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On Simultaneous Data-Based Dimension Reduction and Hidden Phase Identification

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  • 1 Institut für Mathematik, Freie Universität Berlin, Berlin, Germany
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Abstract

A problem of simultaneous dimension reduction and identification of hidden attractive manifolds in multidimensional data with noise is considered. The problem is approached in two consecutive steps: (i) embedding the original data in a sufficiently high-dimensional extended space in a way proposed by Takens in his embedding theorem, followed by (ii) a minimization of the residual functional. The residual functional is constructed to measure the distance between the original data in extended space and their reconstruction based on a low-dimensional description. The reduced representation of the analyzed data results from projection onto a fixed number of unknown low-dimensional manifolds. Two specific forms of the residual functional are proposed, defining two different types of essential coordinates: (i) localized essential orthogonal functions (EOFs) and (ii) localized functions called principal original components (POCs). The application of the framework is exemplified both on a Lorenz attractor model with measurement noise and on historical air temperature data. It is demonstrated how the new method can be used for the elimination of noise and identification of the seasonal low-frequency components in meteorological data. An application of the proposed POCs in the context of the low-dimensional predictive models construction is presented.

Corresponding author address: Illia Horenko, Institut für Mathematik, Freie Universität Berlin, Arnimallee 6, 14195 Berlin, Germany. Email: horenko@math.fu-berlin.de

Abstract

A problem of simultaneous dimension reduction and identification of hidden attractive manifolds in multidimensional data with noise is considered. The problem is approached in two consecutive steps: (i) embedding the original data in a sufficiently high-dimensional extended space in a way proposed by Takens in his embedding theorem, followed by (ii) a minimization of the residual functional. The residual functional is constructed to measure the distance between the original data in extended space and their reconstruction based on a low-dimensional description. The reduced representation of the analyzed data results from projection onto a fixed number of unknown low-dimensional manifolds. Two specific forms of the residual functional are proposed, defining two different types of essential coordinates: (i) localized essential orthogonal functions (EOFs) and (ii) localized functions called principal original components (POCs). The application of the framework is exemplified both on a Lorenz attractor model with measurement noise and on historical air temperature data. It is demonstrated how the new method can be used for the elimination of noise and identification of the seasonal low-frequency components in meteorological data. An application of the proposed POCs in the context of the low-dimensional predictive models construction is presented.

Corresponding author address: Illia Horenko, Institut für Mathematik, Freie Universität Berlin, Arnimallee 6, 14195 Berlin, Germany. Email: horenko@math.fu-berlin.de

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