Univariate Case Research Articles

Lorenz Model Selection

In the paper, we introduce novel model selection measures based on Lorenz zonoids which, differently from measures based on correlations, are based on a mutual notion of variability and are more robust to the presence of outlying observations. By means of Lorenz zonoids, which in the univariate case correspond to the Gini coefficient, the contribution of each explanatory variable to the predictive power of a linear model can be measured more accurately. Exploiting Lorenz zonoids, we develop a Marginal Gini Contribution measure that allows to measure the absolute explanatory power of any covariate, and a Partial Gini Contribution measure that allows to measure the additional contribution of a new covariate to an existing model.

Generalised cepstral models for the spectrum of vector time series

The paper treats the modeling of stationary multivariate stochastic processes via a frequency domain model expressed in terms of cepstrum theory. The proposed model nests the vector exponential model of [20] as a special case, and extends the generalised cepstral model of [36] to the multivariate setting, answering a question raised by the last authors in their paper. Contemporarily, we extend the notion of generalised autocovariance function of [35] to vector time series. Then we derive explicit matrix formulas connecting generalised cepstral and autocovariance matrices of the process, and prove the consistency and asymptotic properties of the Whittle likelihood estimators of model parameters. Asymptotic theory for the special case of the vector exponential model is a significant addition to the paper of [20]. We also provide a mathematical machinery, based on matrix differentiation, and computational methods to derive our results, which differ significantly from those employed in the univariate case. The utility of the proposed model is illustrated through Monte Carlo simulation from a bivariate process characterized by a high dynamic range, and an empirical application on time varying minimum variance hedge ratios through the second moments of future and spot prices in the corn commodity market.

Stable limit theorems on the Poisson space

We prove limit theorems for functionals of a Poisson point process using the Malliavin calculus on the Poisson space. The target distribution is conditionally either a Gaussian vector or a Poisson random variable. The convergence is stable and our conditions are expressed in terms of the Malliavin operators. For conditionally Gaussian limits, we also obtain quantitative bounds, given for the Wasserstein transport distance in the univariate case; and for another probabilistic variational distance in higher dimension. Our work generalizes several limit theorems on the Poisson space, including the seminal works by G. Peccati, J. L. Sole, M. S. Taqqu & F. Utzet. “Stein’s method and normal approximation of Poisson functionals” for Gaussian approximations; and by G. Peccati “The Chen-Stein method for Poisson functionals” for Poisson approximations; as well as the recently established fourth-moment theorem on the Poisson space of C. Dobler & G.Peccati “The fourth moment theorem on the Poisson space”. We give an application to stochastic processes.

Structured Inversion of the Bernstein Mass Matrix

Bernstein polynomials, long a staple of approximation theory and computational geometry, have also increasingly become of interest in finite element methods. Many fundamental problems in interpolation and approximation give rise to interesting linear algebra questions. In [13], we gave block-structured algorithms for inverting the Bernstein mass matrix on simplicial cells but did not study fast algorithms for the univariate case. Here, we give several approaches to inverting the univariate mass matrix based on exact formulae for the inverse; decompositions of the inverse in terms of Hankel, Toeplitz, and diagonal matrices; and a spectral decomposition. In particular, the eigendecomposition can be explicitly constructed in $\mathcal{O}(n^2)$ operations, while its accuracy for solving linear systems is comparable to that of the Cholesky decomposition. Moreover, we study conditioning and accuracy of these methods from the standpoint of the effect of roundoff error in the $L^2$ norm on polynomials, showing that the conditioning in this case is far less extreme than in the standard 2-norm.

Combined multiple testing of multivariate survival times by censored empirical likelihood.

In each study testing the survival experience of one or more populations, one must not only choose an appropriate class of tests, but further an appropriate weight function. As the optimal choice depends on the true shape of the hazard ratio, one is often not capable of getting the best results with respect to a specific dataset. For the univariate case several methods were proposed to conquer this problem. However, most of the interesting datasets contain multivariate observations nowadays. In this work we propose a multivariate version of a method based on multiple constrained censored empirical likelihood where the constraints are formulated as linear functionals of the cumulative hazard functions. By considering the conditional hazards, we take the correlation between the components into account with the goal of obtaining a test that exhibits a high power irrespective of the shape of the hazard ratio under the alternative hypothesis.

