Application of topology in inverse problems: Getting equations from ECG and sound data

Majumdar, Swarnali (2011) Application of topology in inverse problems: Getting equations from ECG and sound data. Doctoral thesis, NIAS.

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Thesis advisorVaidya, PrabhakarUNSPECIFIED
Thesis advisorRamaswamy, MythilyUNSPECIFIED
Abstract: The main topic of this thesis is the theoretical and the practical aspects of inverse problem. By inverse problem we mean getting equations when data is given. For linear systems, inverse problems are studied to a vast extent. In this thesis we discuss non-linear inverse problems. We want to find the equations from data for non-linear systems. Embedding is one of the most popular methods in getting equations from a given time series. We have shown that if a system is embedded in a higher dimension it might create a (stable or unstable) foliation. If data is taken only from a single leaf of the foliation, system identification becomes ambiguous, as there can be many other systems which generate the same data. So if we try to find out the dynamical system from a given time series data, embedding can lead to wrong estimates about the underlying equation. We solve this ambiguity problem for some specific cases by modifying the equation of the foliated dynamical system. We modify the equation in such a way that no matter what initial conditions we choose, the system would asymptotically reach a specific leaf of the foliation. As a consequence the foliation would be collapsed. In reality, the above method might not always work. Hence we consider an alternative approach for finding equations from data which is known as local embedding. By using this method we can completely avoid embedding in high dimension. We cover the manifold with local patches and develop a local dynamics in each patch. Often data points are in high dimension, but the dimension of the manifold where data resides is much lower dimension. In local embedding method, the dimension of the local dynamics is same as the dimension of the manifold where data resides. So we can model the data by a low dimensional dynamical system. Finally, we analyze some practical signals (ECG and sound) by applying various topological ideas and techniques.
Item Type: Thesis (Doctoral)
Additional Information: Thesis submitted to Manipal University, Manipal, India. [Year of Award 2013] [Thesis No. TH14]
Subjects: School of Natural and Engineering Sciences > Mathematical Modeling
Doctoral Programme > Theses
Divisions: Schools > Natural Sciences and Engineering
Date Deposited: 02 Apr 2013 05:26
Last Modified: 18 Feb 2021 11:30
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    Funders: UNSPECIFIED
    Projects: UNSPECIFIED

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