Regulation Regression Methods And Its Application In Chaotic Time Series

In the analysis of chaotic time series, no matter using local linear model to predict chaotic time series or using matrix algorithm to compute Lyapunov exponent spectrum, we often encounter the OLS problem. In the practical problem,when multicollinearity relation exit in variables, the tradition OLS method can produce large bias in result, even becoming too ill-posed to be computed. In this dissertation, through using many modified regularized methods based on OLS regression, combined with the adaptive local linear method, we give the adaptive local linear method based on regularized regression, which is applied to local linear prediction of chaotic time series and the computation of Lyapunov exponent spectrum. First, the introduction of OLS is presented. Afterwards the research actuality of the the modification of the OLS,prediction of chaotic time series,and the computation of Lyapunov exponent spectrum are summarized. Then several prediction methods ,such as local linear prediction, local polynomial prediction, and neural network prediction, and the computational method of the transform matrix in algorithm of Lyapunov exponent spectrum are systemically proposed, from which we get the universal linear regression model in the analysis of chaotic time series. As for the linear regression model, we introduced the regularized estimation method,such as ridge regression estimate,principal components regression estimate and partial least squares regression estimate , which can be expressed using the same formula in the framework of SVD and have smaller MSE than LSE . When it comes to the local prediction of chaotic time series, owing to the fact that the fixed embedding dimension,the number of the nearest neighbours and regularization parameters can cause the added prediction error,we give the adaptive local linear method and discuss the adaptive selection method of regularization parameters, the advantage of which is tested by the Henon map and Lorenz system simulated data and the integrated index in stock market of Shanghai, to show its practicable value. Last, the matrix algorithm of Lyapunov exponent spectrum is computated by regularlized regression method and simulation of Lorenz system shows it can overcome the shortages of the worse computational result when the data matrix is ill-posed.