Multivariate Distribution Research Articles

Regional disparities in home health care utilization for older adults and their associated factors at the secondary medical area level: A Nationwide study in Japan

AimAs aging populations shift health care from hospitals to communities, Japan has implemented policies to promote home health care. This study explored regional differences in home health care recipients among older adults and related factors.MethodsWe used nationwide data from 2020 to describe the proportion of older adults receiving regular home visits and the medical institutions utilized across secondary medical areas: urban, middle and depopulated areas. We examined factors associated with the proportion of patients receiving regular home visits. Exposures included each secondary medical area's medical and long‐term‐care (LTC) resources, adjusted for the older adult population; proportion of single‐person households; and regional factors. We performed a multivariate negative binomial distribution analysis.ResultsA total of 333 secondary medical areas were included. Urban areas had more patients receiving regular home visits, primarily from enhanced home care support clinics/hospitals (HCSCs). Fewer patients received regular home visits in depopulated areas, and conventional HCSCs were more common. Multivariate analysis revealed that the number of conventional HCSCs (coefficient, 0.17 [95% confidence intervals (CI), 0.08 to 0.26]), enhanced HCSCs (coefficient, 0.21 [95% CI, 0.14 to 0.29]) and population density (coefficient, 0.10 [95% CI, 0.02 to 0.19]) were positively associated with higher home‐visit rates. Beds in LTC welfare facilities (coefficient, −0.10 [95% CI, −0.19 to −0.01]) and beds in LTC health facilities (coefficient, −0.09 [95% CI, −0.17 to 0.00]) were negatively associated.ConclusionsPolicies to promote home health care have led to high home‐visit rates in urban areas. Medical and LTC resources and regional factors influence disparities. As Japan's population ages, it is crucial to recognize these disparities and develop medical and LTC systems tailored to each region's characteristics. Geriatr Gerontol Int 2024; ••: ••–••.

The Impact of Non-Normality on Causal Effect Estimates in Structural Equation Models with Ordered Categorical Variables

Multilevel structural equation models (SEM) are widely used for estimating causal effects in social and behavioral sciences. However, the impact of non-normality on these estimates, particularly with ordered categorical variables, remains unclear. This study investigates how non-normality affects causal effect estimation in path and multigroup SEMs with ordered categorical variables. Using the Vine-to-Anything approach, we simulated multivariate distributions by independently varying marginals and copulas. We compared maximum likelihood (ML), robust ML, and diagonally weighted least squares (DWLS) estimators across various non-normal conditions in three simulation studies. Results show that bias in effect estimates critically depends on the combination of copulas and marginal distributions in treatment and control groups. DWLS generally outperformed ML, especially for average treatment effects and interaction effects. However, all methods were sensitive to distributional misspecifications. These findings highlight the importance of considering distributional assumptions when using ordered categorical data in causal inference studies.

Subjective well-being indices in assessing happiness in Russia

Due to the multidimensionality and dynamism of today’s social life, there emerges the need to modify the principles of measuring the effectiveness of socio-economic policy. The article empirically tests and analyzes subjective metrics focused on “failures” in the socio-economic development which are impossible to identify through objective indicators only. These metrics are the subjective well-being indices for the Russian population in terms of gender and age, residence (by federal districts), level of education, marital status, presence of children, subjective health assessments, job and pay satisfaction. The methodological framework resides in Edward Diener’s subjective well-being theory. The method of index numbers and methods of descriptive statistics, including the multivariate frequency distribution of two attributes, are applied. The evidence base is comprised of panel data of The Russia Longitudinal Monitoring Survey – Higher School of Economics (RLMS-HSE) for 2013–2022. According to the research results, the minimum and maximum values of the life satisfaction index were observed in 2016 and 2022, respectively. The affective index of happiness was found to reach higher values compared to the cognitive assessment of it. The findings show that gender, age, region of residence, level of education, as well as satisfaction with health, job, wage and financial status affect the values and dynamics of happiness indices in Russia. A well-being measurement approach covering both objective and sub jective assessments of people’s life and its individual aspects allows identifying gaps in the system of socio-economic support for the population.

