Continuous Multivariate Distributions Research Articles

Generalized Moment Generating Functions for Some Continuous Multivariate Probability Distributions

The traditional moment generating functions of random variables and their probability distributions are known to not exist for all distributions and/or at all points and, where they exist, serious difficult and tedious manipulations are needed for the evaluation of higher central and non-central moments. This paper developed the generalized multivariate moment generating function for some random vectors/matrices and their probability distribution functions with the intention to replace the traditional/conventional moment generating functions due to their simplicity and versatility. The new functions were developed for the multivariate gamma family of distributions, the multivariate normal and the dirrichlet distributions as a binomial expansion of the expected value of an exponent of a random vector/matrix about an arbitrarily chosen constant. The functions were used to generate moments of random vectors/matrices and their probability distribution functions and the results obtained were compared with those from existing traditional/conventional methods. It was observed that the functions generated same results as the traditional/conventional methods; in addition, they generated both central and non-central moments in the same simple way without requiring further tedious manipulations; they gave more information about the distributions, for instance while the traditional method gives skewness and kurtosis values of and respectively for -variate multivariate normal distribution, the new methods gives ((0))p*1 and respectively and; they could generate moments of integral and real powers of random vectors/matrices.

Research and Application of Statistical Method of Data Reduction Based on Empirical Distribution

Data reduction is used to obtain the reduced representation of the data set, which is smaller than the original data, but still maintains the integrity of the original data approximately. Mining on the reduced data set will be more effective and produce the same or almost the same analysis results. A continuous multivariate coupled distribution estimation algorithm with arbitrary distribution is proposed. The distribution is estimated from samples by empirical distribution function, and new individuals are generated by sampling. Secondly, the idea of clustering is introduced into data reduction, and a time dimension reduction method based on clustering is formed. The basic idea of this method is to cluster the time dimension of time series data. In order to verify the feasibility of the two new methods proposed in this paper, a set of simulation experiments are designed in this paper, and representative data are used for data reduction respectively. Experiments show that the two data reduction methods proposed in this paper can not only effectively reduce the amount of data and achieve the purpose of data reduction, but also improve the classification accuracy and have strong practicability.

Mutual information matrix based on asymmetric Shannon entropy for nonlinear interactions of time series

Zhao et al. (Nonlin. Dyn. 88, 477-487, 2017) presented the mutual information matrix (MIM) analysis for the study of nonlinear interactions in multivariate time series as an extension of Random Matrix Theory analysis. They considered the histogram estimation of mutual information based on Shannon entropy for discrete distributions. This paper is motivated by the latter, extending MIM analysis from a nonparametric and probabilistic discrete approach to a parametric and probabilistic continuous approach. Specifically, this paper presents the MIM based on Maximum Likelihood Estimators (MLEs) for flexible and tractable families of continuous multivariate distributions, called multivariate skew-elliptical families of distributions. This method focus on multivariate skew-Gaussian and skew-t distributions that allow modeling skewness and heavy-tails, respectively. Performance of the proposed methodology is illustrated by numerical results given by sinusoidal and vector autoregressive fractionally integrated moving-average models, and applied to a meteorological monitoring network data set. Results show that the consideration of skewness and heavy-tails in the transformed ozone time series produced some differences in the MIM estimations compared with those obtained by applying histogram estimations to transformed data. Given that mutual information index (MII) increases in line with the number of bins for the histogram estimator, the proposed methodology based on MLEs considered more robust estimators with respect to the histogram ones to determine the MII of multivariate time series.

Discrete analogues of continuous multivariate probability distributions

In this note we introduce multivariate extensions to the bivariate discrete analogue to continuous probability distributions proposed in Barbiero [Annals of Operations Research, doi: 10.1007/s10479-019-03388-8, 2019]. We furthermore show that identical properties of the bivariate case also hold for n-variate discrete analogues.

