}\\frac{t_{2n,r}^{\\alpha}t_{2,0}^{\\alpha}}{t_{2\\left( n+1\\right)\n,r}^{\\alpha}},~r=2,3,\\ldots,2n,\\\\\n\\mu_{k,2n+1}^{n+1} & =\\mu_{k,2n-1}^{n}\\frac{t_{2n,2n-1}^{\\alpha}%\nt_{2,2}^{\\alpha}}{t_{2\\left( n+1\\right) ,2n+1}^{\\alpha}}+\\mu_{k,2n}^{n}%\n\\frac{t_{2n,2n}^{\\alpha}t_{2,1}^{\\alpha}}{t_{2\\left( n+1\\right)\n,2n+1}^{\\alpha}},\\\\\n\\mu_{k,2\\left( n+1\\right) }^{n+1} & =\\mu_{k,2n}^{n},\n\\end{align*}\nrespectively. Moreover, due to initial conditions\n(\\ref{trigonometric_initial_conditions}) and simple trigonometric identities\n\\begin{align}\n\\sin\\left( a+b\\right) & =\\sin\\left( a\\right) \\cos\\left( b\\right)\n+\\cos\\left( a\\right) \\sin\\left( b\\right) ,\\label{sin_of_sum}\\\\\n\\cos\\left( a+b\\right) & =\\cos\\left( a\\right) \\cos\\left( b\\right)\n-\\sin\\left( a\\right) \\sin\\left( b\\right) , \\label{cos_of_sum}%\n\\end{align}\none has that%\n\\begin{align*}\n\\sin\\left( \\left( n+1\\right) u\\right) & =\\left( \\sum_{i=0}^{2n}%\n\\lambda_{n,i}^{n}T_{2n,i}^{\\alpha}\\left( u\\right) \\right) \\left(\n\\sum_{j=0}^{2}\\mu_{1,j}^{1}T_{2,j}^{\\alpha}\\left( u\\right) \\right) +\\left(\n\\sum_{i=0}^{2n}\\mu_{n,i}^{n}T_{2n,i}^{\\alpha}\\left( u\\right) \\right)\n\\left( \\sum_{j=0}^{2}\\lambda_{1,j}^{1}T_{2,j}^{\\alpha}\\left( u\\right)\n\\right) \\\\\n& =\\sum_{r=0}^{2\\left( n+1\\right) }\\lambda_{n+1,r}^{n+1}T_{2\\left(\nn+1\\right) ,r}^{\\alpha}\\left( u\\right) ,\\\\\n\\cos\\left( \\left( n+1\\right) u\\right) & =\\left( \\sum_{i=0}^{2n}%\n\\mu_{n,i}^{n}T_{2n,i}^{\\alpha}\\left( u\\right) \\right) \\left( \\sum\n_{j=0}^{2}\\mu_{1,j}^{1}T_{2,j}^{\\alpha}\\left( u\\right) \\right) -\\left(\n\\sum_{i=0}^{2n}\\lambda_{n,i}^{n}T_{2n,i}^{\\alpha}\\left( u\\right) \\right)\n\\left( \\sum_{j=0}^{2}\\lambda_{1,j}^{1}T_{2,j}^{\\alpha}\\left( u\\right)\n\\right) \\\\\n& =\\sum_{r=0}^{2\\left( n+1\\right) }\\mu_{n+1,r}^{n+1}T_{2\\left( n+1\\right)\n,r}^{\\alpha}\\left( u\\right) ,\n\\end{align*}\nwhere%\n\\begin{align*}\n\\lambda_{n+1,0}^{n+1}= & \\lambda_{n,0}^{n}\\mu_{1,0}^{1}+\\mu_{n,0}^{n}%\n\\lambda_{1,0}^{1},\\\\\n\\lambda_{n+1,1}^{n+1}= & \\left( \\lambda_{n,0}^{n}\\mu_{1,1}^{1}+\\mu\n_{n,0}^{n}\\lambda_{1,1}^{1}\\right) \\frac{t_{2n,0}^{\\alpha}t_{2,1}^{\\alpha}%\n}{t_{2\\left( n+1\\right) ,1}^{\\alpha}}+\\left( \\lambda_{n,1}^{n}\\mu_{1,0}%\n^{1}+\\mu_{n,1}^{n}\\lambda_{1,0}^{1}\\right) \\frac{t_{2n,1}^{\\alpha}%\nt_{2,0}^{\\alpha}}{t_{2\\left( n+1\\right) ,1}^{\\alpha}},\\\\\n\\lambda_{n+1,r}^{n+1}= & \\left( \\lambda_{n,r-2}^{n}\\mu_{1,2}^{1}%\n+\\mu_{n,r-2}^{n}\\lambda_{1,2}^{1}\\right) \\frac{t_{2n,r-2}^{\\alpha}%\nt_{2,2}^{\\alpha}}{t_{2\\left( n+1\\right) ,r}^{\\alpha}}+\\left( \\lambda\n_{n,r-1}^{n}\\mu_{1,1}^{1}+\\mu_{n,r-1}^{n}\\lambda_{1,1}^{1}\\right)\n\\frac{t_{2n,r-1}^{\\alpha}t_{2,1}^{\\alpha}}{t_{2\\left( n+1\\right) ,r}%\n^{\\alpha}}\\\\\n& +\\left( \\lambda_{n,r}^{n}\\mu_{1,0}^{1}+\\mu_{n,r}^{n}\\lambda_{1,0}%\n^{1}\\right) \\frac{t_{2n,r}^{\\alpha}t_{2,0}^{\\alpha}}{t_{2\\left( n+1\\right)\n,r}^{\\alpha}},~r=2,3,\\ldots,2n,\\\\\n\\lambda_{n+1,2n+1}^{n+1}= & \\left( \\lambda_{n,2n-1}^{n}\\mu_{1,2}^{1}%\n+\\mu_{n,2n-1}^{n}\\lambda_{1,2}^{1}\\right) \\frac{t_{2n,2n-1}^{\\alpha}%\nt_{2,2}^{\\alpha}}{t_{2\\left( n+1\\right) ,2n+1}^{\\alpha}}+\\left(\n\\lambda_{n,2n}^{n}\\mu_{1,1}^{1}+\\mu_{n,2n}^{n}\\lambda_{1,1}^{1}\\right)\n\\frac{t_{2n,2n}^{\\alpha}t_{2,1}^{\\alpha}}{t_{2\\left( n+1\\right)\n,2n+1}^{\\alpha}},\\\\\n\\lambda_{n+1,2\\left( n+1\\right) }^{n+1}= & \\lambda_{n,2n}^{n}\\mu_{1,2}%\n^{1}+\\mu_{n,2n}^{n}\\lambda_{1,2}^{1}%\n\\end{align*}\nand%\n\\begin{align*}\n\\mu_{n+1,0}^{n+1}= & \\mu_{n,0}^{n}\\mu_{1,0}^{1}-\\lambda_{n,0}^{n}%\n\\lambda_{1,0}^{1},\\\\\n\\mu_{n+1,1}^{n+1}= & \\left( \\mu_{n,0}^{n}\\mu_{1,1}^{1}-\\lambda_{n,0}%\n^{n}\\lambda_{1,1}^{1}\\right) \\frac{t_{2n,0}^{\\alpha}t_{2,1}^{\\alpha}%\n}{t_{2\\left( n+1\\right) ,1}^{\\alpha}}+\\left( \\mu_{n,1}^{n}\\mu_{1,0}%\n^{1}-\\lambda_{n,1}^{n}\\lambda_{1,0}^{1}\\right) \\frac{t_{2n,1}^{\\alpha}%\nt_{2,0}^{\\alpha}}{t_{2\\left( n+1\\right) ,1}^{\\alpha}},\\\\\n\\mu_{n+1,r}^{n+1}= & \\left( \\mu_{n,r-2}^{n}\\mu_{1,2}^{1}-\\lambda\n_{n,r-2}^{n}\\lambda_{1,2}^{1}\\right) \\frac{t_{2n,r-2}^{\\alpha}t_{2,2}%\n^{\\alpha}}{t_{2\\left( n+1\\right) ,r}^{\\alpha}}+\\left( \\mu_{n,r-1}^{n}%\n\\mu_{1,1}^{1}-\\lambda_{n,r-1}^{n}\\lambda_{1,1}^{1}\\right) \\frac\n{t_{2n,r-1}^{\\alpha}t_{2,1}^{\\alpha}}{t_{2\\left( n+1\\right) ,r}^{\\alpha}}\\\\\n& +\\left( \\mu_{n,r}^{n}\\mu_{1,0}^{1}-\\lambda_{n,r}^{n}\\lambda_{1,0}%\n^{1}\\right) \\frac{t_{2n,r}^{\\alpha}t_{2,0}^{\\alpha}}{t_{2\\left( n+1\\right)\n,r}^{\\alpha}},~r=2,3,\\ldots,2n,\\\\\n\\mu_{n+1,2n+1}^{n+1}= & \\left( \\mu_{n,2n-1}^{n}\\mu_{1,2}^{1}-\\lambda\n_{n,2n-1}^{n}\\lambda_{1,2}^{1}\\right) \\frac{t_{2n,2n-1}^{\\alpha}%\nt_{2,2}^{\\alpha}}{t_{2\\left( n+1\\right) ,2n+1}^{\\alpha}}+\\left( \\mu\n_{n,2n}^{n}\\mu_{1,1}^{1}-\\lambda_{n,2n}^{n}\\lambda_{1,1}^{1}\\right)\n\\frac{t_{2n,2n}^{\\alpha}t_{2,1}^{\\alpha}}{t_{2\\left( n+1\\right)\n,2n+1}^{\\alpha}},\\\\\n\\mu_{n+1,2\\left( n+1\\right) }^{n+1}= & \\mu_{n,2n}^{n}\\mu_{1,2}^{1}%\n-\\lambda_{n,2n}^{n}\\lambda_{1,2}^{1},\n\\end{align*}\nrespectively. Summarizing all calculations above, we have proved the next theorem.\n\n\\begin{theorem}\n[Trigonometric basis transformation]%\n\\label{thm:trigonometric_basis_transformation}The matrix form of the linear\ntransformation from the normalized B-basis $\\overline{\\mathcal{T}}_{2\\left( n+1\\right)\n}^{\\alpha}$ to the canonical trigonometric basis $\\mathcal{T}_{2\\left(\nn+1\\right) }^{\\alpha}$ is%\n\\[\n\\left[\n\\begin{array}\n[c]{c}%\n1\\\\\n\\sin\\left( u\\right) \\\\\n\\cos\\left( u\\right) \\\\\n\\vdots\\\\\n\\sin\\left( \\left( n+1\\right) u\\right) \\\\\n\\cos\\left( \\left( n+1\\right) u\\right)\n\\end{array}\n\\right] =\\left[\n\\begin{array}\n[c]{cccccc}%\n1 & 1 & 1 & \\cdots & 1 & 1\\\\\n\\lambda_{1,0}^{n+1} & \\lambda_{1,1}^{n+1} & \\lambda_{1,2}^{n+1} & \\cdots &\n\\lambda_{1,2n+1}^{n+1} & \\lambda_{1,2\\left( n+1\\right) }^{n+1}\\\\\n\\mu_{1,0}^{n+1} & \\mu_{1,1}^{n+1} & \\mu_{1,2}^{n+1} & \\cdots & \\mu\n_{1,2n+1}^{n+1} & \\mu_{1,2\\left( n+1\\right) }^{n+1}\\\\\n\\vdots & \\vdots & \\vdots & & \\vdots & \\vdots\\\\\n\\lambda_{n+1,0}^{n+1} & \\lambda_{n+1,1}^{n+1} & \\lambda_{n+1,2}^{n+1} & \\cdots\n& \\lambda_{n+1,2n+1}^{n+1} & \\lambda_{n+1,2\\left( n+1\\right) }^{n+1}\\\\\n\\mu_{n+1,0}^{n+1} & \\mu_{n+1,1}^{n+1} & \\mu_{n+1,2}^{n+1} & \\cdots &\n\\mu_{n+1,2n+1}^{n+1} & \\mu_{n+1,2\\left( n+1\\right) }^{n+1}%\n\\end{array}\n\\right] \\left[\n\\begin{array}\n[c]{c}%\nT_{2\\left( n+1\\right) ,0}^{\\alpha}\\left( u\\right) \\\\\nT_{2\\left( n+1\\right) ,1}^{\\alpha}\\left( u\\right) \\\\\nT_{2\\left( n+1\\right) ,2}^{\\alpha}\\left( u\\right) \\\\\n\\vdots\\\\\nT_{2\\left( n+1\\right) ,2n+1}^{\\alpha}\\left( u\\right) \\\\\nT_{2\\left( n+1\\right) ,2\\left( n+1\\right) }^{\\alpha}\\left( u\\right)\n\\end{array}\n\\right]\n\\]\nfor all parameters $u\\in\\left[ 0,\\alpha\\right] $.\n\\end{theorem}\n\n\\begin{remark}\nThe matrix of the $\\left( n+1\\right) $th order basis transformation that appears in Theorem \\ref{thm:trigonometric_basis_transformation} can be efficiently calculated by parallel programming since its rows and their entries are independent of each other. Based on the entries of the first and $n$th order transformation matrices that are already calculated in previous steps, each thread block has to build up a single row of the $\\left( n+1\\right) $th order basis transformation matrix, while each thread within a block has to calculate a single entry of the corresponding row.\n\\end{remark}\n\n\\subsection{The hyperbolic case}\n\nIn this case we can proceed as in Subsection\n\\ref{sec:trigonometric_basis_transformation}. Naturally, instead of\ntrigonometric sine, cosine, tangent functions and identities (\\ref{sin_of_sum}%\n) and (\\ref{cos_of_sum}) one has to apply the hyperbolic variant of these\nfunctions and identities, respectively. The only difference consists in a sign\nchange in the hyperbolic counterpart of the identity (\\ref{cos_of_sum}),\nsince\n\\begin{equation}\n\\cosh\\left( a+b\\right) =\\cosh\\left( a\\right) \\cosh\\left( b\\right)\n+\\sinh\\left( a\\right) \\sinh\\left( b\\right) . \\label{cosh_of_sum}%\n\\end{equation}\nLet $k\\in\\left\\{ 0,1,\\ldots,n\\right\\} $ be an arbitrarily fixed natural\nnumber and denote the representations of hyperbolic functions $\\sinh\\left(\nku\\right) $ and $\\cosh\\left( ku\\right) $ in the B-basis (\\ref{Wang_basis})\nby%\n\\begin{equation}\n\\sinh\\left( ku\\right) =\\sum_{i=0}^{2n}\\sigma_{k,i}^{n}H_{2n,i}^{\\alpha\n}\\left( u\\right) ,~u\\in\\left[ 0,\\alpha\\right]\n\\end{equation}\nand%\n\\begin{equation}\n\\cosh\\left( ku\\right) =\\sum_{i=0}^{2n}\\rho_{k,i}^{n}H_{2n,i}^{\\alpha}\\left(\nu\\right) ,~u\\in\\left[ 0,\\alpha\\right]\n\\end{equation}\nrespectively, where coefficients $\\left\\{ \\sigma_{k,i}^{n}\\right\\}\n_{i=0}^{2n}$ and $\\left\\{ \\rho_{k,i}^{n}\\right\\} _{i=0}^{2n}$ are unique\nscalars. The basis transformation from the first order B-basis $\\overline\n{\\mathcal{H}}_{2}^{\\alpha}$ to the first order hyperbolic canonical basis\n$\\mathcal{H}_{2}^{\\alpha}$ can be written in the matrix form%\n\\[\n\\left[\n\\begin{array}\n[c]{c}%\n1\\\\\n\\sinh\\left( u\\right) \\\\\n\\cosh\\left( u\\right)\n\\end{array}\n\\right] =\\left[\n\\begin{array}\n[c]{ccc}%\n\\rho_{0,0}^{1} & \\rho_{0,1}^{1} & \\rho_{0,2}^{1}\\\\\n\\sigma_{1,0}^{1} & \\sigma_{1,1}^{1} & \\sigma_{1,2}^{1}\\\\\n\\rho_{1,0}^{1} & \\rho_{1,1}^{1} & \\rho_{1,2}^{1}%\n\\end{array}\n\\right] \\left[\n\\begin{array}\n[c]{c}%\nH_{2,0}^{\\alpha}\\left( u\\right) \\\\\nH_{2,1}^{\\alpha}\\left( u\\right) \\\\\nH_{2,2}^{\\alpha}\\left( u\\right)\n\\end{array}\n\\right] ,~\\forall u\\in\\left[ 0,\\alpha\\right] ,\n\\]\nwhere%\n\\begin{equation}\n\\left\\{\n\\begin{array}\n[c]{l}%\n\\rho_{0,0}^{1}=\\rho_{0,1}^{1}=\\rho_{0,2}^{1}=1,\\\\\n\\\\\n\\sigma_{1,0}^{1}=0,~\\sigma_{1,1}^{1}=\\tanh\\left( \\frac{\\alpha}{2}\\right)\n,~\\sigma_{1,2}^{1}=\\sinh\\left( \\alpha\\right) ,\\\\\n\\\\\n\\rho_{1,0}^{1}=\\rho_{1,1}^{1}=1,~\\rho_{1,2}^{1}=\\cosh\\left( \\alpha\\right) .\n\\end{array}\n\\right. \\label{hyperbolic_initial_conditions}%\n\\end{equation}\nUsing initial conditions (\\ref{hyperbolic_initial_conditions}) and normalizing\nconstants $\\left[ h_{2,j}^{\\alpha}\\right] _{j=0}^{2},\\left[ h_{2n,i}%\n^{\\alpha}\\right] _{i=0}^{2n}$ and $\\left[ h_{2\\left( n+1\\right)\n,r}^{\\alpha}\\right] _{r=0}^{2\\left( n+1\\right) }$ instead of $\\left[\nt_{2,j}^{\\alpha}\\right] _{j=0}^{2},\\left[ t_{2n,i}^{\\alpha}\\right]\n_{i=0}^{2n}$ and $\\left[ t_{2\\left( n+1\\right) ,r}^{\\alpha}\\right]\n_{r=0}^{2\\left( n+1\\right) }$, respectively, one obtains recursive formulae\nfor the unique order elevated coefficients $\\left[ \\sigma_{k,r}^{n+1}\\right]\n_{k=0,r=0}^{n+1,2\\left( n+1\\right) }$ and $\\left[ \\rho_{k,r}^{n+1}\\right]\n_{k=0,r=0}^{n,2\\left( n+1\\right) }$ in a similar way as it was done in the\ntrigonometric case for constants $\\left[ \\lambda_{k,r}^{n+1}\\right]\n_{k=0,r=0}^{n+1,2\\left( n+1\\right) }$ and $\\left[ \\mu_{k,r}^{n+1}\\right]\n_{k=0,r=0}^{n,2\\left( n+1\\right) }$, respectively, while applying identity\n(\\ref{cosh_of_sum}) for constants $\\left[ \\rho_{n+1,r}^{n+1}\\right]\n_{r=0}^{2\\left( n+1\\right) }$ we have that%\n\\begin{align*}\n\\rho_{n+1,0}^{n+1}= & \\rho_{n,0}^{n}\\rho_{1,0}^{1}+\\sigma_{n,0}^{n}%\n\\sigma_{1,0}^{1},\\\\\n\\rho_{n+1,1}^{n+1}= & \\left( \\rho_{n,0}^{n}\\rho_{1,1}^{1}+\\sigma_{n,0}%\n^{n}\\sigma_{1,1}^{1}\\right) \\frac{h_{2n,0}^{\\alpha}h_{2,1}^{\\alpha}%\n}{h_{2\\left( n+1\\right) ,1}^{\\alpha}}+\\left( \\rho_{n,1}^{n}\\rho_{1,0}%\n^{1}+\\sigma_{n,1}^{n}\\sigma_{1,0}^{1}\\right) \\frac{h_{2n,1}^{\\alpha}%\nh_{2,0}^{\\alpha}}{h_{2\\left( n+1\\right) ,1}^{\\alpha}},\\\\\n\\rho_{n+1,r}^{n+1}= & \\left( \\rho_{n,r-2}^{n}\\rho_{1,2}^{1}+\\sigma\n_{n,r-2}^{n}\\sigma_{1,2}^{1}\\right) \\frac{h_{2n,r-2}^{\\alpha}h_{2,2}^{\\alpha\n}}{h_{2\\left( n+1\\right) ,r}^{\\alpha}}+\\left( \\rho_{n,r-1}^{n}\\rho\n_{1,1}^{1}+\\sigma_{n,r-1}^{n}\\sigma_{1,1}^{1}\\right) \\frac{h_{2n,r-1}%\n^{\\alpha}h_{2,1}^{\\alpha}}{h_{2\\left( n+1\\right) ,r}^{\\alpha}}\\\\\n& +\\left( \\rho_{n,r}^{n}\\rho_{1,0}^{1}+\\sigma_{n,r}^{n}\\sigma_{1,0}%\n^{1}\\right) \\frac{h_{2n,r}^{\\alpha}h_{2,0}^{\\alpha}}{h_{2\\left( n+1\\right)\n,r}^{\\alpha}},~r=2,3,\\ldots,2n,\\\\\n\\rho_{n+1,2n+1}^{n+1}= & \\left( \\rho_{n,2n-1}^{n}\\rho_{1,2}^{1}%\n+\\sigma_{n,2n-1}^{n}\\sigma_{1,2}^{1}\\right) \\frac{h_{2n,2n-1}^{\\alpha}%\nh_{2,2}^{\\alpha}}{h_{2\\left( n+1\\right) ,2n+1}^{\\alpha}}+\\left( \\rho\n_{n,2n}^{n}\\rho_{1,1}^{1}+\\sigma_{n,2n}^{n}\\sigma_{1,1}^{1}\\right)\n\\frac{h_{2n,2n}^{\\alpha}h_{2,1}^{\\alpha}}{h_{2\\left( n+1\\right)\n,2n+1}^{\\alpha}},\\\\\n\\rho_{n+1,2\\left( n+1\\right) }^{n+1}= & \\rho_{n,2n}^{n}\\rho_{1,2}%\n^{1}+\\sigma_{n,2n}^{n}\\sigma_{1,2}^{1}.\n\\end{align*}\nSummarizing all calculations, one can formulate the following theorem.\n\n\\begin{theorem}\n[Hyperbolic basis transformation]\\label{thm:hyperbolic_basis_transformation}%\nThe matrix form of the linear transformation from the normalized B-basis $\\overline\n{\\mathcal{H}}_{2\\left( n+1\\right) }^{\\alpha}$ to the canonical hyperbolic\nbasis $\\mathcal{H}_{2\\left( n+1\\right) }^{\\alpha}$ is%\n\\[\n\\left[\n\\begin{array}\n[c]{c}%\n1\\\\\n\\sinh\\left( u\\right) \\\\\n\\cosh\\left( u\\right) \\\\\n\\vdots\\\\\n\\sinh\\left( \\left( n+1\\right) u\\right) \\\\\n\\cosh\\left( \\left( n+1\\right) u\\right)\n\\end{array}\n\\right] =\\left[\n\\begin{array}\n[c]{cccccc}%\n1 & 1 & 1 & \\cdots & 1 & 1\\\\\n\\sigma_{1,0}^{n+1} & \\sigma_{1,1}^{n+1} & \\sigma_{1,2}^{n+1} & \\cdots &\n\\sigma_{1,2n+1}^{n+1} & \\sigma_{1,2\\left( n+1\\right) }^{n+1}\\\\\n\\rho_{1,0}^{n+1} & \\rho_{1,1}^{n+1} & \\rho_{1,2}^{n+1} & \\cdots &\n\\rho_{1,2n+1}^{n+1} & \\rho_{1,2\\left( n+1\\right) }^{n+1}\\\\\n\\vdots & \\vdots & \\vdots & & \\vdots & \\vdots\\\\\n\\sigma_{n+1,0}^{n+1} & \\sigma_{n+1,1}^{n+1} & \\sigma_{n+1,2}^{n+1} & \\cdots &\n\\sigma_{n+1,2n+1}^{n+1} & \\sigma_{n+1,2\\left( n+1\\right) }^{n+1}\\\\\n\\rho_{n+1,0}^{n+1} & \\rho_{n+1,1}^{n+1} & \\rho_{n+1,2}^{n+1} & \\cdots &\n\\rho_{n+1,2n+1}^{n+1} & \\rho_{n+1,2\\left( n+1\\right) }^{n+1}%\n\\end{array}\n\\right] \\left[\n\\begin{array}\n[c]{c}%\nH_{2\\left( n+1\\right) ,0}^{\\alpha}\\left( u\\right) \\\\\nH_{2\\left( n+1\\right) ,1}^{\\alpha}\\left( u\\right) \\\\\nH_{2\\left( n+1\\right) ,2}^{\\alpha}\\left( u\\right) \\\\\n\\vdots\\\\\nH_{2\\left( n+1\\right) ,2n+1}^{\\alpha}\\left( u\\right) \\\\\nH_{2\\left( n+1\\right) ,2\\left( n+1\\right) }^{\\alpha}\\left( u\\right)\n\\end{array}\n\\right]\n\\]\nfor all parameters $u\\in\\left[ 0,\\alpha\\right] $.\n\\end{theorem}\n\n\\section{Control point based exact description\\label{sec:exact_description}}\n\nUsing curves of the type (\\ref{trigonometric_curve}) or\n(\\ref{hyperbolic_curve}), following subsections provide control point\nconfigurations for the exact description of higher order (mixed partial)\nderivatives of any smooth parametric curve (higher dimensional multivariate\nsurface) specified by coordinate functions given in traditional parametric\nform, i.e., in vector spaces (\\ref{truncated_Fourier_vector_space}) or\n(\\ref{hyperbolic_vector_space}), respectively. The obtained results will also\nbe extended for the control point based exact description of the rational\ncounterpart of these curves and multivariate surfaces. Core properties of this\nsection are formulated by the next lemmas.\n\n\\begin{lemma}\n[Exact description of trigonometric polynomials]\\label{lem:tp}Consider the\ntrigonometric polynomial%\n\\begin{equation}\ng\\left( u\\right) =\\sum_{p\\in P}c_{p}\\cos\\left( pu+\\psi_{p}\\right)\n+\\sum_{q\\in Q}s_{q}\\sin\\left( qu+\\varphi_{q}\\right) ,~u\\in\\left[\n0,\\alpha\\right] ,~\\alpha\\in\\left( 0,\\pi\\right)\n\\label{trigonometric_polynomial}%\n\\end{equation}\nof order at most $n$, where $P,Q\\subset%\n\\mathbb{N}\n$ and $c_{p},\\psi_{p},s_{q},\\varphi_{q}\\in%\n\\mathbb{R}\n$. Then, we have the equality%\n\\[\n\\frac{\\text{\\emph{d}}^{r}}{\\text{\\emph{d}}u^{r}}g\\left( u\\right) =\\sum\n_{i=0}^{2n}d_{i}\\left( r\\right) T_{2n,i}^{\\alpha}\\left( u\\right) ,~\\forall\nu\\in\\left[ 0,\\alpha\\right] ,~\\forall r\\in%\n\\mathbb{N}\n,\n\\]\nwhere trigonometric ordinates $\\left[ d_{i}\\left( r\\right) \\right]\n_{i=0}^{2n}$ are of the form%\n\\begin{align}\nd_{i}\\left( r\\right) = & \\sum_{p\\in P}c_{p}p^{r}\\left( \\mu_{p,i}^{n}%\n\\cos\\left( \\psi_{p}+\\frac{r\\pi}{2}\\right) -\\lambda_{p,i}^{n}\\sin\\left(\n\\psi_{p}+\\frac{r\\pi}{2}\\right) \\right) \\label{trigonometric_ordinates}\\\\\n& +\\sum_{q\\in Q}s_{q}q^{r}\\left( \\lambda_{q,i}^{n}\\cos\\left( \\varphi\n_{q}+\\frac{r\\pi}{2}\\right) +\\mu_{q,i}^{n}\\sin\\left( \\varphi_{q}+\\frac{r\\pi\n}{2}\\right) \\right) .\\nonumber\n\\end{align}\n\n\\end{lemma}\n\n\\begin{proof}\nThe $r$th order derivative of the trigonometric polynomial \\textbf{(}%\n\\ref{trigonometric_polynomial}\\textbf{)} can be written in the form%\n\\begin{align*}\n\\frac{\\text{d}^{r}}{\\text{d}u^{r}}g\\left( u\\right) = & \\sum_{p\\in P}%\nc_{p}p^{r}\\cos\\left( pu+\\psi_{p}+\\frac{r\\pi}{2}\\right) +\\sum_{q\\in Q}%\ns_{q}q^{r}\\sin\\left( qu+\\varphi_{q}+\\frac{r\\pi}{2}\\right) \\\\\n& \\\\\n= & \\sum_{p\\in P}c_{p}p^{r}\\left( \\cos\\left( pu\\right) \\cos\\left(\n\\psi_{p}+\\frac{r\\pi}{2}\\right) -\\sin\\left( pu\\right) \\sin\\left( \\psi\n_{p}+\\frac{r\\pi}{2}\\right) \\right) \\\\\n& +\\sum_{q\\in Q}s_{q}q^{r}\\left( \\sin\\left( qu\\right) \\cos\\left(\n\\varphi_{q}+\\frac{r\\pi}{2}\\right) +\\cos\\left( qu\\right) \\sin\\left(\n\\varphi_{q}+\\frac{r\\pi}{2}\\right) \\right) \\\\\n& \\\\\n= & \\sum_{p\\in P}c_{p}p^{r}\\cos\\left( \\psi_{p}+\\frac{r\\pi}{2}\\right)\n\\cos\\left( pu\\right) -\\sum_{p\\in P}c_{p}p^{r}\\sin\\left( \\psi_{p}+\\frac\n{r\\pi}{2}\\right) \\sin\\left( pu\\right) \\\\\n& +\\sum_{q\\in Q}s_{q}q^{r}\\cos\\left( \\varphi_{q}+\\frac{r\\pi}%\n{2}\\right) \\sin\\left( qu\\right) +\\sum_{q\\in Q}s_{q}q^{r}\\sin\\left(\n\\varphi_{q}+\\frac{r\\pi}{2}\\right) \\cos\\left( qu\\right) \\\\\n& \\\\\n= & \\sum_{p\\in P}c_{p}p^{r}\\cos\\left( \\psi_{p}+\\frac{r\\pi}{2}\\right)\n\\left( \\sum_{i=0}^{2n}\\mu_{p,i}^{n}T_{2n,i}^{\\alpha}\\left( u\\right)\n\\right) -\\sum_{p\\in P}c_{p}p^{r}\\sin\\left( \\psi_{p}+\\frac{r\\pi}{2}\\right)\n\\left( \\sum_{i=0}^{2n}\\lambda_{p,i}^{n}T_{2n,i}^{\\alpha}\\left( u\\right)\n\\right) \\\\\n& +\\sum_{q\\in Q}s_{q}q^{r}\\cos\\left( \\varphi_{q}+\\frac{r\\pi}{2}\\right)\n\\left( \\sum_{i=0}^{2n}\\lambda_{q,i}^{n}T_{2n,i}^{\\alpha}\\left( u\\right)\n\\right) +\\sum_{q\\in Q}s_{q}q^{r}\\sin\\left( \\varphi_{q}+\\frac{r\\pi}%\n{2}\\right) \\left( \\sum_{i=0}^{2n}\\mu_{q,i}^{n}T_{2n,i}^{\\alpha}\\left(\nu\\right) \\right)\n\\end{align*}\nfor all parameters $u\\in\\left[ 0,\\alpha\\right] $, where we have applied\nTheorem \\ref{thm:trigonometric_basis_transformation} for order $n$. Collecting\nthe coefficients of basis functions $\\left\\{ T_{2n,i}^{\\alpha}\\right\\}\n_{i=0}^{2n}$, one obtains the ordinates specified by\n(\\ref{trigonometric_ordinates}).\n\\end{proof}\n\n\\begin{lemma}\n[Exact description of hyperbolic polynomials]\\label{lem:hp}Consider the\nhyperbolic function%\n\\[\ng\\left( u\\right) =\\sum_{p\\in P}c_{p}\\cosh\\left( pu+\\psi_{p}\\right)\n+\\sum_{q\\in Q}s_{q}\\sinh\\left( qu+\\varphi_{q}\\right) ,~u\\in\\left[\n0,\\alpha\\right] ,~\\alpha>0\n\\]\nof order at most $n$, where $P,Q\\subset%\n\\mathbb{N}\n$ and $c_{p},\\psi_{p},s_{q},\\varphi_{q}\\in%\n\\mathbb{R}\n$. Then, one has that%\n\\[\n\\frac{\\text{\\emph{d}}^{r}}{\\text{\\emph{d}}u^{r}}g\\left( u\\right) =\\sum\n_{i=0}^{2n}d_{i}\\left( r\\right) H_{2n,i}^{\\alpha}\\left( u\\right) ,~\\forall\nu\\in\\left[ 0,\\alpha\\right] ,~\\forall r\\in%\n\\mathbb{N}\n,\n\\]\nwhere%\n\\begin{equation}\nd_{i}\\left( r\\right) =\\left\\{\n\\begin{array}\n[c]{ll}%\n{\\displaystyle\\sum\\limits_{p\\in P}}\nc_{p}p^{r}\\left( \\rho_{p,i}^{n}\\cosh\\left( \\psi_{p}\\right) +\\sigma\n_{p,i}^{n}\\sinh\\left( \\psi_{p}\\right) \\right) & \\\\\n+%\n{\\displaystyle\\sum\\limits_{q\\in Q}}\ns_{q}q^{r}\\left( \\sigma_{q,i}^{n}\\cosh\\left( \\varphi_{q}\\right) +\\rho\n_{q,i}^{n}\\sinh\\left( \\varphi_{q}\\right) \\right) , & r=2z,\\\\\n& \\\\%\n{\\displaystyle\\sum\\limits_{p\\in P}}\nc_{p}p^{r}\\left( \\sigma_{p,i}^{n}\\cosh\\left( \\psi_{p}\\right) +\\rho\n_{p,i}^{n}\\sinh\\left( \\psi_{p}\\right) \\right) & \\\\\n+%\n{\\displaystyle\\sum\\limits_{q\\in Q}}\ns_{q}q^{r}\\left( \\rho_{q,i}^{n}\\cosh\\left( \\varphi_{q}\\right) +\\sigma\n_{q,i}^{n}\\sinh\\left( \\varphi_{q}\\right) \\right) , & r=2z+1.\n\\end{array}\n\\right. \\label{hyperbolic_ordinates}%\n\\end{equation}\n\n\\end{lemma}\n\n\\begin{proof}\nUsing the basis transformation formulated in Theorem\n\\ref{thm:hyperbolic_basis_transformation} for order $n$ and applying\nderivative formulae%\n\\begin{align*}\n\\frac{\\text{d}^{r}}{\\text{d}u^{r}}\\cosh\\left( au+b\\right) & =\\left\\{\n\\begin{array}\n[c]{ll}%\na^{r}\\cosh\\left( au+b\\right) , & r=2z,\\\\\n& \\\\\na^{r}\\sinh\\left( au+b\\right) , & r=2z+1,\n\\end{array}\n\\right. \\\\\n& \\\\\n\\frac{\\text{d}^{r}}{\\text{d}u^{r}}\\sinh\\left( au+b\\right) & =\\left\\{\n\\begin{array}\n[c]{ll}%\na^{r}\\sinh\\left( au+b\\right) , & r=2z,\\\\\n& \\\\\na^{r}\\cosh\\left( au+b\\right) , & r=2z+1\n\\end{array}\n\\right.\n\\end{align*}\nwith hyperbolic identities (\\ref{cosh_of_sum}) and%\n\\[\n\\sinh\\left( a+b\\right) =\\sinh\\left( a\\right) \\cosh\\left( b\\right)\n+\\cosh\\left( a\\right) \\sinh\\left( b\\right) ,\n\\]\none can follow the steps of the proof of the previous Lemma \\ref{lem:tp}.\n\\end{proof}\n\n\\subsection{Description of (rational) trigonometric curves and\nsurfaces\\label{sec:exact_description_trigonometric}}\n\nConsider the smooth parametric curve%\n\\begin{equation}\n\\mathbf{g}\\left( u\\right) =\\left[ g^{\\ell}\\left( u\\right) \\right]\n_{\\ell=1}^{\\delta},~u\\in\\left[ 0,\\alpha\\right] ,~\\alpha\\in\\left(\n0,\\pi\\right) \\label{traditional_trigonometric_curve}%\n\\end{equation}\nwith coordinate functions of the form%\n\\[\ng^{\\ell}\\left( u\\right) =\\sum_{p\\in P_{\\ell}}c_{p}^{\\ell}\\cos\\left(\npu+\\psi_{p}^{\\ell}\\right) +\\sum_{q\\in Q_{\\ell}}s_{q}^{\\ell}\\sin\\left(\nqu+\\varphi_{q}^{\\ell}\\right)\n\\]\nand the vector space (\\ref{truncated_Fourier_vector_space}) of order\n\\[\nn\\geq n_{\\min}=\\max\\left\\{ z:z\\in\\cup_{\\ell=1}^{\\delta}\\left( P_{\\ell}\\cup\nQ_{\\ell}\\right) \\right\\} ,\n\\]\nwhere $P_{\\ell},Q_{\\ell}\\subset%\n\\mathbb{N}\n$ and $c_{p}^{\\ell},\\psi_{p}^{\\ell},s_{q}^{\\ell},\\varphi_{q}^{\\ell}\\in%\n\\mathbb{R}\n$.\n\n\\begin{theorem}\n[Control point based exact description of trigonometric curves]%\n\\label{thm:cpbed_trigonometric_curves}The $r$th ($r\\in%\n\\mathbb{N}\n$) order\\ derivative of the curve (\\ref{traditional_trigonometric_curve}) can\nbe written in the form%\n\\[\n\\frac{\\text{\\emph{d}}^{r}}{\\text{\\emph{d}}u^{r}}\\mathbf{g}\\left( u\\right)\n=\\sum_{i=0}^{2n}\\mathbf{d}_{i}\\left( r\\right) T_{2n,i}^{\\alpha}\\left(\nu\\right) ,~\\forall u\\in\\left[ 0,\\alpha\\right] ,\n\\]\nwhere control points $\\mathbf{d}_{i}\\left( r\\right) =\\left[ d_{i}^{\\ell\n}\\left( r\\right) \\right] _{\\ell=1}^{\\delta}$ are determined by coordinates%\n\\begin{align}\nd_{i}^{\\ell}\\left( r\\right) = & \\sum_{p\\in P_{\\ell}}c_{p}^{\\ell}%\np^{r}\\left( \\mu_{p,i}^{n}\\cos\\left( \\psi_{p}^{\\ell}+\\frac{r\\pi}{2}\\right)\n-\\lambda_{p,i}^{n}\\sin\\left( \\psi_{p}^{\\ell}+\\frac{r\\pi}{2}\\right) \\right)\n\\label{trigonometric_control_points}\\\\\n& +\\sum_{q\\in Q_{\\ell}}s_{q}^{\\ell}q^{r}\\left( \\lambda_{q,i}^{n}\\cos\\left(\n\\varphi_{q}^{\\ell}+\\frac{r\\pi}{2}\\right) +\\mu_{q,i}^{n}\\sin\\left(\n\\varphi_{q}^{\\ell}+\\frac{r\\pi}{2}\\right) \\right) .\\nonumber\n\\end{align}\n(In case of zeroth order derivatives we will use the simpler notation\n$\\mathbf{d}_{i}=\\left[ d_{i}^{\\ell}\\right] _{\\ell=1}^{\\delta}=\\left[\nd_{i}^{\\ell}\\left( 0\\right) \\right] _{\\ell=1}^{\\delta}=\\mathbf{d}%\n_{i}\\left( 0\\right) $ for all $i=0,1,\\ldots,2n$.)\n\\end{theorem}\n\n\\begin{proof}\nUsing Lemma \\ref{lem:tp}, the $r$th order derivative of the $\\ell$th\ncoordinate function of the curve (\\ref{traditional_trigonometric_curve}) can\nbe written in the form%\n\\[\n\\frac{\\text{d}^{r}}{\\text{d}u^{r}}g^{\\ell}\\left( u\\right) =\\sum_{i=0}%\n^{2n}d_{i}^{\\ell}\\left( r\\right) T_{2n,i}^{\\alpha}\\left( u\\right)\n,~\\forall u\\in\\left[ 0,\\alpha\\right] ,\n\\]\nwhere the $i$th ordinate has exactly the form of\n(\\ref{trigonometric_control_points}). Repeating this reformulation for all\n$\\ell=1,2,\\ldots,\\delta$ and collecting the coefficients of basis functions\n$\\left\\{ T_{2n,i}^{\\alpha}\\right\\} _{i=0}^{2n}$ one obtains all coordinates\nof control points $\\mathbf{d}_{i}\\left( r\\right) =\\left[ d_{i}^{\\ell\n}\\left( r\\right) \\right] _{\\ell=1}^{\\delta}$ that can be substituted into\nthe description of trigonometric curves of the type (\\ref{trigonometric_curve}).\n\\end{proof}\n\n\\begin{example}\n[Application of Theorem \\ref{thm:cpbed_trigonometric_curves} -- plane\ncurves]Cases (a) and (b) of Fig. \\ref{fig:ed_non_rational_tpc} show the\ncontrol point based exact descriptions of the hypocycloidal arc%\n\\begin{equation}\n\\mathbf{g}\\left( u\\right) =\\left[\n\\begin{array}\n[c]{c}%\ng^{1}\\left( u\\right) \\\\\n\\\\\ng^{2}\\left( u\\right)\n\\end{array}\n\\right] =\\left[\n\\begin{array}\n[c]{l}%\n4\\cos\\left( u-\\frac{\\pi}{3}\\right) +\\cos\\left( 4u-\\frac{\\pi}{3}\\right) \\\\\n\\\\\n4\\sin\\left( u-\\frac{\\pi}{3}\\right) -\\sin\\left( 4u-\\frac{\\pi}{3}\\right)\n\\end{array}\n\\right] ,~u\\in\\left( 0,\\frac{3\\pi}{4}\\right) \\label{hypocycloid}%\n\\end{equation}\nand of the arc\n\\begin{equation}\n\\mathbf{g}\\left( u\\right) =\\left[\n\\begin{array}\n[c]{c}%\ng^{1}\\left( u\\right) \\\\\n\\\\\ng^{2}\\left( u\\right)\n\\end{array}\n\\right] =\\frac{1}{2}\\left[\n\\begin{array}\n[c]{l}%\n\\sin\\left( u-\\frac{\\pi}{12}\\right) +\\sin\\left( 3u-\\frac{\\pi}{4}\\right) \\\\\n\\\\\n\\cos\\left( u-\\frac{\\pi}{12}\\right) -\\cos\\left( 3u-\\frac{\\pi}{4}\\right)\n\\end{array}\n\\right] ,~u\\in\\left( 0,\\frac{2\\pi}{3}\\right) \\label{quadrifolium}%\n\\end{equation}\nof a quadrifolium, respectively.\n\\end{example}\n\n\n\\begin{figure}\n[!h]\n\\begin{center}\n\\includegraphics[\nheight=3.4636in,\nwidth=6.058in\n]%\n{ed_non_rational_tpc.pdf}%\n\\caption{Order elevated control point based exact description of trigonometric\ncurves (\\ref{hypocycloid}) and (\\ref{quadrifolium}) by means of Theorem\n\\ref{thm:cpbed_trigonometric_curves}. (\\emph{a}) A hypocycloidal arc\n($\\alpha=\\frac{3\\pi}{4}$). (\\emph{b}) An arc of a quadrifolium ($\\alpha\n=\\frac{2\\pi}{3}$).}%\n\\label{fig:ed_non_rational_tpc}%\n\\end{center}\n\\end{figure}\n\n\n\\begin{example}\n[Application of Theorem \\ref{thm:cpbed_trigonometric_curves} -- space\ncurve]Fig. \\ref{fig:torus_knot} illustrates the control point based exact\ndescriptions of the arc%\n\\begin{equation}\n\\mathbf{g}\\left( u\\right) =\\left[\n\\begin{array}\n[c]{c}%\ng^{1}\\left( u\\right) \\\\\n\\\\\ng^{2}\\left( u\\right) \\\\\n\\\\\ng^{3}\\left( u\\right)\n\\end{array}\n\\right] =\\left[\n\\begin{array}\n[c]{l}%\n\\frac{1}{2}\\cos\\left( u\\right) +2\\cos\\left( 3u\\right) +\\frac{1}{2}%\n\\cos\\left( 5u\\right) \\\\\n\\\\\n\\frac{1}{2}\\sin\\left( u\\right) +2\\sin\\left( 3u\\right) +\\frac{1}{2}%\n\\sin\\left( 5u\\right) \\\\\n\\\\\n\\sin\\left( 2u\\right)\n\\end{array}\n\\right] ,~u\\in\\left( 0,\\frac{\\pi}{2}\\right) , \\label{torus_knot}%\n\\end{equation}\nof a torus knot.