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Partial least-squares regression


[XL,YL] = plsregress(X,Y,ncomp)
[XL,YL,XS] = plsregress(X,Y,ncomp)
[XL,YL,XS,YS] = plsregress(X,Y,ncomp)
[XL,YL,XS,YS,BETA] = plsregress(X,Y,ncomp,...)
[XL,YL,XS,YS,BETA,PCTVAR] = plsregress(X,Y,ncomp)
[XL,YL,XS,YS,BETA,PCTVAR,MSE] = plsregress(X,Y,ncomp)
[XL,YL,XS,YS,BETA,PCTVAR,MSE] = plsregress(...,param1,val1,param2,val2,...)
[XL,YL,XS,YS,BETA,PCTVAR,MSE,stats] = plsregress(X,Y,ncomp,...)


[XL,YL] = plsregress(X,Y,ncomp) computes a partial least-squares (PLS) regression of Y on X, using ncomp PLS components, and returns the predictor and response loadings in XL and YL, respectively. X is an n-by-p matrix of predictor variables, with rows corresponding to observations and columns to variables. Y is an n-by-m response matrix. XL is a p-by-ncomp matrix of predictor loadings, where each row contains coefficients that define a linear combination of PLS components that approximate the original predictor variables. YL is an m-by-ncomp matrix of response loadings, where each row contains coefficients that define a linear combination of PLS components that approximate the original response variables.

[XL,YL,XS] = plsregress(X,Y,ncomp) returns the predictor scores XS, that is, the PLS components that are linear combinations of the variables in X. XS is an n-by-ncomp orthonormal matrix with rows corresponding to observations and columns to components.

[XL,YL,XS,YS] = plsregress(X,Y,ncomp) returns the response scores YS, that is, the linear combinations of the responses with which the PLS components XS have maximum covariance. YS is an n-by-ncomp matrix with rows corresponding to observations and columns to components. YS is neither orthogonal nor normalized.

plsregress uses the SIMPLS algorithm, first centering X and Y by subtracting off column means to get centered variables X0 and Y0. However, it does not rescale the columns. To perform PLS with standardized variables, use zscore to normalize X and Y.

If ncomp is omitted, its default value is min(size(X,1)-1,size(X,2)).

The relationships between the scores, loadings, and centered variables X0 and Y0 are:

XL = (XS\X0)' = X0'*XS,

YL = (XS\Y0)' = Y0'*XS,

XL and YL are the coefficients from regressing X0 and Y0 on XS, and XS*XL' and XS*YL' are the PLS approximations to X0 and Y0.

plsregress initially computes YS as:

YS = Y0*YL = Y0*Y0'*XS,

By convention, however, plsregress then orthogonalizes each column of YS with respect to preceding columns of XS, so that XS'*YS is lower triangular.

[XL,YL,XS,YS,BETA] = plsregress(X,Y,ncomp,...) returns the PLS regression coefficients BETA. BETA is a (p+1)-by-m matrix, containing intercept terms in the first row:

Y = [ones(n,1),X]*BETA + Yresiduals,

Y0 = X0*BETA(2:end,:) + Yresiduals. Here Yresiduals is the vector of response residuals.

[XL,YL,XS,YS,BETA,PCTVAR] = plsregress(X,Y,ncomp) returns a 2-by-ncomp matrix PCTVAR containing the percentage of variance explained by the model. The first row of PCTVAR contains the percentage of variance explained in X by each PLS component, and the second row contains the percentage of variance explained in Y.

[XL,YL,XS,YS,BETA,PCTVAR,MSE] = plsregress(X,Y,ncomp) returns a 2-by-(ncomp+1) matrix MSE containing estimated mean-squared errors for PLS models with 0:ncomp components. The first row of MSE contains mean-squared errors for the predictor variables in X, and the second row contains mean-squared errors for the response variable(s) in Y.

[XL,YL,XS,YS,BETA,PCTVAR,MSE] = plsregress(...,param1,val1,param2,val2,...) specifies optional parameter name/value pairs from the following table to control the calculation of MSE.


The method used to compute MSE.

  • When the value is a positive integer k, plsregress uses k-fold cross-validation.

  • When the value is an object of the cvpartition class, other forms of cross-validation can be specified.

  • When the value is 'resubstitution', plsregress uses X and Y both to fit the model and to estimate the mean-squared errors, without cross-validation.

The default is 'resubstitution'.


A positive integer indicating the number of Monte-Carlo repetitions for cross-validation. The default value is 1. The value must be 1 if the value of 'cv' is 'resubstitution'.


A structure that specifies whether to run in parallel, and specifies the random stream or streams. Create the options structure with statset. Option fields:

  • UseParallel — Set to true to compute in parallel. Default is false.

  • UseSubstreams — Set to true to compute in parallel in a reproducible fashion. Default is false. To compute reproducibly, set Streams to a type allowing substreams: 'mlfg6331_64' or 'mrg32k3a'.

  • Streams — A RandStream object or cell array consisting of one such object. If you do not specify Streams, plsregress uses the default stream.

[XL,YL,XS,YS,BETA,PCTVAR,MSE,stats] = plsregress(X,Y,ncomp,...) returns a structure stats with the following fields:

  • W — A p-by-ncomp matrix of PLS weights so that XS = X0*W.

  • T2 — The T2 statistic for each point in XS.

  • Xresiduals — The predictor residuals, that is, X0-XS*XL'.

  • Yresiduals — The response residuals, that is, Y0-XS*YL'.


collapse all

Load data on near infrared (NIR) spectral intensities of 60 samples of gasoline at 401 wavelengths, and their octane ratings.

load spectra
X = NIR;
y = octane;

Perform PLS regression with ten components.

[XL,yl,XS,YS,beta,PCTVAR] = plsregress(X,y,10);

Plot the percent of variance explained in the response variable as a function of the number of components.

xlabel('Number of PLS components');
ylabel('Percent Variance Explained in y');

Compute the fitted response and display the residuals.

yfit = [ones(size(X,1),1) X]*beta;
residuals = y - yfit;


[1] de Jong, S. “SIMPLS: An Alternative Approach to Partial Least Squares Regression.” Chemometrics and Intelligent Laboratory Systems. Vol. 18, 1993, pp. 251–263.

[2] Rosipal, R., and N. Kramer. “Overview and Recent Advances in Partial Least Squares.” Subspace, Latent Structure and Feature Selection: Statistical and Optimization Perspectives Workshop (SLSFS 2005), Revised Selected Papers (Lecture Notes in Computer Science 3940). Berlin, Germany: Springer-Verlag, 2006, pp. 34–51.

Introduced in R2008a

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