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Regression modeling via latent predictors
Last modified: 2018-04-26
Abstract
A proposal for multivariate regression modeling based on latent predictors (LPs) is presented. The idea of the proposed model is to predict the responses on LPs which, in turn, are built as linear combinations of disjoint groups of observed covariates. The formulation naturally allows to identify LPs that best predict the responses by jointly clustering the covariates and estimating the regression coefficients of the LPs. Clearly, in this way the LP interpretation is greatly simplified since LPs are exactly represented by a subset of covariates only.
The model is formalized in a maximum likelihood framework which is intuitively appealing for comparisons with other methodologies, for allowing inference on the model parameters and for choosing the number of subsets leading to LPs. An ECM algorithm is proposed for parameter estimation and experiments on simulated and real data show the performance of our proposal.
The model is formalized in a maximum likelihood framework which is intuitively appealing for comparisons with other methodologies, for allowing inference on the model parameters and for choosing the number of subsets leading to LPs. An ECM algorithm is proposed for parameter estimation and experiments on simulated and real data show the performance of our proposal.
References
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\bibitem{DRM} Yuan,   M., Ekici, A., Lu, Z., Monteiro, R.: Dimension reduction and coefficient estimation in multivariate linear regression. J. R. Stat. Soc., Ser. B, Stat. Methodol. \textbf{69}, 329--346 (2007)
\bibitem{FAR} Basilevsky, A.: Factor analysis regression. The Canadian Journal od Statistics, \textbf{9(1)}, 109--117 (1981)
\bibitem{MLRR} Bougeard, S., Hanafi, M., Qannari, E.M.: Multiblock latent root regression: application to epidemiological data. Comput. Stat. \textbf{22(2)}, 209--222 (2007)
\bibitem{CRPLS}Â Â Â Bougeard, S., Hanafi, M., Qannari, E.M.: Continuum redundancy-PLS regression: a simple continuum approach. Comput. Stat. Data Anal. \textbf{52(7)}, 3686--3696 (2008)
\bibitem{CCR} Hotelling, H.: The most predictable criterion. J. Educ. Psychol. \textbf{25}, 139--142 (1935)
\bibitem{PCR2} Izenman, A.J.:Â Reduced-rank regression for the multivariate linear model. J. Multivar. Anal. \textbf{5}, 248--262 (1975)
\bibitem{PCR} Kendall, M.G.: A Course in Multivaraite Analysis. Griffin, London (1957)
\bibitem{MRBOP} Martella, F., Vicari D., Vichi, M.: Partitioning predictors in multivariate regression models. Stat. Comput. \textbf{25(2)}, 261--272 (2013)
\bibitem{CR} Stone, M., Brooks, R.J.: Continuum regression: cross-validated squares partial least squares and principal components regression. J. R. Stat. Soc. b \textbf{52(2)}, 237--269 (1990)
\bibitem{RA}Â Van Den Wollenberg, Â Â Â A.L.: Redundancy analysis an alternative for canonical analysis. Psychometrika \textbf{42(2)}, 207--219 (1977)
\bibitem{PLSR} Wold, H.: Estimation of principal components and relates models by iterative least squares. in Krishnaiaah, P.R. (ed.) Multivariate Analysis, 391--420. Academic Press, New York (1966)
\bibitem{DRM} Yuan,   M., Ekici, A., Lu, Z., Monteiro, R.: Dimension reduction and coefficient estimation in multivariate linear regression. J. R. Stat. Soc., Ser. B, Stat. Methodol. \textbf{69}, 329--346 (2007)
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