Univariate Case Research Articles

Modeling Nonstationary and Asymmetric Multivariate Spatial Covariances via Deformations

Multivariate spatial-statistical models are often used when modeling environmental and socio-demographic processes. The most commonly used models for multivariate spatial covariances assume both stationarity and symmetry for the cross-covariances, but these assumptions are rarely tenable in practice. In this article we introduce a new and highly flexible class of nonstationary and asymmetric multivariate spatial covariance models that are constructed by modeling the simpler and more familiar stationary and symmetric multivariate covariances on a warped domain. Inspired by recent developments in the univariate case, we propose modeling the warping function as a composition of a number of simple injective warping functions in a deep-learning framework. Importantly, covariance-model validity is guaranteed by construction. We establish the types of warpings that allow for cross-covariance symmetry and asymmetry, and we use likelihood-based methods for inference that are computationally efficient. The utility of this new class of models is shown through two data illustrations: a simulation study on nonstationary data and an application on ocean temperatures at two different depths.

A New Stochastic Order of Multivariate Distributions: Application in the Study of Reliability of Bridges Affected by Earthquakes

In this article, we introduce and study a new stochastic order of multivariate distributions, namely, the conditional likelihood ratio order. The proposed order and other stochastic orders are analyzed in the case of a bivariate exponential distributions family. The theoretical results obtained are applied for studying the reliability of bridges affected by earthquakes. The conditional likelihood ratio order involves the multivariate stochastic ordering; it resembles the likelihood ratio order in the univariate case but is much easier to verify than the likelihood ratio order in the multivariate case. Additionally, the likelihood ratio order in the multivariate case implies this ordering. However, the conditional likelihood ratio order does not imply the weak hard rate order, and it is not an order relation on the multivariate distributions set. The new conditional likelihood ratio order, together with the likelihood ratio order and the weak hazard rate order, were studied in the case of the bivariate Marshall–Olkin exponential distributions family, which has a lack of memory type property. At the end of the paper, we also presented an application of the analyzed orderings for this bivariate distributions family to the study of the effects of earthquakes on bridges.

Multivariate conditional aging intensity functions and load-sharing models

The aging intensity functions analyze the aging property quantitatively, in the sense that the larger the aging intensity, the stronger the tendency of aging. They are useful tools to describe reliability properties of distributions. In the literature, the aging intensity functions have been studied in the univariate and bivariate case but without considering the possibility of observing a dynamic history. In this paper, the concept of aging intensity function is extended to the multivariate case by the use of the multivariate conditional hazard rate functions. Some properties of those functions are studied and a focus on the bivariate case is performed. Finally, the multivariate conditional aging intensity functions are studied for the order dependent version of the time-homogeneous load-sharing model and a study on the comparison among surviving components in a system is provided.

Efficient simulation of p-tempered $$\alpha $$-stable OU processes

We develop efficient methods for simulating processes of Ornstein–Uhlenbeck type related to the class of p-tempered \(\alpha \)-stable (\(\textrm{TS}^p_\alpha \)) distributions. Our results hold for both the univariate and multivariate cases and we consider both the case where the \(\textrm{TS}^p_\alpha \) distribution is the stationary law and where it is the distribution of the background driving Lévy process. In the latter case, we also derive an explicit representation for the transition law as this was previous known only in certain special cases and only for \(p=1\) and \(\alpha \in [0,1)\). Simulation results suggest that our methods work well in practice.

Influence of the spatial configuration of available data on hydraulic conductivity estimates for a geostatistical-Kalman filter method

Abstract Accurate hydraulic conductivity estimates are vital for groundwater evaluation. Usually, interpolations of hydraulic conductivity data are needed to obtain spatial estimates over larger areas, but the results present a high uncertainty which can be reduced by adding a secondary variable in the estimation. In this paper, the influence of the number and spatial configuration of hydraulic conductivity (K) and hydraulic head (HH) data on the estimation of K is evaluated using univariate and bivariate geostatistical-Kalman filter approaches (similar to kriging and cokriging, respectively). A synthetic case based on a transient groundwater flow model is used to generate different numbers, spatial arrays, and data. With these data, variogram models for the univariate and bivariate cases were fitted and used to calculate the corresponding covariance matrices for the Kalman filter. The results show that K estimates are more reliable when HH data is added than when only K is used, independently of the number and distribution of the data, since there is a better agreement between the calculated errors and estimate error variances. HH data provides valuable information only where K is not sampled. This evaluation could support the design of optimal sampling strategies to obtain reliable K estimates.

