Regularized Generalized Canonical Correlation Analysis: A Framework for Sequential Multiblock Component Methods

A new framework for sequential multiblock component methods is presented. This framework relies on a new version of regularized generalized canonical correlation analysis (RGCCA) where various scheme functions and shrinkage constants are considered. Two types of between block connections are considered: blocks are either fully connected or connected to the superblock (concatenation of all blocks). The proposed iterative algorithm is monotone convergent and guarantees obtaining at convergence a stationary point of RGCCA. In some cases, the solution of RGCCA is the first eigenvalue/eigenvector of a certain matrix. For the scheme functions x, $${\vert }x{\vert }$$|x|, $$x^{2}$$x2 or $$x^{4}$$x4 and shrinkage constants 0 or 1, many multiblock component methods are recovered.

[1]  H. Hotelling.  Relations Between Two Sets of Variates , 1936 .

[2]  J. Macgregor,et al.  Analysis of multiblock and hierarchical PCA and PLS models , 1998 .

[3]  Achim Kohler,et al.  Connections between multiple co-inertia analysis and consensus principal component analysis , 2011 .

[4]  Jérôme Pagès,et al.  Multiple factor analysis (AFMULT package) , 1994 .

[5]  Robert G. D. Steel,et al.  Minimum Generalized Variance for a set of Linear Functions , 1951 .

[6]  Mohamed Hanafi,et al.  Analysis of K sets of data, with differential emphasis on agreement between and within sets , 2006, Comput. Stat. Data Anal..

[7]  P. Horst.  Generalized canonical correlations and their applications to experimental data. , 1961, Journal of clinical psychology.

[8]  A. L. V. D. Wollenberg.  Redundancy analysis an alternative for canonical correlation analysis , 1977 .

[9]  Mohamed Hanafi,et al.  PLS Path modelling: computation of latent variables with the estimation mode B , 2007, Comput. Stat..

[10]  J. Kettenring,et al.  Canonical Analysis of Several Sets of Variables , 2022 .

[11]  Robert R. Meyer,et al.  Sufficient Conditions for the Convergence of Monotonic Mathematical Programming Algorithms , 1976, J. Comput. Syst. Sci..

[12]  L. Tucker.  An inter-battery method of factor analysis , 1958 .

[13]  L. E. Wangen,et al.  A multiblock partial least squares algorithm for investigating complex chemical systems , 1989 .

[14]  R. P. McDonald,et al.  A unified treatment of the weighting problem , 1968, Psychometrika.

[15]  Marko Sarstedt,et al.  Partial least squares structural equation modeling (PLS-SEM): An emerging tool in business research , 2014 .

[16]  J. Geer.  Linear relations amongk sets of variables , 1984 .

[17]  Paul Horst,et al.  Factor analysis of data matrices , 1965 .

[18]  Yurii Nesterov,et al.  Generalized Power Method for Sparse Principal Component Analysis , 2008, J. Mach. Learn. Res..

[19]  K. Widaman.  Common Factor Analysis Versus Principal Component Analysis: Differential Bias in Representing Model Parameters? , 1993, Multivariate behavioral research.

[20]  Duane T. Wegener,et al.  Evaluating the use of exploratory factor analysis in psychological research. , 1999 .

[21]  R. P. McDonald,et al.  Path Analysis with Composite Variables. , 1996, Multivariate behavioral research.

[22]  K. Strimmer,et al.  Statistical Applications in Genetics and Molecular Biology A Shrinkage Approach to Large-Scale Covariance Matrix Estimation and Implications for Functional Genomics , 2011 .

[23]  Jörg Henseler,et al.  Consistent and asymptotically normal PLS estimators for linear structural equations , 2014 .

[24]  J. Berge,et al.  Generalized approaches to the maxbet problem and the maxdiff problem, with applications to canonical correlations , 1988 .

[25]  T. Dijkstra.  Some comments on maximum likelihood and partial least squares methods , 1983 .

[26]  S. de Jong,et al.  A framework for sequential multiblock component methods , 2003 .

[27]  Michel Tenenhaus,et al.  PLS path modeling , 2005, Comput. Stat. Data Anal..

[28]  D. Chessel,et al.  Analyses de la co-inertie de K nuages de points , 1996 .

[29]  El Mostafa Qannari,et al.  Shedding new light on Hierarchical Principal Component Analysis , 2010 .

[30]  Olivier Ledoit,et al.  A well-conditioned estimator for large-dimensional covariance matrices , 2004 .

[31]  M. Tenenhaus.  Component-based Structural Equation Modelling , 2008 .

[32]  Tormod Næs,et al.  A bridge between Tucker-1 and Carroll's generalized canonical analysis , 2006, Comput. Stat. Data Anal..

[33]  Arthur Tenenhaus,et al.  Regularized generalized canonical correlation analysis for multiblock or multigroup data analysis , 2013, Eur. J. Oper. Res..

[34]  Y. Takane,et al.  Generalized Structured Component Analysis: A Component-Based Approach to Structural Equation Modeling , 2014 .

[35]  J. McKeon,et al.  Canonical analysis : some relations between canonical correlation, factor analysis, discriminant function analysis, and scaling theory , 1965 .

[36]  Svante Wold,et al.  Hierarchical multiblock PLS and PC models for easier model interpretation and as an alternative to variable selection , 1996 .

[37]  Paul Horst,et al.  Relations amongm sets of measures , 1961 .

[38]  E. Qannari,et al.  Deflation strategies for multi-block principal component analysis revisited , 2013 .

[39]  Herman Wold,et al.  Soft modelling: The Basic Design and Some Extensions , 1982 .

[40]  A. Tenenhaus,et al.  Regularized Generalized Canonical Correlation Analysis , 2011, Eur. J. Oper. Res..