Orthogonal Series Research Articles

6524 Sex-Specific Single Cell Genetic Architecture Of Human Islets With And Without Type 2 Diabetes

Abstract Disclosure: M. Qadir: None. R.M. Elgamal: None. P. Kudtarkar: None. K.J. Gaulton: None. F. Mauvais-Jarvis: None. Type 2 diabetes (T2D) is a heterogeneous disease and biological sex affects its pathogenesis including the development of adiposity, insulin resistance, and β cell failure. For example, ketosis-prone diabetes is a phenotypically-defined form of T2D with acute β cell failure and severe insulin deficiency, predominantly observed in men. However, what mechanisms drive male predominance in β cell failure is unknown. To fully understand the sex-based pathogenesis of islet failure in diabetes, the study of sex differences in human islet biology and dysfunction is essential. To decipher sex differences in human islets genetic architecture and function, we performed an orthogonal series of experiments from 15 islets of non-diabetic (ND) donors, using single cell RNAseq (scRNAseq) and single nucleus assay for transposase-accessible chromatin sequencing (snATACseq). We also leveraged 37 ND and T2D donors for scRNAseq data from the human pancreas analysis program (HPAP) dataset. We studied over 200,000 single cells spanning the accessible genome and transcriptome. It is the largest study of sex-based single-cell genomics and transcriptomics of human islets to date. We observe that in cultured human ND islets, in the absence of the in vivo hormonal environment, islet cell transcriptome, and genome accessibility across coding regions are predominantly limited to sex chromosome genes. Of particular interest are sex differences in gene accessibility and expression of the X-linked KDM6A and Y-linked KDM5D demethylase genes in female (F) and male (M) islet cells respectively. Furthermore, analysis of T2D islets revealed a greater number of autosomal differentially expressed genes (DEGs) in F (721 upregulated and 1164 downregulated, FDR < 0.01) compared to M β cells (111 upregulated and 99 downregulated, FDR < 0.01). M β cell DEGs over-representation analysis revealed suppressed insulin secretion and processing pathways, while F β cell DEGs exhibited suppression of oxidative phosphorylation and electron transport chain pathways. Thus, although, sex differences in gene expression and accessibility of cultured ND human islets are restricted to sex chromosome genes, during the transition from ND to T2D, major sex differences in autosomal DEGs appear, with a greater number of autosomal DEGs in F compared to M T2D islets. Presentation: 6/3/2024

The Effect of Unsteady Inlet Boundary Conditions on the Aero-Thermal Behavior of High-Pressure Turbine Vanes – A Numerical Study Using Scale-Resolving Simulations

Abstract This paper presents the application of a novel method to prescribe unsteady boundary conditions to transient, scale-resolving computational fluid dynamics simulations of the high-pressure turbine in modern jet engines. The methodology is based on the compression of the interface data at the combustor–turbine interface, using proper orthogonal decomposition and Fourier series (PODFS). Doing so can reduce the stored data at the interface drastically. The capability of the PODFS method to produce realistic inlet boundary conditions was demonstrated in previous work. Here, the method is applied to a turbine case. The outlet data of a combustor simulation is used to create the PODFS boundary conditions for a scale-resolving simulation of a simplified first nozzle guide vane of the high-pressure turbine. This simulation is compared with simulations with steady-state boundary conditions to show the effect of unsteadiness in the inlet boundary condition on the aerodynamic and thermal behaviors of the turbine. While the aerodynamics show minor sensitivity against the way of applying the inlet boundary conditions, the thermal behavior of the vanes is strongly affected by the modeling of combustor unsteadiness.

Analytical solutions for nonlinear axisymmetric deformations of circular plates by using innovative orthogonal power function series

Analytical solutions for nonlinear axisymmetric deformations of circular plates by using innovative orthogonal power function series

Single cell regulatory architecture of human pancreatic islets suggests sex differences in β cell function and the pathogenesis of type 2 diabetes.

