ESTIMATION FOR PARTIALLY OBSERVED MARKOV-PROCESSES

Many stochastic process models for environmental data sets assume a pr ocess of relatively simple structure which is in some sense partially observed. That is, there is an underlying process (X(n), n greater tha n or equal to 0) or (X(t), t greater than or equal to 0) for which the parameters are of interest and physically meaningful, and an observab le process (Y-n, n greater than or equal to 0) or (Y-t, t greater than or equal to 0) which depends on the X process but not otherwise on th ose parameters. Examples are wide ranging: the Y process may be the X process with missing observations; the Y process may be the X process observed with a noise component; the X process might constitute a rand om environment for the Y process, as with hidden Markov models; the Y process might be a lower dimensional function or reduction of the X pr ocess. In principle, maximum likelihood estimation for the X process p arameters can be carried out by some form of the EM algorithm applied to the Y process data. In the paper we review some current methods for exact and approximate maximum likelihood estimation. We illustrate so me of the issues by considering how to estimate the parameters of a st ochastic Nash cascade model for runoff. In the case of k reservoirs, t he outputs of these reservoirs form a k dimensional vector Markov proc ess, of which only the kth coordinate process is observed usually at a discrete sample of time points.