\n\\end{example}\n\n\n\\begin{figure}\n[!h]\n\\begin{center}\n\\includegraphics[\nheight=3.8553in,\nwidth=4.9882in\n]%\n{torus_knot_n_5_7_9_alpha_pi_per_2_pch_0.pdf}%\n\\caption{Order elevated control point based exact description of an arc\n($\\alpha=\\frac{\\pi}{2}$) of a knot that lies on the surface of the torus\n$\\left[ \\left( R+r\\sin\\left( u_{1}\\right) \\right) \\cos\\left(\nu_{2}\\right) ,\\left( R+r\\sin\\left( u_{1}\\right) \\right) \\sin\\left(\nu_{2}\\right) ,r\\cos\\left( u_{1}\\right) \\right] $, where $\\left( u_{1}%\n,u_{2}\\right) \\in\\left[ 0,2\\pi\\right] \\times\\left[ 0,2\\pi\\right] $, $R=2$\nand $r=1$. Control points were obtained by using Theorem\n\\ref{thm:cpbed_trigonometric_curves}.}%\n\\label{fig:torus_knot}%\n\\end{center}\n\\end{figure}\n\n\nConsider now the rational trigonometric curve%\n\\begin{equation}\n\\mathbf{g}\\left( u\\right) =\\frac{1}{g^{\\delta+1}\\left( u\\right) }\\left[\ng^{\\ell}\\left( u\\right) \\right] _{\\ell=1}^{\\delta},~u\\in\\left[\n0,\\alpha\\right] , \\label{rational_trigonometric_curve_to_be_described}%\n\\end{equation}\ngiven in traditional parametric form\n\\[\ng^{\\ell}\\left( u\\right) =\\sum_{p\\in P_{\\ell}}c_{p}^{\\ell}\\cos\\left(\npu+\\psi_{p}^{\\ell}\\right) +\\sum_{q\\in Q_{\\ell}}s_{q}^{\\ell}\\sin\\left(\nqu+\\varphi_{q}^{\\ell}\\right) ,~P_{\\ell},Q_{\\ell}\\subset%\n\\mathbb{N}\n,~c_{p}^{\\ell},\\psi_{p}^{\\ell},s_{q}^{\\ell},\\varphi_{q}^{\\ell}\\in%\n\\mathbb{R}\n,~\\ell=1,2,\\ldots,\\delta+1,\n\\]\nwhere%\n\\[\ng^{\\delta+1}\\left( u\\right) >0,~\\forall u\\in\\left[ 0,\\alpha\\right] .\n\\]\n\n\n\\begin{algorithm}\n[Control point based exact description of rational trigonometric\ncurves]\\label{alg:cpbed_rational_trigonometric_curves}The process that\nprovides the control point based exact description of the rational curve\n(\\ref{rational_trigonometric_curve_to_be_described}) consists of the following operations:\n\n\\begin{itemize}\n\\item let%\n\\[\nn\\geq n_{\\min}=\\max\\left\\{ z:z\\in\\cup_{\\ell=1}^{\\delta+1}\\left( P_{\\ell}\\cup\nQ_{\\ell}\\right) \\right\\}\n\\]\nbe an arbitrarily fixed order;\n\n\\item apply Theorem \\ref{thm:cpbed_trigonometric_curves} to the pre-image\n\\begin{equation}\n\\mathbf{g}_{\\wp}\\left( u\\right) =\\left[ g^{\\ell}\\left( u\\right) \\right]\n_{\\ell=1}^{\\delta+1},~u\\in\\left[ 0,\\alpha\\right]\n\\label{pre_image_of_rational_trigonometric_curve_to_be_described}%\n\\end{equation}\nof the curve (\\ref{rational_trigonometric_curve_to_be_described}), i.e.,\ncompute control points\n\\[\n\\mathbf{d}_{i}^{\\wp}=\\left[ d_{i}^{\\ell}\\right] _{\\ell=1}^{\\delta+1}\\in%\n\\mathbb{R}\n^{\\delta+1},~i=0,1,\\ldots,2n\n\\]\nfor the exact trigonometric representation of\n(\\ref{pre_image_of_rational_trigonometric_curve_to_be_described}) in the\npre-image space $%\n\\mathbb{R}\n^{\\delta+1}$;\n\n\\item project the obtained control points onto the hyperplane $x^{\\delta+1}=1$\nthat results the control points\n\\[\n\\mathbf{d}_{i}=\\frac{1}{d_{i}^{\\delta+1}}\\left[ d_{i}^{\\ell}\\right] _{\\ell\n=1}^{\\delta}\\in%\n\\mathbb{R}\n^{\\delta},~i=0,1,\\ldots,2n\n\\]\nand weights%\n\\[\n\\omega_{i}=d_{i}^{\\delta+1},~i=0,1,\\ldots,2n\n\\]\nneeded for the rational trigonometric representation\n(\\ref{rational_trigonometric_curve}) of\n(\\ref{rational_trigonometric_curve_to_be_described});\n\n\\item the above generation process does not necessarily ensure the positivity\nof all weights, since the last coordinate of some control points in the\npre-image space $%\n\\mathbb{R}\n^{\\delta+1}$ can be negative; if this is the case, one can increase in $%\n\\mathbb{R}\n^{\\delta+1}$ the order of the trigonometric curve used for the control point\nbased exact description of the pre-image $\\mathbf{g}_{\\wp}$, since -- as\nstated in Remark \\ref{rem:trigonometric_order_elevation} -- order elevation\ngenerates a sequence of control polygons that converges to $\\mathbf{g}_{\\wp}$\nwhich is a geometric object of one branch that does not intersect the\nvanishing plane $x^{\\delta+1}=0\\,$(i.e., the $\\left( \\delta+1\\right) $th\ncoordinate of all its points are of the same sign); therefore, it is\nguaranteed that exists a finite and minimal order $n+z$ ($z\\geq1$) for which\nall weights are positive.\n\\end{itemize}\n\\end{algorithm}\n\n\\begin{example}\n[Application of Algorithm \\ref{alg:cpbed_rational_trigonometric_curves} --\nrational curves]Fig. \\ref{fig:bernoulli_s_lemniscate} shows the control point\nbased description of an arc of the rational trigonometric curve%\n\\begin{equation}\n\\mathbf{g}\\left( u\\right) =\\frac{1}{g^{3}\\left( u\\right) }\\left[\n\\begin{array}\n[c]{c}%\ng^{1}\\left( u\\right) \\\\\n\\\\\ng^{2}\\left( u\\right)\n\\end{array}\n\\right] =\\frac{1}{\\frac{3}{2}-\\frac{1}{2}\\cos\\left( 2u\\right) }\\left[\n\\begin{array}\n[c]{r}%\n\\cos\\left( u\\right) \\\\\n\\\\\n\\frac{1}{2}\\sin\\left( 2u\\right)\n\\end{array}\n\\right] ,~u\\in\\left[ 0,\\frac{2\\pi}{3}\\right] ,\n\\label{Bernoulli_s_lemniscate}%\n\\end{equation}\nalso known as Bernoulli's lemniscate.\n\\end{example}\n\n\n\\begin{figure}\n[!h]\n\\begin{center}\n\\includegraphics[\nheight=3.7239in,\nwidth=6.346in\n]%\n{Bernoulli_s_Lemniscate.pdf}%\n\\caption{Using Algorithm \\ref{alg:cpbed_rational_trigonometric_curves} with\nshape parameter $\\alpha=\\frac{2\\pi}{3}$, the left image shows the control\npoint based exact description of an arc of Bernoulli's lemniscate\n(\\ref{Bernoulli_s_lemniscate}) by means of a fourth order rational\ntrigonometric curve $\\mathbf{g}$ (green) and its pre-image $\\mathbf{g}_{\\wp}$\n(red). Control points $\\mathbf{d}_{i}^{\\wp}=\\left[\n\\protect\\begin{array}\n[c]{cc}%\n\\omega_{i}\\mathbf{d}_{i} & \\omega_{i}%\n\\protect\\end{array}\n\\right] \\in\\mathbb{R} ^{3}$ ($i=0,1,\\ldots,8$) of the pre-image\n$\\mathbf{g}_{\\wp}=\\left[ g^{\\ell}\\left( u\\right) \\right] _{\\ell=1}^{3}$\nwere obtained by the application of Theorem\n\\ref{thm:cpbed_trigonometric_curves}, while control points $\\left[\n\\mathbf{d}_{i}\\right] _{i=0}^{8}$ and weights $\\left[ \\omega_{i}\\right]\n_{i=0}^{8}$ needed for the rational representation\n(\\ref{rational_trigonometric_curve}) were obtained by the central projection\nof points $\\left[ \\mathbf{d}_{i}^{\\wp}\\right] _{i=0}^{8}$ onto the\nhyperplane $x^{3}=1$. The right image also illustrates the control polygons of\nthe second and third order exact rational trigonometric representations of the\nsame arc. (For interpretation of the references to color in this figure\nlegend, the reader is referred to the web version of this paper.)}%\n\\label{fig:bernoulli_s_lemniscate}%\n\\end{center}\n\\end{figure}\n\n\nTheorem \\ref{thm:cpbed_trigonometric_curves} can also be used to provide\ncontrol nets (grids) for the control point based exact description of the\nhigher order mixed partial derivatives of a general class of multivariate surfaces the\nelements of which can be expressed in the form%\n\\begin{equation}\n\\mathbf{s}\\left( \\mathbf{u}\\right) =\\left[\n\\begin{array}\n[c]{cccc}%\ns^{1}\\left( \\mathbf{u}\\right) & s^{2}\\left( \\mathbf{u}\\right) & \\cdots &\ns^{\\delta+\\kappa}\\left( \\mathbf{u}\\right)\n\\end{array}\n\\right] \\in%\n\\mathbb{R}\n^{\\delta+\\kappa},~\\mathbf{u}=\\left[ u_{j}\\right] _{j=1}^{\\delta}\\in\n\\times_{j=1}^{\\delta}\\left[ 0,\\alpha_{j}\\right] ,~\\alpha_{j}\\in\\left(\n0,\\pi\\right) ,~\\kappa\\geq0 \\label{trigonometric_surface_to_be_described}%\n\\end{equation}\nwhere%\n\\[\ns^{\\ell}\\left( \\mathbf{u}\\right) =\\sum_{\\zeta=1}^{m_{\\ell}}\\prod\n_{j=1}^{\\delta}\\left( \\sum_{p\\in P_{\\ell,\\zeta,j}}c_{p}^{\\ell,\\zeta,j}%\n\\cos\\left( pu_{j}+\\psi_{p}^{\\ell,\\zeta,j}\\right) +\\sum_{q\\in Q_{\\ell\n,\\zeta,j}}s_{q}^{\\ell,\\zeta,j}\\sin\\left( qu_{j}+\\varphi_{q}^{\\ell,\\zeta\n,j}\\right) \\right) ,~\\ell=1,2,\\ldots,\\delta+\\kappa\n\\]\nand%\n\\[\nP_{\\ell,\\zeta,j},Q_{\\ell,\\zeta,j}\\subset%\n\\mathbb{N}\n,~m_{\\ell}\\in%\n\\mathbb{N}\n\\setminus\\left\\{ 0\\right\\} ,~c_{p}^{\\ell,\\zeta,j},\\psi_{p}^{\\ell,\\zeta\n,j},s_{q}^{\\ell,\\zeta,j},\\varphi_{q}^{\\ell,\\zeta,j}\\in%\n\\mathbb{R}\n.\n\\]\nIndeed, we have the next theorem.\n\n\\begin{theorem}\n[Control point based exact description of trigonometric surfaces]%\n\\label{thm:cpbed_trigonometric_surfaces}The control point based exact\ndescription of the $\\left( r_{1}+r_{2}+\\ldots+r_{\\delta}\\right) $th order\nmixed partial derivative of the surface\n(\\ref{trigonometric_surface_to_be_described}) fulfills the equality%\n\\begin{equation}\n\\dfrac{\\partial^{r_{1}+r_{2}+\\ldots+r_{\\delta}}}{\\partial u_{1}^{r_{1}%\n}\\partial u_{2}^{r_{2}}\\cdots\\partial u_{\\delta}^{r_{\\delta}}}\\mathbf{s}\\left(\n\\mathbf{u}\\right) =%\n{\\displaystyle\\sum\\limits_{i_{1}=0}^{2n_{1}}}\n{\\displaystyle\\sum\\limits_{i_{2}=0}^{2n_{2}}}\n\\cdots%\n{\\displaystyle\\sum\\limits_{i_{\\delta}=0}^{2n_{\\delta}}}\n\\mathbf{d}_{i_{1},i_{2},\\ldots,i_{\\delta}}\\left( r_{1},r_{2},\\ldots\n,r_{\\delta}\\right)\n{\\displaystyle\\prod\\limits_{j=1}^{\\delta}}\nT_{2n_{j},i_{j}}^{\\alpha_{j}}\\left( u_{j}\\right)\n\\label{exact_trigonometric_surface_represantation}%\n\\end{equation}\nfor all parameter vectors $\\mathbf{u\\in}\\times_{j=1}^{\\delta}\\left[\n0,\\alpha_{j}\\right] $, where%\n\\begin{align*}\nn_{j} & \\geq n_{\\min}^{j}=\\max\\left\\{ z_{j}:z_{j}\\in\\cup_{\\ell=1}%\n^{\\delta+k}\\cup_{\\zeta=1}^{m_{\\ell}}\\left( P_{\\ell,\\zeta,j}\\cup Q_{\\ell\n,\\zeta,j}\\right) \\right\\} ,~j=1,2,\\ldots,\\delta,\\\\\n\\mathbf{d}_{i_{1},i_{2},\\ldots,i_{\\delta}}\\left( r_{1},r_{2},\\ldots\n,r_{\\delta}\\right) & =\\left[ d_{i_{1},i_{2},\\ldots,i_{\\delta}}^{\\ell\n}\\left( r_{1},r_{2},\\ldots,r_{\\delta}\\right) \\right] _{\\ell=1}%\n^{\\delta+\\kappa}\\in%\n\\mathbb{R}\n^{\\delta+\\kappa},\\\\\nd_{i_{1},i_{2},\\ldots,i_{\\delta}}^{\\ell}\\left( r_{1},r_{2},\\ldots,r_{\\delta\n}\\right) & =\\sum_{\\zeta=1}^{m_{\\ell}}\\prod_{j=1}^{\\delta}d_{i_{j}}%\n^{\\ell,\\zeta}\\left( r_{j}\\right) ,~\\ell=1,2,\\ldots,\\delta+\\kappa\n\\end{align*}\nand%\n\\begin{align*}\nd_{i_{j}}^{\\ell,\\zeta}\\left( r_{j}\\right) = & \\sum_{p\\in P_{\\ell,\\zeta,j}%\n}c_{p}^{\\ell,\\zeta,j}p^{r_{j}}\\left( \\mu_{p,i_{j}}^{n_{j}}\\cos\\left(\n\\psi_{p}^{\\ell,\\zeta,j}+\\frac{r_{j}\\pi}{2}\\right) -\\lambda_{p,i_{j}}^{n_{j}%\n}\\sin\\left( \\psi_{p}^{\\ell,\\zeta,j}+\\frac{r_{j}\\pi}{2}\\right) \\right) \\\\\n& +\\sum_{q\\in Q_{\\ell,\\zeta,j}}s_{q}^{\\ell,\\zeta,j}q^{r_{j}}\\left(\n\\lambda_{q,i_{j}}^{n_{j}}\\cos\\left( \\varphi_{q}^{\\ell,\\zeta,j}+\\frac{r_{j}%\n\\pi}{2}\\right) +\\mu_{q,i_{j}}^{n_{j}}\\sin\\left( \\varphi_{q}^{\\ell,\\zeta\n,j}+\\frac{r_{j}\\pi}{2}\\right) \\right) ,\\\\\n\\ell & =1,2,\\ldots,\\delta+\\kappa,~\\zeta=1,2,\\ldots,m_{\\ell},~j=1,2,\\ldots\n,\\delta.\n\\end{align*}\n(In case of zeroth order partial derivatives we will use the simpler notation\n\\[\n\\mathbf{d}_{i_{1},i_{2},\\ldots,i_{\\delta}}=\\left[ d_{i_{1},i_{2}%\n,\\ldots,i_{\\delta}}^{\\ell}\\right] _{\\ell=1}^{\\delta+\\kappa}=\\left[\nd_{i_{1},i_{2},\\ldots,i_{\\delta}}^{\\ell}\\left( 0,0,\\ldots,0\\right) \\right]\n_{\\ell=1}^{\\delta+\\kappa}=\\mathbf{d}_{i_{1},i_{2},\\ldots,i_{\\delta}}\\left(\n0,0,\\ldots,0\\right)\n\\]\nfor all $i_{j}=0,1,\\ldots,2n_{j}$ and $j=1,2,\\ldots,\\delta$.)\n\\end{theorem}\n\n\\begin{proof}\nObserve, by using Lemma \\ref{lem:tp}, that equality%\n\\begin{align*}\n& \\frac{\\partial^{r_{1}+r_{2}+\\ldots+r_{\\delta}}}{\\partial u_{1}^{r_{1}%\n}\\partial u_{2}^{r_{2}}\\cdots\\partial u_{\\delta}^{r_{\\delta}}}s^{\\ell}\\left(\n\\mathbf{u}\\right)= \\\\\n= & \\frac{\\partial^{r_{1}+r_{2}+\\ldots+r_{\\delta}}}{\\partial u_{1}^{r_{1}%\n}\\partial u_{2}^{r_{2}}\\cdots\\partial u_{\\delta}^{r_{\\delta}}}\\sum_{\\zeta=1}^{m_{\\ell}%\n}\\prod_{j=1}^{\\delta}\\left( \\sum_{p\\in P_{\\ell,\\zeta,j}}c_{p}^{\\ell,\\zeta\n,j}\\cos\\left( pu_{j}+\\psi_{p}^{\\ell,\\zeta,j}\\right) +\\sum_{q\\in\nQ_{\\ell,\\zeta,j}}s_{q}^{\\ell,\\zeta,j}\\sin\\left( qu_{j}+\\varphi_{q}%\n^{\\ell,\\zeta,j}\\right) \\right) \\\\\n= & \\sum_{\\zeta=1}^{m_{\\ell}}\\prod_{j=1}^{\\delta}\\frac{\\text{d}^{r_{j}}%\n}{\\text{d}u_{j}^{r_{j}}}\\left( \\sum_{p\\in P_{\\ell,\\zeta,j}}c_{p}^{\\ell\n,\\zeta,j}\\cos\\left( pu_{j}+\\psi_{p}^{\\ell,\\zeta,j}\\right) +\\sum_{q\\in\nQ_{\\ell,\\zeta,j}}s_{q}^{\\ell,\\zeta,j}\\sin\\left( qu_{j}+\\varphi_{q}%\n^{\\ell,\\zeta,j}\\right) \\right) \\\\\n= & \\sum_{\\zeta=1}^{m_{\\ell}}\\prod_{j=1}^{\\delta}\\left( \\sum_{i_{j}%\n=0}^{2n_{j}}d_{i_{j}}^{\\ell,\\zeta}\\left( r_{j}\\right) T_{2n_{j},i_{j}%\n}^{\\alpha_{j}}\\left( u_{j}\\right) \\right) \\\\\n= & \\sum_{i_{1}=0}^{2n_{1}}\\sum_{i_{2}=0}^{2n_{2}}\\cdots\\sum_{i_{\\delta}%\n=0}^{2n_{\\delta}}\\left( \\sum_{\\zeta=1}^{m_{\\ell}}\\prod_{j=1}^{\\delta}%\nd_{i_{j}}^{\\ell,\\zeta}\\left( r_{j}\\right) \\right) T_{2n_{1},i_{1}}%\n^{\\alpha_{1}}\\left( u_{1}\\right) T_{2n_{2},i_{2}}^{\\alpha_{2}}\\left(\nu_{2}\\right) \\cdot\\ldots\\cdot T_{2n_{\\delta},i_{\\delta}}^{\\alpha_{\\delta}%\n}\\left( u_{\\delta}\\right)\n\\end{align*}\nholds for all parameter vectors $\\mathbf{u}=\\left[ u_{j}\\right]\n_{j=1}^{\\delta}\\in\\times_{j=1}^{\\delta}\\left[ 0,\\alpha_{j}\\right] $ and\ncoordinate functions $\\ell=1,2,\\ldots,\\delta+\\kappa$, i.e., $d_{i_{j}}%\n^{\\ell,\\zeta}\\left( r_{j}\\right) $ can be calculated by means of formula\n(\\ref{trigonometric_control_points}) for all indices $\\ell=1,2,\\ldots\n,\\delta+\\kappa$, $\\zeta=1,2,\\ldots,m_{\\ell}$ and $j=1,2,\\ldots,\\delta$.\n\\end{proof}\n\n\\begin{example}\n[Application of Theorem \\ref{thm:cpbed_trigonometric_surfaces} --\nsurfaces]Using trigonometric surfaces of type (\\ref{trigonometric_surface})\nand applying Theorem \\ref{thm:cpbed_trigonometric_surfaces}, Figs.\n\\ref{fig:torus} and \\ref{fig:surface_of_revolution_star} show several control\npoint constellations for the exact description of the toroidal patch%\n\\begin{align}\n\\mathbf{s}\\left( u_{1},u_{2}\\right) & =\\left[\n\\begin{array}\n[c]{c}%\ns^{1}\\left( u_{1},u_{2}\\right) \\\\\n\\\\\ns^{2}\\left( u_{1},u_{2}\\right) \\\\\n\\\\\ns^{3}\\left( u_{1},u_{2}\\right)\n\\end{array}\n\\right] =\\left[\n\\begin{array}\n[c]{c}%\n\\left( 3+\\frac{15}{8}\\left( \\sqrt{5}-1\\right) \\sin\\left( u_{1}\\right)\n\\right) \\cos\\left( u_{2}\\right) \\\\\n\\\\\n\\left( 3+\\frac{15}{8}\\left( \\sqrt{5}-1\\right) \\sin\\left( u_{1}\\right)\n\\right) \\sin\\left( u_{2}\\right) \\\\\n\\\\\n\\frac{15}{8}\\left( \\sqrt{5}-1\\right) \\cos\\left( u_{1}\\right)\n\\end{array}\n\\right] ,\\label{torus}\\\\\n\\left( u_{1},u_{2}\\right) & \\in\\left[ 0,\\frac{3\\pi}{4}\\right]\n\\times\\left[ 0,\\frac{\\pi}{2}\\right] \\nonumber\n\\end{align}\nand of the patch%\n\\begin{align}\n\\mathbf{s}\\left( u_{1},u_{2}\\right) & =\\left[\n\\begin{array}\n[c]{c}%\ns^{1}\\left( u_{1},u_{2}\\right) \\\\\ns^{2}\\left( u_{1},u_{2}\\right) \\\\\ns^{3}\\left( u_{1},u_{2}\\right)\n\\end{array}\n\\right] =\\left[\n\\begin{array}\n[c]{c}%\n\\left( 12+6\\sin\\left( u_{1}\\right) -\\sin\\left( 6u_{1}\\right) \\right)\n\\cos\\left( u_{2}\\right) \\\\\n\\left( 12+6\\sin\\left( u_{1}\\right) -\\sin\\left( 6u_{1}\\right) \\right)\n\\sin\\left( u_{2}\\right) \\\\\n6\\cos\\left( u_{1}\\right) +\\cos\\left( 6u_{1}\\right)\n\\end{array}\n\\right] ,\\label{surface_of_revolution_star}\\\\\n& \\left( u_{1},u_{2}\\right) \\in \\left[ 0,\\frac{\\pi}{2}\\right]\n\\times\\left[ 0,\\frac{2\\pi}{3}\\right] \\nonumber\n\\end{align}\nthat also lies on a surface of revolution generated by the rotation of the\nhypocycloid%\n\\begin{equation}\n\\mathbf{g}\\left( u\\right) =\\left[\n\\begin{array}\n[c]{c}%\ng^{1}\\left( u\\right) \\\\\ng^{2}\\left( u\\right)\n\\end{array}\n\\right] =\\left[\n\\begin{array}\n[c]{c}%\n6\\sin\\left( u\\right) -\\sin\\left( 6u\\right) \\\\\n6\\cos\\left( u\\right) +\\cos\\left( 6u\\right)\n\\end{array}\n\\right] ,~u\\in\\left[ 0,2\\pi\\right] \\label{star_hypocycloid}%\n\\end{equation}\nabout the axis $z$, respectively.\n\\end{example}\n\n\n\\begin{figure}\n[!h]\n\\begin{center}\n\\includegraphics[\nnatheight=4.094900in,\nnatwidth=6.298400in,\nheight=3.6729in,\nwidth=6.3261in\n]%\n{torus_merged+.pdf}%\n\\caption{Control point based exact description of the toroidal patch\n(\\ref{torus}) by means of $3$-dimensional $2$-variate trigonometric surfaces\nof the type (\\ref{trigonometric_surface}). Control nets corresponding to\ndifferent orders were obtained by using Theorem\n\\ref{thm:cpbed_trigonometric_surfaces} with parameter settings $\\delta=2$,\n$\\kappa=1$; $\\alpha_{1}=\\frac{3\\pi}{4}$, $\\alpha_{2}=\\frac{\\pi}{2}$;\n$m_{1}=m_{2}=m_{3}=1$; $P_{1,1,1}=\\left\\{ 0\\right\\} $, $P_{1,1,2}=\\left\\{\n1\\right\\} $, $Q_{1,1,1}=\\left\\{ 1\\right\\} $,~$Q_{1,1,2}=\\varnothing$,\n$c_{0}^{1,1,1}=3$, $c_{1}^{1,1,2}=1$, $s_{1}^{1,1,1}=\\frac{15}{8}\\left(\n\\sqrt{5}-1\\right) $, $\\psi_{0}^{1,1,1}=\\psi_{1}^{1,1,2}=\\varphi_{1}%\n^{1,1,1}=0$; $P_{2,1,1}=\\left\\{ 0\\right\\} $, $P_{2,1,2}=\\varnothing$,\n$Q_{2,1,1}=Q_{2,1,2}=\\left\\{ 1\\right\\} $, $c_{0}^{2,1,1}=3$, $s_{1}%\n^{2,1,1}=\\frac{15}{8}\\left( \\sqrt{5}-1\\right) $, $s_{1}^{2,1,2}=1$,\n$\\psi_{0}^{2,1,1}=\\varphi_{1}^{2,1,1}=\\varphi_{1}^{2,1,2}=0$; $P_{3,1,1}%\n=\\left\\{ 1\\right\\} $, $P_{3,1,2}=\\left\\{ 0\\right\\} $, $Q_{3,1,1}%\n=Q_{3,1,2}=\\varnothing$, $c_{1}^{3,1,1}=\\frac{15}{8}\\left( \\sqrt{5}-1\\right)\n$, $c_{0}^{3,1,2}=1$, $\\psi_{0}^{3,1,1}=\\psi_{1}^{3,1,2}=0$.}%\n\\label{fig:torus}%\n\\end{center}\n\\end{figure}\n\n\\begin{figure}\n[!h]\n\\begin{center}\n\\includegraphics[\nnatheight=2.546900in,\nnatwidth=6.298400in,\nheight=2.6247in,\nwidth=6.3261in\n]%\n{surface_of_revolution_star_merged+.pdf}%\n\\caption{Control point based exact description of the trigonometric patch\n(\\ref{surface_of_revolution_star}) that lies on a surface of revolution\ngenerated by the rotation of the hypocycloid (\\ref{star_hypocycloid}) about\nthe axis $z$. Control nets were generated by formulae described in Theorem\n\\ref{thm:cpbed_trigonometric_surfaces} ($\\delta=2$, $\\kappa=1$; $\\alpha\n_{1}=\\frac{\\pi}{2}$, $\\alpha_{2}=\\frac{2\\pi}{3}$; $m_1=m_2=m_3=1$).}%\n\\label{fig:surface_of_revolution_star}%\n\\end{center}\n\\end{figure}\n\n\n\\begin{example}\n[Application of Theorem \\ref{thm:cpbed_trigonometric_surfaces} -- volumes]%\n$3$-dimensional trigonometric volumes can also be exactly described by means\nof $3$-variate tensor product surfaces of the type\n(\\ref{trigonometric_surface}). Figs.\n\\ref{fig:non_rational_trigonometric_volume} and\n\\ref{fig:non_rational_trigonometric_volume2} illustrate control grids that\ngenerate the volumes%\n\\begin{align}\n\\mathbf{s}\\left( u_{1},u_{2},u_{3}\\right) & =\\left[\n\\begin{array}\n[c]{c}%\ns^{1}\\left( u_{1},u_{2},u_{3}\\right) \\\\\ns^{2}\\left( u_{1},u_{2},u_{3}\\right) \\\\\ns^{3}\\left( u_{1},u_{2},u_{3}\\right)\n\\end{array}\n\\right] \\nonumber\\\\\n& =\\left[\n\\begin{array}\n[c]{c}%\n\\left( 6+\\cos\\left( u_{1}+\\frac{\\pi}{3}\\right) \\right) \\cos\\left(\nu_{2}-\\frac{\\pi}{6}\\right) \\cos\\left( u_{3}+\\frac{\\pi}{3}\\right) \\\\\n\\\\\n\\left( 6+\\cos\\left( u_{1}+\\frac{\\pi}{3}\\right) \\right) \\cos\\left(\nu_{2}-\\frac{\\pi}{6}\\right) \\sin\\left( u_{3}+\\frac{\\pi}{3}\\right) \\\\\n\\\\\n\\cos\\left( u_{1}+\\frac{\\pi}{3}\\right) \\sin\\left( u_{2}-\\frac{\\pi}%\n{6}\\right)\n\\end{array}\n\\right] ,\\label{non_rational_trigonometric_volume1}\\\\\n\\left( u_{1},u_{2},u_{3}\\right) & \\in\\left[ 0,\\frac{\\pi}{2}\\right]\n\\times\\left[ 0,\\frac{\\pi}{2}\\right] \\times\\left[ 0,\\frac{2\\pi}{3}\\right]\n\\nonumber\n\\end{align}\nand%\n\\begin{align}\n\\mathbf{s}\\left( u_{1},u_{2},u_{3}\\right) & =\\left[\n\\begin{array}\n[c]{c}%\ns^{1}\\left( u_{1},u_{2},u_{3}\\right) \\\\\ns^{2}\\left( u_{1},u_{2},u_{3}\\right) \\\\\ns^{3}\\left( u_{1},u_{2},u_{3}\\right)\n\\end{array}\n\\right] \\nonumber\\\\\n& =\\left[\n\\begin{array}\n[c]{c}%\n\\left( 2+\\frac{3}{4}\\sin\\left( u_{1}\\right) -\\frac{1}{4}\\sin\\left(\n3u_{1}\\right) \\right) \\cos\\left( u_{2}\\right) \\left( \\frac{3}{2}-\\frac\n{1}{2}\\cos\\left( 2u_{3}\\right) \\right) \\\\\n\\\\\n\\left( \\frac{5}{2}-\\frac{1}{2}\\cos\\left( 2u_{1}\\right) \\right) \\sin\\left(\nu_{2}\\right) \\left( 1+\\frac{3}{4}\\sin\\left( u_{3}\\right) -\\frac{1}{4}%\n\\sin\\left( 3u_{3}\\right) \\right) \\\\\n\\\\\n\\left( \\frac{1}{2}+\\frac{3}{8}\\cos\\left( u_{1}\\right) +\\frac{1}{2}%\n\\cos\\left( 2u_{1}\\right) +\\frac{1}{8}\\cos\\left( 3u_{1}\\right) \\right)\n\\left( \\frac{3}{2}+\\cos\\left( u_{2}\\right) +\\sin\\left( u_{3}\\right)\n\\right)\n\\end{array}\n\\right] ,\\label{non_rational_trigonometric_volume2}\\\\\n\\left( u_{1},u_{2},u_{3}\\right) & \\in\\left[ 0,\\frac{\\pi}{2}\\right]\n\\times\\left[ 0,\\frac{2\\pi}{3}\\right] \\times\\left[ 0,\\frac{\\pi}{2}\\right]\n,\\nonumber\n\\end{align}\nrespectively.\n\\end{example}\n\n\n\\begin{figure}\n[!h]\n\\begin{center}\n\\includegraphics[\nnatheight=2.801100in,\nnatwidth=6.298400in,\nheight=2.8011in,\nwidth=6.2984in\n]%\n{non_rational_trigonometric_volume_merged+.pdf}%\n\\caption{Different views of the same $3$-dimensional trigonometric volume\n(\\ref{non_rational_trigonometric_volume1}) along with its control grid\ncalculated by means of Theorem \\ref{thm:cpbed_trigonometric_surfaces}\n($\\delta=3$, $\\kappa=0$; $\\alpha_{1}=\\frac{\\pi}{2}$, $\\alpha_{2}=\\frac{\\pi}%\n{2}$, $\\alpha_{3}=\\frac{2\\pi}{3}$; $m_1=m_2=m_3=1$).}%\n\\label{fig:non_rational_trigonometric_volume}%\n\\end{center}\n\\end{figure}\n\n\\begin{figure}\n[!h]\n\\begin{center}\n\\includegraphics[\nnatheight=2.633300in,\nnatwidth=6.138400in,\nheight=2.6602in,\nwidth=6.1661in\n]%\n{non_rational_trigonometric_volume2_merged+.pdf}%\n\\caption{Control point based exact description of the $3$-dimensional\ntrigonometric volume (\\ref{non_rational_trigonometric_volume2}) by means of\n$3$-variate tensor product surfaces of the type (\\ref{trigonometric_surface})\nwith different orders. Control grids were obtained by using Theorem\n\\ref{thm:cpbed_trigonometric_surfaces} ($\\delta=3$, $\\kappa=0$; $\\alpha\n_{1}=\\frac{\\pi}{2}$, $\\alpha_{2}=\\frac{2\\pi}{3}$, $\\alpha_{3}=\\frac{\\pi}{2}%\n$; $m_1=m_2=1$, $m_3=2$).}%\n\\label{fig:non_rational_trigonometric_volume2}%\n\\end{center}\n\\end{figure}\n\n\nTheorem \\ref{thm:cpbed_trigonometric_surfaces} can also be used to provide\ncontrol point based exact description of the rational trigonometric surface%\n\\begin{equation}\n\\mathbf{s}\\left( \\mathbf{u}\\right) =\\frac{1}{s^{\\delta+\\kappa+1}\\left(\n\\mathbf{u}\\right) }\\left[\n\\begin{array}\n[c]{cccc}%\ns^{1}\\left( \\mathbf{u}\\right) & s^{2}\\left( \\mathbf{u}\\right) & \\cdots &\ns^{\\delta+\\kappa}\\left( \\mathbf{u}\\right)\n\\end{array}\n\\right] \\in%\n\\mathbb{R}\n^{\\delta+\\kappa}, \\label{rational_trigonometric_surface_to_be_described}%\n\\end{equation}\nwhere%\n\\begin{align*}\n\\mathbf{u} & =\\left[ u_{j}\\right] _{j=1}^{\\delta}\\in\\times_{j=1}^{\\delta\n}\\left[ 0,\\alpha_{j}\\right] ,~\\alpha_{j}\\in\\left( 0,\\pi\\right)\n,~\\kappa\\geq0,\\\\\ns^{\\ell}\\left( \\mathbf{u}\\right) & =\\sum_{\\zeta=1}^{m_{\\ell}}\\prod\n_{j=1}^{\\delta}\\left( \\sum_{p\\in P_{\\ell,\\zeta,j}}c_{p}^{\\ell,\\zeta,j}%\n\\cos\\left( pu_{j}+\\psi_{p}^{\\ell,\\zeta,j}\\right) +\\sum_{q\\in Q_{\\ell\n,\\zeta,j}}s_{q}^{\\ell,\\zeta,j}\\sin\\left( qu_{j}+\\varphi_{q}^{\\ell,\\zeta\n,j}\\right) \\right) ,\\\\\n\\ell & =1,2,\\ldots,\\delta+\\kappa+1,\\\\\nP_{\\ell,\\zeta,j},Q_{\\ell,\\zeta,j} & \\subset%\n\\mathbb{N}\n,~m_{\\ell}\\in%\n\\mathbb{N}\n\\setminus\\left\\{ 0\\right\\} ,~c_{p}^{\\ell,\\zeta,j},\\psi_{p}^{\\ell,\\zeta\n,j},s_{q}^{\\ell,\\zeta,j},\\varphi_{q}^{\\ell,\\zeta,j}\\in%\n\\mathbb{R}%\n\\end{align*}\nand%\n\\[\ns^{\\delta+\\kappa+1}\\left( \\mathbf{u}\\right) >0,~\\forall\\mathbf{u}\\in\n\\times_{j=1}^{\\delta}\\left[ 0,\\alpha_{j}\\right] .\n\\]\nSimilarly to Algorithm \\ref{alg:cpbed_rational_trigonometric_curves} one can\nformulate the next process.\n\n\\begin{algorithm}\n[Control point based exact description of rational trigonometric\nsurfaces]\\label{alg:cpbed_rational_trigonometric_surfaces}Operations that\nensure the control point based exact description of the surface\n(\\ref{rational_trigonometric_surface_to_be_described}) are as follows:\n\n\\begin{itemize}\n\\item let%\n\\[\nn_{j}\\geq n_{\\min}^{j}=\\max\\left\\{ z_{j}:z_{j}\\in\\cup_{\\ell=1}^{\\delta+\\kappa+1}%\n\\cup_{\\zeta=1}^{m_{\\ell}}\\left( P_{\\ell,\\zeta,j}\\cup Q_{\\ell,\\zeta,j}\\right)\n\\right\\} ,~j=1,2,\\ldots,\\delta\n\\]\nbe arbitrarily fixed orders in directions $u_{1},u_{2},\\ldots,u_{\\delta}$;\n\n\\item apply Theorem \\ref{thm:cpbed_trigonometric_surfaces} to the pre-image\n\\begin{equation}\n\\mathbf{s}_{\\wp}\\left( \\mathbf{u}\\right) =\\left[ s^{\\ell}\\left(\n\\mathbf{u}\\right) \\right] _{\\ell=1}^{\\delta+\\kappa+1}\\in%\n\\mathbb{R}\n^{\\delta+\\kappa+1},~\\mathbf{u}=\\left[ u_{j}\\right] _{j=1}^{\\delta}\\in\n\\times_{j=1}^{\\delta}\\left[ 0,\\alpha_{j}\\right]\n\\label{pre_image_of_rational_trigonometric_surface_to_be_described}%\n\\end{equation}\nof the surface (\\ref{rational_trigonometric_surface_to_be_described}), i.e.,\ncompute control points%\n\\[\n\\mathbf{d}_{i_{1},i_{2},\\ldots,i_{\\delta}}^{\\wp}=\\left[ d_{i_{1},i_{2}%\n,\\ldots,i_{\\delta}}^{\\ell}\\right] _{\\ell=1}^{\\delta+\\kappa+1}\\in%\n\\mathbb{R}\n^{\\delta+\\kappa+1},~i_{j}=0,1,\\ldots,2n_{j},~j=1,2,\\ldots,\\delta\n\\]\nfor the exact trigonometric representation of\n(\\ref{pre_image_of_rational_trigonometric_surface_to_be_described}) in the\npre-image space $%\n\\mathbb{R}\n^{\\delta+\\kappa+1}$;\n\n\\item project the obtained control points onto the hyperplane $x^{\\delta\n+\\kappa+1}=1$ that results the control points\n\\[\n\\mathbf{d}_{i_{1},i_{2},\\ldots,i_{\\delta}}=\\frac{1}{d_{i_{1},i_{2}%\n,\\ldots,i_{\\delta}}^{\\delta+\\kappa+1}}\\left[ d_{i_{1},i_{2},\\ldots,i_{\\delta\n}}^{\\ell}\\right] _{\\ell=1}^{\\delta+\\kappa}\\in%\n\\mathbb{R}\n^{\\delta+\\kappa},~i_{j}=0,1,\\ldots,2n_{j},~j=1,2,\\ldots,\\delta\n\\]\nand weights%\n\\[\n\\omega_{i_{1},i_{2},\\ldots,i_{\\delta}}=d_{i_{1},i_{2},\\ldots,i_{\\delta}%\n}^{\\delta+\\kappa+1},~i_{j}=0,1,\\ldots,2n_{j},~j=1,2,\\ldots,\\delta\n\\]\nneeded for the rational trigonometric representation\n(\\ref{trigonometric_rational_surface}) of\n(\\ref{rational_trigonometric_surface_to_be_described});\n\n\\item if not all weights are positive, try to increase the components of the\norder $\\left( n_{1},n_{2},\\ldots,n_{\\delta}\\right) $ of the trigonometric\nsurface used for the control point based exact description of the pre-image\n$\\mathbf{s}_{\\wp}$ in $%\n\\mathbb{R}\n^{\\delta+\\kappa+1}$ and repeat the previous projectional and weight\ndetermination step.\n\\end{itemize}\n\\end{algorithm}\n\n\\begin{example}\n[Application of Algorithm \\ref{alg:cpbed_rational_trigonometric_surfaces} --\nrational surfaces]Using surfaces of the type\n(\\ref{trigonometric_rational_surface}), Fig.