Uncertainty quantification for seismic response using dimensionality reduction‐based stochastic simulator

AbstractThis paper introduces a stochastic simulator for seismic uncertainty quantification, which is crucial for performance‐based earthquake engineering. The proposed simulator extends the recently developed dimensionality reduction‐based surrogate modeling method (DR‐SM) to address high‐dimensional ground motion uncertainties and the high computational demands associated with nonlinear response history analyses. By integrating physics‐based dimensionality reduction with multivariate conditional distribution models, the proposed simulator efficiently propagates seismic input into multivariate response quantities of interest. The simulator can incorporate both aleatory and epistemic uncertainties and does not assume distribution models for the seismic responses. The method is demonstrated through three finite element building models subjected to synthetic and recorded ground motions. The proposed method effectively predicts multivariate seismic responses and quantifies uncertainties, including correlations among responses.

Scale mixtures of multivariate centered skew-normal distributions

Scale mixtures of multivariate centered skew-normal distributions

Reinforcement Learning for EV Fleet Smart Charging with On-Site Renewable Energy Sources

In 2020, the transportation sector was the second largest source of carbon emissions in the UK and in Newcastle upon Tyne, responsible for about 33% of total emissions. To support the UK’s target of reaching net zero emissions by 2050, electric vehicles (EVs) are pivotal in advancing carbon-neutral road transportation. Optimal EV charging requires a better understanding of the unpredictable output from on-site renewable energy sources (ORES). This paper proposes an integrated EV fleet charging schedule with a proximal policy optimization method based on a framework for deep reinforcement learning. For the design of the reinforcement learning environment, mathematical models of wind and solar power generation are created. In addition, the multivariate Gaussian distributions derived from historical weather and EV fleet charging data are utilized to simulate weather and charging demand uncertainty in order to create large datasets for training the model. The optimization problem is expressed as a Markov decision process (MDP) with operational constraints. For training artificial neural networks (ANNs) through successive transition simulations, a proximal policy optimization (PPO) approach is devised. The optimization approach is deployed and evaluated on a real-world scenario comprised of council EV fleet charging data from Leicester, UK. The results show that due to the design of the rewards function and system limitations, the charging action is biased towards the time of day when renewable energy output is maximum (midday). The charging decision by reinforcement learning improves the utilization of renewable energy by 2–4% compared to the random charging policy and the priority charging policy. This study contributes to the reduction in battery charging and discharging, electricity sold to the grid to create benefits and the reduction in carbon emissions.

ON IMPROVED GAUSSIAN CORRELATION INEQUALITIES FOR SYMMETRICAL -RECTANGLES EXTENDED TO CERTAIN MULTIVARIATE GAMMA DISTRIBUTIONS AND SOME FURTHER PROBABILITY INEQUALITIES

The Gaussian correlation inequality (GCI) for symmetrical n-rectangles is improved if the absolute components have a joint cumulative distribution function (cdf), which is MTP2 (multivariate totally positive of order 2). Inequalities of the given type hold at least for all MTP2-cdfs on or with everywhere positive smooth densities. In particular, at least some infinitely divisible multivariate chi-square distributions (gamma distributions in the sense of Krishnamoorthy and Parthasarathy) with any positive real “degree of freedom” are shown to be MTP2. Moreover, further numerically calculable probability inequalities for a broader class of multivariate gamma distributions are derived. A different improvement for inequalities of the GCI-type, and of a similar type with three instead of two groups of components with more special correlation structures is also obtained. The main idea behind these inequalities is to find for a given correlation matrix with positive correlations, a further correlation matrix with smaller correlations whose inverse is an M-matrix and where the corresponding multivariate gamma distribution function is numerically available.

