The Methods Of Spectrum Density Estimation And Prediction Of Multidimensional ARMA (p, Q) Models

Based on single stationary time series, the modeling, spectral estimation and prediction of multidimensional automatic regression moving average (ARMA(p,q)) models are studied systematically in this paper. On the other hand, some important theorems are also proved.Firstly, It introduce some basal conceptions about multidimensional stationary time series, contained the definition and property of mean vectorn and auto- covariance function. Then the stationary and reversibility of multidimensional ARMA(p,q) models are discussed.Secondly, parameter estimation methods and their deficiencies are obtained by using multidimensional Yule-Walker equation which is similar to single stationary time series. The main methods are as follow: moment method of estimate, least square method and maximum likelihood estimation method. Thus the first step of modeling- parameter estimation is finished.Furthermore, the AIC rule is used to fix the rank of multidimensional ARMA(p,q) models When the model order is unknown.Subsequently, the spectral representation of multidimensional stationary time series is given and the spectral representation theorem is proved, by which the autocovariance matrix and the spectrum density matrix of multidimensional AR(p),MA(q), ARMA(p, q) models are deduced.Finally, Fitting multidimensional ARMA(p,q) model, estimating its spectrum density and doing recursive prediction of multidimensional ARMA(p,q) models, according to multi-dimensional innovation algorithm.