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A generic all-purpose transformation for multivariate modeling through copulas

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Abstract

Copulas have been used in various applications in biomedical sciences and finance. We suggest copulas as the generic all-purpose transformations which can enable one to apply various standard multivariate procedures more efficiently and with better statistical properties and results. More specifically, we consider the problem of transformation of any continuous data to multivariate normality using copulas as a device for defining the transformation. Such a transformation effectively enables us to model a variety of problems involving non-normal data using the classical multivariate statistical techniques. We evaluate and illustrate various applications including those in regression, multicollinearity, principal component analysis, factor analysis, partial least square modeling and structural equation modeling where analyses using the appropriate copula transformations result in substantial improvement in implementation, interpretation, prediction as well as in the corresponding models. A great many datasets available in the literature are analyzed which amply demonstrate the power of such an approach.

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Notes

  1. The variance inflation factor for the ith variable is defined as \((VIF)_i\) = \([\varvec{R}^{-1}_{xx}]_{ii}\) = \(\frac{1}{1-R^2_i},\) where \(\varvec{R}_{xx}\) is sample correlation matrix of the predictor variables and \(R^2_i\) is the \(R^2\) value when \(x_i\) is regressed on \(x_1, x_2, \dots , x_{i-1}, x_{i+1}, \dots ,x_k.\) See [20] for details.

  2. Let \(\lambda _1< \lambda _2< \dots < \lambda _{p}\) be ordered eigenvalues of \(\varvec{X'X},\) where \(\varvec{X}\) is the \(n \times p, p = k+1,\) data matrix on independent variables. Then the ith antieigenvalue of \(\varvec{X'X}\) (Khattree, 2016, [14]) is defined as \(\eta _i^* = 2 \frac{\sqrt{\lambda _i \ \lambda _{p-i+1}}}{\lambda _i + \lambda _{p-i+1}},\)\(i=1,2, \dots , r=[p/2]\) (the integer part of p / 2). Clearly for the fact that geometric mean is smaller than the arithmetic mean, we have \( 0< \eta _i^* < 1.\)

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Acknowledgements

The authors wish to thank the associate editor and two referees for their critical comments leading to the improvement in this work.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Oakland University, Rochester, MI, 48309, USA

    Manoj Bahuguna

  2. Department of Mathematics and Statistics, and Center for Data Science and Big Data Analytics, Oakland University, Rochester, MI, 48309, USA

    Ravindra Khattree

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  1. Manoj Bahuguna
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  2. Ravindra Khattree
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Correspondence to Ravindra Khattree.

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Appendix

Appendix

The data for factor analysis example were simulated using the distributions listed in Table 14.

Table 14 Simulated data set: distribution of various variables
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Bahuguna, M., Khattree, R. A generic all-purpose transformation for multivariate modeling through copulas. Int J Data Sci Anal 10, 1–23 (2020). https://doi.org/10.1007/s41060-019-00198-w

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