\n\\ref{fig:rational_trigonometric_surface}\\ shows the control point based exact\ndescription of the rational trigonometric patch%\n\\begin{equation}\n\\mathbf{s}\\left( u_{1},u_{2}\\right) =\\frac{1}{s^{4}\\left( u_{1}%\n,u_{2}-\\gamma\\right) }\\left[\n\\begin{array}\n[c]{c}%\ns^{1}\\left( u_{1},u_{2}-\\gamma\\right) \\\\\ns^{2}\\left( u_{1},u_{2}-\\gamma\\right) \\\\\ns^{3}\\left( u_{1},u_{2}-\\gamma\\right)\n\\end{array}\n\\right] ,~\\left( u_{1},u_{2}\\right) \\in\\left[ 0,\\frac{\\pi}{4}\\right]\n\\times\\left[ 0,\\frac{\\pi}{3}\\right] , \\label{rational_trigonometric_8}%\n\\end{equation}\nwhere%\n\\begin{align*}\ns^{1}\\left( u_{1},u_{2}\\right) = & \\cos\\left( u_{1}\\right) \\left(\n173\\cos\\left( u_{2}\\right) -10\\sin\\left( 3u_{2}\\right) -10\\sin\\left(\n5u_{2}\\right) +\\cos\\left( 7u_{2}\\right) +\\cos\\left( 9u_{2}\\right) \\right)\n\\\\\n& +\\sin\\left( u_{1}\\right) \\left( -173\\sin\\left( u_{2}\\right)\n+10\\cos\\left( 3u_{2}\\right) -10\\cos\\left( 5u_{2}\\right) +\\sin\\left(\n7u_{2}\\right) -\\sin\\left( 9u_{2}\\right) \\right) ,\\\\\n& \\\\\ns^{2}\\left( u_{1},u_{2}\\right) = & \\cos\\left( u_{1}\\right) \\left(\n173\\sin\\left( u_{2}\\right) -10\\cos\\left( 3u_{2}\\right) +10\\cos\\left(\n5u_{2}\\right) -\\sin\\left( 7u_{2}\\right) +\\sin\\left( 9u_{2}\\right) \\right)\n\\\\\n& +\\sin\\left( u_{1}\\right) \\left( 173\\cos\\left( u_{2}\\right)\n-10\\sin\\left( 3u_{2}\\right) -10\\sin\\left( 5u_{2}\\right) +\\cos\\left(\n7u_{2}\\right) +\\cos\\left( 9u_{2}\\right) \\right) ,\\\\\n& \\\\\ns^{3}\\left( u_{1},u_{2}\\right) = & 20\\cos\\left( u_{1}\\right) \\left(\n5\\cos\\left( u_{2}\\right) +\\sin\\left( 3u_{2}\\right) +\\sin\\left(\n5u_{2}\\right) \\right) \\\\\n& +20\\sin\\left( u_{1}\\right) \\left( 5\\sin\\left( u_{2}\\right)\n+\\cos\\left( 3u_{2}\\right) -\\cos\\left( 5u_{2}\\right) \\right) ,\\\\\n& \\\\\ns^{4}\\left( u_{1},u_{2}\\right) = & 20\\cos\\left( u_{1}\\right) \\left(\n5\\sin\\left( u_{2}\\right) +\\cos\\left( 3u_{2}\\right) -\\cos\\left(\n5u_{2}\\right) \\right) \\\\\n& -20\\sin\\left( u_{1}\\right) \\left( 5\\cos\\left( u_{2}\\right)\n+\\sin\\left( 3u_{2}\\right) +\\sin\\left( 5u_{2}\\right) \\right) +200\n\\end{align*}\nand $\\gamma=\\frac{\\pi}{3}$.\n\\end{example}\n\n\n\\begin{figure}\n[!h]\n\\begin{center}\n\\includegraphics[\nnatheight=2.753600in,\nnatwidth=6.298400in,\nheight=2.7812in,\nwidth=6.3261in\n]%\n{rational_trigonometric_surface_merged+.pdf}%\n\\caption{Control point based exact description of the patch\n(\\ref{rational_trigonometric_8}) by means of $3$-dimensional $2$-variate\nrational trigonometric patches of different orders. Control nets were obtained\nby following the steps of Algorithm\n\\ref{alg:cpbed_rational_trigonometric_surfaces} ($\\delta=2$, $\\kappa=1$;\n$\\alpha_{1}=\\frac{\\pi}{4}$, $\\alpha_{2}=\\frac{\\pi}{3}$; $m_{1}=m_{2}=m_{3}=2$,\n$m_{4}=3$).}%\n\\label{fig:rational_trigonometric_surface}%\n\\end{center}\n\\end{figure}\n\n\n\\subsection{Description of (rational) hyperbolic curves and\nsurfaces\\label{sec:exact_description_hyperbolic}}\n\nAssume that the smooth parametric curve%\n\\begin{equation}\n\\mathbf{g}\\left( u\\right) =\\left[ g^{\\ell}\\left( u\\right) \\right]\n_{\\ell=1}^{\\delta},~u\\in\\left[ 0,\\alpha\\right] ,~\\alpha>0\n\\label{traditional_hyperbolic_curve}%\n\\end{equation}\nhas coordinate functions of the form%\n\\[\ng^{\\ell}\\left( u\\right) =\\sum_{p\\in P_{\\ell}}c_{p}^{\\ell}\\cosh\\left(\npu+\\psi_{p}^{\\ell}\\right) +\\sum_{q\\in Q_{\\ell}}s_{q}^{\\ell}\\sinh\\left(\nqu+\\varphi_{q}^{\\ell}\\right) ,\n\\]\nwhere $P_{\\ell},Q_{\\ell}\\subset%\n\\mathbb{N}\n$ and $c_{p}^{\\ell},\\psi_{p}^{\\ell},s_{q}^{\\ell},\\varphi_{q}^{\\ell}\\in%\n\\mathbb{R}\n$ and consider the vector space (\\ref{hyperbolic_vector_space}) of order\n\\begin{equation}\nn\\geq n_{\\min}=\\max\\left\\{ z:z\\in\\cup_{\\ell=1}^{\\delta}\\left( P_{\\ell}\\cup\nQ_{\\ell}\\right) \\right\\} . \\label{minimal_hyperbolic_order}%\n\\end{equation}\n\n\nUsing Lemma \\ref{lem:hp} and performing calculations similar to the proof of\nTheorem \\ref{thm:cpbed_trigonometric_curves} one obtains the next statement.\n\n\\begin{theorem}\n[Control point based exact description of hyperbolic curves]%\n\\label{thm:cpbed_hyperbolic_curves}For any arbitrarily fixed order\n(\\ref{minimal_hyperbolic_order}) the curve (\\ref{traditional_hyperbolic_curve}%\n) given in traditional hyperbolic parametric form has a unique control point\nbased exact description, more precisely one has that\n\\[\n\\frac{\\text{\\emph{d}}^{r}}{\\text{\\emph{d}}u^{r}}g^{\\ell}\\left( u\\right)\n=\\sum_{i=0}^{2n}d_{i}^{\\ell}\\left( r\\right) H_{2n,i}^{\\alpha}\\left(\nu\\right) ,\n\\]\nwhere%\n\\[\nd_{i}^{\\ell}\\left( r\\right) =\\left\\{\n\\begin{array}\n[c]{ll}%\n{\\displaystyle\\sum\\limits_{p\\in P_{\\ell}}}\nc_{p}^{\\ell}p^{r}\\left( \\rho_{p,i}^{n}\\cosh\\left( \\psi_{p}^{\\ell}\\right)\n+\\sigma_{p,i}^{n}\\sinh\\left( \\psi_{p}^{\\ell}\\right) \\right) & \\\\\n+%\n{\\displaystyle\\sum\\limits_{q\\in Q_{\\ell}}}\ns_{q}^{\\ell}q^{r}\\left( \\sigma_{q,i}^{n}\\cosh\\left( \\varphi_{q}^{\\ell\n}\\right) +\\rho_{q,i}^{n}\\sinh\\left( \\varphi_{q}^{\\ell}\\right) \\right) , &\nr=2z,\\\\\n& \\\\%\n{\\displaystyle\\sum\\limits_{p\\in P_{\\ell}}}\nc_{p}^{\\ell}p^{r}\\left( \\sigma_{p,i}^{n}\\cosh\\left( \\psi_{p}^{\\ell}\\right)\n+\\rho_{p,i}^{n}\\sinh\\left( \\psi_{p}^{\\ell}\\right) \\right) & \\\\\n+%\n{\\displaystyle\\sum\\limits_{q\\in Q_{\\ell}}}\ns_{q}^{\\ell}q^{r}\\left( \\rho_{q,i}^{n}\\cosh\\left( \\varphi_{q}^{\\ell}\\right)\n+\\sigma_{q,i}^{n}\\sinh\\left( \\varphi_{q}^{\\ell}\\right) \\right) , & r=2z+1\n\\end{array}\n\\right.\n\\]\ndenotes the $\\ell$th coordinate of the $i$th control point $\\mathbf{d}_{i}$\nneeded for the hyperbolic curve description (\\ref{hyperbolic_curve}).\n\\end{theorem}\n\n\\begin{example}\n[Application of Theorem \\ref{thm:cpbed_hyperbolic_curves} -- curves]Fig.\n\\ref{fig:equilateral_hyperbola} shows the control point based description of\nthe arc%\n\\begin{equation}\n\\mathbf{g}\\left( u\\right) =\\left[\n\\begin{array}\n[c]{c}%\ng^{1}\\left( u\\right) \\\\\n\\\\\ng^{2}\\left( u\\right)\n\\end{array}\n\\right] =\\left[\n\\begin{array}\n[c]{r}%\n\\sinh\\left( u-\\frac{3}{2}\\right) \\\\\n\\\\\n\\cosh\\left( u-\\frac{3}{2}\\right)\n\\end{array}\n\\right] ,~u\\in\\left[ 0,3\\right] \\label{equilateral_hyperbola}%\n\\end{equation}\nof an equilateral hyperbola.\n\\end{example}\n\n\n\\begin{figure}\n[!h]\n\\begin{center}\n\\includegraphics[\nheight=2.7121in,\nwidth=5.028in\n]%\n{hyperbola_n_1_3_5_alpha_3_pch_m1p5.pdf}%\n\\caption{Using Theorem \\ref{thm:cpbed_hyperbolic_curves}, the image shows the\ncontrol point based exact description of the hyperbolic arc\n(\\ref{equilateral_hyperbola}) by means of hyperbolic curves of the type\n(\\ref{hyperbolic_curve}) of varying order and fixed shape parameter $\\alpha\n=3$.}%\n\\label{fig:equilateral_hyperbola}%\n\\end{center}\n\\end{figure}\n\n\n\\begin{remark}\n[Hyperbolic counterpart of Theorem \\ref{thm:cpbed_trigonometric_surfaces}%\n]Higher order (mixed) partial derivatives of a non-rational higher dimensional\nmultivariate hyperbolic surface can also be exactly described by means of\nTheorem \\ref{thm:cpbed_hyperbolic_curves}; one would simply obtain the\nhyperbolic counterpart of Theorem \\ref{thm:cpbed_trigonometric_surfaces}.\nMoreover, one can combine Theorems \\ref{thm:cpbed_trigonometric_curves} and\n\\ref{thm:cpbed_hyperbolic_curves} in order to exactly describe patches of\nhybrid multivariate surfaces%\n\\[\n\\mathbf{s}\\left( \\mathbf{u}\\right) =\\left[\n\\begin{array}\n[c]{cccc}%\ns^{1}\\left( \\mathbf{u}\\right) & s^{2}\\left( \\mathbf{u}\\right) & \\cdots &\ns^{\\delta+\\kappa}\\left( \\mathbf{u}\\right)\n\\end{array}\n\\right] \\in%\n\\mathbb{R}\n^{\\delta+\\kappa},~\\mathbf{u}=\\left[ u_{j}\\right] _{j=1}^{\\delta}\\in\n\\times_{j=1}^{\\delta}\\left[ 0,\\alpha_{j}\\right] ,~\\alpha_{j}\\in\\left(\n0,\\beta_{j}\\right) ,~\\kappa\\geq0\n\\]\nthat are either trigonometric or hyperbolic in case of a fixed direction\n$u_{j}$, where $\\beta_{j}$ is either $\\pi$ or $+\\infty$ depending on the\ntrigonometric or hyperbolic type of the coordinate function $s^{j}$,\nrespectively. (Along a selected direction each coordinate function must be of\nthe same type).\n\\end{remark}\n\n\\begin{example}\n[Combination of Theorems \\ref{thm:cpbed_trigonometric_curves} and\n\\ref{thm:cpbed_hyperbolic_curves} \\ -- hybrid surfaces]Fig. \\ref{fig:catenoid}\nillustrates the control point based exact description of the patch%\n\\begin{equation}\n\\mathbf{s}\\left( u_{1},u_{2}\\right) =\\left[\n\\begin{array}\n[c]{c}%\ns^{1}\\left( u_{1},u_{2}\\right) \\\\\n\\\\\ns^{2}\\left( u_{1},u_{2}\\right) \\\\\n\\\\\ns^{3}\\left( u_{1},u_{2}\\right)\n\\end{array}\n\\right] =\\left[\n\\begin{array}\n[c]{c}%\n\\left( 1+\\cosh\\left( u_{1}-\\frac{3}{2}\\right) \\right) \\sin\\left(\nu_{2}\\right) \\\\\n\\\\\n\\left( 1+\\cosh\\left( u_{1}-\\frac{3}{2}\\right) \\right) \\cos\\left(\nu_{2}\\right) \\\\\n\\\\\n\\sinh\\left( u_{1}-\\frac{3}{2}\\right)\n\\end{array}\n\\right] ,~\\left( u_{1},u_{2}\\right) \\in\\left[ 0,3\\right] \\times\\left[\n0,\\frac{2\\pi}{3}\\right] \\label{catenoid}%\n\\end{equation}\nthat lies on a surface of revolution (also called hyperboloid) obtained\nby the rotation of the equilateral hyperbolic arc%\n\\begin{equation}\n\\mathbf{g}\\left( u\\right) =\\left[\n\\begin{array}\n[c]{c}%\ng^{1}\\left( u\\right) \\\\\n\\\\\ng^{2}\\left( u\\right)\n\\end{array}\n\\right] =\\left[\n\\begin{array}\n[c]{r}%\n\\cosh\\left( u-\\frac{3}{2}\\right) \\\\\n\\\\\n\\sinh\\left( u-\\frac{3}{2}\\right)\n\\end{array}\n\\right] ,~u\\in\\left[ 0,3\\right] \\label{catenary}%\n\\end{equation}\nalong the axis $z$.\n\\end{example}\n\n\n\\begin{figure}\n[!h]\n\\begin{center}\n\\includegraphics[\nnatheight=3.164400in,\nnatwidth=6.298400in,\nheight=3.192in,\nwidth=6.3261in\n]%\n{catenoid_merged+.pdf}%\n\\caption{Control point based exact description of the hyperboloidal patch\n(\\ref{catenoid}) with hybrid surfaces of different orders. In order to\nformulate the hybrid variant of Theorem \\ref{thm:cpbed_trigonometric_surfaces}%\n, in directions $u_{1}$ and $u_{2}$ the results of Theorems\n\\ref{thm:cpbed_hyperbolic_curves} ($\\alpha_{1}=3$) and\n\\ref{thm:cpbed_trigonometric_curves} ($\\alpha_{2}=\\frac{2\\pi}{3}$) were\napplied ($m_1=m_2=m_3=1$), respectively.}%\n\\label{fig:catenoid}%\n\\end{center}\n\\end{figure}\n\n\n\n\\begin{remark}\n[Hyperbolic counterpart of Algorithms\n\\ref{alg:cpbed_rational_trigonometric_curves} and\n\\ref{alg:cpbed_rational_trigonometric_surfaces}]Any smooth rational hyperbolic\ncurve\/surface that is given in traditional parametric form (with a\nnon-vanishing function in its denominator) can also be exactly described by\nmeans of rational hyperbolic curves\/surfaces of the type\n(\\ref{rational_hyperbolic_curve})\/(\\ref{rational_hyperbolic_surface}); one\nsimply has to apply the hyperbolic counterpart of Algorithms\n\\ref{alg:cpbed_rational_trigonometric_curves} or\n\\ref{alg:cpbed_rational_trigonometric_surfaces}. Moreover, combining the\npresented trigonometric algorithms and their hyperbolic counterparts, higher\ndimensional hybrid multivariate rational surfaces can also exactly described\nby means of multivariate hybrid tensor product surfaces.\n\\end{remark}\n\n\\begin{example}\n[Applying the hyperbolic counterpart of Algorithm\n\\ref{alg:cpbed_rational_trigonometric_curves} ]Cases (a) and (b) of Fig.\n\\ref{fig:ed_rational_hpc}\\ show the control point based exact description of\nthe rational hyperbolic arcs%\n\\begin{equation}\n\\mathbf{g}\\left( u\\right) =\\frac{1}{g^{3}\\left( u\\right) }\\left[\n\\begin{array}\n[c]{c}%\ng^{1}\\left( u\\right) \\\\\ng^{2}\\left( u\\right)\n\\end{array}\n\\right] =\\frac{1}{4+3\\cosh\\left( u-1\\right) +\\cosh\\left( 3u-3\\right)\n}\\left[\n\\begin{array}\n[c]{r}%\n4\\cosh\\left( 2u-2\\right) \\\\\n8\\sinh\\left( u-1\\right)\n\\end{array}\n\\right] ,~u\\in\\left[ 0,3.1\\right] \\label{rational_hyperbolic_curve_1}%\n\\end{equation}\nand%\n\\begin{equation}\n\\mathbf{g}\\left( u\\right) =\\frac{1}{g^{3}\\left( u\\right) }\\left[\n\\begin{array}\n[c]{c}%\ng^{1}\\left( u\\right) \\\\\n\\\\\ng^{2}\\left( u\\right)\n\\end{array}\n\\right] =\\frac{1}{11+4\\cosh\\left( 2u-\\frac{3}{2}\\right) +\\cosh\\left(\n4u-3\\right) }\\left[\n\\begin{array}\n[c]{c}%\n16\\cosh\\left( u-\\frac{3}{4}\\right) \\\\\n\\\\\n4\\sinh\\left( 2u-\\frac{3}{2}\\right)\n\\end{array}\n\\right] ,~u\\in\\left[ 0,2.5\\right] \\label{rational_hyperbolic_curve_2}%\n\\end{equation}\nrespectively. (Note that in both cases $\\lim\\limits_{u\\rightarrow\\pm\\infty\n}\\mathbf{g}\\left( u\\right) =\\mathbf{0}$.)\n\\end{example}\n\n\n\\begin{figure}\n[!h]\n\\begin{center}\n\\includegraphics[\nheight=3.2162in,\nwidth=6.1004in\n]%\n{ed_rational_hpc.pdf}%\n\\caption{Using rational hyperbolic curves of the type\n(\\ref{rational_hyperbolic_curve}), cases (\\emph{a}) and (\\emph{b}) illustrate\nthe control point based exact description of arcs\n(\\ref{rational_hyperbolic_curve_1}) and (\\ref{rational_hyperbolic_curve_2}),\nrespectively. Control polygons were determined by the hyperbolic counterpart\nof Algorithm \\ref{alg:cpbed_rational_trigonometric_curves}.}%\n\\label{fig:ed_rational_hpc}%\n\\end{center}\n\\end{figure}\n\n\n\\begin{example}\n[Applying the hyperbolic counterpart of Algorithm\n\\ref{alg:cpbed_rational_trigonometric_surfaces}]Using surfaces of the type\n(\\ref{rational_hyperbolic_surface}), Fig.\n\\ref{fig:rational_hyperbolic_butterfly} illustrates several control point\nconfigurations for the exact description of the rational hyperbolic surface\npatch%\n\\begin{equation}\n\\mathbf{s}\\left( u_{1},u_{2}\\right) =\\frac{1}{s^{4}\\left( u_{1}%\n,u_{2}\\right) }\\left[\n\\begin{array}\n[c]{c}%\ns^{1}\\left( u_{1},u_{2}\\right) \\\\\ns^{2}\\left( u_{1},u_{2}\\right) \\\\\ns^{3}\\left( u_{1},u_{2}\\right)\n\\end{array}\n\\right] ,~\\left( u_{1},u_{2}\\right) \\in\\left[ 0,6\\right] \\times\\left[\n0,10\\right] ,\\label{rational_hyperbolic_butterfly}%\n\\end{equation}\nwhere%\n\\begin{align*}\ns^{1}\\left( u_{1},u_{2}\\right) & =6\\left( \\cosh\\left( 2u_{1}-2\\right)\n+\\sinh\\left( u_{2}-5\\right) \\right) ,\\\\\ns^{2}\\left( u_{1},u_{2}\\right) & =\\frac{1}{10}\\sinh\\left( u_{1}-1\\right)\n\\cosh\\left( 2u_{2}-10\\right) ,\\\\\ns^{3}\\left( u_{1},u_{2}\\right) & =2\\left( \\sinh\\left( 2u_{1}-2\\right)\n+\\cosh\\left( 2u_{1}-2\\right) \\right) \\cosh\\left( u_{2}-5\\right) ,\\\\\ns^{4}\\left( u_{1},u_{2}\\right) & =275+100\\cosh(2u_{1}-2)+25\\cosh\n(4u_{1}-4).\n\\end{align*}\n\n\\end{example}\n\n\n\\begin{figure}\n[!h]\n\\begin{center}\n\\includegraphics[\nnatheight=5.475100in,\nnatwidth=6.298400in,\nheight=5.5019in,\nwidth=6.3261in\n]%\n{rational_hyperbolic_surface_merged+.pdf}%\n\\caption{Control point based exact description of the patch\n(\\ref{rational_hyperbolic_butterfly}) by means of rational hyperbolic surfaces\nof the type (\\ref{rational_hyperbolic_surface}). Control nets were obtained by\nthe hyperbolic counterpart of Algorithm\n\\ref{alg:cpbed_rational_trigonometric_surfaces} ($\\delta=2$, $\\kappa=1$;\n$\\alpha_{1}=6$, $\\alpha_{2}=10$; $m_1=2$, $m_2=m_3=m_4=1$).}%\n\\label{fig:rational_hyperbolic_butterfly}%\n\\end{center}\n\\end{figure}\n\n\n\n\\begin{example}\n[Hybrid counterpart of Algorithm\n\\ref{alg:cpbed_rational_trigonometric_surfaces} -- hybrid rational\nvolumes]Using multivariate rational tensor product surfaces specified by\nfunctions that are exclusively either trigonometric or hyperbolic in each of\ntheir variables, Fig. \\ref{fig:rational_hybrid_volume} shows the control point\nbased exact description of the $3$-dimensional $3$-variate rational surface\nelement (volume)%\n\\begin{equation}\n\\mathbf{s}\\left( \\mathbf{u}\\right) =\\frac{1}{s^{4}\\left( u_{1},u_{2}%\n,u_{3}\\right) }\\left[\n\\begin{array}\n[c]{c}%\ns^{1}\\left( u_{1},u_{2},u_{3}\\right) \\\\\ns^{2}\\left( u_{1},u_{2},u_{3}\\right) \\\\\ns^{3}\\left( u_{1},u_{2},u_{3}\\right)\n\\end{array}\n\\right] ,~\\left( u_{1},u_{2},u_{3}\\right) \\in\\left[ 0,2\\right]\n\\times\\left[ 0,\\frac{3\\pi}{4}\\right] \\times\\left[ 0,\\frac{\\pi}{2}\\right]\n\\label{rational_hybrid_volume}%\n\\end{equation}\nwhere functions%\n\\begin{align*}\ns^{1}\\left( u_{1},u_{2},u_{3}\\right) = & \\frac{5}{4}\\cosh\\left(\nu_{1}-1\\right) \\cos\\left( u_{2}\\right) \\left( \\frac{3}{2}+\\frac{3}{4}%\n\\sin\\left( u_{3}\\right) -\\frac{1}{2}\\cos\\left( u_{3}\\right) -\\frac{1}%\n{4}\\sin\\left( 3u_{3}\\right) \\right) ,\\\\\ns^{2}\\left( u_{1},u_{2},u_{3}\\right) = & \\cosh\\left( u_{1}-1\\right)\n\\sin\\left( u_{2}\\right) \\left( \\frac{5}{2}+\\frac{3}{4}\\sin\\left(\nu_{3}\\right) -\\frac{1}{4}\\sin\\left( 3u_{3}\\right) \\right) ,\\\\\ns^{3}\\left( u_{1},u_{2},u_{3}\\right) = & -\\frac{5}{4}\\sinh\\left(\nu_{1}-1\\right) \\left( \\frac{7}{4}+\\frac{1}{4}\\cos\\left( 2u_{3}\\right)\n\\right) ,\\\\\ns^{4}\\left( u_{1},u_{2},u_{3}\\right) = & 1-\\frac{3}{32}\\sqrt{2-\\sqrt{2}%\n}\\sin\\left( u_{3}\\right) -\\frac{3}{32}\\sqrt{2+\\sqrt{2}}\\cos\\left(\nu_{3}\\right)\n-\\frac{1}{32}\\sqrt{2+\\sqrt{2}}\\sin\\left( 3u_{3}\\right) -\\frac{1}{32}%\n\\sqrt{2-\\sqrt{2}}\\cos\\left( 3u_{3}\\right)\n\\end{align*}\nare hyperbolic or trigonometric in the first and the last two variables, respectively.\n\\end{example}\n\n\n\\begin{figure}\n[!h]\n\\begin{center}\n\\includegraphics[\nnatheight=2.910100in,\nnatwidth=6.298400in,\nheight=2.9378in,\nwidth=6.3261in\n]%\n{rational_hybrid_volume_merged+.pdf}%\n\\caption{Control point based exact description of the hybrid $3$-dimensional\nrational volume element (\\ref{rational_hybrid_volume}). The control grid was\ncalculated by using the hybrid counterpart of Algorithm\n\\ref{alg:cpbed_rational_trigonometric_surfaces} ($\\delta=3$, $\\kappa=0$;\n$\\alpha_{1}=2$, $\\alpha_{2}=\\frac{3\\pi}{4}$, $\\alpha_{3}=\\frac{\\pi}{2}$; $m_1=m_2=m_3=m_4=1$).}%\n\\label{fig:rational_hybrid_volume}%\n\\end{center}\n\\end{figure}\n\n\n\n\\section{Final remarks\\label{sec:final_remarks}}\n\nSubdivision algorithms of trigonometric and hyperbolic curves detailed in\nSection \\ref{sec:special_parametrizations} can also be easily extended to\nhigher dimensional multivariate (rational) trigonometric or hyperbolic\nsurfaces, respectively. Therefore, similarly to standard rational B\\'{e}zier\ncurves and surfaces that are present in the core of major CAD\/CAM systems, all\nsubdivision based important curve and surface design algorithms (like\nevaluation or intersection detection) can be both mathematically and\nprogrammatically treated in a unified way by means of normalized B-bases\n(\\ref{Sanchez_basis}) and (\\ref{Wang_basis}). Considering the large variety of\n(rational) curves and multivariate surfaces that can be exactly described by\nmeans of control points and the fact that classical (rational) B\\'{e}zier\ncurves and multivariate surfaces are special limiting cases of the\ncorresponding curve and surface modeling tools defined in Sections\n\\ref{sec:special_parametrizations} and \\ref{sec:multivariate_surfaces}, it is\nworthwhile to incorporate all proposed techniques and algorithms presented in\nSection \\ref{sec:exact_description} into CAD systems of our days.\n\n\\begin{acknowledgements}\n\\'{A}goston R\\'{o}th was supported by the European Union and the State of\nHungary, co-financed by the European Social Fund in the framework of\nT\\'{A}MOP-4.2.4.A\/2-11\/1-2012-0001 'National Excellence Program'. All assets,\ninfrastructure and personnel support of the research team led by the author\nwas contributed by the Romanian national grant CNCS-UEFISCDI\/PN-II-RU-TE-2011-3-0047.\n\\end{acknowledgements}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nIn the Cold Dark Matter (CDM) cosmogony, galactic stellar haloes are\nbuilt up in large part from the debris of tidally disrupted satellites\n(e.g. Searle \\& Zinn 1978; White \\& Springel 2000; Bullock \\& Johnston 2005; Cooper {et~al.} 2010). Discovering\nand quantifying halo structures around the Milky Way may provide a\nuseful diagnostic of the Galaxy's merger history\n(e.g. Helmi \\& de~Zeeuw 2000; Johnston {et~al.} 2008; G{\\'{o}}mez \\& Helmi 2010). Upcoming Milky Way\nsurveys (for example with PanSTARRS1, LAMOST, HERMES and the LSST)\nwill provide large datasets in which to search for structure. The\n\\textit{Gaia} mission will determine six-dimensional phase-space\ncoordinates for all stars brighter than $V\\sim17$, from which it\nshould be possible to untangle even well-mixed streams in the nearby\nhalo (G{\\'{o}}mez {et~al.} 2010).\n\nTesting the CDM model by comparing these observations to numerical\nsimulations of stellar halo formation requires a straightforward\ndefinition for the `abundance of substructure', one that can be\nquantified with a method equally applicable to simulations and\nobservations. Algorithms already exist for identifying substructure in\nhuge multidimensional datasets (e.g. Sharma \\& Johnston 2009), such as the\ndata expected from \\textit{Gaia} supplemented by chemical abundance\nmeasurements (Freeman \\& Bland-Hawthorn 2002). These algorithms can\nalso be applied to simulations, although this is not\nstraightforward. One problem is that current (cosmological)\nhydrodynamic simulations still fall short of the star-by-star\n`resolution' of the \\textit{Gaia} data, particularly in the Solar\nneighbourhood (e.g. Brown, Vel{\\'{a}}zquez \\& Aguilar 2005).\n\nIn the outer halo, longer mixing times allow ancient structures to\nremain coherent in configuration space for many gigayears. However, 6D\n\\textit{Gaia} data will be restricted to relatively bright stars. In\nthe near future, studies of the outer halo (beyond $\\sim20$~kpc) will\ncontinue to rely on a more modest number of `tracers' (giant and\nhorizontal branch stars). For these stars, typically only angular\npositions and (more uncertain) estimates of distance and radial\nvelocity are available. In this regime, current simulations contain as\nmany particles as there are (rare) tracer stars in observational\nsamples, and can be compared directly with data that are already\navailable. Here we focus on quantifying the degree of structure in\nrare tracer stars with a generic method, which we apply to\nobservational data and to simulations of Milky Way-like stellar\nhaloes.\n\nMost studies of spatial and kinematic structure in the Milky Way halo\nhave given priority to the discovery of individual overdensities\n(exceptions include Gould 2003, Bell {et~al.} 2008, Xue, Rix \\& Zhao 2009,\nXue {et~al.} 2011 and Helmi {et~al.} 2011a). Relatively few\nhave investigated global statistical quantities for the entire stellar\nhalo, although several authors have suggested an approach based on\nclustering statistics. Re~Fiorentin {et~al.} (2005) analysed the\nvelocity-space clustering of a small number of halo stars in the Solar\nneighbourhood, using a correlation function statistic. Following early\nwork by Doinidis \\& Beers (1989), Brown {et~al.} (2004) examined the angular\ntwo-point correlation function of photometrically selected blue\nhorizontal branch (BHB) stars in the Two Micron All Sky Survey,\nprobing from $\\sim2-9$~kpc. They detected no significant correlations\nat latitudes $|b|\\ga50\\degr$, but did detect correlations on small\nscales ($1\\degr$, $\\sim100$~pc) at lower latitudes, which they\nattributed to structure in the thick disc. Lemon {et~al.} (2004) performed a\nsimilar analysis for nearby F-type stars in the Millennium Galaxy\nCatalogue and found no significant clustering.\n\nStarkenburg {et~al.} (2009) used a correlation function in \\textit{four}\ndimensions to identify substructures in the Spaghetti pencil-beam\nsurvey of the distant halo (Morrison {et~al.} 2000; Dohm-Palmer {et~al.} 2000).\nWith this method they obtained a significant detection of clustering\nand set a lower limit on the number of halo stars in all\nsubstructures. Similarly, Schlaufman {et~al.} (2009) constrained the mass\nfraction of the halo in detectable substructure by estimating the\ncompleteness of their overdensity detection\nalgorithm. Starkenburg {et~al.} and Schlaufman {et~al.}\nconcluded that $>10\\%$ (by number of stars) and $\\sim30\\%$ (by volume)\nof the Milky Way halo belongs to groups meeting their respective\ndefinitions of phase space substructure. These methods were tested on\n`mock catalogues' of tracer stars based on simplified models of the\nstellar halo.\n\nThe work of Xue {et~al.} (2009, 2011) is particularly\n relevant. Xue {et~al.} (2009) considered the pairwise radial velocity\n separation of a sample of 2558 halo BHB stars as a function of their\n separation in space, but found no evidence of clustering. From\n comparisons to the simulations of Bullock \\& Johnston (2005),\n Xue {et~al.} concluded that a pairwise velocity statistic was\n not capable of detecting structure against a more smoothly\n distributed background in phase space (made up from stars in\n phase-mixed streams). However, their observed signal was not\n compared to an expected signal from random realizations. More\n recently, Xue {et~al.} (2011) studied an enlarged catalogue of\n BHBs comprising more than 4000 stars. They quantified clustering\n in this larger sample using a four-dimensional metric\n (similar to that of Starkenburg {et~al.} 2009), finding a significant\n excess of clustering on small scales by comparison to smooth\n models. The conclusions of this more recent study by\n Xue {et~al.} agree with our own, as we discuss further\n in \\mnsec{sec:conclusion}.\n\nThe statistic we develop in this paper also builds on the\n approach of Starkenburg {et~al.} (2009). We define a two-point correlation\n function based on a metric combining pairwise separations in four\n readily obtained phase space observables for halo stars (angular\n position, radial distance and radial velocity). We apply this\n statistic to the catalogue of BHB stars published by\n Xue {et~al.} (2008)\\footnote{The larger sample used by Xue {et~al.} (2011)\n was not publicly available at the time this paper went to press.}\n and demonstrate that a significant clustering signal can be\n recovered from these data.\n\nA clustering metric of the kind we propose can be tuned to probe a\nspecific scale in phase space by adjusting the weight given to each of\nits components. In the stellar halo, however, many `components' may be\nsuperimposed with a complex assortment of scales and morphologies in\nphase space (Johnston {et~al.} 2008; Cooper {et~al.} 2010). For this reason, it is not\nclear, a priori, what sort of signal to expect, or which scales are\nmost relevant. We find that we cannot identify an `optimal'\n metric. Instead, we make a fiducial choice based on the\nself-consistent accreted halo models of Cooper {et~al.} (2010). These\nincorporate an ab initio \\lcdm{} galaxy formation model using\nhigh-resolution cosmological N-body simulations from the Aquarius\nproject (Springel {et~al.} 2008). We apply our fiducial metric to the\ndata and to these models. We find that even though both the metric and\nour choice of scaling are simple, this approach has the power to\ndiscriminate quantitatively between qualitatively different stellar\nhaloes.\n\nWe describe the basis of our method in \\mnsec{sec:method} and the SDSS\nDR6 BHB catalogue of Xue {et~al.} (2008) in \\mnsec{sec:obsdata}. In\n\\mnsec{sec:sims} we describe our simulations\n(\\mnsec{sec:nbodygalform}) and our procedure for constructing mock\ncatalogues (\\mnsec{sec:tracer_stars}). \\mnsec{sec:fiducial} describes\nour choice of a fiducial metric. In \\mnsec{sec:segue} we apply our\nmethod to quantify clustering in the SDSS data and compare this with\nour mock catalogues. Our conclusions are given in\n\\mnsec{sec:conclusion}. \n\n\n\\section{A metric for phase-space distance}\n\\label{sec:method}\n\nThe most readily obtained phase-space observables for halo stars are\ntheir Galactic angular coordinates, $l$ and $b$, heliocentric radial\ndistance, $r_{\\mathrm{hel}}$, and heliocentric line-of-sight velocity,\n$v_{\\mathrm{hel}}$. Using its angular position and distance estimate,\neach star can be assigned a three-dimensional position vector in\ngalactocentric Cartesian coordinates, $\\bmath{r}\\,(X,Y,Z)$, and a\nradial velocity corrected for the Solar and Local Standard of Rest\nmotions, $v$. We begin by defining a scaling relation (metric),\n$\\Delta$, which combines these observables into a simple `phase-space\nseparation' between two stars:\n\n\\begin{equation}\n{\\Delta}_{ij}^2=\n{|\\bmath{r_i}-\\bmath{r_j}|}^2+w_{v}^2({v_{i}-v_{j}})^2.