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  <doc>
    <id>7021</id>
    <completedYear/>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>reportzib</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>--</completedDate>
    <publishedDate>2018-08-24</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Spectral Clustering for Non-reversible Markov Chains</title>
    <abstract language="eng">Spectral clustering methods are based on solving eigenvalue problems for the identification of clusters, e.g. the identification of metastable subsets of a Markov chain. Usually, real-valued eigenvectors are mandatory for this type of algorithms. The Perron Cluster Analysis (PCCA+) is a well-known spectral clustering method of Markov chains. It is applicable for reversible Markov chains, because reversibility implies a real-valued spectrum. We also extend this spectral clustering method to non-reversible Markov chains and give some illustrative examples. The main idea is to replace the eigenvalue problem by a real-valued Schur decomposition. By this extension non-reversible Markov chains can be analyzed. Furthermore, the chains do not need to have a positive stationary distribution. In addition to metastabilities, dominant cycles and sinks can also be identified. This novel method is called GenPCCA  (i.e.&#13;
Generalized PCCA), since it includes the case of non reversible processes. &#13;
We also apply the method to real world eye tracking data.</abstract>
    <identifier type="issn">1438-0064</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-70218</identifier>
    <enrichment key="AcceptedDate">01.07.2018</enrichment>
    <enrichment key="SourceTitle">Comp. Appl. Math., pp 1-16, https://doi.org/10.1007/s40314-018-0697-0</enrichment>
    <author>Konstantin Fackeldey</author>
    <submitter>Konstantin Fackeldey</submitter>
    <author>Alexander Sikorski</author>
    <author>Marcus Weber</author>
    <series>
      <title>ZIB-Report</title>
      <number>18-48</number>
    </series>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>spectral clustering</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Markov chain</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Schur decomposition</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>non-reversible</value>
    </subject>
    <collection role="ccs" number="G.1.3">Numerical Linear Algebra</collection>
    <collection role="pacs" number="05.00.00">Statistical physics, thermodynamics, and nonlinear dynamical systems (see also 02.50.-r Probability theory, stochastic processes, and statistics)</collection>
    <collection role="msc" number="15A21">Canonical forms, reductions, classification</collection>
    <collection role="msc" number="62H30">Classification and discrimination; cluster analysis [See also 68T10]</collection>
    <collection role="institutes" number="num">Numerical Mathematics</collection>
    <collection role="persons" number="fackeldey">Fackeldey, Konstantin</collection>
    <collection role="persons" number="weber">Weber, Marcus</collection>
    <collection role="projects" number="NonequiMSM">NonequiMSM</collection>
    <collection role="persons" number="sikorski">Sikorski, Alexander</collection>
    <file>https://opus4.kobv.de/opus4-zib/files/7021/GenPCCA_FSW.pdf</file>
  </doc>
</export-example>
