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Visualisation of Mahalanobis Distances for Trivariate JOINT Distributions

Author

Listed:
  • Emily Groenewald

    (School of Economics, University of Cape Town, Cape Town, South Africa)

  • Gary Van Vuuren

    (Centre for Business Mathematics and Informatics, North-West University, Potchefstroom, 2351, South Africa)

Abstract

The Mahalanobis distance is a statistical measure used to quantify the distance between elliptic distributions with distinct locations and shared shapes, while accounting for the variables' covariance structure. It is applicable to both estimative and predictive estimation approaches, where variations are limited to location, and it assesses the similarity or dissimilarity between data and the mean (centroid) of a multivariate distribution, within the family of multivariate elliptic distributions. It is thus useful for outlier identification. The aim of the study is to provide, for the first time, a three-dimensional visualisation of the Mahalanobis distance when the underlying framework comprises three jointly connected variables (rather than the standard two variables presented in textbooks). Data with Mahalanobis distances exceeding a predefined threshold, determined using a distribution, are considered outliers. This approach is analogous to identifying outliers for univariate distributions based on critical values derived from confidence levels. While the literature mainly discusses the Mahalanobis distance formulation for bivariate distributions, we extend the discussion to include one additional variable and provide a visualisation of the resulting Mahalanobis distance for a trivariate distribution. An empirical example is presented to illustrate a practical application of a trivariate Mahalanobis distance. Visualising outliers alongside other historical events within three-factor systems can offer valuable insights into the risk profile of the current environment and assess the probability of future extreme events.

Suggested Citation

  • Emily Groenewald & Gary Van Vuuren, 2024. "Visualisation of Mahalanobis Distances for Trivariate JOINT Distributions," International Journal of Economics and Financial Issues, Econjournals, vol. 14(2), pages 203-206, March.
  • Handle: RePEc:eco:journ1:2024-02-21
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    References listed on IDEAS

    as
    1. Han, Chulwoo & Park, Frank C., 2022. "A geometric framework for covariance dynamics," Journal of Banking & Finance, Elsevier, vol. 134(C).
    2. Kay I. Penny, 1996. "Appropriate Critical Values When Testing for a Single Multivariate Outlier by Using the Mahalanobis Distance," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 45(1), pages 73-81, March.
    3. Mark Kritzman & Yuanzhen Li, 2010. "Skulls, Financial Turbulence, and Risk Management," Financial Analysts Journal, Taylor & Francis Journals, vol. 66(5), pages 30-41, September.
    4. Brenton R. Clarke & Andrew Grose, 2023. "A further study comparing forward search multivariate outlier methods including ATLA with an application to clustering," Statistical Papers, Springer, vol. 64(2), pages 395-420, April.
    Full references (including those not matched with items on IDEAS)

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    More about this item

    Keywords

    Mahalanobis Distance; Trivariate; Elliptic Distributions; Outliers;
    All these keywords.

    JEL classification:

    • C38 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Classification Methdos; Cluster Analysis; Principal Components; Factor Analysis
    • G17 - Financial Economics - - General Financial Markets - - - Financial Forecasting and Simulation
    • E37 - Macroeconomics and Monetary Economics - - Prices, Business Fluctuations, and Cycles - - - Forecasting and Simulation: Models and Applications

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