\n\\label{eqn:delta_metric}\n\\end{equation}\n\nHere, $|\\bmath{r_i}-\\bmath{r_j}|$ is the separation of a pair of stars\nin coordinate space (in kiloparsecs), and $v_{i}-v_{j}$ is the\ndifference in their radial velocities (in kilometres per second). The\nscaling factor $w_{v}$ has units of $\\mathrm{kpc\\,km^{{-}1}\\,s}$, such\nthat $\\Delta$ has units of kpc. The choice of $w_{v}$ is arbitrary\nunless a particular `phase space scale' of interest can be\nidentified. This is not straightforward; we discuss some possible\nchoices below.\n \nThe aim of this paper is to explore $\\xi({\\Delta})$, the cumulative\ntwo-point correlation function of halo stars in the metric defined by\nEquation \\ref{eqn:delta_metric}. Throughout, we use the\nestimator \\begin{equation} 1+\\xi({\\Delta}) =\n \\frac{DD(<{\\Delta})}{\\left \\langle RR(<{\\Delta}) \\right\n \\rangle}. \\label{eqn:estimator}\n\\end{equation} Here $DD(<\\Delta)$ counts the number of pairs in the\nsample separated by less than $\\Delta$, and $\\left \\langle RR(<\\Delta)\n\\right \\rangle$ is the equivalent count for pairs of random points\nwithin the survey volume, averaged over several realizations. To\n generate these realizations we `shuffle' the data by randomly\n reassigning $(r_{\\mathrm{hel}},v_{\\mathrm{hel}})$ pairs to different\n $(l,b)$ coordinates drawn from the original catalogue\\footnote{The\n same randomisation procedure was used by\n Starkenburg {et~al.} (2009). Xue {et~al.} (2011) use a similar\n procedure, but for each galaxy they re-assign $r_{\\mathrm{hel}}$\n and $v_{\\mathrm{hel}}$ \\textit{separately} to different $(l,b)$\n coordinates.}.\n\n\nSimilar methods for quantifying the clustering of stars in a\n four-dimensional space are described by Starkenburg {et~al.} (2009, applied to a sample\n of giant stars from the Spaghetti survey) and\n Xue {et~al.} (2011, applied to a sample of BHB stars from\n SDSS). Our $\\Delta$ metric is very similar to the\n $\\delta_{4\\mathrm{d}}$ metric of Starkenburg et al. in the limit of\n small angular separations\\footnote{Starkenburg et al. developed\n their metric with the aim of identifying `true' pairs of stars\n with high confidence. In their definition (equation 1 of\n Starkenburg {et~al.} 2009), the $\\delta_{4\\mathrm{d}}$ separation\n between two stars depends not only on their actual phase-space\n coordinates, but also on how accurately those coordinates are\n measured. For example, moving two stars 10 kpc further apart and\n simultaneously decreasing the error on their distance measurements\n by a factor of 10 (relative to the average of the sample) results\n in the same $\\delta_{4\\mathrm{d}}$. Thus $DD\/RR$ for separations\n in $\\delta_{4\\mathrm{d}}$ is not a straightforward measurement of\n physical clustering.}. We have verified that our metric gives\n similar results when we repeat the analysis of Starkenburg et\n al. using the Spaghetti sample of 101 halo RGB stars. For the rest\n of this paper we will focus on clustering in the SDSS DR6 BHB\n catalogue of Xue {et~al.} (2008).\n\n\n\\section{Observational data}\n\\label{sec:obsdata}\n\nXue {et~al.} (2008) have published a catalogue of 2558 stars from SDSS DR6,\nwhich they identify as high-confidence halo BHBs (contamination\n$<10\\%$) using a combination of colour cuts and Balmer line\ndiagnostics. This sample ranges in distance from $4-60\\,\\mathrm{kpc}$;\na cut on distance error excluded more distant stars observed in\nSDSS. Xue {et~al.} (2008) claim distance errors of $\\sim10\\%$ and radial\nvelocity errors of $5-20\\,\\mathrm{km\\,s^{-1}}$. This catalogue is not\na complete sample of halo BHBs. In particular, Xue {et~al.} note\nthat the targeting of SDSS spectroscopy is biased against the\nfollow-up of more distant stars.\n\nTo isolate stars representing the kinematics of the halo (in order to\nstudy the Galactic circular velocity profile) Xue {et~al.} (2008)\nfurther restricted their sample to stars at heights\n$|z|>4\\,\\mathrm{kpc}$ above the plane (avoiding the thick disc). We\nalso apply this cut (which excludes thick disc stars and low-latitude\nhalo stars alike), leaving 2401 stars in the sample. Finally we\nexclude 9 stars in the sample identified by Xue {et~al.} (2009) as probable\nglobular cluster members. Thus, the sample against which we compare our\nmodels contains 2392 stars from the 2558 stars in the Xue {et~al.} (2008)\ncatalogue. The effects of these refinements to the sample on the\nrecovered \\deltacf{} signal are discussed in\n\\mnsec{sec:obsdata_clustering}.\n\n\\section{Stellar Halo Simulations}\n\\label{sec:sims}\n\n\\subsection{N-body and galaxy formation model}\n\\label{sec:nbodygalform}\n\nThe mock observations that we use to test the \\deltacf{} correlation\nfunction are derived from simulations of the accreted stellar halo\npresented in Cooper {et~al.} (2010). These simulations approximate the\ndynamics of stars in dwarf satellites of Milky Way-like galaxies by\n`tagging' appropriate particles (i.e. those strongly bound within\nsubhaloes) in the Aquarius suite of high-resolution N-body simulations\n(Springel {et~al.} 2008). Each `tag' associates a dark matter (DM)\nparticle with a particular stellar population of a given mass, age\nand metallicity. This `tagging' technique is reasonable in the regime\nof high mass-to-light ratios, which is supported in this case by\nobservations of stellar kinematics in dwarf galaxies\n(e.g. Walker {et~al.} 2009).\n\nThe tagging method has a single free parameter, the fraction of the\nmost-bound particles chosen in each DM halo for each assignment of\nnewly-formed stars (for further details see Cooper {et~al.} 2010). The\nvalue of this parameter (1 per cent) was fixed by requiring the\npopulation of \\textit{surviving} satellites (at the present day) to\nhave a distribution of half-light radius as a function of luminosity\nconsistent with Milky Way and M31 observations\\footnote{The luminosity\n function of surviving satellites in these models also agrees with MW\n and M31 data. This agreement is mostly due to the underlying galaxy\n formation model.}. The Cooper {et~al.} models differ from the\nearlier models of Bullock \\& Johnston (2005) in that they treat the full\ncosmological evolution of all satellites self-consistently in a single\nN-body simulation, and use a comprehensive semi-analytic model of\ngalaxy formation (Bower {et~al.} 2006) constrained by data on large scales\nand compatible with the observed MW satellite luminosity\nfunction. Both the Cooper {et~al.} and the\nBullock \\& Johnston simulations produce highly structured stellar\nhaloes built from the debris of disrupted dwarf galaxies. As we\ndiscuss further below, other halo components formed {\\it in situ} may\nbe present in real galaxies\n(e.g. Abadi, Navarro \\& Steinmetz 2006; Zolotov {et~al.} 2009; Font {et~al.} 2011) but these are expected\nto be more smoothly distributed than the accreted component\n(e.g. Helmi {et~al.} 2011a).\n\n \\begin{figure*}\n\\includegraphics[height=60mmi,clip=True,trim=1cm 0cm 1cm 0 ]{fig1a.pdf}\n\\includegraphics[height=60mm,clip=True]{fig1b.pdf}\n\\includegraphics[height=60mm,clip=True,trim=1cm 0cm 1cm 0 ]{fig1c.pdf}\n\\includegraphics[height=60mm,clip=True, trim=0 0 0 0 ]{fig1d.pdf}\n\n\\caption{\\textit{Left panels:} An example of the sky distribution of\n halo BHB stars in Aq-A from the perspective of a `Solar' observer,\n shown as Mollweide projections in galactic coordinates centred on\n $(l,b)=(0,0)$. Colours indicate the mean heliocentric distance of\n stars in each pixel, in kiloparsecs. Pixels outside the SDSS DR6\n footprint are shown with lower contrast. The upper panel includes\n all BHB stars from $6$--$60$~kpc, the lower panel includes only\n stars in the range $30$--$60$~kpc. Our fiducial choice of the\n Galactic $Z$ axis with respect to the dark halo has been applied,\n but distances in these panels are not convolved with observational\n errors. The normalization of BHB stars per unit stellar mass in the\n halo has been increased in these panels to emphasise the\n distribution of structure. \\textit{Right panels:} Blue histograms\n show the distribution of heliocentric distances (above) and\n heliocentric velocities (below) for the fiducial observer\n corresponding to the left panels, after convolution with\n observational errors (see text). Orange histograms are the\n distributions for stars in the Xue {et~al.} (2008) catalogue. To compare\n the shape of the two distributions, the normalization of the mock\n distributions in these panels has been matched to that of the\n observations for $r<20$~kpc, where the observations are most likely\n to be complete.}\n\\label{fig:fiducial}\n \\end{figure*}\n\nAs in Springel {et~al.} (2008) and Cooper {et~al.} (2010), we refer to our\nsix simulations as haloes \nAq-A, Aq-B, Aq-C, Aq-D, Aq-E and Aq-F. From these simulations, we\nconstruct catalogues of tracer stars (BHB stars, for example) by\nconverting the stellar mass assigned to each dark matter particle into\nan appropriate number of stars.\n\nEach DM particle can give rise to many tracer stars if it is tagged\nwith sufficient stellar mass. We therefore interpolate the positions\nand velocities of these tracer stars between nearby tagged DM\nparticles. To accomplish this, the 32 nearest phase space neighbours\nof each tagged particle are identified using a procedure which we\ndescribe below. The mean dispersion in each of the six phase-space\ncoordinates is then calculated for each particle as an average over\nits neighbours. These dispersions define a 6D ellipsoidal Gaussian\nkernel centred on the particle, from which the positions and\nvelocities of its tracer stars are drawn randomly. Each progenitor\ngalaxy (a set of tagged DM particles accreted by the main `Milky Way'\nhalo as members of a single subhalo) is treated individually in this\nsmoothing operation, i.e. particles are smoothed using only neighbours\nfrom the same progenitor (so there is no `cross talk' between streams\nfrom different progenitors). This procedure can be thought of as a\ncrude approximation to running our original simulation again including\neach tracer star as a test particle.\n\n\n\nThe `distance in phase space' used to identify neighbours in the\ninterpolation scheme is defined by a scaling relation between\ndistances in configuration space and velocity space\\footnote{In this\n part of the calculation, we are only interested in finding\n neighbours, so the absolute values of these distances are not\n important. This scaling of velocity space to configuration space for\n the purpose of resampling the simulations should not be confused\n with the $\\Delta$ metric we define for our analysis of\n clustering.}. For each progenitor, we adopt an individual scaling\nwhich corresponds to making the median pairwise interparticle\nseparation of its particles in configuration space (at $z=0$) equal to\ntheir median separation in velocity space. In practice, the value of\nthis scaling parameter makes very little difference to the results we\npresent, when compared to the extreme choice of selecting only 32\nvelocity or position neighbours (disregarding the other three\ncoordinates in each case). Giving more weight to configuration-space\nneighbours smears out velocity substructure within the debris of a\nprogenitor (for example, where two wraps of a stream pass near one\nanother). Giving more weight to velocity neighbours has the opposite\neffect -- stars can be interpolated over arbitrarily large separations\nin configuration space, but coherent velocity structures are\npreserved. Therefore, the `optimal' choice is the scaling which\nbalances smoothing in configuration space against smoothing in\nvelocity space.\n\nTo quantify this balance between smoothing in configuration and\nvelocity space, we compute six smoothing lengths for each particle,\n$\\epsilon_{x,i}$ and $\\epsilon_{v,i}$, where $i$ represents a single\ndimension in space or velocity. To compute these dispersions,\n we use the 32 nearest phase-space neighbours of each particle. We\ndefine the `optimum' choice of scaling for \\textit{each} progenitor\ngalaxy as that which minimises the quantity\n\n\\begin{equation}\n\\sigma_{\\epsilon}^{2} = \\left (\\frac{1}{\\bar{\\epsilon}_{x,\\mathrm{min}}} \\sum_{i=0}^{3}{\\epsilon_{x,i}} \\right )^{2} + \\left ( \\frac{1}{\\bar{\\epsilon}_{v,\\mathrm{min}}} \\sum_{i=0}^{3}{\\epsilon_{v,i}}\\right )^{2}.\n\\end{equation} This is the sum in quadrature of the mean smoothing lengths in\nconfiguration and velocity space, normalized respectively by\n$\\bar{\\epsilon}_{x,\\mathrm{min}}$, the `minimal' mean smoothing length\nin configuration space (obtained from the 32 nearest configuration space\nneighbours) and $\\bar{\\epsilon}_{v,\\mathrm{min}}$, the `minimal' mean\nsmoothing length in velocity space (obtained from the 32 nearest\nvelocity space neighbours). We find that the scaling obtained by\nmatching the median interparticle separations in position and velocity\nas described above is typically a good approximation to this optimal\nvalue -- a similar result is discussed in more detail by\nMaciejewski {et~al.} (2009).\n\n\nIn the Cooper {et~al.} model, when a given stellar population is\nformed, the most bound 1\\% of DM particles in the corresponding dark\nhalo at that time are chosen as dynamical tracers of that\npopulation. Hence, each DM \\textit{particle} to which stars are\nassigned has an individual mass-to-light ratio, M\/L, which can be as\nhigh as $\\sim1$ (i.e. $M_{\\mathrm{stellar}}\\sim10^{4}\\,M_{\\sun}$) and\nas low as $\\sim10^{-6}$. This will affect the density of stars seeded\nby a DM particle independently of the density of its neighbours in\nphase space (i.e. a low M\/L particle will create a denser `cluster' of\ntracers relative to a high M\/L particle with the same neighbouring\npositions and velocities). We have tested an alternative approach in\nwhich the M\/L of each particle in a given progenitor is resampled by\ndistributing the total stellar mass of the progenitor evenly amongst\nits tagged particles\\footnote{This is almost equivalent to choosing\n M\/L only once, at the time in the simulation when the progenitor\n falls into the main halo (similar to the lower-resolution model of\n De~Lucia \\& Helmi 2008).}. We find that the extra clustering due to a\nfew `hot' particles in our default approach makes no difference to our\nresults. \n\n\n\n\\subsection{Tracer stars and mock observations}\n\\label{sec:tracer_stars}\n\nEach N-body dark matter particle in our simulation contributes a\nnumber of tracer stars to our mock observations, based on the stellar\npopulation with which it has been `tagged'. In the specific case of\nour comparisons to SDSS, these tracers are BHB stars meeting the selection\ncriteria of Xue {et~al.} (2008). Here we assume a global scaling between the\nstellar mass associated with each N-body particle, $M_{\\star}$, and\nthe number of BHBs it contributes to our mock catalogues, i.e.\n$N_{\\mathrm{BHB}} = f_{\\mathrm{BHB}}M_{\\star}$ where\n$f_{\\mathrm{BHB}}$ is the number of tracer stars per unit mass of the\noriginal stellar population\\footnote{We do this because we prefer to\n make a straightforward comparison with the observational data in\n this paper. In principle, the age and metallicity information\n associated with each stellar population in our model could be used\n to populate an individual colour-magnitude diagram for each N-body\n particle, and make a detailed prediction for the appropriate number\n of tracers. The `bias' of BHBs relative to the total stellar mass\n distribution of the halo (Bell {et~al.} 2010) may affect the\n clustering statistic recovered (Xue {et~al.} 2011), but this effect\n is beyond the scope of the present paper. }. For each N-body\nparticle, the actual number of BHB stars generated is drawn from a\nPoisson distribution with mean $N_{\\mathrm{BHB}}$. The correlation\nfunction results described below are not sensitive to the choice of\n$f_{\\mathrm{BHB}}$, provided that the underlying distribution is\nwell-sampled at a given scale. We have therefore selected a fiducial\nvalue of $f_{\\mathrm{BHB}}^{-1} = 3000\\,M_{\\sun}\/\\mathrm{star}$. In\ncreating the mock catalogue, we do not include any stars\ngravitationally bound to satellites. However, we do include stars in\ntheir tidal tails (which, by our definition, are part of the stellar\nhalo).\n\nUsing our simulated BHB catalogues, we create mock observations for\ncomparison to the Xue {et~al.} (2008) data as follows. First we located the\nobserver at a radius $r_{\\sun}=8\\,\\mathrm{kpc}$ from the centre of the\nhalo. For our main comparison to the data, we restrict all observers\nto the same `Galactic plane', with each random vantage point differing\nonly in its azimuthal angle in this plane and in the `polarity' of the\nGalactic rotation axis (the $Z$ coordinate). However, the orientation\nof the rotation axis cannot be directly constrained by the simulation,\nwhich only models the \\textit{accreted} component of the halo and the\nbulge, and not the in situ formation of a stellar disc. As described\nin Cooper {et~al.} (2010), the accreted `bulge' is prolate or mildly\ntriaxial. We define the minor axis of this bulge component\n(conservatively defined by all accreted stars within $r<3$~kpc;\nCooper {et~al.} 2010) as the Galactic $Z$ axis. This axis is essentially\nidentical to the minor axis of the dark halo within $r<3$~kpc. There\nare other plausible choices of Galactic plane (for example, relative\nto the shape or spin vectors of the entire dark halo, rather than the\nstars in its inner regions). However, any choice is somewhat arbitrary\nwithout a self-consistent simulation of disc formation\\footnote{In a\n full hydrodynamic simulation the effects of feedback and adiabatic\n contraction may also make the dark halo itself more spherical\n (e.g. Tissera {et~al.} 2010; Abadi {et~al.} 2010).}.\n\nHaving chosen a location for the observer, we select all tracer stars\nwithin the spectroscopic footprint of SDSS DR6 having galactocentric\ndistance in the range 20--60~kpc (our principal comparison will focus\non the outer halo as defined by this distance range, although we also\nstudy the ranges 5--60~kpc and 5--20~kpc below). Galactic longitude\nand latitude are defined such that $(l,b)=(0,0)$ is the vector\ndirected from the observer to the centre of the halo. We set the\nheliocentric velocity components of each star assuming a Solar motion\nof $U,V,W = (10.0,5.2,7.2)\\:\\mathrm{km\\,s^{{-}1}}$ (Dehnen \\& Binney 1998) and\na velocity of the Local Standard of Rest about the Galactic centre\n$v_{\\mathrm{LSR}}=220\\mathrm{\\:km\\,s^{{-}1}}$. We compute $(X,Y,Z)$\nand $v_{\\mathrm{los}}$ as described by Xue {et~al.} (2008). Finally,\ndistances and velocities are convolved with Gaussian observational\nerrors of $\\sigma_{d}= 10\\%$ and $\\sigma_{v}=15\\;\\mathrm{km\\,s^{-1}}$\nrespectively (Xue {et~al.} 2008).\n\nIn both the mock observations and the real data, the average random\npair count $\\langle RR \\rangle$ is calculated by reshuffling distances\nand velocities among the positions on the sky of stars many times. We find that\nby using 500 random catalogues to calculate $\\langle RR \\rangle$ for\neach mock observation and performing 500 mock observations in each\nhalo, we obtain a sufficiently well-converged estimate of the\ndistribution of \\deltacf{}.\n\n\\fig{fig:fiducial} illustrates the structure of one of our haloes and\nverifies that our mock observations can result in distributions of\nheliocentric distance and heliocentric radial velocity similar to the\nSDSS data of Xue {et~al.} (2008). In this figure we have specifically chosen\nan observer orientation in halo Aq-A such that the distributions of\ndistance and velocity we recover are close to those of the data, after\nconvolution with typical observational errors. We have aligned\nthe Galactic $Z$ axis of the mock observer with the minor axis of \nthe dark halo, as described above. This confines the most prominent\nstructures in the stellar halo to low Galactic latitudes, outside the\nSDSS DR6 spectroscopic footprint. Of course, the simulated haloes are\ninhomogeneous on large scales, and there are many choices of observer\nin each halo that \\textit{do not} resemble the SDSS data\\footnote{As\n discussed by Xue {et~al.} 2008, the completeness of the data declines\n at larger distances ($r_{\\sun}>20$~kpc). Mock catalogue distributions\n that match the observed distributions well at $r_{\\sun}<20$~kpc\n typically show a flatter profile with identifiable overdensities\n (streams) at larger distances. As the SDSS spectroscopic selection\n function for the data we use is difficult to quantify (Xue {et~al.} 2008),\n we do not explore the effects of incompleteness in this paper.}.\n\n\n\\section{Distance - velocity scaling in the $\\Delta$ metric}\n\\label{sec:fiducial}\n\n\\begin{figure}\n\\includegraphics[width=84mm,clip=True]{fig2.pdf}\n\\caption{Correlation functions in spatial separation (blue) and velocity\n separation (red) for stars in halo Aq-A. The velocity separation\n correlation function has been scaled by a factor\n $w_{v}=0.04\\,\\wvunits$ to match the turnover in the configuration\n space separation correlation function.}\n\\label{fig:accretedscale} \n\\end{figure}\n\nBefore \\deltacf{} can be computed, a value must be chosen for the\n velocity-to-distance scaling $w_{v}$ in Equation\n \\ref{eqn:delta_metric}. There is no clearly well-motivated way to\n choose this value, and in the absence of a physical justification,\nit must be treated as a free parameter. The choice of $w_{v}$\ndetermines the scale of substructure to which the correlation function\nis most sensitive. Naively, we expect this to be the typical width and\ntransverse velocity dispersion of a `stream'. It is preferable to fix\nthis parameter in a universal manner that does not depend on the\n details of a particular survey. We make a fiducial choice of\n$w_{v}$ as follows.\n\nIn each simulated halo we adopt the SDSS-like survey configuration\ndescribed in \\mnsec{sec:tracer_stars} (without observational errors or\nassumptions about the orientation of the Galactic plane). We\nconstruct one dimensional distributions of the separation in radial\ndistance and velocity between stars. We generate many random\nrealizations of these distributions by first convolving each simulated\nstar with Gaussian smoothing kernels of width $8\\,\\kpc$ (distance) and\n$80\\,\\kms$ (velocity), and then drawing randomly from these `smoothed'\ndistributions. The smoothing scales were chosen as a compromise\nbetween signal (diminished by oversmoothing) and noise (increased by\nundersmoothing). Using these random realizations we construct\none-dimensional correlation functions for each distribution. These two\ncorrelation functions are shown for halo Aq-A in\n\\fig{fig:accretedscale}. Although the signals are intrinsically weak,\nthey have a very similar shape for both distributions, each with a\ncharacteristic `turnover' scale. Matching this scale in the two\ncorrelation functions corresponds to $w_{v}\\sim0.04\\pm0.01\\,\\wvunits$\nfor the six haloes, which we adopt as a fiducial value. We caution\nthat although the scales on which we match the one-dimensional\ncorrelation functions are somewhat smaller than the smoothing scales\nwe adopt to create the random distributions, this does not guarantee\nthat our choice of $w_{v}$ is unaffected by our choice of smoothing.\n\nClearly, there are other ways of fixing $w_{v}$. In practice, however,\nour conclusions are not highly sensitive to the value of this\nparameter. Values of the order of $w_{v}\\sim0.01$--$1.0\\,\\wvunits$\nresult in very similar $\\xi({\\Delta})$ correlation functions. Values\nlower than $0.01\\,\\wvunits$ recover very little signal. Values above\n$1\\,\\wvunits$ treat $1\\,\\kms$ velocity differences as equivalent to\n$>1\\,\\kpc$ separations in space, and so make the cumulative\ncorrelation function very noisy on small scales for only a marginal\nincrease in the overall signal. (This noise, in turn, increases the\nscatter between signals measured by different observers.) We find that\nour choice of $w_{v}\\sim0.04\\,\\wvunits$ is a reasonable\ncompromise. Our method for choosing $w_{v}$ can be compared with that\nof Starkenburg {et~al.} (2009), who determine the equivalent of $w_{v}$ in\ntheir metric to be the ratio of the Spaghetti survey limits in radial\ndistance and velocity ($0.26\\,\\wvunits$). Either value is acceptable\nto illustrate our approach and compare to simulations. We therefore\nadopt $w_{v}\\sim0.04\\,\\wvunits$.\n\n\\section{Clustering of SDSS BHB stars}\n\\label{sec:segue}\n\n\n\\subsection{Clustering in the Xue et al. sample}\n\\label{sec:obsdata_clustering}\n\n\\begin{figure}\n\\includegraphics[width=84mm,trim= 0.0cm 0cm 0cm 0.0cm,\n clip=True]{fig3.pdf}\n\\caption{\\deltacf{} correlation function for the SDSS BHB sample of\n Xue {et~al.} (2008). Black points (with Poisson error bars) show\n \\deltacf{} computed for all stars at galactocentric distances\n greater than 20~kpc. Grey points show the result for all stars in\n the main sample (galactocentric distances of 4--60~kpc). The blue\n squares include 9 stars suspected to belong to globular clusters,\n while red circles include stars at low Galactic $|z|$ heights\n (possible thick disc stars). Neither of these contributions are\n relevant for the more distant selection shown by the black points.}\n\\label{fig:sdss_real} \n\\end{figure}\n\n\n\\fig{fig:sdss_real} shows \\deltacf{} computed for 2392 BHB stars in\nthe Xue {et~al.} (2008) sample (\\mnsec{sec:obsdata}; grey points). Stars at\nsmall separations in the metric of \\eqn{eqn:delta_metric}\n($\\Delta<4\\,\\mathrm{kpc}$) show significant clustering. The amplitude\nof the signal increases if we restrict the sample to larger\ngalactocentric distances, $r>20$~kpc (black points). At larger\ndistances substructure is expected to be dynamically young and to have\nundergone less phase mixing. Our finding of stronger clustering for\nmore distant halo stars is in qualitative agreement with the results\nof Xue {et~al.} (2011).\n\nAlthough we appear to recover a significant clustering signal in\n\\fig{fig:sdss_real}, we have only one SDSS survey. The observed signal\nmay be an artifact of the particular structures covered by the SDSS\nfootprint. Other parts of the halo may be smoother or more structured,\nor may appear so when viewed from different points around the Solar\ncircle. We will address this issue of sample variance in the following\nsection using our mock catalogues.\n\nWe show two further permutations of the Xue {et~al.} (2008) sample in\n\\fig{fig:sdss_real}. The first of these (red open circles) includes\nstars close to the Galactic plane, $|Z|<4\\,\\mathrm{kpc}$. These were\nexcluded from the main sample of Xue {et~al.} (2008) to excise the thick\ndisc.\n\nAlthough only $\\sim150$ stars are excluded by the cut on\n $|Z|$, they make a substantial difference to the correlation\n function, suppressing the clustering signal on scales below\n $\\Delta<\\sim8\\,\\mathrm{kpc}$. In the SDSS data the majority of\n low-$|Z|$ stars are at small heliocentric radii. These stars\n constitute a foreground `screen' with a relatively smooth\n distribution, which may dilute the signal of correlated stars.\n\nThe final sample shown in \\fig{fig:sdss_real} (blue open squares)\nincludes all stars from the main sample (grey points) and a further\nnine BHB stars identified as globular cluster members by\nXue {et~al.} (2009). Two of these are from one cluster, and seven from\nanother. Including these stars marginally increases the clustering\nsignal in the smallest-separation bin. This shows that the technique\nis sensitive to the clustering of stars on these scales, which\ncorrespond to separations comparable to the distance and velocity\nerrors of the data.