GH Biplot in Reduced-Rank Regression based on Partial Least Squares
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.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.
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
