Nonlinear, multidimensional transformations and their applications to signal processing

Anand PS, Sajini (2015) Nonlinear, multidimensional transformations and their applications to signal processing. Doctoral thesis, NIAS.

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Thesis advisorVaidya, PrabhakarUNSPECIFIED
Abstract: Modeling a system based on time series is a complicated problem in general, especially when the time series is nonlinear and chaotic. The goal of the thesis is to introduce a method of prediction and modeling that exploits the property of recurrence in dynamical systems. A time series is said to be recurrent if keeps on visiting a particular neighborhood in the state space. The thesis demonstrates that the inherent redundancy structure of a well known topological technique known as delay embedding can be coupled with recurrence property to develop a new method of prediction. The modeling procedure empirically finds the recurrence neighborhoods from the signal, which are then subdivided into various equivalence classes based on their recurrence timings. A set of affine maps are then derived across these equivalence classes. This gives is a possibility of simplifying the dynamics in terms of affine transformations in small neighborhoods. The delay-embedding (done in a dimension much higher than the inherent dimension of the dynamics) is used as a scaffolding to analyze the global structure of the system. A projection to a lower dimension was followed to take care of the fundamental issues related to high dimensional models that describe a low dimensional dynamics. Local analysis of the system was done in the low dimensional projected space. A topological conjugacy of the recurrence neighborhoods in both the lower and the higher dimensional spaces are demonstrated. The proposed model uses a nonlinear generalization of a well known linear algebra technique named Singular Value Decomposition (SVD) for data analysis. The method of nonlinear SVD and its uses (i) to determine nonlinearity in a time series and (ii) to empirically arrive at an upper bound for the dimension of a manifold where the data resides are demonstrated. The proposed method of prediction and modeling was used for the analysis of (i) data generated by the Duffing oscillator and (ii) an Electrocardiogram (ECG) record. It is shown that the entire nonlinear structure can be deduced from one or few overlapping neighborhoods for these data. A method of stability analysis by studying the properties of affine maps specific to the neighborhoods are demonstrated for both these data. The thesis gives a theoretical justification for a well known experimental observation that the heart rate variability– a variability in beat-to-beat intervals of the heart is a necessity for healthy functioning of the heart. The relevance and contribution of the introduced method for biomedical signal processing is justified by using it successfully for analyzing a set of multi-channel physiological data.
Item Type: Thesis (Doctoral)
Additional Information: The thesis was submitted to the University of Mysore through the Department of Studies in Mathematics, University of Mysore,Mysore.[Year of Award 2015] [Thesis No. TH19]
Keywords: Time Series Analysis, Dynamical Systems, Signal Value Decomposition, Affine Maps, Heart Rate Variability, Biomedical Signal Processing, Nonlinear Transformations, Multidimensional Transformations
Subjects: School of Natural and Engineering Sciences > Mathematical Modeling
Doctoral Programme > Theses
Divisions: Schools > Natural Sciences and Engineering
Date Deposited: 03 May 2016 10:44
Last Modified: 22 Aug 2023 05:37
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    Funders: UNSPECIFIED
    Projects: UNSPECIFIED

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