Biological sex affects the pathogenesis of type 2 and type 1 diabetes (T2D, T1D) including the development of β cell failure observed more often in males. The mechanisms that drive sex differences in β cell failure is unknown. Studying sex differences in islet regulation and function represent a unique avenue to understand the sex-specific heterogeneity in β cell failure in diabetes. Here, we examined sex and race differences in human pancreatic islets from up to 52 donors with and without T2D (including 37 donors from the Human Pancreas Analysis Program [HPAP] dataset) using an orthogonal series of experiments including single cell RNA-seq (scRNA-seq), single nucleus assay for transposase-accessible chromatin sequencing (snATAC-seq), dynamic hormone secretion, and bioenergetics. In cultured islets from nondiabetic (ND) donors, in the absence of the in vivo hormonal environment, sex differences in islet cell type gene accessibility and expression predominantly involved sex chromosomes. Of particular interest were sex differences in the X-linked KDM6A and Y-linked KDM5D chromatin remodelers in female and male islet cells respectively. Islets from T2D donors exhibited similar sex differences in differentially expressed genes (DEGs) from sex chromosomes. However, in contrast to islets from ND donors, islets from T2D donors exhibited major sex differences in DEGs from autosomes. Comparing β cells from T2D and ND donors revealed that females had more DEGs from autosomes compared to male β cells. Gene set enrichment analysis of female β cell DEGs showed a suppression of oxidative phosphorylation and electron transport chain pathways, while male β cell had suppressed insulin secretion pathways. Thus, although sex-specific differences in gene accessibility and expression of cultured ND human islets predominantly affect sex chromosome genes, major differences in autosomal gene expression between sexes appear during the transition to T2D and which highlight mitochondrial failure in female β cells.

On the resonance frequency of the irregular shaped visco-acoustic Helmholtz cavity problem

In this paper we develop a closed-form mathematical solution for the resonance frequency of the axisymmetric irregular shaped visco-acoustic Helmholtz cavity problem. The solution is based on a conformal mapping that transforms the governing visco-acoustic equations of motion for the irregular cavity geometry into a new domain described by a set of orthogonal curvilinear coordinates. These orthogonal curvilinear coordinates have an important characteristic relative to the transformation of the physical boundary: the mapped domain is constructed such that the locus of all points for the length-wise profile of the physical boundary coincides with a constant value for a (quasi radial) coordinate in the new mapped domain. In these orthogonal curvilinear coordinates we obtain an analytical solution for the visco-acoustic equations of motion and exactly satisfy the no-slip fluid boundary conditions over the irregular shaped cavity walls by matching the coefficients of an orthogonal Legendre polynomial series representation for the non-uniform velocity on the cavity walls. The mathematical solution is an approximation to the extent that the scale factors chosen for the mapping consist of strictly the mutual-coupling terms of the true scale factors. Thus we analyze some results to understand the correlation of the analytical solution frequency response spectra with numerical finite element analyses of the idealized boundary conditions with various irregular shaped cavity geometries. The results for fundamental resonance frequency were found to exhibit good correlation with the results of finite element analyses over a significant range of fluid properties with different irregular cavity geometries. As the primary motivation for the derivation stems from the lack of a rapid yet accurate alternative to computationally intensive MultiPhysics FEA in the formulation of acoustic trendlines for sensor development, the solution’s acoustic impedance was applied as dispersive loading in a simple lumped parameter model of a fluid identification sensor for comparison. The resulting acoustic trendlines of the reduced (5 DOF) model were found to exhibit good correlation with the results of the MultiPhysics FEA (>105 DOF). Thus the solution was found to enable a closed-form analytical method for rapid construction of parametric studies in the practical design of fluid analysis sensors based on complex resonant cavity geometries.

