Discrete Marginal Distributions Research Articles

Abstract # 4281 Subjective social standing is related to inflammation during pregnancy

In this chapter, we review the problem of modeling correlated count data. Among the several methods that can be used for this scope, we focus on the copula approach, illustrating its advantages, but also possible limitations and issues arising in the discrete context if compared to the continuous case. After introducing the basic notions about copulas, the construction of a multivariate joint distribution is discussed and pseudorandom simulation and point estimation of copula-based models for count data are then outlined. Results related to minimum and maximum correlation between two assigned discrete marginal distributions are also described and put in connection with the choice of the copula to be used for modeling correlated counts. A numerical example and an application to a real dataset are provided.

A pathway for multivariate analysis of ecological communities using copulas.

We describe a new pathway for multivariate analysis of data consisting of counts of species abundances that includes two key components: copulas, to provide a flexible joint model of individual species, and dissimilarity‐based methods, to integrate information across species and provide a holistic view of the community. Individual species are characterized using suitable (marginal) statistical distributions, with the mean, the degree of over‐dispersion, and/or zero‐inflation being allowed to vary among a priori groups of sampling units. Associations among species are then modeled using copulas, which allow any pair of disparate types of variables to be coupled through their cumulative distribution function, while maintaining entirely the separate individual marginal distributions appropriate for each species. A Gaussian copula smoothly captures changes in an index of association that excludes joint absences in the space of the original species variables. A permutation‐based filter with exact family‐wise error can optionally be used a priori to reduce the dimensionality of the copula estimation problem. We describe in detail a Monte Carlo expectation maximization algorithm for efficient estimation of the copula correlation matrix with discrete marginal distributions (counts). The resulting fully parameterized copula models can be used to simulate realistic ecological community data under fully specified null or alternative hypotheses. Distributions of community centroids derived from simulated data can then be visualized in ordinations of ecologically meaningful dissimilarity spaces. Multinomial mixtures of data drawn from copula models also yield smooth power curves in dissimilarity‐based settings. Our proposed analysis pathway provides new opportunities to combine model‐based approaches with dissimilarity‐based methods to enhance understanding of ecological systems. We demonstrate implementation of the pathway through an ecological example, where associations among fish species were found to increase after the establishment of a marine reserve.

Simulating Non-Gaussian Stationary Stochastic Processes by Translation Model

The translation model is a useful tool to characterize stochastic processes or random fields. In this paper, this model is extended to simulate stochastic processes with discrete marginal distributions. A theoretical discussion is elaborated on the properties of the correlation distortion function. The spectral representation method is employed to generate the underlying Gaussian process of the translation model. Efficient algorithms are developed to determine the power spectral density function (PSDF) $S_{z}(\omega)$ for Gaussian process. If the marginal distribution and PSDF of the target stochastic process are incompatible, two methods are presented to modify $S_{z}(\omega)$ . Finally, numerical examples are performed to check the proposed methods.

Computationally efficient Bayesian estimation of high-dimensional Archimedean copulas with discrete and mixed margins

Estimating copulas with discrete marginal distributions is challenging, especially in high dimensions, because computing the likelihood contribution of each observation requires evaluating $$2^{J}$$ terms, with J the number of discrete variables. Our article focuses on the estimation of Archimedean copulas, for example, Clayton and Gumbel copulas. Currently, data augmentation methods are used to carry out inference for discrete copulas and, in practice, the computation becomes infeasible when J is large. Our article proposes two new fast Bayesian approaches for estimating high-dimensional Archimedean copulas with discrete margins, or a combination of discrete and continuous margins. Both methods are based on recent advances in Bayesian methodology that work with an unbiased estimate of the likelihood rather than the likelihood itself, and our key observation is that we can estimate the likelihood of a discrete Archimedean copula unbiasedly with much less computation than evaluating the likelihood exactly or with current simulation methods that are based on augmenting the model with latent variables. The first approach builds on the pseudo-marginal method that allows Markov chain Monte Carlo simulation from the posterior distribution using only an unbiased estimate of the likelihood. The second approach is based on a variational Bayes approximation to the posterior and also uses an unbiased estimate of the likelihood. We show that the two new approaches enable us to carry out Bayesian inference for high values of J for the Archimedean copulas where the computation was previously too expensive. The methodology is illustrated through several real and simulated data examples.