\n\n\\subsection{Comparison with Mock Catalogues}\n\\label{sec:mock_clustering}\n\n\\begin{figure}\n\\includegraphics[width=84mm, clip=True]{fig4.pdf}\n\\caption{\\deltacf{} for halo stars of galactocentric distances\n $2020$~kpc is dominated by a single radial stream\n(see figures 6 and 7 of Cooper {et~al.} 2010).\n\nWe find that two haloes, Aq-E and Aq-F (red), are consistent with the\nobserved \\deltacf{} on all scales. The structure of Aq-F is atypical\nfor the sample -- most of its stars are accreted in a late 3:1 merger\nand its surface brightness at the Solar radius is substantially higher\nthan current estimates for the Milky Way halo. In projection, Aq-F\nresembles the `shell'-dominated haloes observed in a number of nearby\nelliptical galaxies. Meanwhile haloes Aq-A (black), Aq-B (cyan) and\nAq-D (green) are marginally inconsistent with the data: below\n$\\Delta\\sim4$~kpc, $\\sim90$ per cent of mock observations in these\nhaloes imply a greater degree of clustering than we find for the Milky\nWay, particularly on small scales. Aq-C (purple) is entirely\ninconsistent with the Milky Way observations on all scales, showing a\nmuch higher degree of clustering. Beyond $20$~kpc, the sky of an\nobserver in Aq-C is dominated by two bright tidal streams on wide\n($\\sim100$~kpc) orbits. Although their orbital planes are\napproximately coincident with our definition of the Galactic plane,\nnevertheless sections of these streams intrude on the SDSS footprint\nat low Galactic latitudes.\n\n\\begin{figure*}\n\\includegraphics[height=65mm,clip=True]{fig5a.pdf}\n\\includegraphics[height=65mm,trim=1.3cm 0 0 0,clip=True]{fig5b.pdf}\n\\includegraphics[height=65mm,trim=1.3cm 0 0 0,clip=True]{fig5c.pdf}\n\\caption{\\deltacf{} for mock observations, as\n \\fig{fig:sdss_mock_pa}. From left to right, we vary our modelling\n assumptions as follows: (a) no restriction on the alignment of the\n Galactic plane with respect to the dark halo (the observer is\n located randomly on a sphere of radius $r_{\\sun}=8$~kpc) and no\nconvolution of the data with the expected observational errors; (b) the\n observer is restricted to the Galactic plane as in\n \\fig{fig:sdss_mock_pa}, but the mock observations are \\textit{not}\n convolved with expected observational errors; (c) as panel (a), but\n mock observations \\textit{are} convolved with errors. Colours are as\n \\fig{fig:sdss_mock_pa}}\n\\label{fig:sdss_mock_no} \n\\end{figure*}\n\nThe DR6 footprint and the cut on extra-planar height in the\nXue {et~al.} (2008) sample exclude stars near the Galactic plane from our\nclustering analysis. \\fig{fig:sdss_mock_no} illustrates how our\ndefinition of the Galactic plane influences the halo clustering\nsignal. In panel (a) the orientation of the Galactic plane with\nrespect to the halo is chosen randomly for each of the 500 mock\nobservers (i.e. observers are distributed over a sphere of radius\n$r_{\\sun}=8$~kpc), whereas in panel (b) the galactic Z direction\nis aligned with the minor axis of the halo for all observers as in\n\\fig{fig:sdss_mock_pa}. To focus on the effects of this alignment, the\ndistances and velocities of stars in these two panels have\n\\textit{not} been convolved with observational errors.\n\nThe systematically higher clustering signals in panel (b) of\n\\fig{fig:sdss_mock_no} suggest that the plane perpendicular to the\nminor axis of the dark matter halo is special. In Cooper {et~al.} (2010) and\nabove, we have noted the strong correlation between the shape of the\ndark halo and the inner regions of the stellar halo. This alignment of\nhalo structure also extends, more loosely, to other prominent stellar\nhalo structures at large distances. An overall flattening of the\nstellar halo arises because our dark matter haloes are embedded in\nlong-lived filaments of the cosmic web. Typically one or two such\nfilaments dominate the infall directions of both satellite galaxies\nand smoothly accreted dark matter, which also contributes to the shape\nof the dark halo\n(e.g. Libeskind {et~al.} 2005; Lovell {et~al.} 2011; Wang {et~al.} 2011; Vera-Ciro {et~al.} 2011). The\ndistribution of stars stripped from infalling satellites echoes the\nlarge-scale correlation of their orbital planes.\n\nBecause of this flattened global structure, the distribution of halo\nstars in our choice of Galactic plane tend to be more smoothly\ndistributed (i.e. this plane contains more diffuse phase-mixed\nmaterial as well as coherent substructure). Panel (b) demonstrates how\nthe `contrast' of small scale substructure in the outer halo is\nenhanced when these smoother components are excluded from the\nclustering analysis (through a combination of the SDSS footprint and\nthe cut on $|Z|$). This is particularly true in the case of Aq-F,\nwhere the majority of the mass in the halo is contributed by one\nextensive and relatively `smooth' component. By contrast, in Aq-C the\naverage clustering amplitude \\textit{decreases} on large scales when\nwe fix the Galactic plane. As noted above, in this case the massive\ncoherent streams that dominate the clustering signal of this halo\nmostly fall outside the SDSS footprint.\n\nPanel (c) of \\fig{fig:sdss_mock_no} shows the randomly aligned\nobservations of panel (a) convolved with observational errors in\ndistance and velocity. These errors `smooth out' the halo, suppress\nthe clustering signal overall and increase the variance between\nobservers on small scales. Again the effect is most pronounced for\nAq-F, where blurring of the dominant smooth component further\ndecreases the contrast of substructure. In most cases these two\neffects (alignment and observational errors) counteract each other to\nproduce the distribution of signals shown in\n\\fig{fig:sdss_mock_pa}. In the case of Aq-E the signal suffers\ndisproportionately from errors in the aligned configuration, perhaps\nbecause this signal is due to a small number of pairs at large\ndistances.\n\nFinally, in \\fig{fig:sdss_mock_distance} we examine differences\nbetween nearby and more distant halo stars (also discussed by\nXue {et~al.} 2011). The left-hand panel corresponds to the full\nrange of the Xue {et~al.} (2008) sample ($520$~kpc. This finding of\nstronger phase space correlations between stars in the outer halo is\nin agreement with that of Xue {et~al.} (2011). \n\nTo test models of the accreted components of stellar haloes and\nunderstand the effects of sample variance, we have computed \\deltacf{}\nfor mock observations constructed from the six $\\Lambda$CDM\nsimulations of Cooper {et~al.} (2010) in which only the stellar haloes\nproduced by disrupted satellites are considered. Our statistic\ndistinguishes quantitatively between these six qualitatively different\nhalo realizations. When only stars with $r>20$~kpc are considered,\nfive of our six simulations show statistically significant\ncorrelations on scales in our metric equivalent to $\\sim 1-8$~kpc (for\nall observers on the Solar circle). Most of the models are consistent\nwith the Milky Way data for the outer halo, $r>30$~kpc. For the inner\nhalo, however, particularly at galactocentric distances smaller than\n20~kpc, the simulations tend to be significantly more strongly\nclustered than the data. One possible explanation for this is a\ndeficiency of smoothly distributed halo stars in the models, perhaps\nattributable to the absence of so-called `in situ' halo stars. These\nstars may be scattered from the Galactic disc, or born on eccentric\norbits (in streams of accreted gas or an unstable cooling flow, for\nexample). Neither of these processes are included in our model of the\naccreted halo.\n\nAlthough it seems reasonable to expect that in situ haloes are\ndistributed with spherical or axial symmetry and have a low degree of\ncoherence in phase space, models of such components and predictions\nfor the fraction of stars they contain are not well constrained. Most\nhypotheses for in situ halo formation limit these stars to an `inner'\nhalo and predict that the accreted component (which we simulate)\ndominates at larger radii\n(e.g. Abadi {et~al.} 2006; Zolotov {et~al.} 2009; Font {et~al.} 2011). However, the fraction\nof the halo formed in situ and the boundary between `inner' and\n`outer' halo are highly model-dependent. Detections of observable\n`dichotomies' in the Milky Way halo (Carollo {et~al.} 2007) are still\ndebated (e.g. Sch{\\\"{o}}nrich, Asplund \\& Casagrande 2011; Beers {et~al.} 2011). It is\npossible to place broad limits on the fraction of stars in a `missing'\nsmooth component, for example by comparing the RMS variation of\nprojected star-counts in our models (Helmi {et~al.} 2011b) to the Milky\nWay (Bell {et~al.} 2008). However, the uncertainties involved are substantial.\n\nAnother factor in the discrepancy between the models and the data may\nbe the absence of a baryonic (disc) contribution to the gravitational\npotential. A massive disc could alter the process of satellite\ndisruption in the inner halo and might make the potential within\n$30$~kpc more spherical (Kazantzidis, Abadi \\& Navarro 2010), possibly distributing\nmore inner halo stars into the SDSS footprint (on the other hand, a\nmore axisymmetric or spherical dark halo might also result in fewer\nchaotic orbits, hence more coherent streams). Because of these\nmodelling uncertainties, our application of the \\deltacf{} statistic\ncan presently serve only as a basic test for the abundance of\nsubstructure in the simulations.\n\nSeveral aspects of our approach could be improved. \nIt seems desirable to use well-measured radial velocity data to\nenhance clustering signals such as our correlation function relative\nto those based on configuration space data alone. However, so far, no\nproposal for \nincluding these velocity data is well-supported by theory. \nHere, we have used a straightforward choice of\nparametrised metric to illustrate the concept of scaling radial\nvelocity separations to `equivalent' configuration space separations,\nand this is empirically useful in recovering a measurable\nsignal. Nevertheless, we have not found any compelling or generic way\nto select the scaling parameter\n($w_{v}$). Improving either the definition of the metric itself or the\nmeans of fixing this parameter is a clear priority for extensions of\nthis approach. A similar issue affects the weighting of velocity\ninformation in clustering algorithms (e.g. Sharma {et~al.} 2010).\n\nFinally, further comparisons between stellar halo models and\nobservational data should also account for the selection effects such\nspectroscopic incompleteness and the potential bias of BHB stars as a\ntracer of the stellar halo (Bell {et~al.} 2010; Xue {et~al.} 2011). For\nstatistical analysis, there is a pressing requirement for\nobservational samples with well-understood selection functions, even\nif they do not probe the most distant halo. The LAMOST Galactic survey\nis likely to be the first to approach this goal.\n\nIn summary, we have taken a first step in adapting a well-studied\ncosmological statistic, the two-point correlation function, to the\nstudy of the Milky Way halo. Our comparisons highlight the complexity\nof statistical analysis in the stellar halo, and the importance of\ninterpreting observational results in the context of realistic models\nof halo assembly. We have compared the SDSS data with the stellar\nhalos produced by disrupted satellites in {\\em ab initio} galaxy\nformation models constructed from the Aquarius N-body simulations of\ngalactic dark halos in the $\\Lambda$CDM cosmology. With further\nrefinements and more data, our statistical approach to quantifying the\nsmoothness of the halo can provide a practical and productive way to\nstudy the structure of the Milky Way halo and compare with theoretical\nexpectations.\n\n\\section*{Acknowledgements}\n\nThe authors thank Heather Morrison and the Spaghetti Survey team for\nmaking their data available prior to publication, and Xiangxiang Xue\nand Sergey Koposov for their assistance. We thank the referee for\ntheir helpful comments which greatly improved the structure of this\npaper. APC acknowledges an STFC studentship and thanks Else\nStarkenburg for useful discussions. SMC acknowledges the support of a\nLeverhulme Research Fellowship. CSF acknowledges a Royal Society\nWolfson Research Merit Award and ERC Advanced Investigator grant\n267291- COSMIWAY. AH acknowledges funding support from the European\nResearch Council under ERC-StG grant GALACTICA-24027. This work was\nsupported in part by an STFC rolling grant to the ICC. The\ncalculations for this paper were performed on the ICC Cosmology\nMachine, which is part of the DiRAC Facility jointly funded by STFC,\nthe Large Facilities Capital Fund of BIS, and Durham University.Fig. 1\nwas produced with the {\\tt HEALPy} implementation of {\\tt HEALPix}\n[http:\/\/healpix.jpl.nasa.gov, G{\\'{o}}rski {et~al.} 2005].\n\n{}","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{introduction}\nIron selenide based superconductors have come to attract particular attention after the discovery that the binary material FeSe avoids the formation of the magnetic order but becomes superconducting at $T_c$~=~8~K. \\cite{hsu2008superconductivity} The value of $T_c$ can be enhanced both by applying pressure, reaching a maximum value of about 37~K,\\cite{medvedev2009electronic,margadonna2009pressure} or in single layer films of FeSe grown on an SrTiO$_3$ substrate, \\cite{qing2012interface,FeSe2013,FeSe2015} and it is suggested that large enhancements of $T_c$ arise due to an increased electron carrier density. \\cite{FeSeCarr} Another method of enhancing the superconductivity is by intercalating alkali metals between FeSe layers, as in the case of $A_x$Fe$_{2-y}$Se$_2$ ($A$~=~K, Rb, Cs,Tl\/Rb, Tl\/K) \\cite{guo2010superconductivity,2011CsSCRep,wang2011superconductivity,2011TlRbSCRep,2011TlKSCRep}. A notable difference from both the iron arsenide based materials and the bulk binary compound FeSe is the absence of the hole pocket in the Fermi surface at the Brillouin zone center.\\cite{qian2011absence} This appears to contradict the s$_{\\pm}$ superconducting state often applied to iron arsenide superconductors, where there is nesting and a sign change of the order parameter between the hole and electron pockets.\\cite{mazin2008unconventional,kuroki2008unconventional} While there have been a variety of alternative proposals for the pairing state, \\cite{mazin2011symmetry,khodas2012interpocket,nica2015orbital} studying the intrinsic superconducting properties is greatly complicated by the clear evidence for phase separation between non-superconducting regions with an ordered arrangement of Fe vacancies and vacancy free superconducting regions. \\cite{li2012phase,PhaseSep2011DL,PhaseSepNMR}.\n\nRecently a new iron selenide based superconductor (Li$_{1-x}$Fe$_x$)OHFeSe ($x\\approx0.2$) was discovered, \\cite{lu2015coexistence} with a high transition temperature of $T_c\\approx$~40~K. It has a quasi-two-dimensional crystal structure, with layers of both (Li$_{1-x}$Fe$_x$)OH and superconducting FeSe. The material displays coexistence between superconductivity and antiferromagnetism, \\cite{lu2015coexistence} while in the phase diagram superconductivity occurs in close proximity to spin-density wave order. \\cite{dong2014phase} In common with $A_x$Fe$_{2-y}$Se$_2$, the hole pocket is also absent,\\cite{niu2015surface,zhao2016common} but the samples are much more homogeneous, indicating that (Li$_{1-x}$Fe$_x$)OHFeSe is a good candidate for probing iron based superconductors without a hole pocket at the zone center. Angle resolved photoemission spectroscopy (ARPES) measurements are consistent with the presence of nodeless superconductivity with a single isotropic energy gap, but disagree over the gap magnitudes.\\cite{niu2015surface,zhao2016common} However, scanning tunneling spectroscopy (STS) studies show two distinct features in the conductance spectra, suggesting the presence of multiple gaps. \\cite{du2016scrutinizing,yan2016surface} Meanwhile the in-plane superfluid density obtained from muon-spin rotation ($\\mu$SR) measurements is consistent with either one or two gaps, but very different behavior is seen in the out-of-plane component, which shows a much more rapid drop of the superfluid density with temperature. \\cite{khasanov2016proximity} Furthermore, inelastic neutron scattering (INS) measurements give evidence for the presence of a spin resonance peak, \\cite{Davies2016spin,pan2016structure} consistent with a sign change of the order parameter across the Fermi surface, while a lack of a sign change was suggested from an STS study, on the basis of quasi-particle interference (QPI) results and the effect of impurities.\\cite{yan2016surface} \n\n\n\nTo further characterize the superconducting gap structure, we report London penetration depth measurements of (Li$_{1-x}$Fe$_x$)OHFeSe single crystals using a tunnel-diode-oscillator (TDO) based technique, from which we obtain the temperature dependence of the London penetration depth shift $\\Delta\\lambda(T)$. The low temperature $\\Delta\\lambda(T)$ gives clear evidence for nodeless superconductivity, while a single-gap isotropic $s$-wave model is unable to account for the superfluid density. The superfluid density is well fitted by a two-gap $s$-wave model, as well as models with anisotropic gaps.\n\n\\begin{figure}[t]\n\\begin{center}\n \\includegraphics[width=\\columnwidth]{Fig1.eps}\n\\end{center}\n\t\\caption{Magnetic susceptibility $\\chi$ of (Li$_{1-x}$Fe$_x$)OHFeSe samples from two batches ($\\#$A and $\\#$B). Both field-cooled (FC) and zero-field-cooled (ZFC) measurements were performed in an applied magnetic field of 1~mT. The inset shows the magnetization as a function of applied field at 2~K. The solid line shows the linear fit at low fields while the arrow marks the position of the lower critical field $H_{c1}$, where there is deviation from linear behavior.}\n \\label{samplecharacterization}\n\\end{figure}\n\\section{Experimental details}\nSingle crystals of (Li$_{1-x}$Fe$_x$)OHFeSe (with $x\\approx0.2$) from two batches prepared by two different groups were measured ($\\#$A and $\\#$B), where the crystals were synthesized following Ref.~\\onlinecite{DongSynth}. Using the parameters from Ref.~\\onlinecite{DongSynth} ($\\rho_0=0.1~$m$\\Omega$-cm and a carrier density of $n=1.04\\times10^{21}$cm$^{-3}$), we estimate a mean free path of $l=12.4$~nm using $l=\\hbar(3\\pi^2)^{\\frac{1}{3}}\/e^2n^{\\frac{2}{3}}\\rho_0$. This is considerably larger than the Ginzburg-Landau coherence length $\\xi_{GL}=2$~nm calculated from an upper critical field of $H_{c2}(0)=79$~T, \\cite{DongSynth} and therefore the material is in the clean limit. Magnetization measurements of samples from both batches were performed utilizing a SQUID magnetometer (MPMS-5T). The temperature dependence of the London penetration depth shift $\\Delta\\lambda(T)~=~\\lambda(T)-\\lambda(0)$ was measured in a ${^3}$He cryostat from 42~K down to a base temperature of about 0.5~K using a tunnel-diode-oscillator based method, with an operating frequency of about 7~MHz. The samples were cut into a regular shape and mounted on a sapphire rod which was inserted into the coil without any contact. A very small ac field of about 2~$\\mu T$ is applied to the sample along the $c$~axis, which is much smaller than the lower critical field $\\mu_0H_{c1}$ and therefore the sample is always in the Meissner state. As such the shift in the resonant frequency from zero temperature $\\Delta f(T)$ is related to the penetration depth shift in the $ab$~plane $\\Delta\\lambda(T)$ via $\\Delta\\lambda(T)$~=~G$\\Delta f(T)$, where the calibration constant $G$ is calculated using the geometry of the coil and sample \\cite{Gfactor}. \n\n\\begin{figure}[t]\n\\begin{center}\n \\includegraphics[width=0.8\\columnwidth]{Fig2.eps}\n\\end{center}\n\t\\caption{Temperature dependence of the London penetration depth $\\Delta\\lambda(T)$ of two samples from (a) batch $\\#$A and (b) batch $\\#$B. The main panels show the low temperature behavior, where the data are fitted below $T_c\/3$ with a fully gapped model and a power law dependence. The insets display the frequency shift $\\Delta f(T)$ from the lowest temperature up to above the superconducting transition temperature, normalized by the value at 40~K.}\n \\label{penetrationdepth}\n\\end{figure}\n\n\\section{Results and discussion}\n\n Both the field-cooled (FC) and zero-field-cooled (ZFC) magnetic susceptibility [$\\chi(T)$] measurements are shown in Fig.~\\ref{samplecharacterization}, from above the superconducting transition temperature down to 2~K under a small applied field of 10~Oe, where corrections for the demagnetization effect were applied. The ZFC $\\chi(T)$ of both samples show sharp superconducting transitions onsetting at around 40~K\nand 37~K for samples $\\#$A and $\\#$B, respectively. At low temperatures the data for $\\#$A flattens, while\nfor $\\#$B there continues to be a slight decrease with decreasing temperature. This indicates the high quality of the single crystals from batch $\\#$A, whereas those from batch $\\#$B show evidence for some inhomogeneity. Meanwhile the superconducting shielding fraction is around 100$\\%$ in both samples. The inset displays the field dependence of the magnetization at 2~K, for a field applied in-plane so that the demagnetization effect is very small. There is a deviation of the low field linear behavior at $\\mu_0H_{c1}\\approx 4.5$~mT, confirming that $H_{c1}$ is significantly larger than the ac field applied in the penetration depth measurements.\n\nFigure~\\ref{penetrationdepth} displays $\\Delta\\lambda(T)$ for single crystal samples from two batches, with samples from $\\#$A and $\\#$B displayed in (a) and (b) respectively. The insets of both panels display the temperature dependence of the frequency shift $\\Delta f(T)$ from above the superconducting transition at 42~K down to 0.5~K. The superconducting transition onsets at respective temperatures of 40~K and 39~K in samples $\\#$A and $\\#$B, while the corresponding endpoints of the transition are around 34~K and 35~K, and the latter values of $T_c$ are used in the subsequent analysis of the superfluid density. The main panels of Fig.~\\ref{penetrationdepth} display the low temperature behavior of $\\Delta\\lambda(T)$. It can be seen in Fig.~\\ref{penetrationdepth}(a) that $\\Delta\\lambda(T)$ for the $\\#$A samples decreases with decreasing temperature before flattening below about 0.1T$_c$, indicating a nodeless gap structure in (Li$_{1-x}$Fe$_x$)OHFeSe with a lack of low energy excitations, which is consistent with previous results. \\cite{niu2015surface,zhao2016common,du2016scrutinizing,yan2016surface,khasanov2016proximity} Furthermore when fitted with a power law dependence $\\Delta\\lambda(T)\\sim T^n$, exponents of $n=2.83$ and $n=2.46$ are obtained for samples $\\#$A-1 and $\\#$B-1 respectively. In the case of nodal superconductors, $n=1$ for line nodes and $n=2$ for point nodes is generally anticipated. While impurity scattering, non-local effects and quantum fluctuations can all lead to $n\\approx2$ at low temperatures for $d$-wave superconductors with line nodes,\\cite{Hirschfeld1993,Kosztin1997,Benfatto2001} our observation that $n$ is significantly larger than two gives further evidence for fully gapped behavior. For a fully-gapped superconductor at $T\\ll T_c$, the penetration depth can be described by $\\Delta\\lambda(T)=\\lambda_{eff}(0)\\sqrt{\\pi\\Delta(0)\/2k_BT}\\textrm{exp}[-\\frac{\\Delta(0)}{k_BT}]$, where $\\Delta(0)$ is the superconducting gap magnitude at zero temperature and $\\lambda_{eff}(0)$ is an effective zero temperature penetration depth. The low temperature data for sample $\\#$A-1 is well fitted with a gap magnitude of $\\Delta(0)=~0.87k_BT_c$. A similar value of $\\Delta(0)=~0.78k_BT_c$ is obtained from the fitting for sample $\\#$B-1, although there is a small deviation of the fitted curve for this sample at the lowest temperatures, where a weak anomaly is observed in the data, the origin of which is not clear. The values of the fitted gaps are much smaller than the value from BCS theory of $1.76k_BT_c$ for weakly coupled isotropic $s$-wave superconductors, suggesting multi-gap superconductivity or gap anisotropy in (Li$_{1-x}$Fe$_x$)OHFeSe. The fitted values of $\\lambda_{eff}(0)$ of 636~\\AA\\ and 1228~\\AA\\ for samples $\\#$A-1 and $\\#$B-1 respectively, are significantly different from the value of $\\lambda(0)\\approx2800$~\\AA\\ from $\\mu$SR measurements. \\cite{khasanov2016proximity}. Such a difference is also expected for multi-gap or anisotropic superconductors. \\cite{Malone2009}\n\n\nIn order to obtain further information about the superconducting pairing state, the normalized superfluid density [$\\rho_s(T)$] was calculated from the penetration depth using $\\rho_s(T)$~=~$[\\lambda(0)$\/$\\lambda(T)]^2$, where $\\lambda(0)\\approx$~2800~\\AA\\ was estimated from $\\mu$SR measurements.\\cite{khasanov2016proximity} Since the samples from batch $\\#$A are higher quality and show a sharper superconducting transition, measurements from this batch were used in the subsequent analysis. The $\\rho_s(T)$ of sample $\\#$A-1 is displayed in Fig.~\\ref{superfluiddensity}. The superfluid density was modelled using \n\\begin{equation}\n\\rho_{\\rm s}(T) = 1 + 2 \\left\\langle\\int_{\\Delta_k}^{\\infty}\\frac{E{\\rm d}E}{\\sqrt{E^2-\\Delta_k^2}}\\frac{\\partial f}{\\partial E}\\right\\rangle_{\\rm FS},\n\\label{equation2}\n\\end{equation}\n\\noindent where $f(E, T)=[1+{\\rm exp}(E\/k_BT)]^{-1}$ is the Fermi function and $\\left\\langle\\ldots\\right\\rangle_{\\rm FS}$ denotes an average over the Fermi surface. The temperature dependence of the gap function $\\Delta_k$ is approximated by \\cite{carrington2003magnetic}\n\\begin{equation}\n\\delta(T)={\\rm tanh}\\left\\{1.82\\left[1.018\\left(T_c\/T-1\\right)\\right]^{0.51}\\right\\},\n\\label{equation3}\n\\end{equation}\n\n\n\\begin{figure}[t]\n\\begin{center}\n \\includegraphics[width=\\columnwidth]{Fig3.eps}\n\\end{center}\n\t\\caption{Normalized superfluid density $\\rho_s$ as a function of the reduced temperature $T\/T_c$ for sample $\\#$A-1 of (Li$_{1-x}$Fe$_x$)OHFeSe. The dashed, dashed-dotted and solid lines show fits to models with a single isotropic $s$-wave gap, two isotropic gaps and the $s\\times\\tau_3$ state respectively. The inset shows $\\rho_s$ upon varying $\\lambda(0)$ by $\\pm30\\%$, along with fits to a two-gap model.}\n \\label{superfluiddensity}\n\\end{figure}\n\n\\noindent As discussed previously, the behavior of $\\Delta\\lambda(T)$ at low temperatures and previous experimental results indicate nodeless superconductivity in (Li$_{1-x}$Fe$_x$)OHFeSe and therefore we fitted $\\rho_s(T)$ with various fully-gapped models. The simplest model is to assume an isotropic superconducting gap, with $\\Delta_k(T)~=~\\Delta(0)\\delta(T)$. The fitted curve for such an isotropic single band $s$-wave model with $\\Delta(0)~=~1.72k_BT_c$ is shown by the dashed line in Fig.