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A Topological Approach of Principal Component Analysis

Received: 10 January 2021     Accepted: 29 January 2021     Published: 20 April 2021
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Abstract

Large datasets are increasingly widespread in many disciplines. The exponential growth of data requires the development of more data analysis methods in order to process information more efficiently. In order to better visualize the data, many methods such as Principal Component Analysis (PCA) and MultiDimensional Scaling (MDS) allow to extract a low-dimensional structure from high-dimensional data set. The proposed approach, called Topological Principal Component Analysis (TPCA), is a multidimensional descriptive method witch studies a homogeneous set of continuous variables defined on the same set of individuals. It is a topological method of data analysis that consists of comparing and classifying proximity measures from among some of the most widely used proximity measures for continuous data. Proximity measures play an important role in many areas of data analysis, the results strongly depend on the proximity measure chosen. So, among the many existing measures, which one is most useful? Are they all equivalent? How to identify the one that is most appropriate to analyze the correlation structure of a set of quantitative variables. TPCA proposes an appropriate adjacency matrix associated to an unknown proximity measure according to the data under consideration, then analyzes and visualizes, with graphic representations, the relationship structure of the variables relating to, the well known PCA problem. Its uses the concept of neighborhood graphs and compares a set of proximity measures for continuous data which can be more-or-less equivalent a topological equivalence criterion between two proximity measures is defined and statistically tested according to the topological correlation between the variables considered. An example on real data illustrates the proposed approach.

Published in International Journal of Data Science and Analysis (Volume 7, Issue 2)
DOI 10.11648/j.ijdsa.20210702.11
Page(s) 20-31
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2021. Published by Science Publishing Group

Keywords

Proximity Measure, Neighborhood Graph, Adjacency Matrix, Topological Equivalence, Correlation Matrix, MDS Graphical Representation

References
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[2] R. Abdesselam, “Selection of proximity measures for a Topological Correspondence Analysis.” In a Book Series, 5th Stochastic Modeling Techniques and Data Analysis, International Conference, Chania, Greece, pp. 11-24, 2018.
[3] R. Abdesselam, “A Topological Discriminant Analysis.” In book Chapter, Vol. 3, Data Analysis and Applications 2: Utilization of Results in Europe and 0ther Topics, ISTE Science Publishing, Wiley, pp. 167-178, 2018.
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[19] J-W. Schneider and P. Borlund, “Matrix comparison, Part 1: Motivation and important issues for measuring the resemblance between proximity measures or ordination results.” In Journal of the American Society for Information Science and Technology, Vol. 58, 11, pp. 1586-1595, 2007.
[20] J-W. Schneider and P. Borlund, “Matrix comparison, Part 2: Measuring the resemblance between proximity measures or ordination results by use of the Mantel and Procrustes statistics.” In Journal of the American Society for Information Science and Technology, Vol. 11, 58, pp. 1596-1609, 2007.
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    Rafik Abdesselam. (2021). A Topological Approach of Principal Component Analysis. International Journal of Data Science and Analysis, 7(2), 20-31. https://doi.org/10.11648/j.ijdsa.20210702.11

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    Rafik Abdesselam. A Topological Approach of Principal Component Analysis. Int. J. Data Sci. Anal. 2021, 7(2), 20-31. doi: 10.11648/j.ijdsa.20210702.11

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    AMA Style

    Rafik Abdesselam. A Topological Approach of Principal Component Analysis. Int J Data Sci Anal. 2021;7(2):20-31. doi: 10.11648/j.ijdsa.20210702.11

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  • @article{10.11648/j.ijdsa.20210702.11,
      author = {Rafik Abdesselam},
      title = {A Topological Approach of Principal Component Analysis},
      journal = {International Journal of Data Science and Analysis},
      volume = {7},
      number = {2},
      pages = {20-31},
      doi = {10.11648/j.ijdsa.20210702.11},
      url = {https://doi.org/10.11648/j.ijdsa.20210702.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijdsa.20210702.11},
      abstract = {Large datasets are increasingly widespread in many disciplines. The exponential growth of data requires the development of more data analysis methods in order to process information more efficiently. In order to better visualize the data, many methods such as Principal Component Analysis (PCA) and MultiDimensional Scaling (MDS) allow to extract a low-dimensional structure from high-dimensional data set. The proposed approach, called Topological Principal Component Analysis (TPCA), is a multidimensional descriptive method witch studies a homogeneous set of continuous variables defined on the same set of individuals. It is a topological method of data analysis that consists of comparing and classifying proximity measures from among some of the most widely used proximity measures for continuous data. Proximity measures play an important role in many areas of data analysis, the results strongly depend on the proximity measure chosen. So, among the many existing measures, which one is most useful? Are they all equivalent? How to identify the one that is most appropriate to analyze the correlation structure of a set of quantitative variables. TPCA proposes an appropriate adjacency matrix associated to an unknown proximity measure according to the data under consideration, then analyzes and visualizes, with graphic representations, the relationship structure of the variables relating to, the well known PCA problem. Its uses the concept of neighborhood graphs and compares a set of proximity measures for continuous data which can be more-or-less equivalent a topological equivalence criterion between two proximity measures is defined and statistically tested according to the topological correlation between the variables considered. An example on real data illustrates the proposed approach.},
     year = {2021}
    }
    

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    T1  - A Topological Approach of Principal Component Analysis
    AU  - Rafik Abdesselam
    Y1  - 2021/04/20
    PY  - 2021
    N1  - https://doi.org/10.11648/j.ijdsa.20210702.11
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    T2  - International Journal of Data Science and Analysis
    JF  - International Journal of Data Science and Analysis
    JO  - International Journal of Data Science and Analysis
    SP  - 20
    EP  - 31
    PB  - Science Publishing Group
    SN  - 2575-1891
    UR  - https://doi.org/10.11648/j.ijdsa.20210702.11
    AB  - Large datasets are increasingly widespread in many disciplines. The exponential growth of data requires the development of more data analysis methods in order to process information more efficiently. In order to better visualize the data, many methods such as Principal Component Analysis (PCA) and MultiDimensional Scaling (MDS) allow to extract a low-dimensional structure from high-dimensional data set. The proposed approach, called Topological Principal Component Analysis (TPCA), is a multidimensional descriptive method witch studies a homogeneous set of continuous variables defined on the same set of individuals. It is a topological method of data analysis that consists of comparing and classifying proximity measures from among some of the most widely used proximity measures for continuous data. Proximity measures play an important role in many areas of data analysis, the results strongly depend on the proximity measure chosen. So, among the many existing measures, which one is most useful? Are they all equivalent? How to identify the one that is most appropriate to analyze the correlation structure of a set of quantitative variables. TPCA proposes an appropriate adjacency matrix associated to an unknown proximity measure according to the data under consideration, then analyzes and visualizes, with graphic representations, the relationship structure of the variables relating to, the well known PCA problem. Its uses the concept of neighborhood graphs and compares a set of proximity measures for continuous data which can be more-or-less equivalent a topological equivalence criterion between two proximity measures is defined and statistically tested according to the topological correlation between the variables considered. An example on real data illustrates the proposed approach.
    VL  - 7
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Author Information
  • Department of Economics and Management, University Lumière of Lyon 2, Lyon, France

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