A Topological Approach of Principal Component Analysis

Rafik Abdesselam

doi:doi:10.11648/j.ijdsa.20210702.11

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A Topological Approach of Principal Component Analysis

Rafik Abdesselam

Published in International Journal of Data Science and Analysis (Volume 7, Issue 2)

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), 2024. Published by Science Publishing Group

Keywords

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

References

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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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ACS 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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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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TY - JOUR
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
DO - 10.11648/j.ijdsa.20210702.11
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
IS - 2
ER -

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Author Information

Rafik Abdesselam

Department of Economics and Management, University Lumière of Lyon 2, Lyon, France

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APA Style

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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ACS 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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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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