| Canonical expansion is an effective tool of studying the second-order random process by decomposing the process into an orthogonal expansion based on the information of the second moment. In essence, it is one of those techniques which can be categorised under the theory of orthogonal functions. The current study is devoted to applying this technique in the optimal estimation of random process according to the principle of Minimum Mean Square Error (MMSE), by constructing both the optimal linear and nonlinear operators through the canonical expansion. The whole theory of canonical expansion is grounded on the theories of linear integral equation and linear algebra. The principle of MMSE results in the Wiener-Hopf equation for the linear estimation, and the regression operator in the non-linear case. Both the estimators can be constructed by the principal components that are generated through the canonical expansion. Numerical experiments show that such a method can give very accurate results for estimations of different time series. Also, the relation and comparison between the linear and non-linear operators are revealed through those numerical examples, in which the noise models are all Gaussian processes. |
