This paper expresses the transition probabilities of a nonstationary
Markov
chain by means of models involving wavelet expansions and then, given part
of a realization of
such a process, proceeds to estimate the coefficients of the expansion and
the
probabilities themselves.
Through choice of the number of and which wavelet terms to include, the
approach provides a flexible method for handling discrete-valued
observations in the nonstationary case.
In particular the method appears useful for detecting abrupt or steady
changes in the structure of Markov chains.
The method is illustrated by means of data sets concerning music, rainfall
and sleep.
In the examples both direct and shruken estimates
are computed. The models include explanatory variables in each case. The
approach is implemented by means of programs for fitting generalized
linear models.
The goodness of fit and the presence of nonstationarity are assessed both
by change of deviance and graphically via periodogram plots of residuals.