Dependency as a measure to estimate the order and the values of Markov processes

Abstract

To evaluate the order and the values of Markov properties of the time series of events, we have proposed a statistical measure “dependency”:D _m = (H ₀ −H _m )/H ₀ , whereH ₀ andH _m are Shannon's entropy and them-th order conditional entropy, respectively. It is indicated that$\tilde D_m = \sum\limits_{v = 1}^m {(\hat D_v - \bar D_v^{sh} } )$ is a better point estimator ofD _m, giving a total value of them-th order Markov process. Here$\hat D_m$ and$\bar D_m^{sh}$ are the estimate ofD _m and the arithmetic mean of$\hat D_m^{sh}$ when them-th order shuffling is made many times for a given observed series, respectively. The value$\hat D_m - \bar D_m^{sh} = d_m$ represents Markov value of the orderm. Under the assumption that the series has continuous variables and the normal distribution, simplified dependency is defined by

, where |S _m | is the determinant of serial correlation coefficients. It is shown that

is practically useful for the estimation of the order and the values of Markov processes with small sample size. It is also indicated that

analysis is basically equivalent to the least mean-square analysis of autoregressive models.