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Theses

For the identification of time-series models. application to arma processes.

Abstract : This thesis is centered on the problem of the identification of time series models with the meeting of two fields of the Statistics, Time Series Analysis and Data Analysis with its descriptive methods. The first stage of our work is to extend to several discrete time series the Jenkins' principal component study developed in the Seventies. Our approach adapts "classic" Principal Component Analysis (PCA) to time series while taking as a starting point the technique Singular Spectrum Analysis (SSA). A principle is deduced and applied to the multidimensional process generating series. A Toeplitz bloc covariance matrix is built around lagged random vectors: it exploits the chronology, the stationarity and the double dimension of the process. Using two corollaries based on the tensorial product of matrices and established by Friedman B. in the Fifties, like the covariance properties of a circular process, we approach the eigenvalues and the eigenvectors of the covariance matrix. The general shape of the principal components of several time series is deduced. In the case of the "independent" processes, a scores property is established and the principal components become moving averages of time series. From the obtained results, we propose a methodology allowing to build reference factorial models on "independent" vector ARMA. The objective is then to project a new series in one of the graphic models for its identification and a first estimate of its parameters. We work within a theoretical framework, then within an experimental framework by simulating samples of stationary, "independent" with symmetrical coefficients AR(1) and MA(1) processes. Based on simulated temporal matrices, several PCA produce good qualities of processes representation, with significant groupings and oppositions preserving the scores property and the eigenvalues symetric behavior. But above all, these factorial models reflect the variability of simulated white noises. Directly based on autocorrelation matrices, PCA give better results whatever the samples except for some processes said "weak". A first reference graphic model ensues with identification and estimation. Description and measure of possible structural changes lead us to introduce oscillators, frequencies and measures of entropy. This is the structural approach. To establish non-linearity between the numerous criteria and to increase the discriminative ability between the series, classifications on MCA are built over measures of entropy and produce outstanding quality of classes' characterization. A second reference graphic model ensues with the class of "weak" processes. This work also makes it possible to deduce a method of time series analysis which combines the usual approach by autocorrelations and a structural approach, less usual, by analysis of oscillators and theory of information, through visualization by factorial methods. The method is applied to simulated AR(2) and MA(2) processes and provides two more reference factorial models.
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Submitted on : Monday, January 29, 2007 - 8:00:00 AM
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  • HAL Id : pastel-00001966, version 1

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Carole Toque. For the identification of time-series models. application to arma processes.. Mathematics [math]. Télécom ParisTech, 2006. English. ⟨pastel-00001966⟩

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