Connecting the multivariate partial least squares with canonical analysis: a path-following approach
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Connecting the multivariate partial least squares with canonical analysis: a path-following approach Lukáš Malec1
· Vladimír Janovský2
Received: 4 December 2018 / Revised: 3 June 2019 / Accepted: 5 August 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract Despite the fact that the regularisation of multivariate methods is a well-known and widely used statistical procedure, very few studies have considered it from the perspective of analytic matrix decomposition. Here, we introduce a link between one variant of partial least squares (PLS) and canonical correlation analysis (CCA) for multiple groups, as well as two groups covered as a special case. A continuation algorithm based on the implicit function theorem is selected, with particular attention paid to potential non-generic points based on real economic data inputs. Both degenerated crossings and multiple eigenvalues are identified on the paths. The theory of Chebyshev polynomials is applied in order to generate novel insights into the phenomenon simply generalisable to a variety of other techniques. Keywords Canonical correlation analysis · Partial least squares · Multi-group case · Analytic singular value decomposition · Analytic eigenvalue decomposition · Multiplicity · Path-following · Economics Mathematics Subject Classification 62H20 · 46N10 · 62P20
1 Introduction Canonical analysis and PLS are essential methods classified under the denominator of multivariate statistics. We focus on a parameterised task described in terms of boundary points corresponding to the PLS and CCA in multi- as well as two-group cases solved by applying a specific continuation approach based on the implicit function
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Lukáš Malec [email protected]
1
Department of Information Technologies and Analytical Methods, University College of Business, Prague, Czech Republic
2
Department of Numerical Analysis, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
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L. Malec, V. Janovský
theorem. The significant advantage of such an algorithm is that it computes a separate factor solution with adaptive step length. From one perspective, the PLS variant can be seen as a fully regularised CCA, in which the optimisation task for maximising the correlation of linear combinations of original variables (latent variates) is substituted by a covariance measure. The starting point of the path-following procedure, PLS-SVD, for the standard two-group variant (Wegelin 2000) is sometimes termed intercorrelations analysis or canonical covariance. In multiple group cases, the realisation of CCA can be solved in many ways (Takane et al. 2008; Tenenhaus and Tenenhaus 2011; Gifi 1990), with the first item cited as the method of choice for transforming an optimisation task into a generalised problem on eigenvalues and eigenvectors of block matrices, resp. the higher dimensional singular value decomposition. This is executed in a similar manner as for other techniques, such as discriminant and factor analyses. However, the two-group case of
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