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Skewness-based projection pursuit: A computational approach

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  • Loperfido, Nicola

Abstract

Projection pursuit is a multivariate statistical technique aimed at finding interesting low-dimensional data projections by maximizing a measure of interestingness commonly known as projection index. Widespread use of projection pursuit has been hampered by the computational difficulties inherent to the maximization of the projection index. The problem is addressed within the framework of skewness-based projection pursuit, focused on data projections with highest third standardized cumulants. First, it is motivated the use of the right dominant singular vector of the third multivariate, standardized moment to start the maximization procedure. Second, it is proposed an iterative algorithm for skewness maximization which relies on the analytically tractable maximization of a third-order polynomial in two variables. Both visual inspection and formal testing based on simulated data clearly suggest that the asymptotic distribution of the maximal skewness achievable by a linear projection of normal data might be skew-normal. The potential of skewness-based projection pursuit for uncovering data structures is illustrated with Olympic decathlon data.

Suggested Citation

  • Loperfido, Nicola, 2018. "Skewness-based projection pursuit: A computational approach," Computational Statistics & Data Analysis, Elsevier, vol. 120(C), pages 42-57.
  • Handle: RePEc:eee:csdana:v:120:y:2018:i:c:p:42-57
    DOI: 10.1016/j.csda.2017.11.001
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    Cited by:

    1. Chunming Zhang & Jimin Ye & Xiaomei Wang, 2023. "A Computational Perspective on Projection Pursuit in High Dimensions: Feasible or Infeasible Feature Extraction," International Statistical Review, International Statistical Institute, vol. 91(1), pages 140-161, April.
    2. Abdi, Me’raj & Madadi, Mohsen & Balakrishnan, Narayanaswamy & Jamalizadeh, Ahad, 2021. "Family of mean-mixtures of multivariate normal distributions: Properties, inference and assessment of multivariate skewness," Journal of Multivariate Analysis, Elsevier, vol. 181(C).
    3. Loperfido, Nicola, 2021. "Some theoretical properties of two kurtosis matrices, with application to invariant coordinate selection," Journal of Multivariate Analysis, Elsevier, vol. 186(C).
    4. Nicola Loperfido & Tomer Shushi, 2023. "Optimal Portfolio Projections for Skew-Elliptically Distributed Portfolio Returns," Journal of Optimization Theory and Applications, Springer, vol. 199(1), pages 143-166, October.
    5. Jorge M. Arevalillo & Hilario Navarro, 2020. "Data projections by skewness maximization under scale mixtures of skew-normal vectors," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 14(2), pages 435-461, June.
    6. Chowdhury, Joydeep & Dutta, Subhajit & Arellano-Valle, Reinaldo B. & Genton, Marc G., 2022. "Sub-dimensional Mardia measures of multivariate skewness and kurtosis," Journal of Multivariate Analysis, Elsevier, vol. 192(C).
    7. Shushi, Tomer, 2018. "A proof for the existence of multivariate singular generalized skew-elliptical density functions," Statistics & Probability Letters, Elsevier, vol. 141(C), pages 50-55.
    8. Lee, Sharon X. & McLachlan, Geoffrey J., 2022. "An overview of skew distributions in model-based clustering," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    9. Nicola Loperfido, 2019. "Finite mixtures, projection pursuit and tensor rank: a triangulation," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 13(1), pages 145-173, March.
    10. Jorge M. Arevalillo & Hilario Navarro, 2021. "Skewness-Kurtosis Model-Based Projection Pursuit with Application to Summarizing Gene Expression Data," Mathematics, MDPI, vol. 9(9), pages 1-18, April.
    11. Dyckerhoff, Rainer & Mozharovskyi, Pavlo & Nagy, Stanislav, 2021. "Approximate computation of projection depths," Computational Statistics & Data Analysis, Elsevier, vol. 157(C).
    12. Daniel A. Griffith, 2022. "Selected Payback Statistical Contributions to Matrix/Linear Algebra: Some Counterflowing Conceptualizations," Stats, MDPI, vol. 5(4), pages 1-16, November.
    13. Ursula Laa & Dianne Cook, 2020. "Using tours to visually investigate properties of new projection pursuit indexes with application to problems in physics," Computational Statistics, Springer, vol. 35(3), pages 1171-1205, September.

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