Optimal feature selection and decision rules in classification problems with time series

The problem to be considered is that of classifying a given time series
Z_{N} = (y(1), \cdots ,y(N))
into one of
r
classes
C_{i}, i= 1, \cdots ,r
. The stochastic process
y(\cdot)
is assumed to obey an autoregressive structure involving a parameter vector
\theta
, whose probability density
p(\theta|C_{i})
depends on the class to which
Z
or
y(\cdot)
belongs. Assuming appropriate expressions for
p(\theta|C_{i})
, it is shown that the probability density of
Z_{N}
characterizing each class, namely
p(Z_{N}|C_{i})
, possesses a vector
\bar{\theta}
of sufficient statistics, i.e., all the information about
Z_{N}
needed for the discrimination between the various classes is contained in the vector
\bar{\theta}=(\bar{\theta}_{1}(Z_{N}), \cdots , \bar{\theta}_{m+1}(Z_{N}))^{T}
, where the functions
\bar{\theta}_{i}(Z_{N}), i=1, \cdots ,m+1
have the same structure for all
N
. Thus the best possible feature set for the problem is
\bar{\theta}
From this is deduced the optimal decision rule to minimize the average probability of error. The optimal feature set and the corresponding optimal decision rule are compared with other feature sets and decision rules mentioned in the literature on speech recognition.