Multivariate Conditional Distribution Research Articles

On convergence and singularity of conditional copulas of multivariate Archimedean copulas, and conditional dependence

The present paper derives an explicit expression for (a version of) every uni- and multivariate conditional distribution (i.e., Markov kernel) of Archimedean copulas and uses this representation to generalize a recently established result, saying that in the class of multivariate Archimedean copulas standard uniform convergence implies weak convergence of almost all univariate Markov kernels, to arbitrary multivariate Markov kernels. Moreover, it is proved that an Archimedean copula is singular if, and only if, almost all uni- and multivariate Markov kernels are singular. These results are then applied to conditional Archimedean copulas which are reintroduced largely from a Markov kernel perspective and it is shown that convergence, singularity and conditional increasingness carry over from Archimedean copulas to their conditional copulas. As a consequence, the surprising fact is established that estimating (the generator of) an Archimedean copula directly yields an estimator of (the generator of) its conditional copula. Building upon that, we sketch the use and estimation of a conditional version of a recently introduced dependence measure as alternative to well-known conditional versions of association measures in order to study the dependence behavior of Archimedean models when fixing covariate values.

Spatial multivariate data imputation using deep learning and lambda distribution

Artificial neural networks (ANNs) are often used to establish a mapping between an input data set and a corresponding output. There are many applications that rely on quantifying the conditional distribution of the output given the input data set. This is often referred to as aleatoric uncertainty associated with variability of the outcome due to inherently random effects. In this paper, deep learning is used to quantify moments of the conditional distribution of a missing variable based on homotopic multivariate observations. The lambda distribution is then used to parametrize the conditional distribution based on the provided moments. Geostatistical quantification of spatial continuity complements the multivariate conditional distribution through Bayesian updating to inform multiple data imputation that accounts for the uncertainty associated with the missing variable(s). Geological data are often incomplete, and data imputation is an essential step to avoid excluding heterotopic data. The proposed data imputation framework trains multi layer perceptron (MLP) neural networks to characterize multivariate relationships inferred from homotopic training data. A case study is conducted using geological data from a lateritic Nickle deposit to demonstrate application of the proposed methodology.

On Reference-based Imputation for Analysis of Incomplete Repeated Binary Endpoints

ABSTRACT Reference-based imputation (RBI) is a popular method for missing data. The methodology is well established for continuous end points but less well developed for repeated binary end points due to the lack of natural multivariate conditional distributions for such end points. In this paper, we propose RBI methods for repeated binary end points based on a multivariate probit model and a logistic model, including jump-to-reference (J2R), copy-reference (CR) and copy-increment-in-reference (CIR). We explore the distribution of the missing binary end points under RBI and propose efficient algorithms to implement the proposed RBI methods. We evaluate the proposed methods by simulations and a data set from a clinical trial.

Dynamic copula-based expectile portfolios

This study investigates expectile Value-at-Risk (EVaR) as a risk measure in dynamic copula-based portfolio optimization, compared with the common variance and CVaR. To estimate the dependence structure between asset returns, the canonical vine copula augmented with the generalized additive models (GAMC-vine) is used. Applying multivariate conditional distributions from the GAMC-vine model, step-ahead asset return forecasts are obtained and used to construct dynamic copula-based EVaR portfolios. Using ten S&P 500 industry sectors, EVaR leads to a min-risk dynamic GAMC-vine portfolio that achieves higher out-of-sample average return and risk-adjusted ratios. Furthermore, EVaR shows a better portfolio ranking than CVaR. Moreover, the copula-based variance and EVaR portfolios show higher-order stochastic dominance compared to CVaR strategies. Finally, a subsample stochastic dominance analysis reveals that, in overall, the risk minimization does not benefit from the choice of risk modeling. However, the dynamic copula model leads to optimal portfolios that dominate the equally weighted benchmark more often compared to those from historical approach.