Vine Copula-Based Classifiers with Applications

AbstractThe vine pair-copula construction can be used to fit flexible non-Gaussian multivariate distributions to a mix of continuous and discrete variables. With multiple classes, fitting univariate distributions and a vine to each class lead to posterior probabilities over classes that can be used for discriminant analysis. This is more flexible than methods with the Gaussian and/or independence assumptions, such as quadratic discriminant analysis and naive Bayes. Some variable selection methods are studied to accompany the vine copula-based classifier because unimportant variables can make discrimination worse. Simple numerical performance metrics cannot give a full picture of how well a classifier is doing. We introduce categorical prediction intervals and other summary measures to assess the difficulty of discriminating classes. Through extensive experiments on real data, we demonstrate the superior performance of our approaches compared to traditional discriminant analysis methods and random forests when features have different dependent structures for different classes.

GenSynthPop: generating a spatially explicit synthetic population of individuals and households from aggregated data

Synthetic populations are representations of actual individuals living in a specific area. They play an increasingly important role in studying and modeling individuals and are often used to build agent-based social simulations. Traditional approaches for synthesizing populations use a detailed sample of the population (which may not be available) or combine data into a single joint distribution, and draw individuals or households from these. The latter group of existing sample-free methods fail to integrate (1) the best available data on spatial granular distributions, (2) multi-variable joint distributions, and (3) household level distributions. In this paper, we propose a sample-free approach where synthetic individuals and households directly represent the estimated joint distribution to which attributes are iteratively added, conditioned on previous attributes such that the relative frequencies within each joint group of attributes are maintained and fit granular spatial marginal distributions. In this paper we present our method and test it for the Zuid-West district of The Hague, the Netherlands, showing that spatial, multi-variable and household distributions are accurately reflected in the resulting synthetic population.

Spectral expansion methods for prediction uncertainty quantification in systems biology

Uncertainty is ubiquitous in biological systems. For example, since gene expression is intrinsically governed by noise, nature shows a fascinating degree of variability. If we want to use a model to predict the behaviour of such an intrinsically stochastic system, we have to cope with the fact that the model parameters are never exactly known, but vary according to some distribution. A key question is then to determine how the uncertainties in the parameters affect the model outcome. Knowing the latter uncertainties is crucial when a model is used for, e.g., experimental design, optimisation, or decision-making. To establish how parameter and model prediction uncertainties are related, Monte Carlo approaches could be used. Then, the model is evaluated for a huge number of parameters sets, drawn from the multivariate parameter distribution. However, when model solutions are computationally expensive this approach is intractable. To overcome this problem, so-called spectral expansion (SE) methods have been developed to quantify prediction uncertainty within a probabilistic framework. Such SE methods have a basis in, e.g., computational mathematics, engineering, physics, and fluid dynamics, and, to a lesser extent, systems biology. The computational costs of SE schemes mainly stem from the calculation of the expansion coefficients. Furthermore, SE effectively leads to a surrogate model which captures the dependence of the model on the uncertainty parameters, but is much simpler to execute compared to the original model. In this paper, we present an innovative scheme for the calculation of the expansion coefficients. It guarantees that the model has to be evaluated only a restricted number of times. Especially for models of high complexity this may be a huge computational advantage. By applying the scheme to a variety of examples we show its power, especially in challenging situations where solutions slowly converge due to high computational costs, bifurcations, and discontinuities.

Reducing semantic ambiguity in domain adaptive semantic segmentation via probabilistic prototypical pixel contrast

