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Stochastic Population ProcessesAnalysis, Approximations, Simulations$
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Eric Renshaw

Print publication date: 2011

Print ISBN-13: 9780199575312

Published to Oxford Scholarship Online: September 2011

DOI: 10.1093/acprof:oso/9780199575312.001.0001

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Modelling bivariate processes

Modelling bivariate processes

(p.331) 7 Modelling bivariate processes
Stochastic Population Processes

Eric Renshaw

Oxford University Press

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

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