A general algorithm for covariance modeling of discrete data

We propose an algorithm that generalizes to discrete data any given covariance modeling algorithm originally intended for Gaussian responses, via a Gaussian copula approach. Covariance modeling is a powerful tool for extracting meaning from multivariate data, and fast algorithms for Gaussian data, such as factor analysis and Gaussian graphical models, are widely available. Our algorithm makes these tools generally available to analysts of discrete data and can combine any likelihood-based covariance modeling method for Gaussian data with any set of discrete marginal distributions. Previously, tools for discrete data were generally specific to one family of distributions or covariance modeling paradigm, or otherwise did not exist. Our algorithm is more flexible than alternate methods, takes advantage of existing fast algorithms for Gaussian data, and simulations suggest that it outperforms competing graphical modeling and factor analysis procedures for count and binomial data. We additionally show that in a Gaussian copula graphical model with discrete margins, conditional independence relationships in the latent Gaussian variables are inherited by the discrete observations. Our method is illustrated with a graphical model and factor analysis on an overdispersed ecological count dataset of species abundances.

System Reliability and Component Importance Under Dependence: A Copula Approach

ABSTRACTSystem reliability and component importance are of great interest in reliability modeling, especially when the components within the system are dependent. We characterize the influence of dependence structures on system reliability and component importance in coherent systems with discrete marginal distributions. The effects of dependence are captured through copula theory. We extend our framework to coherent multi-state system. Applications of the derived results are demonstrated using a Gaussian copula, which yields simple interpretations. Simulations and two examples are presented to demonstrate the importance of modeling dependence when estimating system reliability and ranking of component importance. Proofs, algorithms, code, and data are provided in supplementary materials available online.

On the polynomial solvability of distributionally robust k-sum optimization

In this paper, we define a distributionally robust k-sum optimization problem as the problem of finding a solution that minimizes the worst-case expected sum of up to the k largest costs of the elements in the solution. The costs are random with a joint probability distribution that is not completely specified but rather assumed to be known to lie in a set of probability distributions. For k=1, this reduces to a distributionally robust bottleneck optimization problem while for k=n, this reduces to distributionally robust minimum sum optimization problem. Our main result is that for a Fréchet class of discrete marginal distributions with finite support, the distributionally robust k-sum combinatorial optimization problem is solvable in polynomial time if the deterministic minimum sum problem is solvable in polynomial time. This extends the result of Punnen and Aneja [On k-sum optimization, Oper. Res. Lett. 18(5)(1996), pp. 233–236] from the deterministic to the robust case. We show that this choice of the set of distributions helps preserve the submodularity of the k-sum objective function which is an useful structural property for optimization problems.

Discrete Wasserstein barycenters: optimal transport for discrete data

Wasserstein barycenters correspond to optimal solutions of transportation problems for several marginals, and as such have a wide range of applications ranging from economics to statistics and computer science. When the marginal probability measures are absolutely continuous (or vanish on small sets) the theory of Wasserstein barycenters is well-developed [see the seminal paper (Agueh and Carlier in SIAM J Math Anal 43(2):904–924, 2011)]. However, exact continuous computation of Wasserstein barycenters in this setting is tractable in only a small number of specialized cases. Moreover, in many applications data is given as a set of probability measures with finite support. In this paper, we develop theoretical results for Wasserstein barycenters in this discrete setting. Our results rely heavily on polyhedral theory which is possible due to the discrete structure of the marginals. The results closely mirror those in the continuous case with a few exceptions. In this discrete setting we establish that Wasserstein barycenters must also be discrete measures and there is always a barycenter which is provably sparse. Moreover, for each Wasserstein barycenter there exists a non-mass-splitting optimal transport to each of the discrete marginals. Such non-mass-splitting transports do not generally exist between two discrete measures unless special mass balance conditions hold. This makes Wasserstein barycenters in this discrete setting special in this regard. We illustrate the results of our discrete barycenter theory with a proof-of-concept computation for a hypothetical transportation problem with multiple marginals: distributing a fixed set of goods when the demand can take on different distributional shapes characterized by the discrete marginal distributions. A Wasserstein barycenter, in this case, represents an optimal distribution of inventory facilities which minimize the squared distance/transportation cost totaled over all demands.

Modelling Poisson marked point processes using bivariate mixture transition distributions

A class of bivariate continuous-discrete distributions is proposed to fit Poisson dynamic models in a single unified framework via bivariate mixture transition distributions (BMTDs). Potential advantages of this class over the current models include its ability to capture stretches, bursts and nonlinear patterns characterized by Internet network traffic, high-frequency financial data and many others. It models the inter-arrival times and the number of arrivals (marks) in a single unified model which benefits from the dependence structure of the data. The continuous marginal distributions of this class include as special cases the exponential, gamma, Weibull and Rayleigh distributions (for the inter-arrival times), whereas the discrete marginal distributions are geometric and negative binomial. The conditional distributions are Poisson and Erlang. Maximum-likelihood estimation is discussed and parameter estimates are obtained using an expectation–maximization algorithm, while the standard errors are estimated using the missing information principle. It is shown via real data examples that the proposed BMTD models appear to capture data features better than other competing models.