An extension of holonomic sequences: C2-finite sequences

Holonomic sequences are widely studied as many objects interesting to mathematicians and computer scientists are in this class. In the univariate case, these are the sequences satisfying linear recurrences with polynomial coefficients and also referred to as D-finite sequences. A subclass are C-finite sequences satisfying a linear recurrence with constant coefficients.We investigate the set of sequences which satisfy linear recurrence equations with coefficients that are C-finite sequences and call them C2-finite sequences. These sequences are a natural generalization of holonomic sequences. In this paper, we show that C2-finite sequences form a difference ring and provide methods to compute in this ring.Furthermore, we provide an analogous construction for D2-finite sequences, i.e., sequences satisfying a linear recurrence with holonomic coefficients. We show that these constructions can be iterated and obtain an increasing chain of difference rings.

Two variants of Hilbert’s Nullstellensatz

ABSTRACT We prove two variations of Hilbert’s Nullstellensatz in the univariate case: (1) If h and are polynomials in and there is a positive integer m and a positive constant C such that then there are n polynomials such that with the degree bound (2) If and h are polynomials in and there is a positive integer m and a positive constant C such that then there are n polynomials such that with the degree bound

Nonlinear autocorrelation function of functional time series

In functional time series analysis, the functional autocorrelation function (fACF) plays an important role in revealing the temporal dependence structures underlying the dynamics and identifying the lags at which substantial correlation exists. However, akin to its counterpart in the univariate case, the fACF is restricted by linear structure and can be misleading in reflecting nonlinear temporal dependence. This paper proposes a nonlinear alternative to the fACF for analyzing the temporal dependence in functional time series. We consider linear and nonlinear data generating processes: a functional autoregressive process and a functional generalized autoregressive conditional heteroskedasticity process. We demonstrate that when the process exhibits linear temporal structures, the inference obtained from our proposed nonlinear fACF is consistent with that from the fACF. When the underlying process exhibits nonlinear temporal dependence, our nonlinear fACF has a superior capability in uncovering the nonlinear structure that the fACF misleads. An empirical data analysis highlights its applications in unveiling nonlinear temporal structures in the daily curves of the intraday volatility dynamics of the foreign exchange rate.

Nonparametric bivariate density estimation for censored lifetimes

It is well known that estimation of a bivariate cumulative distribution function of a pair of right censored lifetimes presents challenges unparalleled to the univariate case where a product-limit Kaplan–Meyer’s methodology typically yields optimal estimation, and the literature on optimal estimation of the joint probability density is next to none. The paper, for the first time in the survival analysis literature, develops the theory and methodology of sharp minimax and adaptive nonparametric estimation of the joint density under the mean integrated squared error (MISE) criterion. The theory shows how an underlying joint density, together with the bivariate distribution of censoring variables, affect the estimation, and what and how may or may not be estimated in the presence of censoring. Practical example illustrates the problem.

A disintegration of the Christoffel function

We show that the Christoffel function (CF) factorizes (or can be disintegrated) as the product of two Christoffel functions, one associated with the marginal and the another related to the conditional distribution, in the spirit of “the CF of the disintegration is the disintegration of the CFs”. In the proof one uses an apparently overlooked property (but interesting in its own) which states that any sum-of-squares polynomial is the Christoffel function of some linear form (with a representing measure in the univariate case). The same is true for the convex cone of polynomials that are positive on a basic semi-algebraic set. This interpretation of the CF establishes another bridge between polynomials optimization and orthogonal polynomials.

Tests of serial dependence for multivariate time series with arbitrary distributions

In this paper, one studies tests of serial independence using a fixed number p of consecutive observations from a stationary time series, first in the univariate case, and then in the multivariate case, where even high-dimensional vectors can be used. The common distribution function is not assumed to be continuous, and the test statistics are based on the multilinear copula process. A bootstrap procedure based on multipliers is also proposed and shown to be valid. Tests based on Spearman’s rho and Kendall’s tau statistics are also considered, extending the results known for the case of continuous distributions. Contiguity results are obtained for some specific models and sufficient conditions for consistency of test statistics are stated, as well as a data-driven procedure to select p. Also, numerical experiments are performed to assess the finite sample level and power of the proposed tests. A case study using a time series of Arctic sea ice extent images is used to illustrate the usefulness of the proposed methodologies. The R package MixedIndTests (Nasri et al., 2022) includes all the methodologies proposed in this article.

