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Multidimensional Complex Stationary Centered Gaussian Autoregressive Time Series Machine Learning in Poincaré and Siegel Disks: Application for Audio and Radar Clutter Classification

  • Cabanes, Yann
Publication Date
Mar 31, 2022
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The objective of this thesis is the classification of complex valued stationary centered Gaussian autoregressive time series. We study the case of one-dimensional time series as well as the more general case of multidimensional time series. The contribution of this thesis is both methodological and technical. The methodology presented can be used to represent the probability distributions of the observed time series in a Riemannian manifold in which the classification will be performed. The major steps of our method are: the definition of the space of the coefficients of the parametric model used to represent the considered time series, the estimation of the coefficients of the parametric model from observed time series, to endow the space of the coefficients of the parametric model with a Riemannian metric inspired by information geometry and finally the adaptation of classical machine learning algorithms to the Riemannian manifolds obtained. In the case of multidimensional time series, we will work in a product manifold which involves the Siegel disk (set of complex matrices with singular values strictly lower than 1) endowed with a Riemannian metric. In addition to the methodological contribution mentioned previously, we bring new theoretical tools to classify data in the Siegel manifold: we give the explicit formulas of the Riemannian logarithm map, of the Riemannian exponential map and of the Siegel manifold sectional curvature. Our representation model for complex stationary centered Gaussian autoregressive time series will be applied to simulated time series classification, to radar clutter clustering and to stationary stereo audio time series classification.

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