Integer autoregressive models with structural breaks

Even though integer-valued time series are common in practice, the methods for their analysis have been developed only in recent past. Several models for stationary processes with discrete marginal distributions have been proposed in the literature. Such processes assume the parameters of the model to remain constant throughout the time period. However, this need not be true in practice. In this paper, we introduce non-stationary integer-valued autoregressive (INAR) models with structural breaks to model a situation, where the parameters of the INAR process do not remain constant over time. Such models are useful while modelling count data time series with structural breaks. The Bayesian and Markov Chain Monte Carlo (MCMC) procedures for the estimation of the parameters and break points of such models are discussed. We illustrate the model and estimation procedure with the help of a simulation study. The proposed model is applied to the two real biometrical data sets.

Approximate copula-based estimation and prediction of discrete spatial data

The present paper reports on the use of copula functions to describe the distribution of discrete spatial data, e.g. count data from environmental mapping or areal data analysis. In particular, we consider approaches to parameter point estimation and propose a fast method to perform approximate spatial prediction in copula-based spatial models with discrete marginal distributions. We assess the goodness of the resulting parameter estimates and predictors under different spatial settings and guide the analyst on which approach to apply for the data at hand. Finally, we illustrate the methodology by analyzing the well-known Lansing Woods data set. Software that implements the methods proposed in this paper is freely available in Matlab language on the author’s website.

Copula models for insurance claim numbers with excess zeros and time-dependence

This paper develops two copula models for fitting the insurance claim numbers with excess zeros and time-dependence. The joint distribution of the claims in two successive periods is modeled by a copula with discrete or continuous marginal distributions. The first model fits two successive claims by a bivariate copula with discrete marginal distributions. In the second model, a copula is used to model the random effects of the conjoint numbers of successive claims with continuous marginal distributions. Zero-inflated phenomenon is taken into account in the above copula models. The maximum likelihood is applied to estimate the parameters of the discrete copula model. A two-step procedure is proposed to estimate the parameters in the second model, with the first step to estimate the marginals, followed by the second step to estimate the unobserved random effect variables and the copula parameter. Simulations are performed to assess the proposed models and methodologies.

Copula-based geostatistical modeling of continuous and discrete data including covariates

It is common in geostatistics to use the variogram to describe the spatial dependence structure and to use kriging as the spatial prediction methodology. Both methods are sensitive to outlying observations and are strongly influenced by the marginal distribution of the underlying random field. Hence, they lead to unreliable results when applied to extreme value or multimodal data. As an alternative to traditional spatial modeling and interpolation we consider the use of copula functions. This paper extends existing copula-based geostatistical models. We show how location dependent covariates e.g. a spatial trend can be accounted for in spatial copula models. Furthermore, we introduce geostatistical copula-based models that are able to deal with random fields having discrete marginal distributions. We propose three different copula-based spatial interpolation methods. By exploiting the relationship between bivariate copulas and indicator covariances, we present indicator kriging and disjunctive kriging. As a second method we present simple kriging of the rank-transformed data. The third method is a plug-in prediction and generalizes the frequently applied trans-Gaussian kriging. Finally, we report on the results obtained for the so-called Helicopter data set which contains extreme radioactivity measurements.

Discrete-valued ARMA processes

This paper presents a unified framework of stationary ARMA processes for discrete-valued time series based on Pegram’s [Pegram, G.G.S., 1980. An autoregressive model for multilag markov chains. J. Appl. Probab. 17, 350–362] mixing operator. Such a stochastic operator appears to be more flexible than the currently popular thinning operator to construct Box and Jenkins’ type stationary ARMA processes with arbitrary discrete marginal distributions. This flexibility allows us to yield an ARMA model for time series of binomial or categorical observations as a special case, which was unavailable with the extended thinning operator [Joe, H., 1996. Time series models with univariate margins in the convolution-closed infinitely divisible class. J. Appl. Probab. 33, 664–677] because the binomial/categorical distribution is not infinitely divisible. We also study parameter estimation and comparison with the thinning operator based method, whenever applicable. Real data examples are used to examine and illustrate the proposed method.

Difference Equations for the Higher‐Order Moments and Cumulants of the INAR(1) Model

Recently, as a result of the growing interest in modelling stationary processes with discrete marginal distributions, several models for integer value time series have been proposed in the literature. One of these models is the INteger‐AutoRegressive (INAR) model. Here we consider the higher‐order moments and cumulants of the INAR(1) process and show that they satisfy a set of Yule–Walker type difference equations. We also obtain the spectral and bispectral density functions, thus characterizing the INAR(1) process in the frequency domain. We use a frequency domain approach, namely the Whittle criterion, to estimate the parameters of the model. The estimation theory and associated asymptotic theory of this estimation method are illustrated numerically.