Geometrically designed variable knot splines in generalized (non-)linear models

In this paper we extend the GeDS methodology, recently developed by Kaishev et al. [18] for the Normal univariate spline regression case, to the more general GNM/GLM context. Our approach is to view the (non-)linear predictor as a spline with free knots which are estimated, along with the regression coefficients and the degree of the spline, using a two stage algorithm. In stage A, a linear (degree one) free-knot spline is fitted to the data applying iteratively re-weighted least squares. In stage B, a Schoenberg variation diminishing spline approximation to the fit from stage A is constructed, thus simultaneously producing spline fits of second, third and higher degrees. We demonstrate, based on a thorough numerical investigation that the nice properties of the Normal GeDS methodology carry over to its GNM extension and GeDS favourably compares with other existing spline methods.The proposed GeDS GNM/GLM methodology is extended to the multivariate case of more than one independent variable by utilizing tensor product splines and their related shape preserving variation diminishing property.

The Algebraic Boundary of the Sonc-Cone

In this article, we explore the connections between nonnegativity, the theory of $A$-discriminants, and tropical geometry. For an integral support set $A \subset \mathbb{Z}^n$, we cover the boundary of the sonc-cone by semialgebraic sets that are parametrized by families of tropical hypersurfaces. As an application, we give sufficient conditions for equality between the sonc-cone and the sparse nonnegativity cone for generic support sets, and we describe a semialgebraic stratification of the boundary of the sonc-cone in the univariate case.

One-way MANOVA for functional data via Lawley–Hotelling trace test

Functional data arise from various fields of study and there have been numerous works on their analysis. However, most of existing methods consider the univariate case and methodology for multivariate functional data analysis is rather limited. In this article, we consider testing equality of vectors of mean functions for multivariate functional data, i.e., functional one-way multivariate analysis of variance (MANOVA). To this aim, we study asymptotic null distribution of the functional Lawley–Hotelling trace (FLH) test statistic and approximate it by a Welch–Satterthwaite type χ2-approximation. We describe two approaches to estimating the parameters in the χ2-approximation ratio-consistently. The resulting FLH test has the correct asymptotic level, is root-n consistent in detecting local alternatives, and is computationally efficient. The numerical performance is examined via some simulation studies and application to three real data examples. The proposed FLH test is comparable with four existing tests based on permutation in terms of size control and power. The major advantage is that it is much faster to compute.

Bounds-constrained polynomial approximation using the Bernstein basis

A fundamental problem in numerical analysis and approximation theory is approximating smooth functions by polynomials. A much harder version under recent consideration is to enforce bounds constraints on the approximating polynomial. In this paper, we consider the problem of constructioning such approximations using polynomials in the Bernstein basis. We consider a family of inequality-constrained quadratic programs. In the univariate case, a quadratic cone constraint allows us to search over all nonnegative polynomials of a given degree. In both the univariate and multivariate cases, we consider approximate problems with linear inequality constraints. Additionally, our method can be modified slightly to include equality constraints such as mass preservation.

An extended GCRD algorithm for parametric univariate polynomial matrices and application to parametric Smith form

The first extended greatest common right divisor (GCRD) algorithm for parametric univariate polynomial matrices is presented. The starting point of this GCRD algorithm is the free property of submodules over univariate polynomial rings. We convert the computation of GCRDs to that of free basis for modules and prove that a free basis of the submodule generated by row vectors of input matrices forms just a GCRD of these matrices. The GCRD algorithm is obtained by computing a minimal Gröbner basis for the corresponding submodule since a minimal Gröbner basis of submodules is a free basis for univariate cases. While the key idea of extended algorithm is to construct a special module by adding the unit vectors which can record the representation coefficients. This method based on modules can be naturally generalized to the parametric case because of the comprehensive Gröbner systems for modules. As a consequence, we obtain an extended GCRD algorithm for parametric univariate polynomial matrices. More importantly, we apply the proposed extended GCD algorithm for univariate polynomials (as a special case of matrices) to the computation of Smith normal form, and give the first algorithm for reducing a univariate polynomial matrix with parameters to its Smith normal form.

