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Interpreting Cluster Structure in Waveform Data with Visual Assessment and Dunn’s Index

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Book cover Frontiers in Computational Intelligence

Part of the book series: Studies in Computational Intelligence ((SCI,volume 739))

Abstract

Dunn’s index was introduced in 1974 as a way to define and identify a “best” crisp partition on n objects represented by either unlabeled feature vectors or dissimilarity matrix data. This article examines the intimate relationship that exists between Dunn’s index, single linkage clustering, and a visual method called iVAT for estimating the number of clusters in the input data. The relationship of Dunn’s index to iVAT and single linkage in the labeled data case affords a means to better understand the utility of these three companion methods when data are crisply clustered in the unlabeled case (the real case). Numerical examples using simulated waveform data drawn from the field of neuroscience illustrate the natural compatibility of Dunn’s index with iVAT and single linkage. A second aim of this note is to study customizing the three methods by changing the distance measure from Euclidean distance to one that may be more appropriate for assessing the validity of crisp clusters of finite sets of waveform data. We present numerical examples that support our assertion that when used collectively, the three methods afford a useful approach to evaluation of crisp clusters in unlabeled waveform data.

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Author information

Authors and Affiliations

  1. Institute of Biomaterials and Biomedical Engineering, University of Toronto, Toronto, Canada

    Sara Mahallati, Milos R. Popovic & Taufik A. Valiante

  2. Toronto Rehabilitation Institute, University Health Network, Toronto, Canada

    Sara Mahallati & Milos R. Popovic

  3. Krembil Research Institute, University Health Network, Toronto, Canada

    Taufik A. Valiante

  4. Lyles School of Civil Engineering, Purdue University, West Lafayette, IN, USA

    Dheeraj Kumar

  5. Computer Science and Information Systems Departments, University of Melbourne, Melbourne, Australia

    James C. Bezdek

Authors
  1. Sara Mahallati
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  2. James C. Bezdek
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  3. Dheeraj Kumar
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  4. Milos R. Popovic
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  5. Taufik A. Valiante
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Corresponding author

Correspondence to Sara Mahallati .

Editor information

Editors and Affiliations

  1. Faculty of Computer Science, Otto von Guericke University Magdeburg Faculty of Computer Science, Magdeburg, Germany

    Sanaz Mostaghim

  2. Faculty of Computer Science, Otto von Guericke University Magdeburg, Faculty of Computer Science, Magdeburg, Germany

    Andreas Nürnberger

  3. Department of Computer and Information Science,, University of Konstanz, Department of Computer and Information Science, Konstanz, Germany

    Christian Borgelt

Appendices

Appendix 1. The VAT and IVAT Reordering Algorithms

figure a

A.1 The input matrix D for VAT in line 1 is positive definite and symmetric. Any distance matrix will be of this type, but there are a number of cases that don’t satisfy these constraints. And the size of D can be an issue. This basic version is only useful for fairly small values of n (say, n 10,000 or so). Extensions to rectangular, asymmetric and big data inputs are covered in the notes and remarks for this chapter.

A.2 Prim’s MST algorithm usually starts at either end (i.e., vertex) of a smallest weight edge. Initialization at line 3 starts at the opposite extreme - either end of a largest weight edge. This prevents VAT from a certain type of off-course deviation that is discussed in Bezdek and Hathaway (2002).

A.3 The argmax and argmin function calls in lines 3, 7 and 15 produce sets, not single values. For example, in A4.1 is the set of all ordered pairs (i, j) that have a maximum distance. In case of ties, use a vertex from either end of any one edge in the set.

Appendix 2. Basic Single Linkage Clustering Algorithm

figure b

Appendix 3. Shape Based Distance

figure c

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Mahallati, S., Bezdek, J.C., Kumar, D., Popovic, M.R., Valiante, T.A. (2018). Interpreting Cluster Structure in Waveform Data with Visual Assessment and Dunn’s Index. In: Mostaghim, S., Nürnberger, A., Borgelt, C. (eds) Frontiers in Computational Intelligence. Studies in Computational Intelligence, vol 739. Springer, Cham. https://doi.org/10.1007/978-3-319-67789-7_6

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  • DOI: https://doi.org/10.1007/978-3-319-67789-7_6

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