On minimum volume properties of some confidence regions for multiple multivariate normal means

In a multi-sample model of multivariate normal distributions with covariance matrices being known or known except for unknown multipliers, simultaneous confidence regions for the mean vectors are provided with minimum volume properties. The univariate case with unknown variances is included.

Additive models with trend filtering

We study additive models built with trend filtering, that is, additive models whose components are each regularized by the (discrete) total variation of their $k$th (discrete) derivative, for a chosen integer $k\geq0$. This results in $k$th degree piecewise polynomial components, (e.g., $k=0$ gives piecewise constant components, $k=1$ gives piecewise linear, $k=2$ gives piecewise quadratic, etc.). Analogous to its advantages in the univariate case, additive trend filtering has favorable theoretical and computational properties, thanks in large part to the localized nature of the (discrete) total variation regularizer that it uses. On the theory side, we derive fast error rates for additive trend filtering estimates, and show these rates are minimax optimal when the underlying function is additive and has component functions whose derivatives are of bounded variation. We also show that these rates are unattainable by additive smoothing splines (and by additive models built from linear smoothers, in general). On the computational side, we use backfitting, to leverage fast univariate trend filtering solvers; we also describe a new backfitting algorithm whose iterations can be run in parallel, which (as far as we can tell) is the first of its kind. Lastly, we present a number of experiments to examine the empirical performance of trend filtering.

Chebyshev Multivariate Polynomial Approximation and Point Reduction Procedure

We apply the methods of nonsmooth and convex analysis to extend the study of Chebyshev (uniform) approximation for univariate polynomial functions to the case of general multivariate functions (not just polynomials). First of all, we give new necessary and sufficient optimality conditions for multivariate approximation, and a geometrical interpretation of them which reduces to the classical alternating sequence condition in the univariate case. Then, we present a procedure for verification of necessary and sufficient optimality conditions that is based on our generalization of the notion of alternating sequence to the case of multivariate polynomials. Finally, we develop an algorithm for fast verification of necessary optimality conditions in the multivariate polynomial case.

Image Interpolation Using a Rational Bi-Cubic Ball

This study deals with the application of new rational bi-cubic Ball function with six parameters in image interpolation, especially for the grayscale image. These six free parameters can be modified to get better and quality image resolution, and refine the shape of the interpolating surface. This bivariate rational Ball function has been extended from univariate cases by using a tensor product approach. The proposed scheme is tested for image upscaling with factors of two and four through an efficient algorithm. The effectiveness of the proposed scheme is measured by using an image quality assessment (IQA), such as peak-signal-to-noise-ratio (PSNR), root mean square error (RMSE) or feature similarity (FSIM) index. Numerical and graphical results with comparisons against some existing scheme are presented by using MATLAB. The proposed scheme resulted in higher PSNR and FSIM, and smaller RMSE. Thus, the new rational bi-cubic Ball with six parameters is better than the existing scheme via an efficient algorithm.

Sufficient dimension folding in regression via distance covariance for matrix‐valued predictors

AbstractIn modern data, when predictors are matrix/array‐valued, building a reasonable model is much more difficult due to the complicate structure. However, dimension folding that reduces the predictor dimensions while keeps its structure is critical in helping to build a useful model. In this paper, we develop a new sufficient dimension folding method using distance covariance for regression in such a case. The method works efficiently without strict assumptions on the predictors. It is model‐free and nonparametric, but neither smoothing techniques nor selection of tuning parameters is needed. Moreover, it works for both univariate and multivariate response cases. In addition, we propose a new method of local search to estimate the structural dimensions. Simulations and real data analysis support the efficiency and effectiveness of the proposed method.

Convex envelope of bivariate cubic functions over rectangular regions

In recent years many papers have derived polyhedral and non-polyhedral convex envelopes for different classes of functions. Except for the univariate cases, all these classes of functions share the property that the generating set of their convex envelope is a subset of the border of the region over which the envelope is computed. In this paper we derive the convex envelope over a rectangular region for a class of functions which does not have this property, namely the class of bivariate cubic functions without univariate third-degree terms.

