Encyclopedia of Systems Biology

2013 Edition
| Editors: Werner Dubitzky, Olaf Wolkenhauer, Kwang-Hyun Cho, Hiroki Yokota

Partial Least-Squares Regression (PLSR)

  • Daniel V. Guebel
  • Néstor V. Torres
Reference work entry
DOI: https://doi.org/10.1007/978-1-4419-9863-7_1274



Any exercise of mathematical regression aims to describe the behavior of a group of dependent variables (Y, responses block) as a function of a group of independent variables (X, predictors block). However, when there are more variables than observations or the predictors are correlated, ordinary multiple linear regressions (MLR) are not feasible. Partial-Least-Squares Regression (PLSR) provides a much more predictive linear-relationship, even in the case of a rank-deficient X matrix, and allows the simultaneous decomposition of X and Y blocks, thus facilitating a better understanding of the underlying structure (Geladi and Kowalski, 1986; Trygg, 2002; Höskuldsson 2004; Jørgensen and Goegebeur 2007; Varmuza and Filzmoser 2009).

PLSR Model

Any PLSR model is described by the following equations:
$$ {X_{{\rm n,p}}} = {\hat{X}_{{\rm n,p}}}+ {E_{{\rm n,p}}} = {T_{{\rm n,k}}}P_{{\rm k,p}}^{\rm T} + {E_{{\rm n,p}}} $$
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  1. Abdi H (2003) Partial least squares (PLS) regression. In: Lews-Beck M, Bryman A, Futing T (eds) Encyclopedia of Social Sciences Research Methods. Sage, Thousand Oaks, pp 792–795Google Scholar
  2. Brereton R (2003) Chemometrics: data analysis for the laboratory to chemistry plant. Wiley, New York, pp 298–338Google Scholar
  3. Geladi P, Kowalski B (1986) Partial least-squares regression: A tutorial. Analytica Chimica Acta 185:1–17Google Scholar
  4. Höskuldsson A (2004) PLS regression and the covariance. Homepage of Chemometrics. Editorial July. http://acc.umu.se/~tnktg/ Chemometrics/Editorial. Accessed 2 Sept 2011
  5. Jørgensen B, Goegebeur Y (2007) partial least squares regression (model PLS2). In: Multivariate data analysis and chemometrics. Departament Statistics. Syddansk University. Denmark. http://statmaster.sdu.dk/courses/STO2/module08/module.pdf. Accessed 2 Sept 2011
  6. Miyashita Y, Itozawa T, Katsumi H, Sasaki SI (1990) Comments on the NIPALS algorithm. J Chemom 4:97–100Google Scholar
  7. Trygg J (2002) Have you ever wondered why PLS sometimes needs more than one component for a single –y vector? Homepage of Chemometrics. Editorial February. Available at: http://www.chemometrics.se/editorial/feb2002.html
  8. Varmuza K, Filzmoser P (2009) Introduction to multivariate statistical analysis in chemometrics. Taylor and Francis Group/CRC Press, Boca Raton, pp 150–162 and pp 177–179Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2013

Authors and Affiliations

  1. 1.Department of Systems Biology and BioinformaticsInstitute of Computer Sciences, University of RostockRostockGermany
  2. 2.Department of Biochemistry and Molecular BiologyUniversity of La LagunaSan Cristóbal de La Laguna, Islas CanariasSpain