Probabilistic neural networks for incremental learning over time-varying streaming data with application to air pollution monitoring

This paper proposes a novel algorithm for incremental learning over streaming data in a non-stationary environment. The idea refers to the applicability of Probabilistic Neural Networks (PNNs), commonly used as a fast and robust method for solving classification problems in a stationary scenario. It is well known that PNNs have solid mathematical foundations but they fail in the non-stationary scenario and are not able to deal with concept drift. In this paper, the method based on the orthogonal series expansions of unknown probability densities is proposed. This approach significantly extends the applicability of the PNNs. The nonparametric procedures called Incremental Probabilistic Neural Networks (IPNNs) for tracking drifting probability densities or conditional class densities, in the case of classification, are presented. We prove their convergence as the stream data size is growing to infinity. More precisely, we show convergence in probability and with probability one as the sample size tends to infinity. Compared to the existing literature, almost exclusively based on various heuristics, the proposed method is mathematically justified and has solid theoretical foundations. Simulations performed on synthetic and air pollution data confirm the effectiveness of the algorithm.

Real space iterative reconstruction for vector tomography (RESIRE-V)

Tomography has had an important impact on the physical, biological, and medical sciences. To date, most tomographic applications have been focused on 3D scalar reconstructions. However, in some crucial applications, vector tomography is required to reconstruct 3D vector fields such as the electric and magnetic fields. Over the years, several vector tomography methods have been developed. Here, we present the mathematical foundation and algorithmic implementation of REal Space Iterative REconstruction for Vector tomography, termed RESIRE-V. RESIRE-V uses multiple tilt series of projections and iterates between the projections and a 3D reconstruction. Each iteration consists of a forward step using the Radon transform and a backward step using its transpose, then updates the object via gradient descent. Incorporating with a 3D support constraint, the algorithm iteratively minimizes an error metric, defined as the difference between the measured and calculated projections. The algorithm can also be used to refine the tilt angles and further improve the 3D reconstruction. To validate RESIRE-V, we first apply it to a simulated data set of the 3D magnetization vector field, consisting of two orthogonal tilt series, each with a missing wedge. Our quantitative analysis shows that the three components of the reconstructed magnetization vector field agree well with the ground-truth counterparts. We then use RESIRE-V to reconstruct the 3D magnetization vector field of a ferromagnetic meta-lattice consisting of three tilt series. Our 3D vector reconstruction reveals the existence of topological magnetic defects with positive and negative charges. We expect that RESIRE-V can be incorporated into different imaging modalities as a general vector tomography method. To make the algorithm accessible to a broad user community, we have made our RESIRE-V MATLAB source codes and the data freely available at https://github.com/minhpham0309/RESIRE-V.

Transverse vibration of plate with multiple curved through cracks

A modeling method for transverse vibration of cracked plate containing multiple internal curved through cracks is presented and experimental studies are performed to validate its effectiveness. The cracked plate is assumed to be divided into several rectangular and curved trapezoidal domains in the global coordinates system. For each sub-domain, the transverse displacement is expanded as Chebyshev orthogonal polynomials series in local coordinates system. As for the shared edges between neighboring two sub-domains, the local collocation method is adopted and a number of Chebyshev zero points are selected to ensure the displacement continuity condition. To verify the convergence and correctness of the presented model and solutions, the natural frequencies, mode shapes and dynamic responses for several cases are compared with reference results and experimental results, with good agreement observed. In addition, the influence of several significant parameters of cracks are investigated, including the number, location and geometrical shape of the cracks. It is shown that the crack inside the plate presents less influence than that at the boundary on the mode change of the plate. The investigations in the study implies that the present model method is effective and convenient for plate with multiple curved through cracks.

Fourier series for the multidimensional-matrix functions of the vector variable

In the article, the theory of the Fourier series on the orthogonal multidimensional-matrix (mdm) polynomials is developed. The known results from the theory of the orthogonal polynomials of the vector variable and the Fourier series are given and the new results are presented. In particular, the known results of the Fourier series theory are extended to the case of the mdm functions, what allows us to solve more general approximation problems. The general case of the approximation of the mdm function of the vector argument by the Fourier series on the orthogonal mdm polynomials is realized programmatically as the program function and its efficiency is confirmed. The analytical expressions for the coefficients of the second degree orthogonal polynomials and Fourier series for possible analytical studies are obtained.