~\\ref{superfluiddensity}, where although there is reasonable agreement above 0.5$T_c$, there is a significant discrepancy in the intermediate temperature range between 0.2$T_c$ and 0.5$T_c$. This difference arises due to the data dropping more quickly than the calculated $\\rho_s(T)$, indicating the presence of an additional lower energy scale in the gap structure, which is consistent with the smaller gap value obtained from fitting $\\Delta\\lambda(T)$ at low temperatures. Therefore the data are fitted with a two-gap model, as has been applied to many iron based superconductors. For this model the total superfluid density is given by the weighted sum of two components with different gaps,\n\\begin{equation}\n\\rho_{\\rm s}(T) = \\alpha\\rho{_{\\rm s}^1}(\\Delta_k^1, T) + (1-\\alpha)\\rho{_{\\rm s}^2}(\\Delta_k^2, T),\n\\label{equation4}\n\\end{equation}\n\n\\noindent where $\\rho_s^i (i=1, 2)$ is the normalized superfluid density with a gap function $\\Delta_k^i (i=1, 2)$ and $\\alpha$ is the relative weight for the component $\\rho_s^1$ ($0\\leq\\alpha\\leq1$). The fitting result is shown in Fig.~\\ref{superfluiddensity} by the dashed-dotted line, which agrees well with the data across the whole temperature range. The fitted parameters are $\\Delta_1(0)=0.8k_BT_c$, $\\Delta_2(0)=1.9k_BT_c$ and $\\alpha=0.13$, where the value of the small gap is close to that found from fitting $\\Delta\\lambda(T)$, which is further consistent with two-gap superconductivity. In order to consider possible uncertainties in the calibration constant $G$ or $\\lambda(0)$, in the inset we show $\\rho_s(T)$ upon varying $\\lambda(0)$ by $\\pm30\\%$. The data can still be fitted by a two gap model with slightly different parameters of $\\Delta_1(0)=0.8k_BT_c$, $\\Delta_2(0)=1.6k_BT_c$ and $\\alpha=0.15$ for $\\lambda(0)$~=~2000~\\AA ~and $\\Delta_1(0)=0.8k_BT_c$, $\\Delta_2(0)=2.1k_BT_c$ and $\\alpha=0.12$ for $\\lambda(0)$~=~3600~\\AA. The data were also well fitted with an anisotropic single band model with $\\Delta_k(T,\\phi)=\\Delta(0)(1+r{\\rm cos}2\\phi)\\delta(T)$, using $\\Delta(0)=1.32k_BT_c$ and $r=0.65$, which is not shown for the sake of clarity.\n\nARPES measurements indicate that the Fermi surface consists of electron pockets at the Brillouin zone corners, without the presence of hole pockets at the zone center \\cite{niu2015surface,zhao2016common}. From a recent STM study it was proposed on the basis of QPI measurements, as well as examining the effects of impurities, that there is no sign change of the superconducting gap across the Fermi surface \\cite{yan2016surface}, in which case such a two gap $s$-wave model readily explains the data. However INS measurements show evidence for a spin resonance peak,\\cite{Davies2016spin,pan2016structure} which indicates that there is a sign change of the order parameter between regions of the Fermi surface connected by the resonance wave vector. This would be incompatible with two-gap $s$-wave superconductivity with no sign change and are also difficult to account for with the $s_{\\pm}$ state proposed for many iron arsenide superconductors, due to both the lack of a hole pocket at the zone center and a different nesting wave vector for the spin resonance. \\cite{mazin2008unconventional,kuroki2008unconventional} On the other hand a different sign changing $s_{\\pm}$ state has been suggested for $A_x$Fe$_{2-y}$Se$_2$ (A~=~K, Rb, Cs), \\cite{khodas2012interpocket,mazin2011symmetry} with a very similar Fermi surface to (Li$_{1-x}$Fe$_x$)OHFeSe.\n\nAnother proposed pairing state for nodeless sign changing superconductivity in iron based superconductors with only electron pockets is an orbital selective $s\\times\\tau_3$ state. \\cite{nica2015orbital} In this scenario, intraband pairing has $d_{x^2-y^2}$ symmetry, while the interband pairing has $d_{xy}$ symmetry. As a result, the zeroes of the intraband and interband pairing are offset by an angle of $\\pi\/4$ and therefore the gap remains nodeless. A simple model for the gap function of this state taking into account the Fermi surface is $\\Delta_k(T,\\phi)$=[($\\Delta_1(0)$)$^2$+($\\Delta_2(0)$sin($\\phi$))$^2$]$^{1\/2}$$\\delta(T)$ \\cite{nica2015orbital,SiPrivate}. As shown in Fig.~\\ref{superfluiddensity}, this model can also well fit the experimental data, with fitted parameters of $\\Delta_1(0)=1.05k_BT_c$ and $\\Delta_2(0)=3.2k_BT_c$. In this case the gap minimum $\\Delta_1(0)$ is slightly larger than the smaller gap from the two-gap $s$-wave fit. Therefore the data can be well accounted for by fitting with models with either two-gaps or a strong gap anisotropy and our results are compatible with two-gap behavior, anisotropic $s$-wave superconductivity or an orbital selective $s\\times\\tau_3$ state. It is often very difficult to distinguish between scenarios with multiple gaps and those where there is one anisotropic gap. While the isotropic nature of the gap inferred from ARPES would favor the two-gap scenario over an anisotropic gap, this still requires further study. \\cite{niu2015surface,zhao2016common} Furthermore, the superfluid density is only sensitive to the gap magnitude rather than the phase and different measurements are required to clarify the presence of a sign change and to determine the nature of the pairing state.\n\nWe note that there have been conflicting reports about the nature of the gap structure of (Li$_{1-x}$Fe$_x$)OHFeSe, with only a single gap being resolved from ARPES measurements \\cite{niu2015surface,zhao2016common}, while two gaps are found from STS \\cite{du2016scrutinizing,yan2016surface} and the in-plane superfluid density from $\\mu$SR measurements is compatible with both single-gap and two-gap models. \\cite{khasanov2016proximity} The gap values we obtain from fitting the in-plane superfluid density are smaller than those reported from previous measurements, particularly in the case of the smaller gap. Evidence for this smaller gap is clearly observed from our measurements of $\\Delta\\lambda(T)$ and $\\rho_s$ at low temperatures, which may be a result of the high sensitivity of the TDO-based technique. A further reason for the discrepancies could be due to the non-stoichiometric nature of (Li$_{1-x}$Fe$_x$)OHFeSe, where the exact composition and homogeneity may influence the doping level, $T_c$ and the gap magnitudes. In addition, the doping level can be different between the surface and in the bulk, \\cite{niu2015surface} and therefore probes which primarily measure the surface properties may give different results. Indeed different results have been found from different measurements of the gap structure of other iron-based superconductors, \\cite{evtushinsky2009momentum} and significant sample dependence has been suggested for Ba$_{1-x}$K$_x$Fe$_2$As$_2$.\\cite{hashimoto2009microwave} \n\n\n\\section{summary}\n\nWe have measured the temperature dependence of the London penetration depth $\\Delta\\lambda(T)$ and the derived superfluid density $\\rho_s(T)$ of the recently discovered high-$T_c$ iron-based superconductor (Li$_{1-x}$Fe$_x$)OHFeSe. The behavior of $\\Delta\\lambda(T)$ at low temperatures gives clear evidence for nodeless superconductivity, with a relatively small energy gap, while the analysis of $\\rho_s(T)$ is consistent with both two-gap superconductivity and models with significant gap anisotropy such as an orbital selective $s\\times\\tau_3$ or anisotropic $s$-wave state.\n\nWe thank Q. Si, E. M. Nica, R. Yu and X. Lu for helpful discussions and valuable suggestions. This work was supported by the National Natural Science Foundation of China (No.11574370), National Key Research and Development Program of China (No. 2016YFA0300202), and the Science Challenge Project of China. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Kronecker tensor structures for complex fields}\n\\label{App:ProjA}\nIn this appendix we demonstrate how a product of two operators can be decomposed onto irreps of $U(m)\\times U(n)$ in the picture where we work with complex fields. For simplicity we will start with just $U(n)$. The generalization to $U(m)\\times U(n)$ is trivial. The main utility of working with complex fields is that the projectors take their simplest form possible, namely as combinations of Kronecker deltas with the least number of indices possible. The real field picture can also have its projectors expressed in terms of Kronecker deltas, albeit at the cost of more indices. We hope that presenting our projectors in three different pictures will make our work more intuitive. In order to capture all irreps that can appear in the real field notation, and hence not miss any information in the bootstrap algorithm, we need to consider two OPEs, namely $\\Phi_i^\\dagger \\times \\Phi_j$ and $\\Phi_i \\times \\Phi_j$. Below we show how they may be decomposed onto irreps:\n\\begin{equation}\n\\begin{split}\n \\Phi_i^\\dagger \\times \\Phi_j &\\sim \\Big(\\Phi_i^\\dagger \\Phi_j - \\frac{1}{n}\\delta_{ij} \\Phi_k^\\dagger \\Phi_k \\Big) +\\frac{1}{n}\\delta_{ij} \\Phi_k^\\dagger \\Phi_k\\,, \\\\\n \\Phi_i \\times \\Phi_j &\\sim (\\Phi_i \\Phi_j + \\Phi_j \\Phi_i) + (\\Phi_i \\Phi_j - \\Phi_j \\Phi_i)\\,.\n\\end{split}\n\\label{complexOPE}\n\\end{equation}\nThe first line in \\eqref{complexOPE} shows the decomposition into the adjoint ($R$) and singlet ($S$) representations, where as the second line shows the decomposition into the symmetric ($T$) and antisymmetric representations ($A$). To read off the projectors from \\eqref{complexOPE} it is useful to remember\n\\begin{equation}\n\\begin{aligned}\n O^X_{ij} \\sim P^X_{ijkl} \\Phi_k \\Phi_l\\,,\\qquad\n O^X_{ij} \\sim P^X_{ijkl} \\Phi^\\dagger_k \\Phi_l\\,,\n\\end{aligned}\n\\label{projdefinitionscomplex}\n\\end{equation}\nwhere $O^X$ is the exchanged field in some irrep, e.g.\\ $O^T_{12} \\sim \\Phi_1 \\Phi_2 + \\Phi_2 \\Phi_1$. The first relation in \\eqref{projdefinitionscomplex} can be used to read off the $T$ and $A$ projectors, whereas the second can be used for $S$ and $R$. Notice that we have implicitly assumed that fields are inserted at different positions in order for antisymmetric irreps to not vanish identically. The projectors can be read off as\\footnote{Note that we take the correlator to be $\\langle \\Phi_i^\\dagger \\Phi_j \\Phi_k \\Phi_l^\\dagger \\rangle$ which is why the projector of the adjoint representation is equal to $P^R_{ijkl} =\\delta_{ik}\\delta_{jl}-\\frac{1}{n}\\delta_{ij}\\delta_{kl}$ instead of $P^R_{ijkl} =\\delta_{il}\\delta_{jk}-\\frac{1}{n}\\delta_{ij}\\delta_{kl}$.}\n\\begin{align}\n P^S_{ijkl} &=\\frac{1}{n}\\delta_{ij}\\delta_{kl}\\,,\\qquad\n P^R_{ijkl} =\\delta_{ik}\\delta_{jl}-\\frac{1}{n}\\delta_{ij}\\delta_{kl}\\,, \\\\\n P^T_{ijkl} &=\\frac{1}{2}(\\delta_{ik}\\delta_{jl} +\\delta_{il}\\delta_{jk})\\,,\\qquad\n P^A_{ijkl}= \\frac{1}{2}(\\delta_{ik}\\delta_{jl} -\\delta_{il}\\delta_{jk})\\,.\n\\label{complexprojectorsUn}\n\\end{align}\nThe dimensions of the corresponding irreps are $(1,(n-1)(n+1),\\frac12n(n+1),\\frac12n(n-1))$. As the reader may have observed from the main text, or the next appendix, when going from the complex field picture to the real field picture the above dimensions get multiplied by a factor of $2$. For example $\\frac{n(n+1)}{2}$ becomes $n(n+1)$. This is because each of the initial $\\frac{n(n+1)}{2}$ complex elements contains two real elements. The last step now is to write down the projectors for $U(m) \\times U(n)$. This is trivial, in the sense that they are just products of $U(m)$ with $U(n)$ projectors. We have\n\\begin{equation}\n\\begin{aligned}\n P^{S}_{ijklmnop} &=P^S_{ijkl}P^S_{mnop} \\,,\\quad\n P^{RS}_{ijklmnop} =P^R_{ijkl}P^S_{mnop} \\,,\\quad\n P^{SR}_{ijklmnop} =P^S_{ijkl}P^R_{mnop} \\,,\\\\\n P^{RR}_{ijklmnop} &=P^R_{ijkl}P^R_{mnop} \\,,\\quad\n P^{TT}_{ijklmnop} =P^T_{ijkl}P^T_{mnop} \\,,\\quad\n P^{TA}_{ijklmnop} =P^T_{ijkl}P^A_{mnop} \\,,\\\\\n P^{AT}_{ijklmnop} &=P^A_{ijkl}P^T_{mnop} \\,,\\quad\n P^{AA}_{ijklmnop} =P^A_{ijkl}P^A_{mnop}\\,.\n\\end{aligned}\n\\label{complexprojectorsUnII}\n\\end{equation}\nThe sum rules that can be derived with the above projectors can be checked to be completely equivalent to those derived from the projectors of real fields outlined in the main text. Another observation is that, compared to real fields, we do not need separate projectors for even and odd spins.\n\n\\section{Kronecker tensor structures for real fields}\n\\label{App:ProjB}\nThe projectors that correspond to a four-point function of real fields can be intuitively presented in terms of Kronecker deltas if we add an additional index. This form is useful since one may directly extract the form of exchanged operators, as we will show. The form of exchanged operators is useful to know since it can guide us with respect to assumptions we may impose. Also, we expect it to be easier to work with in a mixed correlator system. We start by labeling the real and complex parts of an operator $\\Phi_i$ with an index (we start with $U(n)$ for simplicity)\n\\begin{equation}\n\\Phi_i = \\phi^1_{i} + i\\hspace{1pt}\\phi^2_{i}\\,,\n\\label{complextoreal}\n\\end{equation}\nwhere the upper case $\\Phi$ denotes the complex operator and the lower case $\\phi$ denote real fields. We must now simply plug in \\eqref{complextoreal} to the expressions for the representations of the previous appendix. For simplicity we will do this for the singlet representation, and then quote the results for rest of the representations. Note that implicitly we consider the two external fields of the OPE at different positions, for otherwise the antisymmetric combinations would vanish identically. We have\n\\begin{equation}\n \\Phi_i^\\dagger \\Phi_i = (\\phi^1_i \\phi^1_i + \\phi^2_i \\phi^2_i ) + i (\\phi^1_i \\phi^2_i - \\phi^2_i \\phi^1_i)\\,,\n\\end{equation}\nwhere the first parenthesis corresponds to what was called $S_{\\text{even}}$ in the main text, and the second parenthesis corresponds to what was called $S_{\\text{odd}}$. As expected $S_{\\text{odd}}$ vanishes identically if we don't insert powers of derivatives between the operators. The projectors are now very straightforward to write down by recalling the relation\n\\begin{equation}\n O^X_{ij;ab} = P^X_{ijkl;abcd}\\hspace{1pt} \\phi^a_i \\phi^b_j\\,,\n\\label{projrelation}\n\\end{equation}\nwhere $X$ stands for some specific irrep and indices from the beginning of the latin alphabet take the values $1,2$. Notice that \\eqref{projrelation} is simply the statement that projectors must project products of operators onto irreps. We have\n\\begin{equation}\n\\begin{split}\n P^{S_{\\text{even}}}_{ijkl;abcd} &= \\frac{1}{2n} \\delta_{ab}\\delta_{cd}\\delta_{ij}\\delta_{kl}\\,, \\\\\n P^{S_{\\text{odd}}}_{ijkl;abcd} &= \\frac{1}{2n}(\\delta_{ac}\\delta_{bd}-\\delta_{ad}\\delta_{bc})\\delta_{ij}\\delta_{kl}\\,.\n\\end{split}\n\\end{equation}\nIndeed, one may confirm that, for example,\n\\begin{equation}\n O^{S_{\\text{even}}}_{11;11}\\sim(\\phi^1_i \\phi^1_i + \\phi^2_i \\phi^2_i ) \\sim P^{S_{\\text{even}}}_{11kl;11cd} \\hspace{1pt}\\phi^{c}_k \\phi^{d}_l\\,.\n\\end{equation}\nThis procedure can be repeated for the rest of the irreps. The resulting projectors are\n\\begin{equation}\n\\begin{split}\n P^{S_{\\text{even}}}_{ijkl;abcd} &= \\frac{1}{2n} \\delta_{ab}\\delta_{cd}\\delta_{ij}\\delta_{kl}\\,, \\\\\n P^{S_{\\text{odd}}}_{ijkl;abcd} &= \\frac{1}{2n}(\\delta_{ac}\\delta_{bd}-\\delta_{ad}\\delta_{bc})\\delta_{ij}\\delta_{kl}\\,,\\\\\n P^{R_{\\text{even}}}_{ijkl;abcd} &= \\tfrac{1}{2}\\delta_{ab}\\delta_{cd}\\Big(\\delta_{ik}\\delta_{jl}-\\frac{1}{n}\\delta_{ij}\\delta_{kl}\\Big)\\,,\\\\\n P^{R_{\\text{odd}}}_{ijkl;abcd} &= \\tfrac{1}{2}(\\delta_{ac}\\delta_{bd}-\\delta_{ad}\\delta_{bc})\\Big(\\delta_{ik}\\delta_{jl}-\\frac{1}{n}\\delta_{ij}\\delta_{kl}\\Big)\\,,\\\\\n P^{T_{\\text{even}}}_{ijkl;abcd} &= \\tfrac{1}{4}(\\delta_{ac}\\delta_{bd}+\\delta_{ad}\\delta_{bc}-\\delta_{ab}\\delta_{cd})(\\delta_{ik}\\delta_{jl}+\\delta_{il}\\delta_{jl})\\,,\\\\\n P^{A_{\\text{odd}}}_{ijkl;abcd} &= \\tfrac{1}{4}(\\delta_{ac}\\delta_{bd}+\\delta_{ad}\\delta_{bc}-\\delta_{ab}\\delta_{cd})(\\delta_{ik}\\delta_{jl}-\\delta_{il}\\delta_{jl})\\,.\n\\end{split}\n\\end{equation}\nThe dimensions of the corresponding irreps are $(1,1,(n-1)(n+1),(n-1)(n+1),n(n+1),n(n-1))$.\n\nUsing the above expressions, it is now trivial to write down the $U(m)\\times U(n)$ projectors:\n\\begin{align}\n P^{S_{\\text{even}}}_{ijklmnop;abcdefgh} &= P^{S_{\\text{even}}}_{ijkl;abcd}\\hspace{1pt} P^{S_{\\text{even}}}_{mnop;efgh}+ P^{S_{\\text{odd}}}_{ijkl;abcd}\\hspace{1pt} P^{S_{\\text{odd}}}_{mnop;efgh}\\,,\\nonumber\\\\\n P^{S_{\\text{odd}}}_{ijklmnop;abcdefgh} &= P^{S_{\\text{even}}}_{ijkl;abcd}\\hspace{1pt} P^{S_{\\text{odd}}}_{mnop;efgh}+ P^{S_{\\text{odd}}}_{ijkl;abcd}\\hspace{1pt} P^{S_{\\text{even}}}_{mnop;efgh}\\,,\\nonumber\\\\\n P^{RS_{\\text{even}}}_{ijklmnop;abcdefgh} &= P^{R_{\\text{even}}}_{ijkl;abcd}P^{S_{\\text{even}}}_{mnop;efgh} + P^{R_{\\text{odd}}}_{ijkl;abcd}P^{S_{\\text{odd}}}_{mnop;efgh}\\,,\\nonumber\\\\\n P^{RS_{\\text{odd}}}_{ijklmnop;abcdefgh} &= P^{R_{\\text{even}}}_{ijkl;abcd}P^{S_{\\text{odd}}}_{mnop;efgh} + P^{R_{\\text{odd}}}_{ijkl;abcd}P^{S_{\\text{even}}}_{mnop;efgh}\\,,\\nonumber\\\\\n P^{SR_{\\text{even}}}_{ijklmnop;abcdefgh} &= P^{S_{\\text{even}}}_{ijkl;abcd}P^{R_{\\text{even}}}_{mnop;efgh} + P^{S_{\\text{odd}}}_{ijkl;abcd}P^{R_{\\text{odd}}}_{mnop;efgh}\\,,\\nonumber\\\\\n P^{SR_{\\text{odd}}}_{ijklmnop;abcdefgh} &= P^{S_{\\text{even}}}_{ijkl;abcd}P^{R_{\\text{odd}}}_{mnop;efgh} + P^{S_{\\text{odd}}}_{ijkl;abcd}P^{R_{\\text{even}}}_{mnop;efgh}\\,,\\nonumber\\\\\n P^{RR_{\\text{even}}}_{ijklmnop;abcdefgh} &= P^{R_{\\text{even}}}_{ijkl;abcd}P^{R_{\\text{even}}}_{mnop;efgh} + P^{R_{\\text{odd}}}_{ijkl;abcd}P^{R_{\\text{odd}}}_{mnop;efgh}\\,,\\nonumber\\\\\n P^{RR_{\\text{odd}}}_{ijklmnop;abcdefgh} &= P^{R_{\\text{even}}}_{ijkl;abcd}P^{R_{\\text{odd}}}_{mnop;efgh} + P^{R_{\\text{odd}}}_{ijkl;abcd}P^{R_{\\text{even}}}_{mnop;efgh}\\,,\\nonumber\\\\\n P^{TT_{\\text{even}}}_{ijklmnop;abcdefgh} &= P^{T_{\\text{even}}}_{ijkl;abcd}P^{T_{\\text{even}}}_{mnop;efgh}\\,,\\nonumber\\\\\n P^{TA_{\\text{odd}}}_{ijklmnop;abcdefgh} &= P^{T_{\\text{even}}}_{ijkl;abcd}P^{A_{\\text{odd}}}_{mnop;efgh}\\,,\\nonumber\\\\\n P^{AT_{\\text{odd}}}_{ijklmnop;abcdefgh} &= P^{A_{\\text{odd}}}_{ijkl;abcd}P^{T_{\\text{even}}}_{mnop;efgh}\\,,\\nonumber\\\\\n P^{AA_{\\text{even}}}_{ijklmnop;abcdefgh} &= P^{A_{\\text{odd}}}_{ijkl;abcd}P^{A_{\\text{odd}}}_{mnop;efgh}\\,.\n\\end{align}\nFrom these expressions we can also see explicitly that when $m=n$, if we choose to consider the two $U(n)$ symmetries as indistinguishable (which we remind the reader is not strictly necessary), the $RS$ irreps become the same as the $SR$ irreps. The same also happens for $TA$ and $AT$.\n\n\\section{Numerical parameters}\n\\label{App:C}\nFor most of our plots, the bounds are obtained with the use of\n\\texttt{PyCFTBoot}~\\cite{Behan:2016dtz} and \\texttt{SDPB}~\\cite{Landry:2019qug}. We use the numerical parameters $\\texttt{m\\_max}=6, \\texttt{n\\_max}=9, \\texttt{k\\_max}=36$ in \\texttt{PyCFTBoot}, and we include spins up to $\\texttt{l\\_max}=26$. The binary precision for the produced \\texttt{xml} files is 896 digits. \\texttt{SDPB} is run with the options \\texttt{-}\\texttt{-precision=896}, \\texttt{-}\\texttt{-detectPrimalFeasibleJump}, \\texttt{-}\\texttt{-detectDualFeasibleJump} and default values for other parameters. We refer to this set of parameters as ``\\!\\emph{A}''. Unless otherwise stated, our plots are ran with parameters ``\\!\\emph{A}''.\n\nFor some of the plots we used $\\texttt{m\\_max}=5, \\texttt{n\\_max}=7, \\texttt{l\\_max}=36, \\texttt{k\\_max}=42$ and\n$\\texttt{m\\_max}=6, \\texttt{n\\_max}=9, \\texttt{l\\_max}=36, \\texttt{k\\_max}=42$, referred to as ``\\!\\emph{B}'' and ``\\hspace{-1pt}\\emph{C}\\hspace{1pt}'' respectively. Lastly, we also used \\texttt{qboot} \\cite{Go:2020ahx}, with $\\Lambda=15$, $\\texttt{n\\_max}=500$, $\\nu\\texttt{\\_max}=25$ and $l=\\{0\\text{--}49,55,56, 59, 60, 64, 65, 69, 70, 74, 75,\\\\ 79, 80, 84, 85, 89, 90\\}$ referred to as ``\\hspace{-1pt}\\emph{D}\\hspace{0.5pt}''.\n\n\n\n\n\n\\end{appendices}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nIn recent years, data involving large numbers of features have become increasingly prevalent. Broadly speaking, there are two main approaches to analyzing such data: \\textit{large-scale testing} and \\textit{regression modeling}. The former entails conducting separate tests for each feature, while the later considers all features simultaneously in a single model.\nA major advance in large-scale testing has been the development of methods for estimating \\textit{local false discovery rates}, which provide an assessment of the significance of individual features while controlling the false discovery rate across the multiple tests.\nWe present here an approach for extending local false discovery rates to penalized regression models such as the lasso, thereby quantifying each feature's importance in a way that has been absent in the field of penalized regression until now.\n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=.95\\textwidth]{Fig1}\n\\caption{\\label{Fig:Example} The rows of this figure correspond to two simulated datasets. The column on the left shows the usual lasso coefficient path, while the column on the right displays our method's feature specific local false discovery rates ($\\mfdr$) along the lasso path. The triangles prior to the start of the mfdr path are the traditional local false discovery rate estimates resulting from large-scale testing. Along the mfdr path dashed lines indicate the portion of the path where a feature is inactive ($\\bh = 0$) in the model. The vertical dotted line shows the value of the penalty parameter, $\\lambda$, chosen by cross validation.}\n\\end{figure}\n\nFigure~\\ref{Fig:Example} shows two simulated examples to illustrate the information provided by local false discovery rates in the penalized regression context. In both examples, there are two features causally related to the outcome, two features that are correlated with the causal variables, and 96 features that are purely noise. The panels on the left show the lasso coefficient estimates returned by standard software packages such as \\texttt{glmnet} \\citep{Friedman2010}, while the panels on the right display the local false discovery rate, with a dotted line at $\\lambda_{\\CV}$, the point along the path which minimizes cross validation error. In the upper left panel, the model at $\\lambda_{\\CV}$ contains several noise variables, but the two causal features clearly stand out from others in the coefficient path. The $\\mfdr$ plot in the upper right panel confirms this visual assessment -- the two causal features have much lower false discovery rates than the other features. The dataset in the second row demonstrates a more challenging case. Here, it is not obvious from the coefficient path which features are significant and which may be noise. The $\\mfdr$ plot in the lower right panel lends clarity to this situation, showing at $\\lambda_{\\CV}$ that the truly causal features are considerably less likely to be false discoveries.\n\nThe two $\\mfdr$ paths of Figure~\\ref{Fig:Example} also illustrate the connection between the $\\mfdr$ approach and traditional large-scale testing approach to local false discovery rates. These univariate local false discovery rates are denoted by the triangles prior to the start of the $\\mfdr$ path, and are equivalent to the $\\mfdr$ estimates at the beginning of the $\\mfdr$ path when no features are active in the model. Initially, each method identifies both causal features, along with some of the correlated features, as important. However as $\\lambda$ decreases, and the causal features become active in the model, the regression-based $\\mfdr$ method reveals that the correlated features are only indirectly related to the outcome.\n\nHaving presented an initial case for the utility of $\\mfdr$ and an illustration of the connections it shares with both traditional local false discovery rates and lasso regression, we structure the remainder of the paper as follows: Section~\\ref{Sec:mfdr_back} gives a more formal introduction to false discovery rate approaches in the context of both large-scale testing and model based approaches to high dimensional data. Section~\\ref{Sec:method} introduces our lasso based $\\mfdr$ estimator in the linear regression setting and extends the approach to a more general class of penalized likelihood models including penalized logistic and Cox regression models. Section~\\ref{Sec:loc_sims} studies the $\\mfdr$ approach using simulation, comparing it to existing methods commonly used in high-dimensional analysis, and Section~\\ref{Sec:loc_case} explores two real data case studies where the method proves to be useful.\n\n\\section{Background}\n\\label{Sec:mfdr_back}\n\nIn the context of both large-scale testing and model-based approaches, this paper will focus on false discovery rates, a common approach to inference in high-dimensional data analysis. There are two main types of false discovery rates: tail-area approaches, which describe the expected rate of false discoveries for all features beyond a given threshold, and local approaches, which describe the density of false discoveries at a specific point. We adopt the general convention throughout, used by many other authors, of using Fdr to refer to tail-area approaches, and fdr to refer to local approaches, reflecting the traditional use of $F$ and $f$ to refer to distribution and density functions. Both Fdr and fdr have been well studied in the realm of large-scale testing. The seminal Fdr procedure \\citep{BH_1995} remains a widely popular approach to Fdr and has led to many extensions and related approaches \\citep[][and many others]{Storey2004, Genovese2004, Strimmer2008}. Using an empirical Bayes framework, \\citet{Efron2001} introduced the idea of local false discovery rates, a proposal which has also been extended in many ways \\citep[e.g.,][]{Muralidharan2010,Stephens2017}. In this section, we provide a brief overview of false discovery rate estimation from an empirical Bayes estimation perspective; for more information, including other ways of motivating FDR estimation and control, see \\citet{Farcomeni2008} and \\citet{Efron_book}.\n\nRecently, false discovery rates have been considered in the realm of high-dimensional modeling, although thus far this research has focused exclusively on tail-area approaches. The majority of this work has been concentrated on lasso regression \\citep{tibshirani_1996}, a popular modeling approach which naturally performs variable selection by using $L_1$ regularization. The issue of false discovery rate control is particularly important in this case, as false discoveries can be quite prevalent for lasso models \\citep{Su2017}.\n\nThe false discovery rate control provided by large-scale testing approaches is marginal in the sense that a feature $X_j$ is considered a false discovery only if that feature is marginally independent of the outcome $Y$: $X_j \\independent Y$. In regression, where many features are being considered simultaneously, the issue is more complicated and can involve various kinds of conditional independence. For example, we can adopt a \\textit{fully conditional} perspective, which considers a feature $X_j$ to be a false discovery if it is independent of the outcome conditional upon all other features: $X_j \\independent Y | X_{k \\neq j}$. Several approaches to controlling the fully conditional false discovery rate have been proposed, including procedures based on the bootstrap \\citep{Dezeure2017}, de-biasing \\citep{Javanmard2014}, sample splitting \\citep{Wasserman2009,Meinshausen2009}, and the knock-off filter \\citep{barber2015, candes2018}.