S126. THE RELATION OF THE PSYCHOSIS CONTINUUM WITH SCHIZOPHRENIA POLYGENIC RISK SCORE AND CANNABIS USE

BackgroundThere has been much debate about whether research into psychosis should be conducted using symptom dimensions as opposed to diagnostic categories. Indeed, dimensions, like categories, may be practical but arbitrary tools for research and clinical practice; hence, they should not be based on psychometric data only. The aim of this study was to externally validate empirically derived symptom dimensions using combined genetic and environmental data. Specifically, we examined the hypothesis that the continuous multivariate distribution of psychosis is a function of cannabis use and genetic liability to schizophrenia, as summarised by polygenic risk score (SZ-PRS).MethodsAs part of the European Network of National Schizophrenia Networks Studying Gene-Environment Interactions (EU-GEI) study, we analysed a large multinational sample of First Episode Psychosis patients (FEP) and population controls, with available genotype and psychopathology information. Using item response modelling in Mplus, we estimated a bifactor model of psychotic symptoms in FEP, and of psychotic experiences in controls. Using PRSice, we built SZ-PRS by weighting individuals’ risk variants by the log(odds ratio), where the odds ratio was extracted from the latest summary statistics of Psychiatric Genomic Consortium mega-analyses on schizophrenia. Finally, we used linear regression to test the combined associations of the positive symptom/experience dimensions with SZ-PRS and daily/current cannabis use, separately in FEP and controls, after covarying for 10 ancestry principal components, sex, age, and primary diagnosis.ResultsThe continuous distribution of psychosis was represented by two bi-factor models composed of 1) in FEP, one general psychosis factor and five specific dimensions; 2) in controls, one general psychosis factor and three specific dimensions.Linear regression modelling showed that in 617 FEP, both daily cannabis use (B=0.31; 95%CI 0.11 to 0.52; p=0.002) and SZ-PRS (B=0.22; 95%CI 0.04 to 0.39; p=0.014) were independently associated with the dimension of positive symptoms. Similar results were found in 979 population controls, with a positive association of both current use of cannabis (B=0.26, 95%CI 0.06 to 0.46; p=0.011) and SZ-PRS (B=0.13, 95%CI 0.02 to 0.25; p=0.022) with the dimension of psychotic experiences.DiscussionWe found two factors associated with the latent dimensional structure of psychosis. SZ risk variants and cannabis use independently map onto specific dimensions of positive symptoms, contributing to variation across the psychosis continuum. Our study supports the theory that psychotic experiences in the general population are biologically similar to clinical psychotic symptoms.

Goodness-of-fit testing for copulas: A distribution-free approach

Consider a random sample from a continuous multivariate distribution function $F$ with copula $C$. In order to test the null hypothesis that $C$ belongs to a certain parametric family, we construct an empirical process on the unit hypercube that converges weakly to a standard Wiener process under the null hypothesis. This process can therefore serve as a ‘tests generator’ for asymptotically distribution-free goodness-of-fit testing of copula families. We also prove maximal sensitivity of this process to contiguous alternatives. Finally, we demonstrate through a Monte Carlo simulation study that our approach has excellent finite-sample performance, and we illustrate its applicability with a data analysis.

Approximation and sampling of multivariate probability distributions in the tensor train decomposition

General multivariate distributions are notoriously expensive to sample from, particularly the high-dimensional posterior distributions in PDE-constrained inverse problems. This paper develops a sampler for arbitrary continuous multivariate distributions that is based on low-rank surrogates in the tensor train format, a methodology that has been exploited for many years for scalable, high-dimensional density function approximation in quantum physics and chemistry. We build upon recent developments of the cross approximation algorithms in linear algebra to construct a tensor train approximation to the target probability density function using a small number of function evaluations. For sufficiently smooth distributions, the storage required for accurate tensor train approximations is moderate, scaling linearly with dimension. In turn, the structure of the tensor train surrogate allows sampling by an efficient conditional distribution method since marginal distributions are computable with linear complexity in dimension. Expected values of non-smooth quantities of interest, with respect to the surrogate distribution, can be estimated using transformed independent uniformly-random seeds that provide Monte Carlo quadrature or transformed points from a quasi-Monte Carlo lattice to give more efficient quasi-Monte Carlo quadrature. Unbiased estimates may be calculated by correcting the transformed random seeds using a Metropolis–Hastings accept/reject step, while the quasi-Monte Carlo quadrature may be corrected either by a control-variate strategy or by importance weighting. We show that the error in the tensor train approximation propagates linearly into the Metropolis–Hastings rejection rate and the integrated autocorrelation time of the resulting Markov chain; thus, the integrated autocorrelation time may be made arbitrarily close to 1, implying that, asymptotic in sample size, the cost per effectively independent sample is one target density evaluation plus the cheap tensor train surrogate proposal that has linear cost with dimension. These methods are demonstrated in three computed examples: fitting failure time of shock absorbers; a PDE-constrained inverse diffusion problem; and sampling from the Rosenbrock distribution. The delayed rejection adaptive Metropolis (DRAM) algorithm is used as a benchmark. In all computed examples, the importance weight-corrected quasi-Monte Carlo quadrature performs best and is more efficient than DRAM by orders of magnitude across a wide range of approximation accuracies and sample sizes. Indeed, all the methods developed here significantly outperform DRAM in all computed examples.