Nonparametric density estimation over its unknown support for right censored data

The long-standing problem of nonparametric density estimation of the lifetime of interest over its unknown support, when only right-censored observations are available, is considered. Two cases, when the support of the lifetime of interest is a proper subset of the support of censoring variable and when the supports are the same, are explored. An orthogonal series density estimation, over a random interval defined by a specially chosen sequence of order statistics, is proposed and rates of the mean integrated squared error convergence are evaluated.

ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell ^1$$\\end{document}-summability and Fourier series of B-splines with respect to their knots

We study the ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell ^1$$\\end{document}-summability of functions in the d-dimensional torus Td\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb {T}}}^d$$\\end{document} and so-called ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell ^1$$\\end{document}-invariant functions. Those are functions on the torus whose Fourier coefficients depend only on the ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell ^1$$\\end{document}-norm of their indices. Such functions are characterized as divided differences that have cosθ1,…,cosθd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\cos {\ heta }_1,\\ldots ,\\cos {\ heta }_d$$\\end{document} as knots for (θ1…,θd)∈Td\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({\ heta }_1\\,\\ldots , {\ heta }_d) \\in {{\\mathbb {T}}}^d$$\\end{document}. It leads us to consider the d-dimensional Fourier series of univariate B-splines with respect to its knots, which turns out to enjoy a simple bi-orthogonality that can be used to obtain an orthogonal series of the B-spline function.

Characterization of anisotropy in basin-scale subsurface using teleseismic receiver function analysis

Teleseismic receiver function (RF) analysis offers a passive-source analogue to detect impedance boundaries using the converted body waves generated by earthquakes. While the technique traditionally has targeted deep earth structures such as the Moho and transition zones, there is growing interest in assessing its applicability in basin-scale seismic characterization, ultimately aimed for onshore commercial integration as a cost-effective complement to existing active-source seismic surveys. Specifically, the conventional broadband seismometers used in global observational seismic experiments are not only logistically simple in terms of data acquisition, but they also record ground motions in three mutually orthogonal time series, enabling effective detection of shear waves and directional variations of observed signals. Here, we perform teleseismic RF analysis to detect shear-wave anisotropy and related symmetry axes orientations in a basin setting, using open-source seismic data recorded at 55 closely spaced seismic stations in the LaBarge Passive Seismic Experiment deployed in Wyoming between November 2008 and June 2009. We find that the strengths and geometry of the observed anisotropy are variable along the array. Significantly, not only can anisotropy effectively delineate subsurface fault interfaces, it can also substantiate and reveal additional interpretable signals that are otherwise disregarded. The estimated fast axes orientations compare favorably with the complex fracture systems documented in the region. Finally, we show that P-to-S amplitude variations with P incidence are systematic and modelable using existing computational tools, offering an opportunity to develop an analysis technique similar to amplitude variation with offset with the products of RF analysis.

Algebra of Orthogonal Series

Algebra of Orthogonal Series

A general unsteady Fourier solution for orthotropic heat transfer in 2D functionally graded cylinders

In the present research, an analytical solution is presented for unsteady heat transfer in a two‐dimensional finite cylinder made of functionally graded Materials (FGM). The governing equation, as an unsteady partial differential equation (PDE), is solved with general boundary conditions (BCs). The thermal conductivity coefficients are considered as a power function of the radius, both in the longitudinal and radial directions of the FGM cylinder. The presence of non‐constant coefficients and general BCs will confront us with the challenge of solving an unsteady problem with high complexity. A combination of Laplace transform, Fourier transform, and meromorphic functions are utilized to deal with this complex PDE. The Laplace transform was applied to transform to the frequency domain, the Sturm–Liouville equation was used to extract the orthogonal basis Fourier series, and the meromorphic function was deployed to transform back from the frequency domain to the time domain. Results were well verified against previous research works. Results showed that the proposed analytical solution could well predict the temperature distribution. The current analytical solution is useful to better understand unsteady heat transfer in FGMs.