Diagonal sections of copulas, multivariate conditional hazard rates and distributions of order statistics for minimally stable lifetimes

Abstract As a motivating problem, we aim to study some special aspects of the marginal distributions of the order statistics for exchangeable and (more generally) for minimally stable non-negative random variables T 1, ..., Tr. In any case, we assume that T 1, ..., Tr are identically distributed, with a common survival function ̄G and their survival copula is denoted by K. The diagonal sections of K, along with ̄G, are possible tools to describe the information needed to recover the laws of order statistics. When attention is restricted to the absolutely continuous case, such a joint distribution can be described in terms of the associated multivariate conditional hazard rate (m.c.h.r.) functions. We then study the distributions of the order statistics of T 1, ..., Tr also in terms of the system of the m.c.h.r. functions. We compare and, in a sense, we combine the two different approaches in order to obtain different detailed formulas and to analyze some probabilistic aspects for the distributions of interest. This study also leads us to compare the two cases of exchangeable and minimally stable variables both in terms of copulas and of m.c.h.r. functions. The paper concludes with the analysis of two remarkable special cases of stochastic dependence, namely Archimedean copulas and load sharing models. This analysis will allow us to provide some illustrative examples, and some discussion about peculiar aspects of our results.

BlackBox: Generalizable reconstruction of extremal values from incomplete spatio-temporal data

We describe our submission to the Extreme Value Analysis 2019 Data Challenge in which teams were asked to predict extremes of sea surface temperature anomaly within spatio-temporal regions of missing data. We present a computational framework which reconstructs missing data using convolutional deep neural networks. Conditioned on incomplete data, we employ autoencoder-like models as multivariate conditional distributions from which possible reconstructions of the complete dataset are sampled using imputed noise. In order to mitigate bias introduced by any one particular model, a prediction ensemble is constructed to create the final distribution of extremal values. Our method does not rely on expert knowledge in order to accurately reproduce dynamic features of a complex oceanographic system with minimal assumptions. The obtained results promise reusability and generalization to other domains.

Stochastic Simulation of Daily Suspended Sediment Concentration Using Multivariate Copulas

An estimation of daily suspended sediment concentration (SSC) is required for water resource and environmental management. The traditional methods for simulating daily SSC focus on modeling the SSCs themselves, whereas the cross-correlation structure between SSC and streamflow has received only minor attention. To address this issue, we propose a stochastic method to generate long-term daily SSC using multivariate copula functions that account for temporal and cross dependences in daily SSCs. We use the conditional copula method to construct daily multivariate distributions to alleviate the complications and workload of parameter estimations using high-dimensional copulas. The observed daily streamflow and SSC data are normalized using the normal quantile transform method to relax the computationally intensive model of building daily marginal distributions. Daily SSCs can thus be simulated through the multivariate conditional distribution using previous daily SSC and concurrent daily streamflow values. The proposed method is rigorously examined by application to a case study at the Pingshan station in the Jinsha River Basin, China, and compared with the bivariate copula method. The results show that the proposed method has a high degree of accuracy, in preserving the statistics and temporal correlation of daily SSC observations, and better preserves the lag-0 cross correlation compared with the bivariate copula method. The multivariate copula framework proposed here can accurately and efficiently generate long-term daily SSC data for water resource and environmental management, which play a critical role in accurately estimating the frequency and magnitude of extreme SSC events.

Bayesian Mixture Modeling for Multivariate Conditional Distributions

We present a Bayesian mixture model for estimating the joint distribution of mixed ordinal, nominal, and continuous data conditional on a set of fixed variables. The modeling strategy is motivated by applied contexts in marketing and the social sciences, in particular data fusion and the analysis of stratified or quota samples. The model uses multivariate normal and categorical mixture kernels for the random variables. It induces dependence between the random and fixed variables through the means of the multivariate normal mixture kernels and via a truncated local Dirichlet process. The latter encourages observations with similar values of the fixed variables to share mixture components. We illustrate use of the model for missing data imputation, in particular data fusion of two surveys, and for the analysis of stratified or quota samples. The data fusion example suggests that the model can estimate underlying relationships in the data and the distributions of the missing values more accurately than several other approaches, including a mixture model applied to the random and fixed variables jointly. We also use the model to analyze consumers’ reading behaviors from a quota sample, i.e., a sample where the empirical distribution of some variables is fixed by design and so should not be modeled as random, conducted by the book publisher HarperCollins.

Identification of behavioural model input data sets for WWTP uncertainty analysis.

Uncertainty analysis is important for wastewater treatment plant (WWTP) model applications. An important aspect of uncertainty analysis is the identification and proper quantification of sources of uncertainty. In this contribution, a methodology to identify an ensemble of behavioural model representations (combinations of input data, model structure and parameter values) is presented and evaluated. The outcome is a multivariate conditional distribution of input data that is used for generating samples of likely inputs (such as Monte Carlo input samples) to perform WWTP model uncertainty analysis. This article presents an approach to verify uncertainty distributions of input data (otherwise often assumed) by using historical observations and actual plant data.

