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Connecting the multivariate partial least squares with canonical analysis: a path-following approach. (English) Zbl 1474.62203

Summary: 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.

MSC:

62H20 Measures of association (correlation, canonical correlation, etc.)
46N10 Applications of functional analysis in optimization, convex analysis, mathematical programming, economics
62P20 Applications of statistics to economics

Software:

PMA; fda (R); MATCONT; Chebfun
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References:

[1] Batles, Z.; Trefethen, L., An extension of Matlab to continuous functions and operators, SIAM J Sci Comput, 25, 5, 1743-1770 (2004) · Zbl 1057.65003
[2] Buja A (1985) Theory of bivariate ACE, Technical report no. 74. Department of Statistics, University of Washington
[3] Bunse-Gerstner, A.; Byers, R.; Mehrmann, V.; Nichols, NK, Numerical computation of an analytic singular value decomposition of a matrix valued function, Numer Math, 60, 1-39 (1991) · Zbl 0743.65035
[4] Czech Statistical Office (2017) CSO database (downloaded 2017, October 25). Czech Statistical Office, Prague, Czech Republic. Retrieved from: http://www.czso.cz/csu/czso/statistics
[5] Dhooge, A.; Govaerts, W.; Kuznetsov, YuA, Matcont: a Matlab package for numerical bifurcation analysis of ODEs, ACM Trans Math Softw, 29, 2, 141-164 (2003) · Zbl 1070.65574
[6] Dieci, L.; Pugliese, A., Singular values of two-parameter matrices: an algorithm to accurately find their intersection, Math Comput Simul, 79, 4, 1255-1269 (2008) · Zbl 1162.65328
[7] European Statistical Office (2017) Eurostat database (downloaded 2017, November 3). European Statistical Office, Luxembourg, Luxembourg. Retrieved from: http://ec.europa.eu/eurostat/data/database
[8] Gifi, A., Nonlinear multivariate analysis (1990), New York: Wiley, New York · Zbl 0697.62048
[9] Golub, GH; van Loan, CF, Matrix computations (1996), Baltimore: The Johns Hopkins University Press, Baltimore
[10] Hoerl, AE; Kennard, RW, Ridge regression: applications to non-orthogonal problems, Technometrics, 12, 69-82 (1970) · Zbl 0202.17206
[11] Janovská, D.; Janovský, V., The analytic SVD: on the non-generic points on the path, Electron Trans Numer Anal, 37, 70-86 (2010) · Zbl 1205.65154
[12] Janovská, D.; Janovský, V.; Tanabe, K., An algorithm for computing the analytic singular value decomposition, World Acad Sci Eng Technol, 2, 11, 115-120 (2008)
[13] Kato, T., Perturbation theory for linear operators (1976), New York: Springer, New York
[14] Krantz, S.; Parks, H., A primer of real analytic functions (2002), New York: Birkhauser, New York · Zbl 1015.26030
[15] Lax, PD, Linear algebra and its applications (2013), New York: Wiley, New York
[16] Leissa, W., On a curve veering aberration, J Appl Math Phys, 25, 99-111 (1974) · Zbl 0293.65083
[17] Malec L (2014) Studying economics and tourism industry relations by smooth partial least squares method depending on parameter. 17th applications of mathematics and statistics in economics. Poland, Jerzmanowice, pp 173-179
[18] Malec, L., Some remarks on the functional relation between canonical correlation analysis and partial least squares, J Stat Comput Simul, 86, 12, 2379-2391 (2016) · Zbl 07184738
[19] Malec, L.; Abrhám, J., Determinants of tourism industry in selected European countries: a smooth partial least squares approach, Econ Res, 29, 1, 66-84 (2016)
[20] Mehrmann, V.; Rath, W., Numerical methods for the computation of analytic singular value decompositions, Electron Trans Numer Anal, 1, 72-88 (1993) · Zbl 0809.65034
[21] Nakatsukasa Y, Noferini V, Trefethen N (2016) Computing the analytic SVD (downloaded 2018, March 23). University of Oxford and Chebfun developers. Retrieved from http://www.chebfun.org/examples/linalg/AnalyticSVD.html
[22] Oelker, M-R; Tutz, G., A uniform framework for the combination of penalties in generalized structured models, Adv Data Anal Classif, 11, 1, 97-120 (2017) · Zbl 1414.62321
[23] Parkhomenko, E.; Tritchler, D.; Beyene, J., Sparse canonical correlation analysis with application to genomic data integration, Stat Appl Genet Mol Biol, 8, 1-34 (2009) · Zbl 1276.92071
[24] Ramsay, JO; Silverman, BW, Applied functional data analysis: methods and case studies (2002), New York: Springer, New York
[25] Seber, GAF, Multivariate observations (2004), New York: Wiley, New York
[26] SenGupta, A., Generalized correlations in the singular case, J Stat Plan Inference, 28, 241-245 (1991) · Zbl 0724.62062
[27] Takane, Y.; Hwang, H.; Abdi, H., Regularized multiple-set canonical correlation analysis, Psychometrika, 73, 753-775 (2008) · Zbl 1284.62750
[28] Tenenhaus, A.; Tenenhaus, M., Regularized generalized canonical correlation analysis, Psychometrika, 76, 257-284 (2011) · Zbl 1284.62753
[29] Trefethen, LN, Approximation theory and approximation practice (2013), Philadelphia: SIAM, Philadelphia
[30] Vinod, HD, Canonical ridge and econometrics of joint production, J Econom, 4, 2, 147-166 (1976) · Zbl 0331.62079
[31] Wegelin JA (2000) A survey of partial least squares (PLS) methods, with emphasis on two-block case, Technical report no. 371. Department of Statistics, University of Washington
[32] Witten, DM; Tibshirani, R.; Hastie, T., A penalized matrix decomposition, with applications to sparse principal components and canonical correlation analysis, Biostatistics, 10, 3, 515-534 (2009) · Zbl 1437.62658
[33] Wright, K., Differential equations for the analytic singular value decomposition of a matrix, Numer Math, 63, 283-295 (1992) · Zbl 0756.65060
[34] Young LJ, Hwang MC (2001) Curve veering phenomena one-dimensional eigenvalue problems. In: Proceedings of the 18th national conference of the Chinese Society of Mechanical Engineers. Taiwan, Taipei, pp 239-246
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