Discrete Bivariate Distributions Generated By Copulas: DBEEW Distribution

Copulas are the most powerful tools in constructing continuous multivariate distributions given marginals and certain dependence structures. In this paper, we shall propose a general and novel method of generating discrete bivariate distributions using copulas. The advantage of our method is that, contrary to the standard methods, we do not need to have the joint distribution of the base variables, but we only need the marginal distributions. In particular, we shall concentrate on generating a new family of discrete bivariate exponentiated extended Weibull (DBEEW) distribution by a Cuadras–Auge copula. We shall study several mathematical properties of a DBEEW distribution. Three special submodels of our new proposed distribution are studied in more detail. Finally, estimation of parameters for these three submodels is investigated and their potential and flexibility are examined using a real data set.

Subsampling (weighted smooth) empirical copula processes

A key tool to carry out inference on the unknown copula when modeling a continuous multivariate distribution is a nonparametric estimator known as the empirical copula. One popular way of approximating its sampling distribution consists of using the multiplier bootstrap. The latter is however characterized by a high implementation cost. Given the rank-based nature of the empirical copula, the classical empirical bootstrap of Efron does not appear to be a natural alternative, as it relies on resamples which contain ties. The aim of this work is to investigate the use of subsampling in the aforementioned framework. The latter consists of basing the inference on statistic values computed from subsamples of the initial data. One of its advantages in the rank-based context under consideration is that the formed subsamples do not contain ties. Another advantage is its asymptotic validity under minimalistic conditions. In this work, we show the asymptotic validity of subsampling for several (weighted, smooth) empirical copula processes both in the case of serially independent observations and time series. In the former case, subsampling is observed to be substantially better than the empirical bootstrap and equivalent, overall, to the multiplier bootstrap in terms of finite-sample performance.

Data-Driven Risk-Averse Two-Stage Optimal Stochastic Scheduling of Energy and Reserve With Correlated Wind Power

This paper proposes a data-driven optimization method to solve the integrated energy and reserve dispatch problem with variable and correlated renewable energy generation. The proposed method applies the kernel density estimation to establish an ambiguity set of continuous multivariate probability distributions and the optimization model for the integrated dispatch is formulated as a combination of stochastic and robust optimization problems. First, a risk-averse two-stage stochastic optimization model is formulated to hedge the distributional uncertainty. Next, the second-stage worst case expectation is evaluated, using the equivalent model reformulation, as a combination of conditional value-at-risk (CVaR) and the extreme cost in the worst case scenario. The CVaR is calculated using a scenario-based stochastic optimization problem. After describing the wind power correlation in ellipsoidal uncertainty sets, the robust optimization problem for finding the worst case cost is cast into a mixed-integer second-order cone programming problem. Finally, the column-and-constraint generation method is employed to solve the proposed risk-averse two-stage problem. The proposed method is tested on the 6-bus and IEEE 118-bus systems and validated by comparing the results with those of conventional stochastic and robust optimization methods.

A novel scale-space approach for multinormality testing and the k-sample problem in the high dimension low sample size scenario.

Two classical multivariate statistical problems, testing of multivariate normality and the k-sample problem, are explored by a novel analysis on several resolutions simultaneously. The presented methods do not invert any estimated covariance matrix. Thereby, the methods work in the High Dimension Low Sample Size situation, i.e. when n ≤ p. The output, a significance map, is produced by doing a one-dimensional test for all possible resolution/position pairs. The significance map shows for which resolution/position pairs the null hypothesis is rejected. For the testing of multinormality, the Anderson-Darling test is utilized to detect potential departures from multinormality at different combinations of resolutions and positions. In the k-sample case, it is tested whether k data sets can be said to originate from the same unspecified discrete or continuous multivariate distribution. This is done by testing the k vectors corresponding to the same resolution/position pair of the k different data sets through the k-sample Anderson-Darling test. Successful demonstrations of the new methodology on artificial and real data sets are presented, and a feature selection scheme is demonstrated.

A new generalisation of multivariate Wigner distribution of kind-2

In this paper authors proposed a new generalization of family of Sarmanov type Continuous multivariate symmetric probability distributions. Specifically, a new generalization of Multivariate Wigner distribution of kind-2 from the univariate case. Further, we find its Cumulation, Marginal, Conditional distributions, Generating functions and also discussed its special case. The authors derived the generating functions of this distribution in terms of Bessel function. The special cases include the transformation of Multivariate Wigner distribution of Kind-2 into Multivariate log-Wigner distribution of kind-2 and Multivariate Inverse-Wigner distribution of Kind-2. It is found that the conditional variance of Multivariate conditional Wigner distribution is homoscedastic and the population correlation co-efficient among the random variables are similar to Pearson’s population product moment correlation co-efficient. Area values of the bivariate Wigner surface also extracted and visualized.

