Modelling bivariate processes

Eric Renshaw

in Stochastic Population Processes

Published in print February 2011 | ISBN: 9780199575312
Published online September 2011 | e-ISBN: 9780191728778 | DOI:
Modelling bivariate processes

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This chapter examines the general bivariate process, and illustrates the basic approaches involved by first developing a simple process for which the preceding methods of solution do carry across. The univariate saddlepoint approximation is then extended to cover multivariate processes, with cumulant truncation being covered in some detail. A bivariate process of considerable practical importance involves employing total counts as a second variable, especially since these can generate intriguing effects in which the structure of the occupation probabilities depends on whether the population size is odd or even. Various examples are presented, including a family of processes that generates different probability distributions which have the same moment structure. Moreover, although complex systems often exhibit extremely rich dynamic behaviour, gaining a direct understanding of the underlying structure may not be possible if the system remains hidden and observations can only be made on external counts. The chapter shows that a surprisingly high level of analytic information can still be gained from the counts alone, and demonstrates how to employ Markov chain Monte Carlo techniques in such situations.

Keywords: univariate saddlepoint approximation; multivariate processes; cumulant truncation; Markov chain Monte Carlo techniques

Chapter.  30363 words.  Illustrated.

Subjects: Applied Mathematics

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