Domain adaptation aims to reduce the model degradation on the target domain caused by the domain shift between the source and target domains. Although encouraging performance has been achieved by combining contrastive learning with the self-training paradigm, they suffer from ambiguous scenarios caused by scale, illumination, or overlapping when deploying deterministic embedding. To address these issues, we propose probabilistic prototypical pixel contrast (PPPC), a universal adaptation framework that models each pixel embedding as a probability via multivariate Gaussian distribution to fully exploit the uncertainty within them, eventually improving the representation quality of the model. In addition, we derive prototypes from probability estimation posterior probability estimation which helps to push the decision boundary away from the ambiguity points. Moreover, we employ an efficient method to compute similarity between distributions, eliminating the need for sampling and reparameterization, thereby significantly reducing computational overhead. Further, we dynamically select the ambiguous crops at the image level to enlarge the number of boundary points involved in contrastive learning, which benefits the establishment of precise distributions for each category. Extensive experimentation demonstrates that PPPC not only helps to address ambiguity at the pixel level, yielding discriminative representations but also achieves significant improvements in both synthetic-to-real and day-to-night adaptation tasks. It surpasses the previous state-of-the-art (SOTA) by +5.2% mIoU in the most challenging daytime-to-nighttime adaptation scenario, exhibiting stronger generalization on other unseen datasets. The code and models are available at https://github.com/DarlingInTheSV/Probabilistic-Prototypical-Pixel-Contrast.

TCDM: Transformational Complexity Based Distortion Metric for Perceptual Point Cloud Quality Assessment.

The goal of objective point cloud quality assessment (PCQA) research is to develop quantitative metrics that measure point cloud quality in a perceptually consistent manner. Merging the research of cognitive science and intuition of the human visual system (HVS), in this article, we evaluate the point cloud quality by measuring the complexity of transforming the distorted point cloud back to its reference, which in practice can be approximated by the code length of one point cloud when the other is given. For this purpose, we first make space segmentation for the reference and distorted point clouds based on a 3D Voronoi diagram to obtain a series of local patch pairs. Next, inspired by the predictive coding theory, we utilize a space-aware vector autoregressive (SA-VAR) model to encode the geometry and color channels of each reference patch with and without the distorted patch, respectively. Assuming that the residual errors follow the multi-variate Gaussian distributions, the self-complexity of the reference and transformational complexity between the reference and distorted samples are computed using covariance matrices. Additionally, the prediction terms generated by SA-VAR are introduced as one auxiliary feature to promote the final quality prediction. The effectiveness of the proposed transformational complexity based distortion metric (TCDM) is evaluated through extensive experiments conducted on five public point cloud quality assessment databases. The results demonstrate that TCDM achieves state-of-the-art (SOTA) performance, and further analysis confirms its robustness in various scenarios.

Bivariate Tail Conditional Co-Expectation for elliptical distributions

In this paper, we consider a random vector X=(X1,X2) following a multivariate Elliptical distribution and we provide an explicit formula for E(X|X≤X˜), i.e., the expected value of the bivariate random variable X conditioned to the event X≤X˜, with X˜∈R2. Such a conditional expectation has an intuitive interpretation in the context of risk measures. Specifically, E(X|X≤X˜) can be interpreted as the Tail Conditional Co-Expectation of X (TCoES). Our main result analytically proves that for a large number of Elliptical distributions, the TCoES can be written as a function of the probability density function of the Skew Elliptical distributions introduced in the literature by the pioneering work of Azzalini (1985). Some numerical experiments based on empirical data show the usefulness of the obtained results for real-world applications.

Explicit bivariate simplicial depth

The simplicial depth (SD) is a celebrated tool defining elements of nonparametric and robust statistics for multivariate data. While many properties of SD are well-established, and its applications are abundant, explicit expressions for SD are known only for a handful of the simplest multivariate probability distributions. This paper deals with SD in the plane. It (i) develops a one-dimensional integral formula for SD of any properly continuous probability distribution, (ii) gives exact explicit expressions for SD of uniform distributions on (both convex and non-convex) polygons in the plane or on the boundaries of such polygons, and (iii) discusses several implications of these findings to probability and statistics: (a) An upper bound on the maximum SD in the plane, (b) an implication for a test of symmetry of a bivariate distribution, and (c) a connection of SD with the classical Sylvester problem from geometric probability.

