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Scale Independent Principal Component Analysis and Factor Analysis with Preserved Inherent Variability of the Indicators

Received: 2 February 2017     Accepted: 17 February 2017     Published: 2 March 2017
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Abstract

Principal Component Analysis (PCA) and Factor Analysis (FA) are common multivariate techniques used for dimensionality reduction. With these techniques it is expected to identify actual number of dimensions while accounting almost all observed variability. Standard PCA is based either on correlation matrix (CORM) or covariance matrix (COVM). When it is based on CORM, scale dependency can be removed but inherent variability cannot be preserved. On the other hand, when PCA is based on COVM, inherent variability can be preserved but scale dependency cannot be removed. As a solution to this issue, this paper suggests scaling each indicator by its mean, resulting in new mean equal to 1 and standard deviation equal to the coefficient of variance (CV). This leads to PCs, which are scale independent while retaining the observed variability. The computation of PCs and factors under the suggested method is derived in the study. The procedure is illustrated using the lowest level administrative division census data of Western province of Sri Lanka.

Published in American Journal of Theoretical and Applied Statistics (Volume 6, Issue 2)
DOI 10.11648/j.ajtas.20170602.13
Page(s) 90-94
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), 2017. Published by Science Publishing Group

Keywords

Scaling Indicators, Coefficient of Variation, Multivariate Techniques, Dimensional Reduction, Computation of PCAs and Factors

References
[1] Silva, G. (2000) Construction of Human Development Indices using Multivariate Techniques, PGIA, University of Peradeniya.
[2] The Organization for Economic Co-operation and Development, (2004) The OECD-JRC Handbook on Practices for Developing Composite Indicators. https://stats.oecd.org/glossary/detail.asp?ID=6278.
[3] Jolliffe, T. (2002) Principal Component Analysis, Springer Verlag, New York.
[4] Josseph, F. et al. (2010) Multivariate Data Analysis. A Global Perspective, Pearson Education Inc, New Jersey.
[5] Fernando, S., Samita, S and Abenayake R (2011). Modified factor analysis to construct composite indices: Illustration on Urbanization index. Tropical Agricultural Research, 24, 271–281.
[6] Tuan, A. and Magi, S. (2009) Principal Component Analysis: Final Paper in Financial Pricing. National Cheng Kung University, 3-26.
[7] Farrugia. N, (2007) conceptual issues in constructing composite indices. Occasional papers on islands and small states, 2/2007.
[8] Yutaka, K., Yusuke, M. and Shohei, S. (2003) factor rotation and ica. 4th International Symposium on Independent Component Analysis and Blind Signal Separation (ICA2003), Japan.
[9] Delchambre, L. (2014) Weighted principal component analysis: a weighted covariance eigen decomposition approach, University of Liege, Belgium.
[10] Abdi, H., and Williams, J., Principal component analysis John Wiley & Sons; WIREs Comp Stat 2010 2 433–459 2010.
[11] http://www.theanalysisfactor.com/factor-analysis-1-introduction/.
[12] The Organization for Economic Co-operation and Development, (2008) Handbook on constructing composite indicators: methodology and user guide.
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  • APA Style

    Priyadarshana Dharmawardena, Raphel Ouseph Thattil, Sembakutti Samita. (2017). Scale Independent Principal Component Analysis and Factor Analysis with Preserved Inherent Variability of the Indicators. American Journal of Theoretical and Applied Statistics, 6(2), 90-94. https://doi.org/10.11648/j.ajtas.20170602.13

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    ACS Style

    Priyadarshana Dharmawardena; Raphel Ouseph Thattil; Sembakutti Samita. Scale Independent Principal Component Analysis and Factor Analysis with Preserved Inherent Variability of the Indicators. Am. J. Theor. Appl. Stat. 2017, 6(2), 90-94. doi: 10.11648/j.ajtas.20170602.13

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    AMA Style

    Priyadarshana Dharmawardena, Raphel Ouseph Thattil, Sembakutti Samita. Scale Independent Principal Component Analysis and Factor Analysis with Preserved Inherent Variability of the Indicators. Am J Theor Appl Stat. 2017;6(2):90-94. doi: 10.11648/j.ajtas.20170602.13

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  • @article{10.11648/j.ajtas.20170602.13,
      author = {Priyadarshana Dharmawardena and Raphel Ouseph Thattil and Sembakutti Samita},
      title = {Scale Independent Principal Component Analysis and Factor Analysis with Preserved Inherent Variability of the Indicators},
      journal = {American Journal of Theoretical and Applied Statistics},
      volume = {6},
      number = {2},
      pages = {90-94},
      doi = {10.11648/j.ajtas.20170602.13},
      url = {https://doi.org/10.11648/j.ajtas.20170602.13},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20170602.13},
      abstract = {Principal Component Analysis (PCA) and Factor Analysis (FA) are common multivariate techniques used for dimensionality reduction. With these techniques it is expected to identify actual number of dimensions while accounting almost all observed variability. Standard PCA is based either on correlation matrix (CORM) or covariance matrix (COVM). When it is based on CORM, scale dependency can be removed but inherent variability cannot be preserved. On the other hand, when PCA is based on COVM, inherent variability can be preserved but scale dependency cannot be removed. As a solution to this issue, this paper suggests scaling each indicator by its mean, resulting in new mean equal to 1 and standard deviation equal to the coefficient of variance (CV). This leads to PCs, which are scale independent while retaining the observed variability. The computation of PCs and factors under the suggested method is derived in the study. The procedure is illustrated using the lowest level administrative division census data of Western province of Sri Lanka.},
     year = {2017}
    }
    

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    T1  - Scale Independent Principal Component Analysis and Factor Analysis with Preserved Inherent Variability of the Indicators
    AU  - Priyadarshana Dharmawardena
    AU  - Raphel Ouseph Thattil
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    T2  - American Journal of Theoretical and Applied Statistics
    JF  - American Journal of Theoretical and Applied Statistics
    JO  - American Journal of Theoretical and Applied Statistics
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    EP  - 94
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    UR  - https://doi.org/10.11648/j.ajtas.20170602.13
    AB  - Principal Component Analysis (PCA) and Factor Analysis (FA) are common multivariate techniques used for dimensionality reduction. With these techniques it is expected to identify actual number of dimensions while accounting almost all observed variability. Standard PCA is based either on correlation matrix (CORM) or covariance matrix (COVM). When it is based on CORM, scale dependency can be removed but inherent variability cannot be preserved. On the other hand, when PCA is based on COVM, inherent variability can be preserved but scale dependency cannot be removed. As a solution to this issue, this paper suggests scaling each indicator by its mean, resulting in new mean equal to 1 and standard deviation equal to the coefficient of variance (CV). This leads to PCs, which are scale independent while retaining the observed variability. The computation of PCs and factors under the suggested method is derived in the study. The procedure is illustrated using the lowest level administrative division census data of Western province of Sri Lanka.
    VL  - 6
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Author Information
  • Department of Census and Statistics, Battaramulla, Sri Lanka

  • Postgraduate Institute of Agriculture, University of Peradeniya, Peradeniya, Sri Lanka

  • Postgraduate Institute of Agriculture, University of Peradeniya, Peradeniya, Sri Lanka

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