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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>
  <doc>
    <id>7040</id>
    <completedYear/>
    <publishedYear>2018</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst>6376</pageFirst>
    <pageLast>6391</pageLast>
    <pageNumber/>
    <edition/>
    <issue>5</issue>
    <volume>37</volume>
    <type>article</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>--</completedDate>
    <publishedDate>--</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., generalized PCCA), since it includes the case of non-reversible processes. We also apply the method to real-world eye-tracking data.</abstract>
    <parentTitle language="eng">Computational and Applied Mathematics</parentTitle>
    <identifier type="doi">https://doi.org/10.1007/s40314-018-0697-0</identifier>
    <enrichment key="PeerReviewed">yes</enrichment>
    <enrichment key="AcceptedDate">2018/08/22</enrichment>
    <enrichment key="PreprintUrn">urn:nbn:de:0297-zib-70218</enrichment>
    <author>Konstantin Fackeldey</author>
    <submitter>Marcus Weber</submitter>
    <author>Alexander Sikorski</author>
    <author>Marcus Weber</author>
    <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>Non-reversible</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Schur decomposition</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>GenPCCA</value>
    </subject>
    <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="EyeTracking">EyeTracking</collection>
    <collection role="projects" number="SFB1114-A5">SFB1114-A5</collection>
    <collection role="persons" number="sikorski">Sikorski, Alexander</collection>
  </doc>
  <doc>
    <id>5573</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>2015-08-19</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Finding dominant structures of nonreversible Markov processes</title>
    <abstract language="eng">Finding metastable sets as dominant structures of Markov processes has been shown to be especially useful in modeling interesting slow dynamics of various real world complex processes. Furthermore, coarse graining of such processes based on their dominant structures leads to better understanding and dimension reduction of observed systems. However, in many cases, e.g. for nonreversible Markov processes, dominant structures are often not formed by metastable sets but by important cycles or mixture of both. This paper aims at understanding and identifying these different types of dominant structures for reversible as well as nonreversible ergodic Markov processes. Our algorithmic approach generalizes spectral based methods for reversible process by using Schur decomposition techniques which can tackle also nonreversible cases. We illustrate the mathematical construction of our new approach by numerical experiments.</abstract>
    <identifier type="issn">1438-0064</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-55739</identifier>
    <identifier type="doi">10.1137/15M1032272</identifier>
    <enrichment key="SourceTitle">Appeared in: Multiscale Modeling and Simulation 14(4): 1319-1340</enrichment>
    <author>Natasa Djurdjevac Conrad</author>
    <submitter>Erlinda Koernig</submitter>
    <author>Marcus Weber</author>
    <author>Christof Schütte</author>
    <series>
      <title>ZIB-Report</title>
      <number>15-40</number>
    </series>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>nonreversible Markov processes</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>metastable sets</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>cycle decomposition</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Schur decomposition</value>
    </subject>
    <collection role="msc" number="60J20">Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) [See also 90B30, 91D10, 91D35, 91E40]</collection>
    <collection role="msc" number="65C40">Computational Markov chains</collection>
    <collection role="msc" number="82C26">Dynamic and nonequilibrium phase transitions (general)</collection>
    <collection role="institutes" number="num">Numerical Mathematics</collection>
    <collection role="persons" number="schuette">Schütte, Christof</collection>
    <collection role="persons" number="weber">Weber, Marcus</collection>
    <collection role="projects" number="BMS-Nielsen">BMS-Nielsen</collection>
    <collection role="projects" number="EyeTracking">EyeTracking</collection>
    <collection role="projects" number="NonequiMSM">NonequiMSM</collection>
    <collection role="persons" number="natasa.conrad">Conrad, Natasa</collection>
    <file>https://opus4.kobv.de/opus4-zib/files/5573/ZIB-Report_15-40.pdf</file>
  </doc>
</export-example>