\n\nAn alternative for penalized models is the \\textit{pathwise conditional} perspective. Pathwise approaches focus on the point in the regularization path at which feature $j$ first becomes active and condition only on the other variables present in the model (denote this set $M_j$) when assessing whether or not variable $j$ is a false discovery: $X_j \\independent Y | M_j$. The methods of \\citet{CovTest} and \\citet{Selective_Inference} used in conjunction with the sequential stopping rule of \\citet{GSell2016} allow for control over the pathwise conditional Fdr. \n\nLess restrictive approaches to false discovery rates for penalized regression models have also been proposed. \\citet{Breheny2019} developed an analytic method which bounds the \\textit{marginal} false discovery rate (mFdr) of penalized linear regression models. \\citet{Miller2019} extended this approach to a more general class of penalized likelihood models, while \\citet{HuangFDR} addressed a similar question using a Monte Carlo approach.\n\nIn this paper we combine the marginal perspective on false discoveries employed by \\citet{Breheny2019} and \\citet{Miller2019} with the idea of local false discovery rates. The resulting method provides a powerful new inferential tool for penalized regression models, allowing one to assess each individual feature's probability of having been selected into the model by chance alone.\n\n\\subsection{Large-scale testing, Fdr, and fdr}\n\nConsider data of the usual form $(\\y, \\X)$, where $\\y$ denotes the response for $i = \\{1, \\ldots, n\\}$ independent observations and $\\X$ is an $n$ by $p$ matrix containing the values of $j = \\{1, \\ldots, p\\}$ explanatory features. We presume that only a subset of the available features have a non-null relationship with the outcome, and the goal of our analysis is to correctly identify those important features.\n\nLet $\\beta_j$ denote the effect of feature $j$ in relation to $\\y$; this paper focuses on regression coefficients, but $\\beta_j$ could also represent a difference in means or some other measure. Large-scale univariate testing considers $p$ separate null hypotheses, $H_j:\\beta_j = 0$, each corresponding to a single feature, and conducts a univariate test on each of those hypotheses. These tests are usually performed by scaling the estimated effects $\\{\\bh_1, \\bh_2, \\ldots, \\bh_p\\}$ by their standard errors $\\{\\hat{s}_1, \\hat{s}_2, \\ldots, \\hat{s}_p\\}$ to obtain test statistics $\\{t_1, t_2, \\ldots, t_p\\}$ which are used to calculate p-values $\\{p_1, p_2, \\ldots, p_p\\}$. Alternatively, these test statistics can be converted to $z$-values defined by $z_j = \\Phi^{-1}(F_t(t_j))$, where $\\Phi$ is the standard normal CDF, so that the $z$-values for features with true null hypotheses will follow a N(0,1) distribution.\n\nThe false discovery rate is a tool for meaningfully aggregating the results of these tests while quantifying the expected proportion of false positives. In this paper, our primary focus is on local false discovery rates, or estimates of $\\Pr(H_j | \\bh_j, \\hat{s}_j) = \\Pr(H_j | z_j)$, the probability that feature $j$ has a null relationship with the outcome given the observed data.\n\nA natural framework for estimating this probability is to assume that features arise from two classes with prior probability $\\pi_0$ and $\\pi_1$, with $f_0(z)$ denoting the density of the $z$-values for features in the null class and $f_1(z)$ denoting the density for features in the non-null class, with the mixture density $f(z) = \\pi_0f_0(z) + \\pi_1f_1(z)$ giving the marginal distribution of $z$. In addition, let $\\cZ$ denote any subset of the real line, $F_0(\\cZ) = \\int_{\\cZ}^{}f_0(z)dz$ denote the probability of observing $z \\in \\cZ$ for a null feature and $F_1(\\cZ) = \\int_{\\cZ}^{}f_1(z)dz$ the probability for a non-null feature\n\nGiven the framework described in the preceding paragraph, applying Bayes' rule yields\n\\begin{align} \\label{eq:FdrBayes}\n\\Pr(\\text{Null}|z \\in \\cZ) = \\frac{\\pi_0F_0(\\cZ)}{F(\\cZ)} = \\Fdr(\\cZ),\n\\end{align}\nwhere $F(\\cZ) = \\pi_0F_0(\\cZ) + \\pi_1F_1(\\cZ)$. In a typical application, $\\cZ$ would denote a tail area such as $z > 3$ or $\\abs{z} > 3$, and would allow the analyst to control the Fdr through the choice of $\\cZ$.\n\nInstead of focusing on the tail area condition $z \\in \\cZ$, we may alternatively consider the limit as $\\cZ$ approaches the single point $z_j$, in which case the distribution functions become density functions and we have\n\\begin{align} \\label{eq:fdrzch2}\n\\Pr(\\text{Null}|z = z_j) = \\frac{\\pi_0f_0(z_j)}{f(z_j)} = \\fdr(z_j).\n\\end{align}\nThis quantity is typically referred to as the ``local'' false discovery rate, and describes feature $j$ specifically, as opposed to the collection of features whose $z$-statistics fall in $\\cZ$.\n\nThere are several important connections between Fdr and fdr \\citep{Efron_locfdr}. Of particular interest is the relationship\n\\begin{equation} \\label{eq:Fdr-fdr}\n\\Fdr(\\cZ) = \\Ex\\big(\\fdr(z)| z \\in \\cZ\\big).\n\\end{equation}\nIn words, the Fdr of the set of features whose normalized test statistics fall in the tail region $\\cZ$ is equal to the average fdr of the features in $\\cZ$. This ensures that selecting individual features using a threshold $fdr(z) < \\alpha$ also limits Fdr below $\\alpha$ for the entire set of features defined by the threshold. Additional detail on the links between Fdr and fdr can be found in \\citet{Efron_locfdr} and \\citet{Strimmer2008}.\n\nBroadly speaking, there are two main approaches to estimating local false discovery rates given the observed collection $\\{z_j\\}_{j=1}^p$. Recall that by construction, $f_0$ is the density function of the standard normal distribution; thus, $\\pi_0$ and $f$ are the only quantities that must be estimated. The first approach, originally proposed in \\citep{Efron_locfdr} but extended and modified by many authors since then, focuses on estimating the marginal density $f$. Replacing $\\pi_0$ with its upper bound of 1, and estimating $f(z)$ using any nonparametric density estimation method (e.g., kernel density estimation), we have\n\\begin{align} \\label{eq:fdrhat1}\n\\widehat{\\fdr}(z_j) = \\frac{f_0(z_j)}{\\hat{f}(z_j)}.\n\\end{align}\n\nAlternatively, one can explicitly model the mixture distribution of $z$ (in this approach, there is typically one null distribution and many non-null distributions), obtaining the estimates $\\hat{\\pi}_0, \\hat{\\pi}_1, \\hat{\\pi}_2, \\ldots$. The estimated local fdr is therefore\n\\begin{align} \\label{eq:fdrhat2}\n\\widehat{\\fdr}(z_j) = \\frac{\\hat{\\pi}_0f_0(z_j)}{\\sum_{k=0}^K\\hat{\\pi}_k\\hat{f}_k(z_j)},\n\\end{align}\nwhere $K$ is the number of non-null mixture components and $\\hat{f}$ must be estimated for the non-null components; estimation of $\\pi_k$ and $f_k$ is typically accomplished with maximum likelihood via EM algorithm. This approach was originally proposed by \\citet{Muralidharan2010}, but as with the marginal density approach, has been explored by many other authors since then. In particular, we focus on a mixture model proposed by \\citet{Stephens2017}, which requires that all non-null components have a mode of 0 (Stephens refers to this as the ``unimodal assumption''). One attractive aspect of this model, which is implemented in the R package \\texttt{ashr}, is that the resulting $fdr$ is a monotone function of as $z$ moves away from 0 in either direction; this is typically not true for marginal density estimates of the form \\eqref{eq:fdrhat1}.\n\nIn the sections that follow we refer to fdr estimates based on \\eqref{eq:fdrhat2} as the ``ashr'' approach, and fdr estimates found using \\eqref{eq:fdrzch2} as the ``density'' approach. Both ashr and density approaches estimate the posterior probability, given the observed data, that feature $j$ is a false discovery, but their assumptions can lead to different estimates. We further discuss the relative strengths and weaknesses of these two approaches in Section~\\ref{Sec:density_est}.\n\n\\subsection{Penalized regression and mFdr}\n\\label{Sec:background_mfdr}\n\nIn contrast with the univariate nature of large-scale testing, regression models simultaneously relate all of the explanatory features in $\\X$ with $\\y$ using a probability model involving coefficients $\\bb$. In what follows we assume the columns of $\\X$ are standardized such that each variable has a mean of $0$ and $\\sum_i \\x_{ij}^2 = n$. The fit of a regression model can be summarized using the log-likelihood, which we denote $\\ell(\\bb|\\X,\\y)$. In the classical setting, $\\bb$ is estimated by maximizing $l(\\bb|\\X,\\y)$. However, this approach is unstable when $p > n$ unless an appropriate penalty is imposed on the size of $\\bb$.\nIn the case of the lasso penalty, estimates of $\\bb$ are found by minimizing the objective function:\n\\al{eq:obj}{\nQ(\\bb|X,\\y) = -\\frac{1}{n} \\ell(\\bb|X,\\y) + \\lambda||\\bb||_1\n}\n\nThe maximum likelihood estimate is found by setting the score, $\\u(\\bb) = \\nabla \\ell(\\bb|\\X,\\y)$, equal to zero. The lasso estimate, $\\bbh$, can be found similarly, although allowances must be made for the fact that the penalty function is typically not differentiable. These penalized score equations are known as the Karush-Kuhn-Tucker (KKT) conditions in the convex optimization literature, and are both necessary and sufficient for a solution $\\bbh$ to minimize $Q(\\bb|\\X,\\y)$. \n\nAn important property of the lasso is that it naturally performs variable selection. The lasso estimates are sparse, meaning that $\\bh_j = 0$ for a large number of features, with only a subset of the available features being active in the model. The regularization parameter $\\lambda$ governs the degree of sparsity with smaller values of $\\lambda$ leading to more variables having non-zero coefficients.\n\nThe KKT conditions can be used to to develop an upper bound for the number of features expected to be selected in a the lasso model by random chance. Heuristically, if feature $j$ is marginally independent of $\\y$, then $Pr(\\bh_j \\neq 0)$ is approximately equal to $Pr(\\tfrac{1}{n}\\abs{u_j(\\bb)} > \\lam)$, where $\\u_j$ denotes the $j^{th}$ component of the score function (gradient of the log-likelihood). Classical likelihood theory provides asymptotic normality results and allows for estimation of this tail probability, which in turn provides a bound on the mFdr. For additional details and proofs, see \\citet{Miller2019}.\n\nThis approach provides an overall assessment of model selection, but it does not offer any specific information about individual features.\nIt is often the case that among the selected features, some appear to be clearly related to the outcome while others are of borderline significance.\nFor example, as suggested by \\eqref{eq:Fdr-fdr}, we may select two features, one with a 1\\% fdr and the other with a 39\\% fdr, but the overall Fdr of the model is 20\\%. Providing this level of feature-specific inference is the major motivation for estimating local false discovery rates for penalized regression.\n\nThe ability to provide feature-specific false discovery rates also allows one to overcome the tension between predictive accuracy and selection reliability. For lasso models, it is typically the case that the number of features that can be selected under Fdr restrictions is much smaller than the number of active features in model that achieves maximum predictive performance as determined by cross-validation.\nThis poses something of a dilemma, as we must choose between a model with sub-optimal predictions and one with a high proportion of false discoveries.\nLocal fdr, however, allows us to use the most predictive model while retaining the ability to identify features that are unlikely to be false discoveries.\n\n\\section{Estimating mfdr}\n\\label{Sec:method}\n\nWe begin by mentioning that the elements of $\\bbh$ are not directly suitable for local false discovery rate estimation. In particular, for most choices of $\\lam$, $\\bh_j$ is exactly zero for many features, making it impossible to construct statistics with a N(0,1) distribution under the null. Instead, we use the KKT conditions, which mathematically characterize feature selection at a given value of $\\lambda$, to construct normally distributed statistics appropriate for the given model. Section~\\ref{Sec:linear} addresses linear regression, while Section~\\ref{Sec:glm_cox} addresses GLM and Cox regression models.\n\n\\subsection{Linear regression} \n\\label{Sec:linear}\n\nConsider the linear regression setting:\n\\begin{align*}\n\\y = \\X \\bb + \\pmb{\\epsilon}, \\qquad \\epsilon_i \\sim N(0, \\sigma^2).\n\\end{align*}\nAs mentioned in Section~\\ref{Sec:background_mfdr}, the lasso solution, $\\bbh$, is mathematically characterized by the KKT conditions, which are given by \\citep{lasso_kkt}:\n\\begin{alignat*}{2}\n\\frac{1}{n}\\x_j^T(\\y - \\X\\bbh) &= \\lambda \\textrm{ sign}(\\bh_j) \\qquad & & \\text{for all } \\bh_j \\ne 0 \\\\\n\\frac{1}{n}\\x_j^T(\\y - \\X\\bbh) &\\leq \\lambda & & \\text{for all } \\bh_j = 0.\n\\end{alignat*}\n\nWe define the partial residual as $\\rj = \\y - \\X_{-j}\\bbh_{-j}$ where the subscript $-j$ indicates the removal of the $j^{th}$ feature. Using this definition it follows directly from the KKT conditions that:\n\\begin{alignat*}{2}\n\\frac{1}{n}|\\x_j^T \\rj| &> \\lambda \\qquad && \\text{for all } \\bh_j \\ne 0 \\\\\n\\frac{1}{n}|\\x_j^T \\rj| &\\leq \\lambda && \\text{for all } \\bh_j = 0.\n\\end{alignat*}\nThe quantity $\\frac{1}{n}\\x_j^T \\rj$ governs the selection of the $j^{th}$ feature: if its absolute value is large enough, relative to $\\lambda$, feature $j$ is selected.\nIn this manner, $\\frac{1}{n}\\x_j^T \\rj$ can be considered analogous to a test statistic in the hypothesis testing framework.\n\n\nIn the special case of orthonormal design where $\\frac{1}{n}\\X^T \\X = \\I$, it is straightforward to show that $\\frac{1}{n}\\x_j^T \\rj \\sim N(\\beta_j, \\sigma^2\/n)$ \\citep{Breheny2019}. Under the null hypothesis that $\\beta_j = 0$, this result can be used to construct the normalized test statistic\n\\begin{align} \\label{eq:fdr_linearch2}\nz_j = \\frac{\\frac{1}{n}\\x_j^T \\rj}{\\hat{\\sigma}\/\\sqrt{n}},\n\\end{align}\nwhere $\\hat{\\sigma}$ is an estimate of $\\sigma$; for the sake of simplicity, we use the residual sum of squares divided by the model degrees of freedom \\citep{zou2007}, but many other possibilities exist \\citep{reid2016}. These statistics are then used to estimate local false discovery rates using either \\eqref{eq:fdrhat1} or \\eqref{eq:fdrhat2}.\n\nIn practice, the design matrix will not be orthonormal and the result $\\frac{1}{n}\\x_j^T \\rj \\sim N(\\beta_j, \\sigma^2\/n)$ will not hold exactly. Nevertheless, it still holds approximately under reasonable conditions. To understand these conditions we explore the relationship between $\\frac{1}{n}\\X^T \\X$ and $z_j$:\n\\begin{equation}\n\\begin{aligned}\n\\label{eq:remainder}\n\\frac{1}{n}\\x_j^T \\rj &= \\frac{1}{n}\\x_j^T (\\X\\bb + \\bep - \\X_{-j}\\bbh_{-j}) \\\\\n&= \\frac{1}{n}\\x_j^T\\bep + \\beta_j + \\frac{1}{n}\\x_j^T \\X_{-j} (\\bb_{-j} - \\bbh_{-j}).\n\\end{aligned}\n\\end{equation}\nThe component $\\frac{1}{n}\\x_j^T\\bep + \\beta_j$ is unaffected by the structure of $\\frac{1}{n}\\X^T \\X$; thus the estimator in \\eqref{eq:fdr_linearch2} will be accurate in situations where the final term, $\\frac{1}{n}\\x_j^T \\X_{-j} (\\bb_{-j} - \\bbh_{-j})$, is negligible (in orthonormal designs, this term is exactly zero). If feature $j$ is independent of all other features, then $\\frac{1}{n}\\x_j^T \\X_{-j}$ will converge to zero as $n$ increases, making the term asymptotically negligible provided $\\sqrt{n}(\\bb_{-j} - \\bbh_{-j})$ is bounded in probability.\n\nIf pairwise correlations exist between features, $\\frac{1}{n}\\x_j^T \\X_{-j}$ will not converge to zero, and the null distribution of $z_j$ will not follow a standard normal distribution; in particular, as discussed in \\citet{Breheny2019}, its distribution will have thinner tails than a standard normal distribution. This will causes the local mfdr estimates to be somewhat conservative in the presence of strong correlation; this phenomenon is explored in depth in Section~\\ref{Sec:loc_sims}.\n\nWhen $\\frac{1}{n}\\x_j^T \\rj \\sim N(\\beta_j, \\sigma^2\/n)$ holds exactly, the mfdr estimator of \\eqref{eq:fdr_linearch2} shares an important relationship with the mFdr estimator proposed by \\citet{Breheny2019}, captured in the following theorem, whose proof appears in the appendix:\n\n\\begin{theorem}\n\\label{Thm:Efdr}\nLet $\\cM_{\\lam}$ denote the set of nonzero coefficients selected by a lasso model, and let $C_j = n^{-1}\\x_j\\rj$ denote the random variable governing the selection of a given feature. If $C_j$ has density $g = \\pi_0 g_0 + (1 - \\pi_0) g_1$, where $g_0$ is the $\\Norm(0, \\sigma^2\/n)$ density, then\n\\begin{align*}\n\\mFDR(\\cM_\\lambda) = \\EX\\left\\{\\mfdr(\\tfrac{C_j}{\\sigma\/\\sqrt{n}}) | j \\in \\cM_\\lambda\\right\\}.\n\\end{align*}\n\\end{theorem}\n\nNoting that $z_j$ from \\eqref{eq:fdr_linearch2} is a standardized version of $c_j$, dividing by $\\sigma\/\\sqrt{n}$ in order to have unit variance, Theorem~\\ref{Thm:Efdr} states that, on average, the marginal false discovery rate of a model is the average local false discovery rate of its selections. Alternatively, this result implies that the expected number of false discoveries in a model can be decomposed into the sum of each selected feature's mfdr.\n\nThe above theorem assumes known values for $\\pi_0$, $\\sigma$, and $f$; when these quantities are estimated from the data, the equality no longer holds. In our experience, the average mfdr is typically close to the mFdr, although this depends on how the above quantities are estimated.\n\n\\subsection{GLM and Cox models}\n\\label{Sec:glm_cox}\n\nWe now consider the more general case where $\\y$ need not be normally distributed. Specifically we focus our attention on binary outcomes (logistic regression) and survival outcomes (Cox regression), although the approach is general and can also be applied to other likelihood based models.\n\nSimilar to the linear regression setting, we can develop a local false discovery rate estimator by studying minimization of the objective function, $Q(\\bb|X,\\y)$, as defined in \\eqref{eq:obj}. When $\\y$ is not normally distributed, $\\ell(\\bb|X,\\y)$ is no longer a quadratic function. However, we can construct a quadratic approximation by taking a Taylor series expansion of $\\ell(\\bb|X,\\y)$ about a point $\\bbt$. In the context of this approach it is useful to work in terms of the linear predictor $\\be = \\X\\bb$ (and $\\tilde{\\be} = \\X\\bbt$), noting that we can equivalently express the likelihood in terms of $\\be$ such that:\n\\begin{align*}\n\\ell(\\bb|X,\\y) & \\approx l(\\bbt) + (\\bb - \\bbt)^T l'(\\bbt) + \\frac{1}{2}(\\bb - \\bbt)^T l''(\\bbt) (\\bb - \\bbt) \\\\\n& = \\frac{1}{2}(\\yt - \\be)^T f''(\\bet) (\\yt - \\be) + \\text{const}.\n\\end{align*}\nHere $\\yt = \\bet - f''(\\bet)^{-1} f'(\\bet)$ serves as a pseudo-response in the weighted least squares expression.\nThe KKT conditions here are very similar to those in the linear regression setting, differing only by the inclusion of a weight matrix $\\W = f''(\\bet)$ and the fact that $\\y$ has been replaced by $\\yt$. \n\nProceeding similarly to Section~\\ref{Sec:linear}, we define the partial pseudo-residual $\\rj = \\yt - \\X_{-j}\\bbh_{-j}$, which implies:\n\\begin{alignat*}{2}\n\\frac{1}{n}|\\x_j^T \\W \\rj| &> \\lambda \\qquad & &\\text{for all } \\bh_j \\ne 0 \\\\\n\\frac{1}{n}|\\x_j^T \\W \\rj| &\\leq \\lambda & &\\text{for all } \\bh_j = 0.\n\\end{alignat*}\n\\citet{Miller2019} show that, under appropriate regularity conditions,\n\\begin{align}\\label{eq:glm_fdr}\nz_j=\\frac{\\tfrac{1}{n}\\x_j^T\\W\\rj}{\\hat{s}_j\/\\sqrt{n}} \\inD N(0, 1),\n\\end{align}\nwhere $\\hat{s}_j = \\sqrt{\\x_j^T\\W\\x_j\/n}$. As in Section~\\ref{Sec:linear}, these statistics can be used to estimate local false discovery rates using either \\eqref{eq:fdrhat1} or \\eqref{eq:fdrhat2}.\n\nFor the most part, the regularity conditions required for \\eqref{eq:glm_fdr} to hold are the same as those required for asymptotic normality in classical likelihood theory, with one additional requirement. Just as the mfdr estimator in Section~\\ref{Sec:linear} holds under feature independence, the estimator in Equation~\\ref{eq:glm_fdr} relies on an assumption of vanishing correlation; consequently it will be accurate in the case of independent features but tends to result in conservative false discovery rate estimates when features are correlated. The assumption of vanishing correlation is unlikely to be literally true in practice; its purpose is to establish a hypothetical worst-case scenario in terms of false discovery selection so that a fdr can be estimated. We investigate how robust the estimator is to this assumption in Section~\\ref{Sec:loc_sims}.\n\n\\section{Simulation studies}\n\\label{Sec:loc_sims}\n\nIn this section we conduct a series of simulations studying the behavior of the mfdr estimators resulting from \\eqref{eq:fdr_linearch2} and \\eqref{eq:glm_fdr}. We investigate both the internal validity of the method in terms of whether the estimate accurately reflects the probably that a feature is a false discovery as well as compare the results of mfdr-based inference with other approaches to inference in the high-dimensional setting.\n\nFor the mfdr approach, we present results for two different values of $\\lambda$. The first, $\\lambda_{\\CV}$, characterizes the model with the lowest cross-validated error. The second, $\\lambda_{1\\SE}$, characterizes the most parsimous model within one standard error of the lowest cross-validated error. Unless otherwise indicated, we estimate mfdr using the approach of \\citet{Stephens2017} as implemented in the R package \\texttt{ashr}.\n\nWe generate data from three models: linear regression, logistic regression, and Cox regression. For each model we present results for two data-generating scenarios we refer to as ``Assumptions Met'', where the underlying assumption of independent features and furthermore $n > p$, which improves asymptotic approximations, and ``Assumptions Violated'', where features are correlated in a manner consistent with real data and $p > n$.\n\n\\textbf{Assumptions Met:} In this scenario, $n > p$ and all features are independent of each other. Both of these factors should lead to a small remainder term in \\eqref{eq:remainder}.\n\n\\begin{itemize}\n\\item $n = 1000, p=600$\n\\item Covariate values $x_{ij}$ independently generated from the standard normal distribution.\n\\item Response variables are generated as follows:\n\\begin{itemize}\n\\item Linear regression, $\\y = \\X\\bb + \\bep$ where $\\epsilon_i \\sim N(0,\\sigma^2)$, $\\bb_{1:60} = 4$, and $\\bb_{61:600} = 0$, and $\\sigma = \\sqrt{n}$\n\\item Logistic regression, $y_i \\sim \\textrm{Bin}\\bigg(1, \\pi_i = \\frac{\\exp(\\x_i^T \\bb)} {1 + \\exp(\\x_i^T \\bb)}\\bigg)$, $\\bb_{1:60} = .15$, and $\\bb_{61:600} = 0$\n\\item Cox regression, $y_i \\sim \\textrm{Exp}\\big(\\exp(\\x_i^T \\bb)\\big)$, $\\bb_{1:60} = .15$, and $\\bb_{61:600} = 0$, and 10\\% random censoring\n\\end{itemize}\n\\end{itemize}\n\n\n\\textbf{Assumptions Violated:} In this scenario, we impose an association structure motivated by the causal diagram below.\n\\begin{center}\n\\begin{tikzpicture}[node distance=1cm]\n\n\\node(b)[text centered] {$B$};\n\\node(u)[below of = b, text centered] {$ $};\n\\node(a)[left of = u, text centered, xshift = -1.5cm] {$A$};\n\\node(c)[right of = u, text centered, xshift = 1.5cm] {$C$};\n\\node(y)[below of = u, text centered] {$Y$};\n\\draw [arrow] (a) -- (b);\n\\draw [arrow] (a) -- (y);\n\n\\end{tikzpicture} \\\\\n\\end{center}\nHere, variable $A$ has a direct causal relationship with the outcome variable $Y$, variable $B$ is correlated with $Y$ through its relationship with $A$, but is not causally related, and variable $C$ is unrelated to all of the other variables and the outcome. In terms of the false discovery perspectives introduced in Section~\\ref{Sec:mfdr_back}, all of the perspectives agree that $A$ would never be a false discovery and that $C$ would always be a false discovery. However, selecting $B$ is considered a false discovery by the fully conditional perspective, but not the marginal perspective. From the pathwise conditional perspective, whether $B$ is a false discovery depends on whether $A$ has entered the model or not.\n\n\\begin{itemize}\n\\item $n = 200, p=600$\n\\item Covariates generated with the following dependence structure:\n\\begin{itemize}\n\\item 6 causative features ($A$), which are independent of each other\n\\item 54 correlated features ($B$), grouped such that 9 are related to each causative feature with $\\rho = 0.5$\n\\item 540 noise features ($C$), which are correlated with each other by an autoregressive correlation structure where $\\textrm{Cor}(\\x_j, \\x_k) = 0.8^{|j - k|}$\n\\end{itemize}\n\\item Response variables ($Y$) are generated from the same models described in the Assumptions Met scenario; however, $\\bb$ differs to reflect the change in sample size:\n\\begin{itemize}\n\\item Linear regression, $\\bb_{1:6} = (6,-6,5,-5,4,-4)$, and $\\bb_{7:600} = 0$, and $\\sigma = \\sqrt{n}$\n\\item Logistic regression, $\\bb_{1:6} = (1.1,-1.1,1,-1,.9,-.9)$, and $\\bb_{7:600} = 0$\n\\item Cox regression, $\\bb_{1:6} = (.6, -.6, .5, -.5, .4, -.4)$, and $\\bb_{7:600} = 0$, and 10\\% random censoring\n\\end{itemize}\n\\end{itemize}\n\nTo summarize each of these scenarios in terms of the diagram above, the Assumptions Met scenario consists of 60 features akin to variable $A$, 0 features akin to variable $B$, and 540 features akin to variable $C$, while the Assumptions Violated scenario consists of 6 features akin to variable $A$, 54 features akin to variable $B$, and 540 features akin to variable $C$. The Assumptions Violated scenario also imposes an autoregressive correlation structure on the noise features, undermining the assumption of vanishing correlation.\n\nFor comparison, we also include results for the traditional univariate approach to fdr throughout our simulations, defining the univariate procedure to consist of fitting a univariate regression model to each of the $j \\in \\{1, \\ldots, p\\}$ features, extracting the test statistic, $t_j$, corresponding to the test on the hypothesis that $\\beta_j = 0$, then normalizing these test statistics such that $z_j = \\Phi^{-1}(Pr(T < t_j))$. The \\texttt{ashr} package was then used to calculate local false discovery rates.\n\n\\subsection{Calibration}\n\nHow well do our proposed estimates reflect the true probability that a feature is purely noise (i.e., unrelated to the outcome either directly or indirectly)? We address this question through calibration plots comparing our mfdr estimates to the observed proportion of noise features across the full spectrum of false discovery rates. For example, an estimate of $0.2$ is well-calibrated if 20\\% of features with $\\widehat{\\text{mfdr}} = .2$ are observed to be false discoveries. \n\n\\begin{figure}[hbt]\n\\centering\n\\includegraphics[width=.85\\textwidth]{Fig2}\n\\caption{\\label{Fig:Calibr} The expected proportion of false discoveries at a given estimated local false discovery rate after smoothing for the linear regression setting. Univariate estimates along with two lasso estimates (at different values of $\\lam$) are shown.\n \n}\n\\end{figure}\n\nFigure~\\ref{Fig:Calibr} displays results for the linear regression setting for both scenarios.\nStrip plots of individual mfdr estimates are provided along with smoothed estimates of the calibration relationships and a 45 degree line is provided for reference.\nSimilar results are available in Supplemental Material for logistic and Cox models; the same patterns hold for all three models.