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Partial least-squares (PLS) regression

`[`

also returns:`XL`

,`YL`

,`XS`

,`YS`

,`BETA`

,`PCTVAR`

,`MSE`

,`stats`

] = plsregress(`X`

,`Y`

,`ncomp`

)

The predictor scores

`XS`

. Predictor scores are PLS components that are linear combinations of the variables in`X`

.The response scores

`YS`

. Response scores are linear combinations of the responses with which the PLS components`XS`

have maximum covariance.A matrix

`BETA`

of coefficient estimates for PLS regression.`plsregress`

adds a column of ones in the matrix`X`

to compute coefficient estimates for a model with constant terms (intercept).The percentage of variance

`PCTVAR`

explained by the regression model.The estimated mean squared errors

`MSE`

for PLS models with`ncomp`

components.A structure

`stats`

that contains the PLS weights,*T*^{2}statistic, and predictor and response residuals.

`[`

specifies options using one or more name-value arguments in addition to any of the
input argument combinations in previous syntaxes. The name-value arguments specify
`XL`

,`YL`

,`XS`

,`YS`

,`BETA`

,`PCTVAR`

,`MSE`

,`stats`

] = plsregress(___,`Name,Value`

)`MSE`

calculation parameters. For example,
`'cv',5`

calculates the `MSE`

using 5-fold
cross-validation.

`plsregress`

uses the SIMPLS algorithm [1].
The function first centers `X`

and `Y`

by
subtracting the column means to get the centered predictor and response variables
`X0`

and `Y0`

, respectively. However, the function
does not rescale the columns. To perform PLS regression with standardized variables, use
`zscore`

to normalize `X`

and `Y`

(columns of `X0`

and `Y0`

are centered to have mean 0 and scaled to have standard deviation 1).

After centering `X`

and `Y`

,
`plsregress`

computes the singular value decomposition (SVD) on
`X0'*Y0`

. The predictor and response loadings
`XL`

and `YL`

are the coefficients obtained
from regressing `X0`

and `Y0`

on the predictor score
`XS`

. You can reconstruct the centered data `X0`

and `Y0`

using `XS*XL'`

and `XS*YL'`

,
respectively.

`plsregress`

initially computes `YS`

as
`YS = Y0*YL`

. By convention [1],
however, `plsregress`

then orthogonalizes each column of
`YS`

with respect to preceding columns of
`XS`

, so that `XS'*YS`

is a lower triangular
matrix.

[1] de Jong, Sijmen. “SIMPLS: An Alternative Approach to
Partial Least Squares Regression.” *Chemometrics and Intelligent
Laboratory Systems* 18, no. 3 (March 1993): 251–63. https://doi.org/10.1016/0169-7439(93)85002-X.

[2] Rosipal, Roman, and Nicole 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, vol. 3940, pp. 34–51. https://doi.org/10.1007/11752790_2.

[3] Chong, Il-Gyo, and Chi-Hyuck
Jun. “Performance of Some Variable Selection Methods When Multicollinearity Is Present.”
*Chemometrics and Intelligent Laboratory Systems*
78, no. 1–2 (July 2005) 103–12. https://doi.org/10.1016/j.chemolab.2004.12.011.

`pca`

| `regress`

| `sequentialfs`