Estimation of the Hubble constant using Gaussian process regression and viable alternatives

A well-known problem in cosmology is the ‘Hubble tension’ problem, i.e. different estimates for the Hubble constant H0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_0$$\\end{document} are not concordant with each other. This work investigates different statistical methods for estimating this value using cosmic chronometer, type Ia supernova and baryonic acoustic data. We start by making use of methods already established in the literature for this purpose, namely Gaussian process regression and Markov chain Monte Carlo (MCMC) methods based on the concordance Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varLambda $$\\end{document}CDM model. We also consider two novel approaches; the first makes use of non-parametric MCMC inference on the hyperparameters of a Gaussian process kernel, independently of any cosmological model. The second approach is Student’s t-process regression, which is a generalised version of Gaussian process regression that makes use of the multivariate Student’s t-distribution instead of the multivariate Gaussian distribution. We also consider variants of these two methods which account for heteroscedasticity within the data. A comparison of the different approaches is made. In particular, the model-independent techniques investigated mostly agree with predictions based on the Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varLambda $$\\end{document}CDM model. Moreover, Gaussian process regression is highly sensitive to the prior specification, while Student’s t-process regression and the heteroscedastic variants of both methods are more robust to this. Student’s t-process regression and both heteroscedastic models suggest a lower value of the Hubble constant. Across all the estimates obtained for the Hubble constant within this work, the median value is H^0med=68.85±1.67\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hat{H}_0^{med} = 68.85 \\pm 1.67$$\\end{document} km s-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{-1}$$\\end{document} Mpc-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{-1}$$\\end{document}.

Multivariate skew-normal distribution for modelling skewed spatial data

Multivariate spatial data are commonly modelled using the shared spatial component and multivariate intrinsic conditional autoregressive (MICAR) models where the spatial random variables are assumed to be normally distributed. However, the normality assumption may not be always right as the spatially structured component may show non-normal distributions. We present, multivariate skew-normal spatial distribution in the modelling of multivariate conditional autoregressive models. Simulations and an application to estimate district HIV rates in South Africa are used for illustrating the capabilities of the proposed multivariate skewed spatial model. The estimation is done in a Bayesian framework. A comparison between our suggested approach and the common MICAR model is made using conditional predictive ordinate (CPO). The CPO values indicate that our suggested approach is better than the MICAR model for predicting the outcome variables of both the simulated and HIV data.

Exploring the synergy of guided numeric and text analysis in e-commerce: a comprehensive investigation into univariate and multivariate distributions

This research adopts a holistic approach to analyze customer reviews in the e-commerce industry by utilizing a combined approach of numerical and text analysis. Specifically, this study integrates univariate, multivariate, and sentiment analysis to gain comprehensive insights into product preferences and customer satisfaction. The methodology includes a detailed examination of univariate distributions to uncover numerical trends in product ratings and preferences. Multivariate distributions are explored to understand the complex relationships between related variables. Sentiment analysis is performed using the Sentiment Intensity Analyzer to categorize reviews into positive, neutral, and negative sentiments. Additionally, N-gram analysis is applied to both recommended and non-recommended reviews to identify key themes, such as dissatisfaction with product size and satisfaction with fit. Logistic regression and naive Bayes models are employed to classify sentiment, with logistic regression achieving high accuracy on both training (91.3%) and validation data (89.2%). This research highlights the significant role of product recommendations as indicators of positive sentiment, while product ratings reveal the complexity in consumer judgment. The study contributes significantly to understanding the dynamics of customer reviews in the e-commerce industry, providing a solid foundation for smarter decision-making to improve customer experience and product quality.

Moments of doubly truncated multivariate normal mean-mixture distributions

Recently, considerable interest has been in deriving explicit expressions for moment of different truncated multivariate distributions. This is due to their usefulness in developing efficient EM-algorithm for the maximum likelihood estimation of the corresponding distributional parameters. One rich family of distributions is the multivariate normal mean-mixture distribution as it includes a number of important multivariate distributions as special cases. In this article, we derive exact expressions for the first two moments of doubly truncated multivariate normal mean-mixture distributions, from which results for various special cases are then deduced.