Simplified Matrix Methods for Multivariate Edgeworth Expansions

Simplified matrix methods are used to analyze the higher order asymptotic properties of $$k\times 1$$ sample averages. Kronecker differentiation is used to define $$k^{j }\times 1$$ , j’th order moments, $$\mu _j$$ , cumulants $$\kappa _j$$ and Hermite polynomials, $$H_j$$ . These are then used to derive valid multivariate Edgeworth expansions of arbitrary order having the same form as the standard univariate case: $$p(x) = \phi (x)[1 + N^{-1/2} \kappa _{3}' H_{ 3}(x) /6 +{ N^{-1} } ( 3 { \kappa _{4}' }{ } H_{ 4}(x) + \kappa _3'^{ \otimes 2} H_{ 6}(x) )/72+\cdots ]$$ . All the usual steps in the development of a valid Edgeworth expansion are shown to be easily derived using matrix algebra.

Bayesian nonparametric analysis of multivariate time series: A matrix Gamma Process approach

Many Bayesian nonparametric approaches to multivariate time series rely on Whittle’s Likelihood, involving the second order structure of a stationary time series by means of its spectral density matrix. In this work, we model the spectral density matrix by means of random measures that are constructed in such a way that positive definiteness is ensured. This is in line with existing approaches for the univariate case, where the normalized spectral density is modeled similar to a probability density, e.g. with a Dirichlet process mixture of Beta densities. We present a related approach for multivariate time series, with matrix-valued mixture weights induced by a Hermitian positive definite Gamma process. The latter has not been considered in the literature, allows to include prior knowledge and possesses a series representation that will be used in MCMC methods. We establish posterior consistency and contraction rates and small sample performance of the proposed procedure is shown in a simulation study and for real data.

On the positivity and magnitudes of Bayesian quadrature weights

This article reviews and studies the properties of Bayesian quadrature weights, which strongly affect stability and robustness of the quadrature rule. Specifically, we investigate conditions that are needed to guarantee that the weights are positive or to bound their magnitudes. First, it is shown that the weights are positive in the univariate case if the design points locally minimise the posterior integral variance and the covariance kernel is totally positive (e.g. Gaussian and Hardy kernels). This suggests that gradient-based optimisation of design points may be effective in constructing stable and robust Bayesian quadrature rules. Secondly, we show that magnitudes of the weights admit an upper bound in terms of the fill distance and separation radius if the RKHS of the kernel is a Sobolev space (e.g. Matérn kernels), suggesting that quasi-uniform points should be used. A number of numerical examples demonstrate that significant generalisations and improvements appear to be possible, manifesting the need for further research.

Fused Density Estimation: Theory and Methods

Summary We introduce a method for non-parametric density estimation on geometric networks. We define fused density estimators as solutions to a total variation regularized maximum likelihood density estimation problem. We provide theoretical support for fused density estimation by proving that the squared Hellinger rate of convergence for the estimator achieves the minimax bound over univariate densities of log-bounded variation. We reduce the original variational formulation to transform it into a tractable, finite dimensional quadratic program. Because random variables on geometric networks are simple generalizations of the univariate case, this method also provides a useful tool for univariate density estimation. Lastly, we apply this method and assess its performance on examples in the univariate and geometric network setting. We compare the performance of various optimization techniques to solve the problem and use these results to inform recommendations for the computation of fused density estimators.

Graph-based multivariate conditional autoregressive models

ABSTRACTThe conditional autoregressive model is a routinely used statistical model for areal data that arise from, for instances, epidemiological, socio-economic or ecological studies. Various multivariate conditional autoregressive models have also been extensively studied in the literature and it has been shown that extending from the univariate case to the multivariate case is not trivial. The difficulties lie in many aspects, including validity, interpretability, flexibility and computational feasibility of the model. In this paper, we approach the multivariate modelling from an element-based perspective instead of the traditional vector-based perspective. We focus on the joint adjacency structure of elements and discuss graphical structures for both the spatial and non-spatial domains. We assume that the graph for the spatial domain is generally known and fixed while the graph for the non-spatial domain can be unknown and random. We propose a very general specification for the multivariate conditional modelling and then focus on three special cases, which are linked to well-known models in the literature. Bayesian inference for parameter learning and graph learning is provided for the focused cases, and finally, an example with public health data is illustrated.