\n\nWhen all assumptions are met, the mfdr estimates are very accurate at both $\\lambda_{\\CV}$ and $\\lambda_{1\\SE}$, showing essentially perfect correspondence with the empirical proportion of false discoveries. The traditional univariate method also appears to be very well calibrated, only slightly underestimating the proportion of noise features at the upper end of the mfdr spectrum.\n\nIn the Assumptions Violated scenario, the lasso mfdr estimates tend to be accurate in the regions near 0 and 1, which are typically of greatest practical interest. In between, the estimates based on $\\lambda_{\\CV}$ are conservative. For example, among features with an estimated mfdr of 35\\%, only about 20\\% were actually noise features. The lasso mfdr estimates based on $\\lambda_{1\\SE}$ and the univariate estimates, on the other hand, slightly underestimated the probability that a given feature was a false discovery.\n\nIt is worth noting that the presence of correlated ``B'' variables in the Violated scenario complicates the assessment of calibration, as features cannot be unambiguously separated into noise and signal. Since our method is based upon the marginal perspective, we do not treat the ``B'' variables as false (noise) discoveries here. However, as one would expect, they tend to have much higher mfdr estimates than the causative (``A'') variables; this should be kept in mind when interpreting these calibration plots.\n\nThe three methods depicted in Figure~\\ref{Fig:Calibr} show markedly different potential with respect to classifying features as noise versus signal. At $\\lambda_{\\CV}$, the mfdr estimates of noise features are tightly clustered near 1 and the mfdr estimates of causal features are tightly clustered near 0, as seen in the strip plots at the top and bottom of the figure, respectively. In contrast, the traditional univariate method yields far more intermediate estimates for variables of both types. In other words, the lasso estimates allow one to much more confidently identify signals compared to a univariate analysis. As expected, the results for $\\lambda_{1\\SE}$ fall somewhere in between the results at $\\lambda_{\\CV}$ and the univariate approach.\n\n\\begin{table}[!htb]\n\\centering\n\\caption{\\label{Table:hard} Local false discovery rate accuracy results for the Assumptions Violated scenario. Features are binned based upon their estimated fdr and the observed proportion of noise variables in each bin is reported in the body of the table for each method.}\n\\begin{tabular}{ l c c c c c }\n\\hline\nLinear & (0, 0.2] & (0.2, 0.4] & (0.4, 0.6] & (0.6, 0.8] & (0.8, 1] \\\\\n\\hline\nUnivariate $\\widehat{\\text{fdr}}$ & 0.11 & 0.40 & 0.63 & 0.85 & 0.95 \\\\ \n$\\widehat{\\mfdr}$ at $\\lambda_{1\\SE}$ & 0.03 & 0.36 & 0.60 & 0.84 & 0.92 \\\\ \n$\\widehat{\\mfdr}$ at $\\lambda_{\\CV}$ & 0.03 & 0.25 & 0.32 & 0.49 & 0.91 \\\\ \n\\hline\nLogistic & (0, 0.2] & (0.2, 0.4] & (0.4, 0.6] & (0.6, 0.8] & (0.8, 1] \\\\\n\\hline\nUnivariate $\\widehat{\\text{fdr}}$ & 0.15 & 0.45 & 0.68 & 0.87 & 0.95 \\\\ \n$\\widehat{\\mfdr}$ at $\\lambda_{1\\SE}$ & 0.00 & 0.05 & 0.15 & 0.29 & 0.91 \\\\ \n$\\widehat{\\mfdr}$ at $\\lambda_{\\CV}$ & 0.00 & 0.00 & 0.07 & 0.31 & 0.91 \\\\ \n\\hline \nCox & (0, 0.2] & (0.2, 0.4] & (0.4, 0.6] & (0.6, 0.8] & (0.8, 1] \\\\\n\\hline\nUnivariate $\\widehat{\\text{fdr}}$ & 0.08 & 0.39 & 0.65 & 0.86 & 0.95 \\\\ \n$\\widehat{\\mfdr}$ at $\\lambda_{1\\SE}$ & 0.00 & 0.00 & 0.07 & 0.29 & 0.91 \\\\ \n$\\widehat{\\mfdr}$ at $\\lambda_{\\CV}$ & 0.00 & 0.00 & 0.00 & 0.26 & 0.91 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nTable~\\ref{Table:hard} displays an alternative representation of the calibration results for the Assumptions Violated scenario. Here, features are sorted into five equally spaced bins based upon their estimated local false discovery rate under a given procedure. Within in each bin we calculate the proportion of noise (i.e., ``C'') variables; for a well-calibrated estimator, the proportion of noise features within a bin should remain within the range of the bin. For example, in the linear regression case, 25\\% of the features with $\\widehat{\\mfdr}$ estimates between 0.2 and 0.4 at $(\\lam_\\CV)$ were truly noise. Overall, the mfdr estimates at $\\lam_\\CV$ were always either well-calibrated or conservative, with the estimates for logistic and Cox regression more conservative than those for linear regression. In comparison, the univariate approach occasionally underestimated the false discovery probability. Again, results for $\\lam_{1\\SE}$ were intermediate between the other two approaches.\n\n\\subsection{Power compared to univariate fdr}\n\nIn this section we compare the number of mfdr feature selections of each type, $A$, $B$, and $C$, for the two previously described scenarios to the selections based on univariate local false discovery rates.\n\nUsing a local false discovery rate threshold of $0.1$, Figure~\\ref{Fig:power} shows that the regression-based mfdr approach leads to increased selection of causally important variables. At $\\lambda_{\\CV}$, the lasso mfdr approach selects 61\\% more causal (``A'') variables than the univariate approach in the Assumptions Met scenario. In the Assumptions Violated scenario, lasso mfdr ($\\lam_\\CV$) remains slightly more powerful than the univariate approach, selecting on average 5\\% more $A$ variables (3.63 vs. 3.45) than the univariate approach.\n\n\\begin{figure} [htb]\n\\centering\n\\includegraphics[width=.8\\textwidth]{Fig3}\n\\caption{\\label{Fig:power} The average number of features of each type with estimated local false discovery rates of less than 0.10 for each method in the two scenarios.}\n\\end{figure}\n\nIn addition to improving the power to detect causal variables, the mfdr approach drastically reduces the amount of correlated, non-causal features with low local false discovery rate estimates. This is most notable when comparing mfdr at $\\lambda_{\\CV}$ with univariate fdr, where the number of $B$ variables with fdr $< 0.1$ is \\textit{36 times higher} for the univariate approach.\nFurthermore, in the presence of correlated (``B'') features, a univariate approach also selects 6.6 times as many features that are purely noise (``C'') when compared to mfdr at $\\lambda_{\\CV}$.\nThus, the penalized regression mfdr approach proposed here results in selecting increased numbers of causally important variables while also reducing the number of correlated and noise features selected.\n\nThe results shown in Figure~\\ref{Fig:power} use a somewhat arbitrary threshold of $0.1$.\nTo illustrate performance over the entire spectrum of classification thresholds, we also performed an ROC analysis. Specifically, we considered the number of false positives, defined as noise features classified as significant at a given threshold, and false negatives, defined as important features classified as noise at a given threshold, and assessed discriminatory power using the area under the ROC curve (AUC). For the Assumptions Violated scenario, we omit $B$ variables from these calculations.\nAt $\\lambda_{\\CV}$, the mfdr approach results in average AUC values of 0.936 and 0.990, respectively, for the Met and Violated Scenarios. This is an improvement over the average AUC values of 0.908 and 0.966 for the univariate procedure, further demonstrating the advantages of regression-based mfdr over univariate approaches.\n\n\n\\subsection{Comparisons with other inferential approaches for penalized regression}\n\nTheorem~\\ref{Thm:Efdr} indicates that mfdr can be used to control mFdr, motivating a comparison of the mfdr method and existing Fdr control approaches for lasso regression models. In this simulation, we use mfdr to select features with $\\widehat{\\mfdr} < 0.1$. This is a conservative approach to mFdr control -- recalling the relationship between mfdr and mFdr given in Theorem~\\ref{Thm:Efdr}, if $\\widehat{\\mfdr} < 0.1$ for every feature, then the average mfdr will be $\\ll 0.1$ -- but serves to illustrate the most salient differences between local mfdr and other inferential approaches.\n\nWe compare our results with the selective inference approach of \\citet{Selective_Inference} using the ForwardStop rule \\citep{GSell2016}, which controls the pathwise-wise Fdr at 10\\%, the repeated sample splitting method as implemented by the {\\tt hdi} package \\citep{Dezeure2015}, which controls the fully conditional Fdr at 10\\%, and the (Model-X) knock-off filter method implented in the {\\tt knockoff} package \\citep{candes2018}, which also controls the fully conditional Fdr at 10\\%.\n\n\\begin{table}[!htb]\n\\centering\n\\caption{\\label{Tab:SelectiveInference} Simulation results comparing the average number of selections of causal, correlated, and noise variables, as well as the proportion of noise variable selections, for various model-based false discovery rate control procedures. The ``exact'', ``spacing'', ``mod-spacing'', and ``covtest'' methods are related tests performed by the {\\tt selectiveInference} package. Noise rate here refers to the fraction of selected variables that come from the ``Noise'' group of features (i.e., the mFdr).}\n\\begin{tabular}{l r r r r @{\\hskip 0.5in} r r r}\n& \\multicolumn{4}{c}{Assumptions Violated} & \\multicolumn{3}{c}{Assumptions Met}\\\\\n\\hline\n& Causal & Correlated & Noise & Noise rate & Causal & Noise & Noise rate\\\\\n& (of 6) & (of 54) & (of 540) & (mFdr) & (of 60) & (of 540) & (mFdr)\\\\\n\\hline\nmfdr (CV) & 3.84 & 0.60 & 0.14 & 3.0 \\% & 16.25 & 0.6 & 4.0 \\% \\\\\nmulti-split & 2.03 & 0.05 & 0 & 0\\% & 10.53 & 0 & 0 \\% \\\\\nknock-off & 0.42 & 0.16 & 0.02 & 5.0 \\% & 3.50 & 0 & 0\\% \\\\\nexact & 0.84 & 0.060 & 0 & 0\\% & 0.84 & 0 & 0\\%\\\\\nspacing & 1.45 & 0.070 & 0 & 0 \\% & 0.98 & 0 & 0\\% \\\\\nmod-spacing & 1.45 & 0.10 & 0 & 0 \\% & 0.98 & 0 & 0\\% \\\\\ncovtest & 1.43 & 0.10 & 0 & 0 \\% & 0.97 & 0 & 0\\% \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nTable~\\ref{Tab:SelectiveInference} shows the average number of causal and correlated features selected along with the false discovery rate for the aforementioned approaches. As they are intended to, the conditional Fdr approaches greatly limit the number of correlated, non-causal (type B) features present in the model compared to the proposed marginal fdr approach. However, this added control comes at a considerable cost in terms of power to discover causal features of interest. Overall, using pathwise or fully conditional approaches resulted in discovering at least 50\\% fewer causative features, and with the conditional approaches discovering up to 95\\% fewer features than the marginal approach in some cases. These results demonstrate that conditional Fdr control approaches tend to be much more conservative than marginal approaches, even in moderate dimensions.\n\n\\subsection{Differences in the ashr and density approaches}\n\\label{Sec:density_est}\n\nIn Sections \\ref{Sec:mfdr_back} and \\ref{Sec:method}, we discussed two methods of local false discovery rate estimation, the ``ashr'' approach, which estimates fdr using the posterior distribution of feature effects under a mixture model \\citep{Stephens2017}, and the ``density'' approach, which estimates fdr as the ratio of the theoretical null to an empirically estimated mixture density. It is beyond the scope of this paper to exhaustively review the differences of these approaches, but we did explore their performance in the simulations described in section~\\ref{Sec:loc_sims}. Overall, the approaches provide very similar results in most cases (see Supplemental Material), although each one has strengths and weaknesses.\n\nThe primary weakness of the mixture modeling approach is that when a very high proportion of features are null (i.e., when $\\pi_0 \\approx 1$), there is very little information with which to estimate the non-null mixture components. In particular, it is not uncommon for the mixture modeling approach used by \\texttt{ashr} to estimate $\\pi_0=1$ even when non-null features are present, especially when the signal is weak or the sample size is small. Certainly, this does not invalidate the procedure -- one could argue that it is wise to be cautious in the presence of weak signals -- although it does contribute, in some scenarios, to the method being conservative.\n\n\\begin{figure} [htb]\n\\centering\n\\includegraphics[width=.95\\textwidth]{Fig5}\n\\caption{\\label{Fig:density_comparison}The relationship between feature $z$-values and their estimated local false discovery rates in the assumptions violated simulation scenario when using the ashr and density approaches. The right panel shows the empirical distribution of $z$-values (histogram and rug plot), the estimated mixture density using kernel density estimation (solid line), and the theoretical N(0,1) null distribution (dashed line). The left panel shows the relationship between $z$-values and mfdr estimates for each method.}\n\\end{figure}\n\nThe primary weakness of the density modeling approach is that the resulting mfdr is not a monotone function of the normalized test statistic $z$ and as a result, is somewhat prone to artifacts as shown in Figure~\\ref{Fig:density_comparison}. The figure depicts results from a single dataset simulated under the assumptions violated scenario. Due to the numerous pairwise correlations between noise features the null distribution of $z$ is more tightly concentrated around zero than the theoretical $N(0,1)$ suggests. For the density modeling approach, this has the undesirable consequence that features with $z$-values near zero have fdr estimates smaller than those of features with $z$-values further away from zero. The mixture modeling approach of \\texttt{ashr}, which uses unimodal mixture components, protects against this and leads to a monotonic relationship between a feature's local false discovery rate estimate and how far its $z$-value is from zero.\n\nFor this reason, we consider it safer in general to use the \\texttt{ashr} approach and make it the default method in the \\texttt{ncvreg} package (provided the \\texttt{ashr} package is installed). However, as the figure suggests, the two approaches tend to produce very similar results in the tails of the $z$ distribution. If one is careful to avoid artifacts away from the tails, both methods provide valid results; in the next section, we present analyses of real data using both approaches.\n\n\\section{Case studies}\n\\label{Sec:loc_case}\n\n\\subsection{BRCA1 Gene Expression}\n\nOur first case study examines the gene expression of breast cancer patients from The Cancer Genome Atlas (TCGA) project. The data set is publicly available at \\url{http:\/\/cancergenome.nih.gov}, and consists of 17,814 gene expression measures for 536 subjects. One of these genes, BRCA1, is a tumor suppressor that plays a critical role in the development of breast cancer. When BRCA1 is under-expressed the risk of breast cancer is significantly increased, which makes genes that are related to BRCA1 expression interesting candidates for future research.\n\nOne would expect a large number of genes to have indirect relationships with BRCA1, and relatively few genes to directly affect BRCA1 expression, as in the following diagram:\n\\begin{center}\n\\begin{tikzpicture}[node distance=1cm]\n\n\\node(b)[text centered] {Correlated Gene};\n\\node(u)[below of = b, text centered] {$ $};\n\\node(a)[left of = u, text centered, xshift = -1.5cm] {Promoter\/Repressor Gene};\n\\node(y)[below of = u, text centered] {BRCA1};\n\\draw [arrow] (a) -- (b);\n\\draw [arrow] (a) -- (y);\n\\draw [dashed] (b) to [bend left](y);\n\n\\end{tikzpicture} \\\\\n\\end{center}\nUnsurprisingly, at an fdr threshold of $0.10$, the univariate approach selects 8,431 genes, clearly picking up on a large number of indirect associations.\n\nAlternatively, we may use lasso regression to jointly model the relationship between BCRA1 and the remaining 17,813 genes. Here, cross validation selects a model containing 96 features. This model, however, has a high false discovery rate -- the average mfdr of these features, estimated using the ashr approach, is $0.760$. One can lower the false discovery rate by choosing a larger value of $\\lam$ and selecting fewer features; for example, at $\\lam_{1\\SE}$, the model selects 49 features with an average mfdr of 0.026. However, this smaller model is considerably less accurate, raising the cross validation error by 11\\%. Using the feature-specific inference that mfdr provides, however, we can base our analysis on the model with the greatest predictive accuracy and still identify which of the 96 features are likely to be false discoveries. In this example, 16 of those genes have local false discovery rates under 10\\%.\n\n\\begin{table}[!htb]\n\\centering\n\\caption{\\label{Table:BRCA1}The top 10 selected genes from the univariate approach and their local false discovery rate estimates (ashr) at each $\\lambda$ value.}\n\\begin{tabular}{l c |c c c }\nGene & Chromosome & Univariate $\\widehat{\\fdr}$ & $\\widehat{\\mfdr}$ at $\\lambda_{1\\SE}$ & $\\widehat{\\mfdr}$ at $\\lambda_{\\CV}$ \\\\\n\\hline\nC17orf53 & 17 &$<$0.0001 & 0.0005 & 0.31987 \\\\\nTUBG1 & 17 &$<$0.0001 & 0.0310 & 0.3507 \\\\\nDTL & 1 &$<$0.0001 & $<$0.0001 & $<$0.0001 \\\\\nVPS25 & 17 &$<$0.0001 & $<$0.0001 & 0.0211 \\\\\nTOP2A & 17 &$<$0.0001 & 0.0002 & 0.0020 \\\\\nPSME3 & 17 &$<$0.0001& $<$0.0001 & 0.0004 \\\\\nTUBG2 & 17 &$<$0.0001 & 0.0508 & 0.3599 \\\\\nTIMELESS & 12 &$<$0.0001& 0.0123 & 0.3196 \\\\\nNBR2 & 17 &$<$0.0001 & $<$0.00001 & $<$0.00001\\\\\nCCDC43 & 17 &$<$0.0001 & 0.0169 & 0.3326 \\\\\n\\end{tabular}\n\\end{table}\n\nTable~\\ref{Table:BRCA1} displays the 10 genes with the lowest univariate local false discovery rates, along with their mfdr estimates at $\\lambda_{1\\SE}$ and $\\lambda_{\\CV}$ using the ashr approach. Similar results were found at both $\\lambda$ values using the density approach. Many of the genes with the lowest fdr estimates according to univariate analysis have biological roles with no apparent connection to BRCA1, but are located near BRCA1 on chromosome 17 and therefore all correlated with each other: TUBG1, TUBG2, NBR2, VPS25, TOP2A, PSME3, and CCDC43.\nAlmost all of these genes are estimated to have much higher false discovery rates in the simultaneous regression model than in the univariate approach.\n\nOther selections that have low local false discovery rates in both the univariate and lasso approaches have very plausible relationships with BRCA1. For example PSME3 encodes a protein that is known to interact with p53, a protein that is widely regarded as playing a crucial role in cancer formation \\citep{zhang_p53}. Another example is DTL, which interacts with p21, another protein known to have a role in cancer formation \\citep{DTL_p21}. These results demonstrate the potential of the mfdr approach to identify more scientifically relevant relationships by reducing the number of features only indirectly associated with the outcome.\n\n\\subsection{Lung Cancer Survival}\n\n\\citet{Shedden2008} studied the survival of 442 early-stage lung cancer subjects. Researchers collected expression data for 22,283 genes as well as information regarding several clinical covariates: age, race, gender, smoking history, cancer grade, and whether or not the subject received adjuvant chemotherapy. The goal of our analysis is to identify genetic features that are associated with survival after adjusting for the clinical covariates. \n\nWe first analyze the data using the traditional univariate fdr approach, which is based upon the test statistics from 22,283 separate Cox regression models. Each of these models contains a single genetic feature in addition to the clinical covariates. Note that although these models contain more than one variable, we will refer to this as the ``univariate approach'' to indicate how the high-dimensional features are being treated.\n\nWe compare results from the univariate approach with the proposed local mfdr approach. Here, the clinical covariates are included in the model as unpenalized covariates along with the 22,283 features, to which a lasso penalty is applied. We consider both cross validation and the one standard error rule as methods of selecting $\\lambda$ and estimate mfdr using the ``density\" approach. Cross validation selects $\\lambda=0.095$, while the 1SE approach suggests $\\lambda=0.155$, corresponding with 43 and 1 genetic features being selected, respectively.\n\n\\begin{table}[!htb] \n\\centering\n\\caption{\\label{Tab:Shedden}Local false discovery rate estimates of the top ten features, when performing univariate testing, for the Shedden survival data.}\n\\begin{tabular}{l l | l l c | l l c }\nUnivariate fdr & & mfdr at $\\lambda_{1\\SE}$ & & & mfdr at $\\lambda_{\\CV}$ & & \\\\\n\\hline \nFeature & $\\widehat{\\text{fdr}}$ & Feature & $\\widehat{\\mfdr}$ & $\\bh_j \\ne 0$ & Feature & $\\widehat{\\mfdr}$ & $\\bh_j \\ne 0$ \\\\\n\\hline\nZC2HC1A & 0.0010 & FAM117A & 0.0576 & * & FAM117A & 0.0678 &* \\\\\nFAM117A & 0.0014 & TERF1 & 0.1938 & & NUDT6 & 0.4238 & * \\\\\nSCGB1D1 & 0.0016 & PTGER3 & 0.1951 & & RAB2A & 0.8658 &* \\\\\nCHEK1 & 0.0022 & CDC42 & 0.1955 & & MAP1A & 0.8658 & * \\\\\nHILPDA & 0.0027 & BHLHB9 & 0.1996 & & RHOA &0.8658 & * \\\\\nCSRP1 & 0.0039 & NDST1 & 0.2005 & & PLP1 & 0.8658& * \\\\\nPDPK1 & 0.0050 & CPT1A& 0.2137 & & GUK1 & 0.8658 & * \\\\\nBSDC1 & 0.0051 & AFFX-M27830 & 0.2161 & & PREP & 0.8658 & * \\\\\nXPNPEP1 & 0.0051 & BSDC1 & 0.2233 & & SCN7A & 0.8658 & * \\\\\nARHGEF2 & 0.0051 & ETV5 & 0.2269 & & BTBD1 & 0.8658 & * \\\\\n\\end{tabular}\n\\end{table}\n\nTable~\\ref{Tab:Shedden} displays the ten features with the lowest local false discovery rates for the univariate and lasso mfdr approaches. Here, we present mfdr estimates based on the density modeling approach. Using the \\texttt{ashr} approach, results are very similar at $\\lambda_{1\\SE}$, but at $\\lambda_{\\CV}$, all features were estimated to have local false discovery rates near 1 using \\texttt{ashr}. As discussed in Section~\\ref{Sec:density_est}, this may result from not having enough features to estimate the non-null mixture components.\n\nWe observe one feature, FAM117A, stands out in all approaches, albeit with different estimates. We also notice the estimates for the univariate fdr tend to be smaller than the mfdr approach at $\\lambda_{1\\SE}$, which in turn tend to be smaller than those at $\\lambda_{\\CV}$. This illustrates a key aspect of local mfdr in practice: although the development of the fdr estimator is concerned only with marginal false discoveries, the regression model is certainly making conditional adjustments. At $\\lam_{\\max}$, the lasso mfdr and univariate fdr are equivalent, but as $\\lam$ decreases and the model grows larger, more extensive conditional adjustments are being performed.\n\n\\begin{figure} [!htb]\n\\centering\n\\includegraphics[width=.85\\textwidth]{Fig4}\n\\caption{\\label{locfdr_shedden_density} The mixture density estimates, $\\hat{f}(z)$, for the different methods applied to the Shedden data. We observe that the distribution of test statistics more closely resembles the null as more features are adjusted for by the model. }\n\\end{figure}\n\nFigure~\\ref{locfdr_shedden_density} shows the estimated marginal density, $\\hat{f}$, for each method. With the univariate approach, which does not account for any correlations between features, we see that the distribution of univariate test statistics is quite different from the null distribution. In the lasso model at $\\lambda_{1\\SE}$ , the model adjusts for gene FAM117A and consequently, the distribution narrows relative to that of the univariate approach. When cross validation is used to select $\\lambda$, the model adjusts for 43 genes and the distribution narrows even further to the point where it closely resembles the null.\nAs predictors enter the model and help to explain the outcome, the residuals (or pseudo-residuals) increasingly resemble white noise and exhibit no correlation with the remaining features.\n\n\\section{Discussion}\n\nLocal approaches to marginal false discovery rates for penalized regression models provides a very useful way of quantifying the reliability of individual feature selections after a model is fit. The estimator can be quickly computed, even in high dimensions, using quantities that are easily obtained from the fitted model. This makes it a convenient and informative way to carry out inference after fitting a lasso model. The method is currently implemented in the {\\tt summary} function of the R package {\\tt ncvreg} \\citep{Breheny2011}. By default {\\tt summary} reports local false discovery rates for all features selected at a given value of $\\lambda$, but includes options to report all variables that meet a specified mfdr threshold or to report a specified number of features in order of mfdr significance. For more information on using {\\tt ncvreg} to calculate mfdr, see the package vignette or the online documentation at \\href{http:\/\/pbreheny.github.io\/ncvreg}{http:\/\/pbreheny.github.io\/ncvreg}.\n\nLike any estimate, the local mfdr has limitations. Although it has clear advantages over univariate hypothesis testing in many cases, a regression approach is not practical in many situations in which high-throughput testing arises, such as two-group comparisons with $n<5$ in each group. Likewise, the local mfdr is far more powerful than other approaches to inference for the lasso such as selective inference and sample splitting, but this is because it controls a weaker notion of fdr control -- namely, it can only claim to limit the number of selections that are purely noise and does not attempt to eliminate features that are marginally associated with the outcome. Finally, although we have introduced causal ideas and diagrams to motivate ideas here, any attempt to infer causal relationships from observational data in practice should be taken with a grain of salt.\n\nNevertheless, the local mfdr approach that we propose here addresses a critical need for feature-specific inference in high-dimensional penalized regression models, especially in the GLM and Cox regression settings where few other options have been proposed.\n\n\\section{Appendix}\n\\subsection{Proof of Theorem~\\ref{Thm:Efdr}}\n\n\\begin{proof}\n\n Let $Z=\\sqrt{n}C_j\/\\sigma$, so that $Z$ has density $f=\\pi_0 f_0 + (1-\\pi_0)f_1$, with $f_0$ the standard normal density, and let $\\cZ = (-\\infty, -\\lam\\sqrt{n}\/\\sigma] \\cup [\\lam\\sqrt{n}\/\\sigma], \\infty)$, so that $Z \\in \\cZ$ is equivalent to $C_j \\in \\cM_\\lam$. Now,\n \\begin{align*}\n \\EX\\left\\{\\mfdr(\\tfrac{C_j}{\\sigma\/\\sqrt{n}}) | c_j \\in \\cM_\\lam\\right\\} &= \\EX\\left\\{\\mfdr(Z) | Z \\in \\cZ\\right\\} \\\\\n &= \\int_\\cZ \\frac{\\mfdr(z)f(z)}{F(\\cZ)}\\,\\mathrm{d}z\\\\\n \\intertext{(where $F(\\cZ)$ denotes the probability assigned to the set $\\cZ$ by the distribution function $F$)}\n &= \\int_\\cZ \\frac{\\pi_0f_0(z)}{f(z)}\\frac{f(z)}{F(\\cZ)} \\,\\mathrm{d}z\\\\\n &= \\frac{\\pi_0F_0(\\cZ)}{F(\\cZ)} \\\\\n &= \\frac{2\\pi_0\\Phi(-\\lam\\sqrt{n}\/\\sigma)}{F(\\cZ),}\n \\end{align*}\n where $\\Phi$ denotes the standard Gaussian CDF. The mFdr estimator consists of replacing $F(\\cZ)$ with its empirical estimate $\\abs{\\cM_\\lam}\/p$, in which case the above quantity yields the one given in Section 2.1 of \\cite{Breheny2019}.\n\\end{proof}\n\n\n\\section*{Reproducibility} A repository containing code to reproduce all results in this paper is located at \\url{https:\/\/github.com\/remiller1450\/loc-mfdr-paper}.\n\n\\bibliographystyle{ims}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}