GH Biplot in Reduced-Rank Regression based on Partial Least Squares

  • Willín Gabriel Álvarez Universidad Regional Amazónica Ikiam
  • Victor John Griffin Universidad de Carabobo
Keywords: GH biplot, reduced-rank regression, partial least squares, singular value decomposition.

Abstract

One of the challenges facing statisticians is to provide tools to enable researchers to interpret and present their data and conclusions in ways easily understood by the scientific community. One of the tools available for this purpose is a multivariate graphical representation called reduced rank regression biplot. This biplot describes how to construct a graphical representation in nonsymmetric contexts such as approximations by least squares in multivariate linear regression models of reduced rank. However multicollinearity invalidates the interpretation of a regression coefficient as the conditional effect of a regressor, given the values of the other regressors, and hence makes biplots of regression coefficients useless. So it was, in the search to overcome this problem, Alvarez and Griffin  presented a procedure for coefficient estimation in a multivariate regression model of reduced rank in the presence of multicollinearity based on PLS (Partial Least Squares) and generalized singular value decomposition. Based on these same procedures, a biplot construction is now presented for a multivariate regression model of reduced rank in the presence of multicollinearity. This procedure, called PLSSVD GH biplot, provides a useful data analysis tool which allows the visual appraisal of the structure of the dependent and independent variables. This paper defines the procedure and shows several of its properties. It also provides an implementation of the routines in R and presents a real life application involving data from the FAO food database to illustrate the procedure and its properties.

Author Biography

Victor John Griffin, Universidad de Carabobo
Mathematics DepartmentFaculty of Science and Technology

References

Álvarez W. and Griffin V. (2016).Estimation procedure for reduced rank regression, PLSSVD. Stat., Optim. Inf. Comput., Vol. 4, pp 107-117.

Anderson T., (1999). Asymptotic Distibution of the Reduced Rank Regression estimator under general conditions. The Annals of Statistcs, 27, 1141-1154.

Anderson T., (1951). Estimating Linear Restrictions on Regression Coefficients for Multivariate Normal Distributions. Ann. Math. Statist, 22, 327-351

Cárdenas, O. and Galindo, P. (2004). Biplots con información externa basados en modelos bilineales generalizados. Universidad Central de Venezuela-Consejo de Desarrollo Cientíco y Humanístico.

Cárdenas, O., Galindo, P., Vicente-Villardon J.L. (2007).Los métodos Biplot: Evolucón y aplicaciones. Revista Venezolana de Análisis de Coyuntura. Vol. XIII, num. 1, enero-julio, pp. 279 - 303. Caracas, Venezuela.

Davies, P. and Tso M., (1982). Procedures for reduced rank regression. Applied Statistic 31,244-255

Eckart, C. and Young, G, (1936). The Approximation de One Matrix for Another of Lower Rank. Psychometrika 1, 211-218.

Gabriel, K., (1971). The Biplot-graphic display of matrices with applications to principal component analysis. Biometrika, 58, 453-467.

Gabriel (1981), Biplot display of multivariate matrices for inspection of data and diagnosis, Barnet V. (ed.), Interpreting Multivariate Data, 147-173, Wiley, London.

Gabriel, K. R., Odoroff, C. L. (1990), Biplots in biomedical research, Statistics in Medicine 9 (5): 469-485.

Höskuldsson, A., (1988). PLS regression methods. Journal of Chemometrics 2, 211- 228.

Izenman, A. J., (1975). Reduced rank regression for the multivariate linear model. Journal of Multivaiate Analysis 5, 248-264

Lauro, N. C. (2004) .Non Symmetrical Correspondence Analysis and Related Methods. Department of Mathematics and Statistics. University of Naples Federico II. Italy

Montgomery D., Peck E. and Vining G. (2001). Introducción al análisis de regresión. John Wiley and Sons

Phatak A. and De Jong S. (1997): The geometry of partial least squares. Journal of Chemometrics, Vol. 11:311-338

Strauss, J., Gabriel, K. R. (1979), Do psychiatric patients fit their diagnosis, pattern of symtomatology as described with the biplot, Journal of Nervous and Mental Disease. 167: 105-113.

Tenenhaus, M. (1998). La régression pls, théorie et pratique. Éditions Technip. Paris.

Ter Braak, C. and Looman, C. (1994). Biplots in reduced rank regression. Biometrical Journal 8, 983-1003.

Ter Braak, C. (1990). Interpreting canonical correlation analysis through biplots of structural correlations and weights. Psychometrika, 55, 519-531

Tsianco, M., Gabriel, K. R. (1984), Modeling temperature data: An ilustration of the use of biplot and bimodels in non-linear modeling, Journal of Climate and Applied Meteorology 23: 787-799.

Valencia, J., Díaz-LLanos F. and Calleja S. (2004). Métodos de predicción en situaciones límite. Editorial La Muralla, S.A.

Published
2021-07-10
How to Cite
Alvarez, W., & Griffin, V. J. (2021). GH Biplot in Reduced-Rank Regression based on Partial Least Squares. Statistics, Optimization & Information Computing, 9(3), 717-734. https://doi.org/10.19139/soic-2310-5070-1112
Section
Research Articles