Cross-Sectional Learning of Extremal Dependence Among Financial Assets

We propose a novel probabilistic model to facilitate the learning of multivariate tail dependence of multiple financial assets. Our method allows one to construct from known random vectors, e.g., standard normal, sophisticated joint heavy-tailed random vectors featuring not only distinct marginal tail heaviness, but also flexible tail dependence structure. The novelty lies in that pairwise tail dependence between any two dimensions is modeled separately from their correlation, and can vary respectively according to its own parameter rather than the correlation parameter, which is an essential advantage over many commonly used methods such as multivariate $t$ or elliptical distribution. It is also intuitive to interpret, easy to track, and simple to sample comparing to the copula approach. We show its flexible tail dependence structure through simulation. Coupled with a GARCH model to eliminate serial dependence of each individual asset return series, we use this novel method to model and forecast multivariate conditional distribution of stock returns, and obtain notable performance improvements in multi-dimensional coverage tests. Besides, our empirical finding about the asymmetry of tails of the idiosyncratic component as well as the market component is interesting and worth to be well studied in the future.

A cause-specific hazard spatial frailty model for competing risks data

Competing risks data is often grouped into clusters, such as geographic regions. Shared frailty models can be used to model the correlation between individuals within each cluster. For the better fitting of the model in the competing risks data, one random effect can be used for every type of event in each cluster. But the correlation between random effects in each cluster should also be taken into account. On the other hand, there may also be a spatial correlation between random effects during geographic regions. In this paper, we use cause-specific hazard spatial frailty model with multivariate conditional autoregressive distribution for frailties via Bayesian approach. Simulation studies are used to assess the regression coefficient estimators as well as the variance and correlation of random effects within the clusters. We apply the proposed model to the analysis of the gastrointestinal cancer data of Iran’s National Institute of Health Research and the Louisiana breast cancer data from the Surveillance Epidemiology and End Results database of the National Cancer Institute.

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 multivariate generalized log-double exponential family of distribution of kind-2

This paper proposed a Kind-2 distribution of the multivariate generalization of log-double exponential family of distributions. Further, the constants of the proposed distribution namely it's Marginal, Multivariate conditional distributions and generating functions also expressed in a closed series expansion form. Many special cases of this distribution are derived and the location, scale, shape parameters determines the family of some existing known distributions such as familiar bivariate, mixtures of bivariate distributions, generalized and multivariate distributions are mushroomed as subclasses of the proposed multivariate generalized log-double exponential family of distribution of kind-2.

A Spatial Survival Model in Presence of Competing Risks for Iranian Gastrointestinal Cancer Patients

Background:Gastrointestinal cancer is one of the common causes of death from cancer in Iran. Survival analysis is usually used to detect prognostic factors of time to death from gastrointestinal cancers. The use of ordinary survival models, in the presence of competing risks and/or when data is collected within geographic areas, may lead to distorting the results. Therefore, the aim of this study is to use the spatial survival models in the presence of competing risks to assess the risk factors affecting the survival time of gastrointestinal cancer patients.Methods:The data in this study was collected from 602 patients who were diagnosed with gastrointestinal cancer in Golestan and Mazandaran provinces registered in Iran’s National Institute of Health Research from 2002 through 2007 and were followed up to July 2017. The data was analyzed using the cause-specific hazard frailty model with multivariate conditional autoregressive distribution for frailties in the presence of competing risks (death from gastrointestinal cancer, heart disease, and other causes) via OpenBUGS software.Results:The hazard of death from gastrointestinal cancer in men patients, patients who lived in rural areas, patients whose relatives did not have a history of cancer, patients who did not undergo surgery, and patients with gastric cancer was significantly higher than others. Based on the deviance information criterion (DIC), frailty models and spatial frailty models seemed better than no-frailty model and non-spatial frailty model, respectively.Conclusions:This study showed that the use of the spatial frailty term in the model helps better fit the model. Also, the spatial pattern in the figures suggests the necessity of presence of some still missing, spatially varying covariates relevant for time to death from gastrointestinal cancer, heart disease, or other causes.

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.