Variational Autoencoders and Generative Adversarial Networks for Multivariate Scenario Generation

When making decisions with lasting implications over a medium to long timeframe, it is essential to consider not only the most probable scenario, possibly obtained through a forecasting model, but also a range of potential outcomes. This approach allows for effective risk mitigation across a spectrum of scenarios, including less probable ones, and enhances the resilience of planning strategies. In this paper, we demonstrate the development of a generative model capable of learning the multivariate joint probability distribution of link speeds on a road network, using real sensor data. To further enhance the performance of our Generative Adversarial Network model, we employed a Variational AutoEncoder for pre-training the generator network. Experimental results, conducted on three distinct benchmark datasets, highlight the potential of the proposed model in generating new scenario samples of multivariate variables. The Wasserstein distance between the generated distribution and the real data, confirms the good performance of our model compared to state-of-the-art models, based on copulae. The proposed model has shown its ability to generate scenarios that preserve correlations among variables, while producing samples that faithfully represent the empirical marginal distributions.

Copula Approximate Bayesian Computation Using Distribution Random Forests

Ongoing modern computational advancements continue to make it easier to collect increasingly large and complex datasets, which can often only be realistically analyzed using models defined by intractable likelihood functions. This Stats invited feature article introduces and provides an extensive simulation study of a new approximate Bayesian computation (ABC) framework for estimating the posterior distribution and the maximum likelihood estimate (MLE) of the parameters of models defined by intractable likelihoods, that unifies and extends previous ABC methods proposed separately. This framework, copulaABCdrf, aims to accurately estimate and describe the possibly skewed and high-dimensional posterior distribution by a novel multivariate copula-based meta-t distribution based on univariate marginal posterior distributions that can be accurately estimated by distribution random forests (drf), while performing automatic summary statistics (covariates) selection, based on robustly estimated copula dependence parameters. The copulaABCdrf framework also provides a novel multivariate mode estimator to perform MLE and posterior mode estimation and an optional step to perform model selection from a given set of models using posterior probabilities estimated by drf. The posterior distribution estimation accuracy of the ABC framework is illustrated and compared with previous standard ABC methods through several simulation studies involving low- and high-dimensional models with computable posterior distributions, which are either unimodal, skewed, or multimodal; and exponential random graph and mechanistic network models, each defined by an intractable likelihood from which it is costly to simulate large network datasets. This paper also proposes and studies a new solution to the simulation cost problem in ABC involving the posterior estimation of parameters from datasets simulated from the given model that are smaller compared to the potentially large size of the dataset being analyzed. This proposal is motivated by the fact that, for many models defined by intractable likelihoods, such as the network models when they are applied to analyze massive networks, the repeated simulation of large datasets (networks) for posterior-based parameter estimation can be too computationally costly and vastly slow down or prohibit the use of standard ABC methods. The copulaABCdrf framework and standard ABC methods are further illustrated through analyses of large real-life networks of sizes ranging between 28,000 and 65.6 million nodes (between 3 million and 1.8 billion edges), including a large multilayer network with weighted directed edges. The results of the simulation studies show that, in settings where the true posterior distribution is not highly multimodal, copulaABCdrf usually produced similar point estimates from the posterior distribution for low-dimensional parametric models as previous ABC methods, but the copula-based method can produce more accurate estimates from the posterior distribution for high-dimensional models, and, in both dimensionality cases, usually produced more accurate estimates of univariate marginal posterior distributions of parameters. Also, posterior estimation accuracy was usually improved when pre-selecting the important summary statistics using drf compared to ABC employing no pre-selection of the subset of important summaries. For all ABC methods studied, accurate estimation of a highly multimodal posterior distribution was challenging. In light of the results of all the simulation studies, this article concludes by discussing how the copulaABCdrf framework can be improved for future research.