Parametric Vine Copula Framework in the Trivariate Probability Analysis of Compound Flooding Events

The interaction between oceanographic, meteorological, and hydrological factors can result in an extreme flooding scenario in the low-lying coastal area, called compound flooding (CF) events. For instance, rainfall and storm surge (or high river discharge) can be driven by the same meteorological forcing mechanisms, tropical or extra-tropical cyclones, resulting in a CF phenomenon. The trivariate distributional framework can significantly explain compound events’ statistical behaviour reducing the associated high-impact flood risk. Resolving heterogenous dependency of the multidimensional CF events by incorporating traditional 3D symmetric or fully nested Archimedean copula is quite complex. The main challenge is to preserve all lower-level dependencies. An approach based on decomposing the full multivariate density into simple local building blocks via conditional independence called vine or pair-copulas is a much more comprehensive way of approximating the trivariate flood dependence structure. In this study, a parametric vine copula of a drawable (D-vine) structure is introduced in the trivariate modelling of flooding events with 46 years of observations of the west coast of Canada. This trivariate framework searches dependency by combining the joint impact of annual maximum 24-h rainfall and the highest storm surge and river discharge observed within the time ±1 day of the highest rainfall event. The D-vine structures are constructed in three alternative ways by permutation of the conditioning variables. The most appropriate D-vine structure is selected using the fitness test statistics and estimating trivariate joint and conditional joint return periods. The investigation confirms that the D-vine copula can effectively define the compound phenomenon’s dependency. The failure probability (FP) method is also adopted in assessing the trivariate hydrologic risk. It is observed that hydrologic events defined in the trivariate case produce higher FP than in the bivariate (or univariate) case. It is also concluded that hydrologic risk increases (i) with an increase in the service design life of the hydraulic facilities and (ii) with a decrease in return periods.

On Fourier phases and their relevance for nonlinear time series analysis

Many processes in nature are governed by nonlinear mechanisms that are frequently superposed by pronounced linear components. To characterize such complex dynamics it is desirable to disentangle linear and nonlinear features in empirical data. Quantitative nonlinear measures are also influenced by linear properties of the signals and, for the univariate case, Fourier transform surrogates do not represent properly the null hypothesis of zero nonlinear features. Here we elaborate on the application of a recently published method in Fourier space that does not suffer from comparison with inappropriate surrogate data and that reveals with high sensitivity correlations between Fourier phases and amplitudes. In addition, we propose a simple pre-processing procedure that avoids the mixing of linear and nonlinear features when applying conventional nonlinear measures, which can drastically increase the sensitivity of the statistical evaluation and avoids numerical problems associated with surrogate data. We test our proposal on data derived from numerical models and we analyze electroencephalographic recordings from epilepsy patients as well as heart rate signals from healthy subjects before and during meditation.

Best orthogonal trigonometric approximations of the Nikol'skii-Besov-type classes of periodic functions of one and several variables

We obtained the exact order estimates of the best orthogonal trigonometric approximations of periodic functions of one and several variables from the Nikol'skii-Besov-type classes $B^{\omega}_{1,\theta}$ ($B^{\Omega}_{1,\theta}$ in the multivariate case $d\geq2$) in the space $B_{\infty,1}$. We observe that in the multivariate case the orders of mentioned approximation characteristics of the functional classes $B^{\Omega}_{1,\theta}$ are realized by their approximations by step hyperbolic Fourier sums that contain the necessary number of harmonics. In the univariate case, an optimal in the sense of order estimates for the best orthogonal trigonometric approximations of the corresponding functional classes are the ordinary partial sums of their Fourier series. As a consequence of the obtained results, the exact order estimates of the orthowidths of the classes $B^{\omega}_{1,\theta}$ ($B^{\Omega}_{1,\theta}$ for $d\geq2$) in the space $B_{\infty,1}$ are also established. Besides, we note that in the univariate case, in contrast to the multivariate one, the estimates of the considered approximation characteristics do not depend on the parameter $\theta$.

Monitoring the multivariate coefficient of variation in presence of autocorrelation with variable parameters control charts

ABSTRACT The coefficient of variation is a very important process parameter in many processes. A few control charts have been considered so far for monitoring its multivariate counterpart, i.e., the multivariate coefficient of variation (MCV). In addition, autocorrelation is very likely to occur in processes with high sampling frequency. Hence, designing suitable control charts and investigating the effect of autocorrelation on these charts is necessary. However, no control chart has been developed so far for the coefficient of variation that is capable of accounting for autocorrelation in either univariate or multivariate cases. This paper fills the gap by developing multivariate Shewhart-type control charts to monitor MCV with different autocorrelation structures for the observations: vector autoregressive, vector moving average, and vector mixed autoregressive and moving average. In addition, we add variable parameters adaptive features to the Shewhart-type scheme, in order to improve its performance. We develop a Markov chain model to get the statistical performance measures; then, we perform extensive numerical analyses to evaluate the effect of autocorrelation on adaptive and non-adaptive charts in the presence of downward and upward MCV shifts. Finally, we present an illustrative example from a healthcare process to show the implementation of this scheme in real practice.