Thesis Abstract:
We make use of the state-space model in three separate statistical problems:
time series discrimination, signal comparison testing, and model selection. We
propose and investigate two discriminant procedures for classifying a
state-space process of unknown origin into one of several known populations.
Both approaches employ a nearest neighbor classification rule (Cover and Hart,
1967) and utilize discriminant functions based on Kullback (1968) information
measures. Our first procedure classifies state-space processes based on
the mechanism generating the states, and our second procedure classifies
state-space processes based on the latent trajectory of the states. The
first discriminant function assesses the discrepancy between the Kalman smoothed
values (Kalman, 1960; Jazwinski, 1970) produced under two fitted models: the
model associated with the unclassified series and a model associated with one
of the classified series. The second discriminant function compares the
smoothed version of the unclassified series to that of a classified
series. We provide interpretable representations for the discriminants,
and develop formulae for easy computation of the functions in the context of
the EM algorithm. Both methods are evaluated in extensive simulations and the
second procedure is illustrated with an application to the stock market. We
derive a procedure for testing if the unobserved states of two state-space
processes follow the same latent trajectory. Our test statistic, which
considers the point-wise progression of two state-space series, has an
approximate chi-square distribution under the null hypothesis and the
assumption of Gaussian measurement noise. The test is shown to perform
well in simulations, and is illustrated with data from climatology. We develop a model selection criterion useful
in the state-space framework. Our criterion is based on estimating
Kullback's directed divergence between the data-generating density and an
estimated model density. To obtain an approximately unbiased estimate of
the directed divergence, we employ
Keywords: Kullback-Leibler information, nearest-neighbor discrimination, time series analysis, time series discrimination, state-space model, signal extraction, model
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