A new generalisation of Sam-Solai’s multivariate additive gumbel min distribution

The paper proposed a new generalization of bounded Continuous multivariate symmetric probability distributions. More specifically, a new generalization of Sam - Solai’s Multivariate additive Gumbel Min distribution. Further, the authors find its Marginal, Multivariate Conditional distributions, Multivariate Generating functions, Multivariate survival, hazard functions and also discussed it’s special cases which includes the transformation of Sam-Solai’s Multivariate additive Gumbel Min distribution into Multivariate additive log - Gumbel Min distribution, Multivariate additive Gumbel max distribution. Moreover, the bivariate correlation between two Gumbel min variables and Co-variances were derived.

Multivariate reliability modelling based on dependent dynamic shock models

• This paper develops new classes of multivariate reliability models with positive dependence. • The developed novel methodology is also used to generate multivariate exponential distributions. • Multivariate dependence and ageing properties of the developed models are discussed. • This paper uses advanced shock models to generate multivariate reliability models. In this paper, we stochastically model positively dependent multivariate reliability distributions based on stochastically dependent dynamic shock models. In the first part, we consider a shock model with delayed failures. This shock model will be used to construct a class of absolutely continuous multivariate reliability distributions. Explicit parametric forms for the multivariate reliability functions are suggested. Multivariate ageing properties and dependence structures of the class are discussed as well. In the second part, we obtain two types of absolutely continuous multivariate exponential distributions based on further generalized shock models.

Multivariate quantile mapping bias correction: an N-dimensional probability density function transform for climate model simulations of multiple variables

Most bias correction algorithms used in climatology, for example quantile mapping, are applied to univariate time series. They neglect the dependence between different variables. Those that are multivariate often correct only limited measures of joint dependence, such as Pearson or Spearman rank correlation. Here, an image processing technique designed to transfer colour information from one image to another—the N-dimensional probability density function transform—is adapted for use as a multivariate bias correction algorithm (MBCn) for climate model projections/predictions of multiple climate variables. MBCn is a multivariate generalization of quantile mapping that transfers all aspects of an observed continuous multivariate distribution to the corresponding multivariate distribution of variables from a climate model. When applied to climate model projections, changes in quantiles of each variable between the historical and projection period are also preserved. The MBCn algorithm is demonstrated on three case studies. First, the method is applied to an image processing example with characteristics that mimic a climate projection problem. Second, MBCn is used to correct a suite of 3-hourly surface meteorological variables from the Canadian Centre for Climate Modelling and Analysis Regional Climate Model (CanRCM4) across a North American domain. Components of the Canadian Forest Fire Weather Index (FWI) System, a complicated set of multivariate indices that characterizes the risk of wildfire, are then calculated and verified against observed values. Third, MBCn is used to correct biases in the spatial dependence structure of CanRCM4 precipitation fields. Results are compared against a univariate quantile mapping algorithm, which neglects the dependence between variables, and two multivariate bias correction algorithms, each of which corrects a different form of inter-variable correlation structure. MBCn outperforms these alternatives, often by a large margin, particularly for annual maxima of the FWI distribution and spatiotemporal autocorrelation of precipitation fields.

The empirical beta copula

Given a sample from a continuous multivariate distribution F , the uniform random variates generated independently and rearranged in the order specified by the componentwise ranks of the original sample look like a sample from the copula of F . This idea can be regarded as a variant on Baker’s [J. Multivariate Anal. 99 (2008) 2312–2327] copula construction and leads to the definition of the empirical beta copula. The latter turns out to be a particular case of the empirical Bernstein copula, the degrees of all Bernstein polynomials being equal to the sample size. Necessary and sufficient conditions are given for a Bernstein polynomial to be a copula. These imply that the empirical beta copula is a genuine copula. Furthermore, the empirical process based on the empirical Bernstein copula is shown to be asymptotically the same as the ordinary empirical copula process under assumptions which are significantly weaker than those given in Janssen, Swanepoel and Veraverbeke [J. Statist. Plann. Inference 142 (2012) 1189–1197]. A Monte Carlo simulation study shows that the empirical beta copula outperforms the empirical copula and the empirical checkerboard copula in terms of both bias and variance. Compared with the empirical Bernstein copula with the smoothing rate suggested by Janssen et al., its finite-sample performance is still significantly better in several cases, especially in terms of bias.

