A copula-based approximation to Markov chains

The Markov chain is well studied and widely applied in many areas.For some Markov chains,it is infeasible to obtain the explicit expressions of their corresponding finite-dimensional distributions and sometimes it is time-consuming for computation.In this paper,we propose an approximation method for Markov chains by applying the copula theory.For this purpose,we first discuss the checkerboard copula-based Markov chain,which is the Markov chain generated by the family of checkerboard copulas.This Markov chain has some appealing properties,such as self-similarity in copulas and having explicit forms of finite-dimensional distributions.Then we prove that each Markov chain can be approximated by a sequence of checkerboard copula-based Markov chains,and the error bounds of the approximate distributions are provided.Employing the checkerboard copula-based approximation method,we propose a sufficient condition for the geometric β-mixing of copula-based Markov chains.This condition allows copulas of Markov chains to be asymmetric.Finally,by applying the approximation method,analytical recurrence formulas are also derived for computing approximate distributions of both the first passage time and the occupation time of a Markov chain,and numerical results are listed to show the approximation errors.

A copula-based approach to accommodate intra-household interaction in workers' daily maintenance activity stop generation modeling

A new stop frequency model was developed to predict the daily number of maintenance activity stops made by individual household heads during a typical weekday. This new model was based on the modification of a conventional multivariate ordered probit(MOP) model by maintaining the probit assumption for the marginal distributions while introducing nonnormal dependence among the error terms using copula functions. Therefore, the copulabased MOP model would relieve the restriction of imposing joint normality on the error terms in the conventional MOP model. The new MOP model would not only account for the intrahousehold interactions in stop-making decisions, but also allow the best functional form to be determined for representing dependencies among household heads. Using the New York Metropolitan Transportation Council’s 2010/2011 regional household travel survey data, the copula-based MOP model was employed to examine stop-making behavior for individual household heads residing in New York City and its adjacent counties in Mid-Hudson Valley and New Jersey. Empirical results provided useful insights into the observed effects of sociodemographics, land use density, transportation service, and work schedule together with potential unobserved common effects on the inter-relatedness of spousal stop-making decisions at the household level. The results show that the MOP model with a Clayton copula structure provides the best data fits and there is a very strong positive dependence among error terms of stop-making equations. Furthermore, the dependence among the maintenance activity propensities of household heads is asymmetric, with a stronger tendency of household heads to simultaneously have low maintenance activity levels than to simultaneously have high maintenance activity levels.