A Nonlinear Generalization of Singular Value Decomposition and Its Applications to Mathematical Modeling and Chaotic Cry
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A Nonlinear Generalization of Singular Value Decomposition and Its Applications to Mathematical Modeling and Chaotic Cryptanalysis Prabhakar G. Vaidya · Sajini Anand P. S · Nithin Nagaraj
Received: 11 May 2009 / Accepted: 7 January 2010 / Published online: 23 January 2010 © Springer Science+Business Media B.V. 2010
Abstract Singular Value Decomposition (SVD) is a powerful tool in linear algebra and has been extensively applied to Signal Processing, Statistical Analysis and Mathematical Modeling. We propose an extension of SVD for both the qualitative detection and quantitative determination of nonlinearity in a time series. The method is to augment the embedding matrix with additional nonlinear columns derived from the initial embedding vectors and extract the nonlinear relationship using SVD. The paper demonstrates an application of nonlinear SVD to identify parameters when the signal is generated by a nonlinear transformation. Examples of maps (Logistic map and Henon map) and flows (Van der Pol oscillator and Duffing oscillator) are used to illustrate the method of nonlinear SVD to identify parameters. The paper presents the recovery of parameters in the following scenarios: (i) data generated by maps and flows, (ii) comparison of the method for both noisy and noise-free data, (iii) surrogate data analysis for both the noisy and noise-free cases. The paper includes two applications of the method: (i) Mathematical Modeling and (ii) Chaotic Cryptanalysis. Keywords Nonlinear time series analysis · Singular value decomposition · Chaotic Cryptanalysis · Mathematical Modeling Mathematics Subject Classification (2000) 37N30 · 46N40 · 62J02 1 Introduction Singular Value Decomposition (SVD) along with its related variations known as Principal Component Analysis and Independent Component Analysis are powerful techniques for P.G. Vaidya · S. Anand P. S () National Institute of Advanced Studies, Indian Institute of Science Campus, Bangalore 560 012, India e-mail: [email protected] P.G. Vaidya e-mail: [email protected] N. Nagaraj Amrita School of Engg. Amrita Viswa Vidyapeetham, Kollam, Kerala 690525, India e-mail: [email protected]
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data analysis in linear algebra which have found lot of applications in various fields such as signal processing, statistical analysis, biomedical engineering, genetics analysis, mathematical and statistical modeling, graph theory, psychology etc. [1–8]. Historically SVD has been used for finding the dimension of a linear system as it gives statistically independent set of variables which could span the state space. One SVD based method known as singular spectrum analysis are used for detecting nonlinearity in a qualitative manner [9]. An abrupt decrease in the profile of the singular spectrum is an indication of lower dimensional determinism. But this method fails to distinguish a chaotic time-series from its surrogates [10, 11]. Surrogates are stochastic counterparts of the data which has the same power spectrum. Bhattacharya et al. proposed a method of quadrat
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