Orthogonal Polynomials and Fourier Series for Functions of Vector Variable: Multidimensional-Matrix Approach

In the article, the theory of the Fourier series on the orthogonal multidimensional-matrix polynomials is developed. The known results from the theory of the orthogonal polynomials of the vector variable and the Fourier series are given and the new results are presented. In particular, the known results of the Fourier series are extended to the case of the multidimensional-matrix functions, what allows us to solve more general approximation problems. The general case of the approximation of the multidimensional-matrix function of the vector argument by the Fourier series on the orthogonal multidimensional-matrix polynomials is realized programmatically as the program function and its efficiency is confirmed. The analytical expressions for the coefficients of the second degree orthogonal polynomials and Fourier series for the potential studies are obtained.

A hierarchical Bayesian modeling framework for identification of Non-Gaussian processes

Non-Gaussian processes are frequently encountered in engineering problems, posing a challenge when it comes to identification. The main challenge in the identification arises from the fact that a non-Gaussian process can be treated as a collection of infinite dimensional non-Gaussian variables. The application of the hierarchical Bayesian modeling (HBM) framework is constrained due to the inherent complexity of dimensionality and non-Gaussian characteristics associated with these variables. To tackle the issue of dimensionality, the improved orthogonal series expansion (iOSE) representing a non-Gaussian process by time functions with non-Gaussian coefficients, which are readily obtained from discretizing the process at some specific time points, is introduced within the HBM framework. In particular, the iOSE is embedded to convert the identification of a non-Gaussian process into a finite number of non-Gaussian coefficients. Regarding their non-Gaussian characteristics, polynomial chaos expansion (PCE) is used to quantify the uncertainty of the non-Gaussian coefficients with parameters in PCE treated as hyper parameters to be estimated by the HBM framework. The proposed framework is applicable to the identification of both stationary and nonstationary non-Gaussian processes. The effectiveness of non-Gaussian process quantification by the proposed framework is demonstrated using simulated data of a non-stationary extreme value process. Theapplicability of this approach for non-Gaussian process identification is validated by accurately identifying a stochastic load in a structural dynamic problem. Furthermore, it is successfully applied to the reconstruction of random mode shapes of a building arising from different environmental conditions.

Continuous time-resolved estimated synthetic 4D-CTs for dose reconstruction of lung tumor treatments at a 0.35 T MR-linac

Objective. To experimentally validate a method to create continuous time-resolved estimated synthetic 4D-computed tomography datasets (tresCTs) based on orthogonal cine MRI data for lung cancer treatments at a magnetic resonance imaging (MRI) guided linear accelerator (MR-linac). Approach. A breathing porcine lung phantom was scanned at a CT scanner and 0.35 T MR-linac. Orthogonal cine MRI series (sagittal/coronal orientation) at 7.3 Hz, intersecting tumor-mimicking gelatin nodules, were deformably registered to mid-exhale 3D-CT and 3D-MRI datasets. The time-resolved deformation vector fields were extrapolated to 3D and applied to a reference synthetic 3D-CT image (sCTref), while accounting for breathing phase-dependent lung density variations, to create 82 s long tresCTs at 3.65 Hz. Ten tresCTs were created for ten tracked nodules with different motion patterns in two lungs. For each dataset, a treatment plan was created on the mid-exhale phase of a measured ground truth (GT) respiratory-correlated 4D-CT dataset with the tracked nodule as gross tumor volume (GTV). Each plan was recalculated on the GT 4D-CT, randomly sampled tresCT, and static sCTref images. Dose distributions for corresponding breathing phases were compared in gamma (2%/2 mm) and dose–volume histogram (DVH) parameter analyses. Main results. The mean gamma pass rate between all tresCT and GT 4D-CT dose distributions was 98.6%. The mean absolute relative deviations of the tresCT with respect to GT DVH parameters were 1.9%, 1.0%, and 1.4% for the GTV D 98%, D 50%, and D 2%, respectively, 1.0% for the remaining nodules D 50%, and 1.5% for the lung V 20Gy. The gamma pass rate for the tresCTs was significantly larger (p < 0.01), and the GTV D 50% deviations with respect to the GT were significantly smaller (p < 0.01) than for the sCTref. Significance. The results suggest that tresCTs could be valuable for time-resolved reconstruction and intrafractional accumulation of the dose to the GTV for lung cancer patients treated at MR-linacs in the future.

