A Statistical Analysis of Nonstationary Random Noise in View of Temporal Change of Cumulants and its Application to Dynamical Prediction of L_α

Abstract

In this paper, firstly, a unified theory of the statistical treatment of the probability distribution function is introduced in the case when a general random noise of arbitrary distribution type exhibits the nonstationary property with arbitrary temporal change of various cumulants by generalizing previous studies of nonstationary properties with fluctuation of mean value and/or variance. As the result for the purpose of finding an effect of nonstationality due to temporal change of cumulants on the output probability distribution, the explicit expressions of cumulative distribution function and probability density function in the general form of statistical expansion series, taking the stationary term into the first term, are derived. The nonstationary effect caused by the change of various cumulants is concretely reflected in each expansion coefficient of the second and higher terms in the above expansion expression. The validity of the above theoretical result is also supported experimentally by the nonstationary random traffic noise observed in Hiroshima city (cf. Fig. 1). Next, a new approach toward the dynamical prediction problem of L_α is considered by use of the above universal expression of probability density function for arbitrary nonstationary random noise level. The usefulness of our theoretical prediction method is confirmed experimentally by application to an actual nonstationary random traffic noise level (cf. Fig. 2 and Table 1). The brief summary of our theoretical consideration is given below. The characteristic function for the general nonstationary random noise level with various temporal changes of cumulants ig given by Eq. (1) in consideration of the multi-variate probability density function of cumulants. In Eq. (2), using the usual integration by parts under Eq. (5), we can derive the relationship shown in Eq. (7). Thus, the probability density function for the nonstationary random noise level can be found in the general form of statistical expansion series, taking the stationary term into the first term (cf. Eq. (8) and Eq. (9) for a general case with dimensionless variable). The nonstationary effect caused by the change of various cumulants on the noise level distribution form is concretely reflected in each expansion coefficient of the expansion expression (cf. Eq. (4)). For several special cases, the expression (8) is quite in agreement with those of the corresponding expressions derived in the previous papers (cf. Eqs. (13) and (14)). When each cumulant is predicted by auto-regressive model under the error criteria shown in Eq. (16), the regression coefficient can be calculated by Eqs. (17) and (18) in the case when the third order auto-regressive model is employed. When Eq. (19) is considered under the criteria (20) as the other prediction method, then each expansion coefficient can be calculated from Eqs. (21) and (22).