Coherent pairs of bivariate orthogonal polynomials

Coherent pairs of measures were introduced in 1991 and constitute a very useful tool in the study of Sobolev orthogonal polynomials on the real line. In this work, coherence and partial coherence in two variables appear as the natural extension of the univariate case. Given two families of bivariate orthogonal polynomials expressed as polynomial systems, they are a partial coherent pair if a polynomial of the second family can be given as a linear combination of the first partial derivatives of (at most) three consecutive polynomials of the first family. A full coherent pair is a pair of families of bivariate orthogonal polynomials related by means of partial coherent relations in each variable. Consequences of this kind of relations concerning both families of bivariate orthogonal polynomials are studied. Finally, some illustrative examples are provided.

SPACE PATTERN OF FOREST SPECIES AND ITS RELATIONSHIP WITH AGRICULTURAL FACTORS AGROFLORESTAL SYSTEM

The objective of this work was to describe the spatial distribution pattern of the arboreal and regenerating forest stratum and its relationship with edaphic factors in a multistratified agroforestry system without management. The data were obtained by means of a census of the arboreal and regenerating plants, in which the variables diameter were measured at 1.30m of the soil and total height, besides the botanical identification. The plants of the arboreal stratum were grouped in diameter and height classes and the regenerating plants in height classes. In both strata, the horizontal structure indicators were estimated and the species of higher Importance Value Index (IVI) were selected to perform the spatial analysis. The K Ripley function was applied in the univariate case to test the hypothesis of complete randomness in the groups and in the arboreal and regenerating species of higher IVI. In the bivariate case, the Ripley K function was applied to test the hypothesis of complete spatial independence between groups or aggregate pattern species, with edaphic attributes. The results indicated random spatial pattern for most of the tested groups, with only the species Gliricidia sepium and Archontophoenix alexandrae with aggregate spatial pattern, up to 30m distance, in the arboreal and regenerating strata, respectively. However, no spatial relationship was observed between the pattern of distribution of the species and the soil factors of agroforestry system (SAF). In general, it is concluded that edaphic factors were fundamental in the development of plants, but not for the formation of aggregates

A note on application of kernel derivatives in density estimation with the univariate case

This paper considers the application of kernel density derivatives to real life data. Kernel density derivatives estimation is very fundamental and critical in statistical data analysis especially for exploratory and visualization purposes. As a result of the wide range of its applications, appropriate estimation of the kernel derivatives from its function and locating some statistical features such as bumps and modes of a set of observation is of a great importance. We consider the first and second derivative of the Gaussian kernel and compare their results in terms of performance using asymptotic mean integrated squared error as the error criterion function. The results of the comparison shows that as the derivative of the kernel function increases, the AMISE decreases with an increase in the smoothing parameter.

Sparse polynomial interpolation: sparse recovery, super-resolution, or Prony?

We show that the sparse polynomial interpolation problem reduces to a discrete super-resolution problem on the $n$-dimensional torus. Therefore the semidefinite programming approach initiated by Cand\`es \& Fernandez-Granda \cite{candes_towards_2014} in the univariate case can be applied. We extend their result to the multivariate case, i.e., we show that exact recovery is guaranteed provided that a geometric spacing condition on the supports holds and the number of evaluations are sufficiently many (but not many). It also turns out that the sparse recovery LP-formulation of $\ell_1$-norm minimization is also guaranteed to provide exact recovery {\it provided that} the evaluations are made in a certain manner and even though the Restricted Isometry Property for exact recovery is not satisfied. (A naive sparse recovery LP-approach does not offer such a guarantee.) Finally we also describe the algebraic Prony method for sparse interpolation, which also recovers the exact decomposition but from less point evaluations and with no geometric spacing condition. We provide two sets of numerical experiments, one in which the super-resolution technique and Prony's method seem to cope equally well with noise, and another in which the super-resolution technique seems to cope with noise better than Prony's method, at the cost of an extra computational burden (i.e. a semidefinite optimization).