A relationship between two innovation representations is discussed in a stationary process, and a numerical algorithm of a new type innovation model is introduced for the time series analysis. Coefficient matrices of both innovation models are derived from correlation functions by using the singular value decomposition method of Hankel matrix and solving a Riccati type equation. Zeros of both models are also examined, since they have an important role not only in the analysis of AR model, but also in system diagnosis.

An AR model is one of linear stochastic equations, which is characterized by the eigenvalues and eigenvectors. Since the poles in the AR model correspond to the eigenvalues in linear equations, the weights of poles in the AR model correspond to the eigenvectors in linear equations. The AR type model has essentially two types poles; system poles and virtual poles corresponding to system zeros. These poles can be distinguished by observing the weight of each pole in the partial fraction expansion of the AR model transfer function. The rules for separation of AR poles are: (a) If the weight of an AR pole is constant for AR model order change, the AR pole is a system pole. (b) If the weight of that is inversely proportional to the AR model order, the AR pole is one of virtual poles.

A stochastic system represented by an input-output model can be described by mainly two different types of state space representation. Corresponding to state space representations innovation models are examined. The relationship between both representations is made clear systematically. An easy transformation between them is presented. Zeros of innovation models are the same as those of an ARMA model which is stochastically equivalent to innovation models, and related to stable eigenvalues of generalized eigenvalue problem of matrix Riccati equation.

A new blind identification method of transfer functions between variables in feedback systems is introduced for single sweep type of MEG data. The method is based on the viewpoint of stochastic/statistical inverse problems. The required conditions of the model are stationary and linear Gaussian processes. Raw MEG data of the brain activities are heavily contaminated with several noises and artifacts. The elimination of them is a crucial problem especially for the method. Usually, these noises and artifacts are removed by notch and high-pass filters which are preset automatically. In the present paper, we will try two types of more careful preprocessing procedures for the identification method to obtain impulse functions. One is a careful notch filtering and the other is a blind source separation method based on temporal structure. As results, identifiably of transfer functions and their impulse responses are improved in both cases. Transfer functions and impulse responses identified between MEG sensors are obtained by using the method in Appendix A, when eyes are closed with rest state. Some advantages of the blind source separation method are discussed.