Gaussian closure technique applied to the hysteretic Bouc model with non-zero mean white noise excitation

Devices that emit structure-borne sound are commonly decoupled by elastic components to shield the environment from acoustical noise and vibrations. The elastic elements often have a hysteretic behavior that is typically neglected. In order to take hysteretic behavior into account, Bouc developed a differential equation for such materials, especially joints made of rubber or equipped with dampers. In this work, the Bouc model is solved by means of the Gaussian closure technique based on the Kolmogorov equation. Kolmogorov developed a method to derive probability density functions for arbitrary explicit first-order vector differential equations under white noise excitation using a partial differential equation of a multivariate conditional probability distribution. Up to now no analytical solution of the Kolmogorov equation in conjunction with the Bouc model exists. Therefore a wide range of approximate solutions, especially the statistical linearization, were developed. Using the Gaussian closure technique that is an approximation to the Kolmogorov equation assuming a multivariate Gaussian distribution an analytic solution is derived in this paper for the Bouc model. For the stationary case the two methods yield equivalent results, however, in contrast to statistical linearization the presented solution allows to calculate the transient behavior explicitly. Further, stationary case leads to an implicit set of equations that can be solved iteratively with a small number of iterations and without instabilities for specific parameter sets.

A New Time Series Model for Transaction Data and Marked Point Processes

We propose a new time series model for the bivariate/multivariate transaction data, where model their joint transition dynamics by directly modeling the conditional distribution of the current observation to be a mixture of conditional distributions given the past p-lags information. Each joint multivariate conditional distribution in the mixture is constructed via a certain class of Copula. Such model can successfully capture some distinguished nonlinear non-Gaussian time series features, such as the irregular bursts or jumps that are so widely observed in the transaction data set and other more general marked point processes. We illustrate the new time series model and related methods by modeling three pieces of IBM stock transaction data, and show its advantages over some benchmark models.

A New Generalisation of Sam-Solai's Multivariate Additive Uniform Distribution

This paper proposed an alternate generalization of Sam-Solai's Multivariate additive Uniform distribution from the continuous univariate distribution. Further,we find its Marginal, Multivariate Conditional distributions, Multivariate Generating functions, Multivariate survival, hazard functions and also discussed it's special cases. The special cases include the transformation of Sam-Solai's Multivariate additive Uniform distribution into Multivariate standard additive uniform distribution and Multivariate exponential distribution. Moreover, the bivariate correlation between any two standard uniform random variables was found to be approximately -0.33.

A Multivariate Generalized Double Exponential Family of Distribution of Kind-1

This paper proposed a generalization of double exponential family of Sarmanov type Continuous multivariate symmetric probability distribution with reference to a new form of Sam-Solai's Multivariate Generalized double exponential family of distributions from univariate case. Further, we found it's Marginal, Multivariate conditional distributions, generating functions in a closed series expression form and also discussed its special cases. The location, scale and shape parameters played a significance role and it determines the family of some existing known and familiar Bivariate, mixtures of bivariate distributions, Generalized and multivariate distributions as subclass of the proposed multivariate generalized double exponential family of distribution.

Vine-copula GARCH model with dynamic conditional dependence

Constructing multivariate conditional distributions for non-Gaussian return series has been a major research agenda recently. Copula GARCH models combine the use of GARCH models and a copula function to allow flexibility on the choice of marginal distributions and dependence structures. However, it is non-trivial to define multivariate copula densities that allow dynamic dependent structures in returns. The vine-copula method has been gaining attention recently in that a multi-dimensional density can be decomposed into a product of conditional bivariate copulas and marginal densities. The dependence structure is interpreted individually in each copula pair. Yet, most studies have only focused on time varying correlation. A vine-copula GARCH model with dynamic conditional dependence is proposed. A generic approach to specifying dynamic conditional dependence using any dependence measures is developed. The characterization also induces multivariate conditional dependence dynamically through vine decomposition. The main idea is to incorporate dynamic conditional dependence, such as Kendall’s tau and rank correlation, not to mention linear correlation, in each bivariate copula pair. The estimation is conducted through a sequential approach. Simulation experiments are performed and five Hong Kong blue chip stock data from January 2004 to December 2011 are studied. Using t and two Archimedean copulas, it is revealed that Kendall’s tau and linear correlation of the stock returns vary over time, indicating the presence of time varying properties in dependence.