Stationary Random Field Estimation Theory And Application

The time series ARMA model with its simplicity, flexibility, intuitive features andeasy operation are very popular. Their theoretical system has developed very perfect.And they are very widely used in data processing, data modeling and informationforecast etc. But, with the need of the deep development of the mathematical theory andthe improvement of industrial production, time series analysis and its modeling methodhas not already been satisfied with the people’s needs. Therefore, it is very nature toextend from the single index’ stochastic process (time series) to several indexes’random process and several indexes’ random process is also random field.But, the extension from time series to random fields is very difficult, so far, there hasnot been an authority book on random field. Because in several indexes’ case,"time"has not the deterministic "past" and "future" in general. So we need to define the orderamong space points. The difficulties of extension are just the order. Now the severalexisting random fields have one same weakness: the distance to the current dot hashuge difference from people’s thinking habits. Thus, the research of random field is veryslow, and its applications are very limited.Based on the different "past" and "future",there exist different forecasting theories,such as a quarter forecasting theory, half plane forecasting theory, asymmetric halfplane forecasting theory, and a minimum variance estimation theory. First, this papersummarizes the research status of stationary random fields, introduces the principle andconclusions of the four random fields, and compares the advantages and disadvantagesof them. Secondly, this paper proved from the two way of theory and application thatthe covariance matrix of Yule-Walker equation of AR model of the stationary randomfield is positive definite, AR model parameters can only be determined by the equation.When in application proving the covariance matrix of Yule-Walker equation is positivedefinite matrix, use the expansion method, namely between two rows of data insert intoa certain length of zero, so as to avoid different covariance intertwined. This is aninnovative point of this paper. The significance of the conclusions is: on the one hand, prove the Yule-Walker equation is an effective method, AR model only parameters canbe obtained by the equation; On the other hand, the data we deal always is verycomplicated, how can we effectively make use of Yule-Walker equation so that it has asolution when modeling? This paper has found such a way. Although the conclusionsare proved in the two indicators random field of a quarter plane, they can be extended toother random fields and many indicators random fields.