General location model with factor analyzer covariance matrix structure and its applications

General location model (GLOM) is a well-known model for analyzing mixed data. In GLOM one decomposes the joint distribution of variables into conditional distribution of continuous variables given categorical outcomes and marginal distribution of categorical variables. The first version of GLOM assumes that the covariance matrices of continuous multivariate distributions across cells, which are obtained by different combination of categorical variables, are equal. In this paper, the GLOMs are considered in both cases of equality and unequality of these covariance matrices. Three covariance structures are used across cells: the same factor analyzer, factor analyzer with unequal specific variances matrices (in the general and parsimonious forms) and factor analyzers with common factor loadings. These structures are used for both modeling covariance structure and for reducing the number of parameters. The maximum likelihood estimates of parameters are computed via the EM algorithm. As an application for these models, we investigate the classification of continuous variables within cells. Based on these models, the classification is done for usual as well as for high dimensional data sets. Finally, for showing the applicability of the proposed models for classification, results from analyzing three real data sets are presented.

Scenario generation for stochastic optimization problems via the sparse grid method

We study the use of sparse grids in the scenario generation (or discretization) problem in stochastic programming problems where the uncertainty is modeled using a continuous multivariate distribution. We show that, under a regularity assumption on the random function involved, the sequence of optimal objective function values of the sparse grid approximations converges to the true optimal objective function values as the number of scenarios increases. The rate of convergence is also established. We treat separately the special case when the underlying distribution is an affine transform of a product of univariate distributions, and show how the sparse grid method can be adapted to the distribution by the use of quadrature formulas tailored to the distribution. We numerically compare the performance of the sparse grid method using different quadrature rules with classic quasi-Monte Carlo (QMC) methods, optimal rank-one lattice rules, and Monte Carlo (MC) scenario generation, using a series of utility maximization problems with up to 160 random variables. The results show that the sparse grid method is very efficient, especially if the integrand is sufficiently smooth. In such problems the sparse grid scenario generation method is found to need several orders of magnitude fewer scenarios than MC and QMC scenario generation to achieve the same accuracy. It is indicated that the method scales well with the dimension of the distribution—especially when the underlying distribution is an affine transform of a product of univariate distributions, in which case the method appears scalable to thousands of random variables.

Monitoring nonlinear profiles adaptively with a wavelet-based distribution-free CUSUM chart

A wavelet-based distribution-free tabular CUSUM chart based on adaptive thresholding, is designed for rapidly detecting shifts in the mean of a high-dimensional profile whose noise components have a continuous nonsingular multivariate distribution. First computing a discrete wavelet transform of the noise vectors for randomly sampled Phase I (in-control) profiles, uses a matrix-regularization method to estimate the covariance matrix of the wavelet-transformed noise vectors; then, those vectors are aggregated (batched) so that the non-overlapping batch means of the wavelet-transformed noise vectors have manageable covariances. Lower and upper in-control thresholds are computed for the resulting batch means of the wavelet-transformed noise vectors using the associated marginal Cornish–Fisher expansions that have been suitably adjusted for between-component correlations. From the thresholded batch means of the wavelet-transformed noise vectors, Hotelling’s -type statistics are computed to set the parameters of a CUSUM procedure. To monitor shifts in the mean profile during Phase II (regular) operation, computes a similar Hotelling’s -type statistic from successive thresholded batch means of the wavelet-transformed noise vectors using the in-control thresholds; then applies the CUSUM procedure to the resulting -type statistics. Experimentation with several normal and non-normal test processes revealed that outperformed existing non-adaptive profile-monitoring schemes.

Two-step Dirichlet random walks

Random walks of n steps taken into independent uniformly random directions in a d-dimensional Euclidean space (d⩾2), which are characterized by a sum of step lengths which is fixed and taken to be 1 without loss of generality, are named “Dirichlet” when this constraint is realized via a Dirichlet law of step lengths. The latter continuous multivariate distribution, which depends on n positive parameters, generalizes the beta distribution (n=2). It is simply obtained from n independent gamma random variables with identical scale factors. Previous literature studies of these random walks dealt with symmetric Dirichlet distributions whose parameters are all equal to a value q which takes half-integer or integer values. In the present work, the probability density function of the distance from the endpoint to the origin is first made explicit for a symmetric Dirichlet random walk of two steps. It is valid for any positive value of q and for all d⩾2. The latter pdf is used in turn to express the related density of a random walk of two steps whose step length is distributed according to an asymmetric beta distribution which depends on two parameters, namely q and q+s where s is a positive integer.