By Iain L. MacDonald, Walter Zucchini
This booklet describes various hidden Markov types and issues out the place they come up and the way to estimate parameters of the version. It additionally issues out the place they come up in a common demeanour and the way the versions can be utilized in purposes. it isn't speculated to be a mathematically rigorous remedy of the topic for which one may still glance in different places just like the booklet through R.J.Elliott, L.Aggoun and J.B.Moore (1995): Hidden Markov types: Estimation and keep watch over. Springer-Verlag. one could learn. however it lacks intensity to a undeniable quantity and isn't entire sufficient to fulfill all sorts of wishes.
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Additional resources for Hidden Markov and other models for discrete-valued time series
Sample text
We suppose now that the 2m-component random vectors J(n); defined by BIVARIATE GEOMETRIC MODELS 35 satisfy the requirements and m L P((Ji(n, 1), ... , Ji(n, my) ej) ' I - (Yi(n) j=l for i == 1,2 and all n, where ej J;as its jth component equal to one and the other m- 1 components equal to zero. That is, (Ji(n, 1), ... , Ji(n, my) is either all zeros (with probability (Yi(n)), a single one and m - 1 zeros (with probability 1 -- (Yi (n)). Define also, fot q - 1,2, ... ( h(no,q) , ° ). J 2 (n, q) The BGAR(m) process is then defined by G (n) == ,,{' M(n) ~r;=1 C(n, q)G(n - q) +' N(n) n == 0,1, ..
G. Brockwell and Davis, 1991, p. 239). == 0, we have for kEN l P(k) == LAgP(k - g)Q. 5) g=l It is not always possible to solve these equations uniquely, . but Raftery gives (separatelyfor l == 2, 3 and > 4) sufficient conditions on the parameters Ai' and qj k for uniqueness. 5) a system of equations for the autocorrelations Pk Corr( St, St+k) which resembles the Yule-Walker equations to sorrie extent and may be solved uniquely in certain special cases only. He considers in detail the autocorrelation behaviour of his model when m == 3, l == 2, 'if ..
8, Parameter~drivenmodels We now consider models of the kInd described by Cox {1981) as 'parameter-driven'. Conditional on some unobserved 'parameter process', the observations in such a model are independent, with distribution determined by the current state of theparalneter process. For instance, the conditional distribution of an observation could be Poisson, with mean Al or A2 depending on whether a two-state parameter process is ,in state 1 or state 2. Alternatively, the mean could be provided by some positive-valued process having (say) gamma or lognormal marginal distribution.
