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

摘要

To evaluate the order and the values of Markov properties of the time series of events, we have proposed a statistical measure “dependency”:Dm= (H0−Hm)/H0, whereH0andHmare Shannon's entropy and them-th order conditional entropy, respectively. It is indicated that$$tilde D_m = sumlimits_{v = 1}^m {(hat D_v - bar D_v^{sh} } )$$is a better point estimator ofDm, giving a total value of them-th order Markov process. Here$$hat D_m $$and$$bar D_m^{sh} $$are the estimate ofDmand 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 Sm is the determinant of serial correlation coefficients. It is shown thatis practically useful for the estimation of the order and the values of Markov processes with small sample size. It is also indicated thatanalysis is basically equivalent to the least mean-square analysis of autoregressive m