**Page description appears here**

“The partial least squares algorithm: a truncated Cayley-Hamilton series approximation used to solve the regression problem”

Authors: David Di Ruscio,
Affiliation: Telemark University College
Reference: 1998, Vol 19, No 3, pp. 117-140.

     Valid XHTML 1.0 Strict

Keywords: Least squares, Cayley-Hamilton, PLS

Abstract: In this paper it is shown that the PLS algorithm for univariate data is equivalent to using a truncated Cayley-Hamilton polynomial expression of degree 1 less than a less than r for the matrix inverse inv(X“X) in R^(rxr) used to compute the LS solution. Furthermore, the a coefficients in this polynomial are computed as the LS optimal solution (minimizing parameters) to the prediction error. The resulting solution is non-iterative. The solution can be expressed in terms of one matrix inverse and is given by BPLS = Ka * inv(Ka“*X“*X*Ka)*Ka“*X“*Y where Ka in R^(rxr) is the controllability (Krylov) matrix for the pair (X“X,X“Y).

PDF PDF (3422 Kb)        DOI: 10.4173/mic.1998.3.1

DOI forward links to this article:
  [1] Maryam Ghadrdan, Chriss Grimholt and Sigurd Skogestad (2013), doi:10.1021/ie400542n

[1] BURNHAM, A.J., VIVEROS, IC MACGREGOR, J.F. (1996). Frameworks for Latent Variable Multivariate Regression, Journal of Chemometrics, Vol. 10, pp. 31-45.
[2] DE MOOR, B. DAVID, J. (1996). Total least squares and the algebraic Riccati equation, Katholieke Universiteit Leuven, B-3001 Leuven, Belgium. Internal report.
[3] DI RUSCIO, D. (1997). On subspace identification of the extended observability matrix, In: Proceedings of the 1997 IEEE Conference on Decision and Control, San Diego, California, December 10-12.
[4] FIERRO, R.D., GOLUB, G.H., HANSEN, P.C. O“LEARY, D.P. (1997). Regularization by truncated total least squares, Siam Journal on Scientific Computing, Vol. 18, No. 4, pp. 1223-1241 doi:10.1137/S1064827594263837
[5] FRANK, L.E. FRIEDMAN, J.H. (1993). A Statistical View of Some Chemometrics Regression Tool, Technometrics, Vol. 35, No. 2, pp. 109-135 doi:10.2307/1269656
[6] HANSEN, P.C. (1992). Regularization Tools, A Matlab Package for Analysis and Solution of Discrete Ill-Posed Problems. Danish Computing Centre for Research and Education, DK-2800 Lyngby, Denmark.
[7] HELLAND, I.S. (1988). On the Structure of Partial Least Squares Regression, Commun. in Stat. Simulation and Computation, Vol. 17, No. 2, pp. 581-607 doi:10.1080/03610918808812681
[8] HÖSKULDSSON, A. (1996). Prediction Methods in Science and Technology, COLOURSCAN Warsaw, ISBN 87-985941-0-9.
[9] HÖSKULDSSON, A. (1988). PLS regression methods, Journal of Chemometrics, Vol. 2, pp. 211-228 doi:10.1002/cem.1180020306
[10] JOHANSEN, T.A. (1997). On Tikhonov Regularization, Bias and Variance, In Nonlinear System Identification. Automatica, Vol. 33, No. 3, pp. 441-446 doi:10.1016/S0005-1098(96)00168-9
[11] MANNE, R. (1987). Analysis of Two Partial-Least-Squares Algorithms for Multivariate Calibration, Chemometrics and Intelligent Laboratory Systems, Vol. 2, pp. 187-197 doi:10.1016/0169-7439(87)80096-5
[12] MARTENS, H. NĘS, T. (1989). Multivariate Calibration, John Wiley and Sons Ltd.
[13] TER BRAAK, C.J. DE JONG, S. (1998). The objective function of Partial Least Squares, Journal of Chemometrics, Vol. 12, pp. 41-54.
[14] MARTENS, H. NĘS, T. (1985). Comparison of Prediction Methods for Multicolincar Data, Commun. in Stat. Simulation and Computation, Vol. 14, No. 3, pp. 544-576.
[15] WOLD, H. (1966). Non-linear estimation by iterative least squares procedures, Research papers in statistics, Ed. F. David. Wiley, New York, pp. 411-444.
[16] WOLD, H. (1975). Soft modeling by latent variables: the Non-linear Iterative Partial Least Squares Approach, In: Perspectives in Probability and Statistics, Editor J. Gani. London, Academic Press.
[17] WOLD, H. (1985). Partial Least Squares, In: Encyclopedia of statistics sciences, Editors S. Kotz and N.L. Johnson. Wiley, Vol. 6, pp. 581-591.
[18] LORBER, A., LAWRENCE, E.W. KOWALSKI B.R. (1987). A Theoretical Foundation for the PLS Algorithm, Journal of Chemometrics, Vol. 1, pp. 19-31 doi:10.1002/cem.1180010105
[19] VETTER, W.J. (1973). Matrix calculus operations and Taylor expansions, Siam review, Vol. 15, No. 2, pp. 352-369 doi:10.1137/1015034

  title={{The partial least squares algorithm: a truncated Cayley-Hamilton series approximation used to solve the regression problem}},
  author={Di Ruscio, David},
  journal={Modeling, Identification and Control},
  publisher={Norwegian Society of Automatic Control}


May 2016: MIC reaches 2000 DOI Forward Links. The first 1000 took 34 years, the next 1000 took 2.5 years.

July 2015: MIC's new impact factor is now 0.778. The number of papers published in 2014 was 21 compared to 15 in 2013, which partially explains the small decrease in impact factor.

Aug 2014: For the 3rd year in a row MIC's impact factor increases. It is now 0.826.

Dec 2013: New database-driven web-design enabling extended statistics. Article number 500 is published and MIC reaches 1000 DOI Forward Links.

Jan 2012: Follow MIC on your smartphone by using the RSS feed.


July 2011: MIC passes 1000 ISI Web of Science citations.

Mar 2010: MIC is now indexed by DOAJ and has received the Sparc Seal seal for open access journals.

Dec 2009: A MIC group is created at LinkedIn and Twitter.

Oct 2009: MIC is now fully updated in ISI Web of Knowledge.