The University of Liverpool
Department of Mathematical Sciences
Division of Statistics and Probability
SEMINAR
The Evolution of Batch-Immigration-Death Processes Subject to Counts
Eric Renshaw, University of Strathclyde
Wednesday, 2nd March 2005, 2pm
The Whittaker Room (211)
Abstract:
We consider the general question of whether it is possible to determine the
fundamental structure of a hidden stochastic process purely from counts of
escaping individuals. This is of immense importance in fields such as
quantum optics, where externally based radiation elucidates the nature of
the underlying electromagnetic radiation process. The problem is initially
established in terms of a single/paired immigration-death process, in which
batches of i immigrants enter the population at rate, and each individual
dies independently at rate . This construct gives rise to a dramatic
odd-even (i.e. saw-tooth) effect in the probability, moment and
autocorrelation structure, and is directly associated with non-classical
light. That the approach can be extended to the k-tuple immigration
scenario is demonstrated through the development of the triple-immigration
case. The strong link between population and counting structures is
highlighted through two specific examples, involving k- and Schoenberg-
batch immigration. By introducing births we then show that the technique is
equally successful in exposing hidden multiplicative effects. The analyses
presented uncover novel and highly illuminating features, and emphasise the
potential of this population-counting construct for expanding into more
complex multi-type situations such as predator-prey, competition and
general epidemic processes. Finally, we introduce a MCMC approach, and show
that considerable caution is required due to the presence of improper
distributions and the effect of the initial configuration on the estimated
parameter values. Such problems can be avoided by developing a more
mathematically amenable likelihood via a parallel limiting process.
Following the talk, tea and biscuits will be available in Room 304
ALL WELCOME
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Ingrid Harper
University of Liverpool
Department of Mathematical Sciences
Division of Statistics and Probability
Peach Street
Liverpool L69 7ZL
Tel: 0151 794 4751
Fax: 0151 794 4754
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