plsregress
Partial least-squares regression
Syntax
[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,...)
Description
[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(..., specifies
optional parameter name/value pairs from the following table to control
the calculation of param1,val1,param2,val2,...)MSE.
| Parameter | Value |
|---|---|
'cv' | The method used to compute
The default is |
'mcreps' | A positive integer indicating the number of Monte-Carlo
repetitions for cross-validation. The default value is |
options | A structure that specifies whether to run in parallel, and specifies the random stream
or streams. Create the
To compute in parallel, you need Parallel Computing Toolbox™ |
[XL,YL,XS,YS,BETA,PCTVAR,MSE,stats] = plsregress(X,Y,ncomp,...) returns a
structure stats with the following fields:
W— A p-by-ncompmatrix of PLS weights so thatXS = X0*W.T2— The T2 statistic for each point inXS.Xresiduals— The predictor residuals, that is,X0-XS*XL'.Yresiduals— The response residuals, that is,Y0-XS*YL'.
Examples
Perform Partial Least-Squares Regression
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.
plot(1:10,cumsum(100*PCTVAR(2,:)),'-bo'); 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; stem(residuals) xlabel('Observation'); ylabel('Residual');
References
[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.