Semi-parametric single-index predictive regression models with cointegrated regressors

This paper considers the estimation of a semi-parametric single-index regression model that allows for nonlinear predictive relationships. This model is useful for predicting financial asset returns, whose observed behaviour resembles a stationary process, if the multiple nonstationary predictors are cointegrated. The presence of cointegrated regressors imposes a single-index structure in the model, and this structure not only balances the nonstationarity properties of the multiple predictors with the stationarity properties of asset returns but also avoids the curse of dimensionality associated with nonparametric regression function estimation. An orthogonal series expansion is used to approximate the unknown link function for the single-index component. We consider the constrained nonlinear least squares estimator of the single-index (or the cointegrating) parameters and the plug-in estimator of the link function, and derive their asymptotic properties. In an empirical application, we find some evidence of in-sample nonlinear predictability of U.S. stock returns using cointegrated predictors. We also find that the single-index model in general produces better out-of-sample forecasts than both the historical average benchmark and the linear predictive regression model.

On Traces of Linear Operators with Symmetrized Volterra-Type Kernels

A solution to the trace convergence problem, which arises in proving the mean-square convergence for the approximation of iterated Stratonovich stochastic integrals, is proposed. This approximation is based on the representation of factorized Volterra-type functions as the orthogonal series. Solving the trace convergence problem involves the theory of trace class operators for symmetrized Volterra-type kernels. The main results are primarily focused on the approximation of iterated Stratonovich stochastic integrals, which are used to implement numerical methods for solving stochastic differential equations based on the Taylor–Stratonovich expansion.

On the use of L-functionals in regression models

Abstract In this article, we survey and unify a large class or L L -functionals of the conditional distribution of the response variable in regression models. This includes robust measures of location, scale, skewness, and heavytailedness of the response, conditionally on covariates. We generalize the concepts of L L -moments (G. Sillito, Derivation of approximants to the inverse distribution function of a continuous univariate population from the order statistics of a sample, Biometrika 56 (1969), no. 3, 641–650.), L L -skewness, and L L -kurtosis (J. R. M. Hosking, L-moments: analysis and estimation of distributions using linear combinations or order statistics, J. R. Stat. Soc. Ser. B Stat. Methodol. 52 (1990), no. 1, 105–124.) and introduce order numbers for a large class of L L -functionals through orthogonal series expansions of quantile functions. In particular, we motivate why location, scale, skewness, and heavytailedness have order numbers 1, 2, (3,2), and (4,2), respectively, and describe how a family of L L -functionals, with different order numbers, is constructed from Legendre, Hermite, Laguerre, or other types of polynomials. Our framework is applied to models where the relationship between quantiles of the response and the covariates follows a transformed linear model, with a link function that determines the appropriate class of L L -functionals. In this setting, the distribution of the response is treated parametrically or nonparametrically, and the response variable is either censored/truncated or not. We also provide a framework for asymptotic theory of estimates of L L -functionals and illustrate our approach by analyzing the arrival time distribution of migrating birds. In this context, a novel version of the coefficient of determination is introduced, which makes use of the abovementioned orthogonal series expansion.