dkplsr {rchemo}

Direct KPLSR Models

Description

Direct kernel PLSR (DKPLSR) (Bennett & Embrechts 2003). The method builds kernel Gram matrices and then runs a usual PLSR algorithm on them. This is faster (but not equivalent) to the "true" NIPALS KPLSR algorithm such as described in Rosipal & Trejo (2001).

Usage

dkplsr(X, Y, weights = NULL, nlv, kern = "krbf", ...)

## S3 method for class 'Dkpls'
transform(object, X, ..., nlv = NULL)  

## S3 method for class 'Dkpls'
coef(object, ..., nlv = NULL)  

## S3 method for class 'Dkplsr'
predict(object, X, ..., nlv = NULL)  

Arguments

X

For the main function: Matrix with the training X-data (n, p). — For auxiliary functions: A matrix with new X-data (m, p) to consider.

Y

Matrix with the training Y-data (n, q).

weights

vector of weights (n, 1) to apply to the training observations. Internally, weights are "normalized" to sum to 1. Default to NULL (weights are set to 1 / n).

nlv

For the main function: The number(s) of LVs to calculate. — For auxiliary functions: The number(s) of LVs to consider.

kern

Name of the function defining the considered kernel for building the Gram matrix. See krbf for syntax, and other available kernel functions (krbf, kpol, ktanh).

...

Optional arguments to pass in the kernel function defined in kern (e.g. gamma for krbf, gamma and coef0 for ktanh, gamma and coef0 and degree for kpol).

object

For auxiliary functions: A fitted model, output of a call to the main function.

Value

For dkplsr:

X

Matrix with the training X-data (n, p).

fm

List with the outputs of the PLSR ((T): the X-score matrix (n,nlv); (P): the X-loadings matrix (p,nlv); (R): The PLS projection matrix (p,nlv); (W): The X-loading weights matrix (p,nlv); (C): The Y-loading weights matrix; (TT): the X-score normalization factor; (xmeans): the centering vector of X (p,1); (ymeans): the centering vector of Y (q,1); (weights): the weights vector of X-variables (p,1); (U): intermediate output.

K

kernel Gram matrix

kern

kernel function

dots

Optional arguments passed in the kernel function

For transform.Dkplsr : A matrix (m, nlv) with the projection of the new X-data on the X-scores

For predict.Dkplsr:

pred

A list of matrices (m, q) with the Y predicted values for the new X-data

K

kernel Gram matrix (m, nlv), with values for the new X-data

For coef.Dkplsr:

int

matrix (1,nlv) with the intercepts

B

matrix (n,nlv) with the coefficients

Note

The second example concerns the fitting of the function sinc(x) described in Rosipal & Trejo 2001 p. 105-106

References

Bennett, K.P., Embrechts, M.J., 2003. An optimization perspective on kernel partial least squares regression, in: Advances in Learning Theory: Methods, Models and Applications, NATO Science Series III: Computer & Systems Sciences. IOS Press Amsterdam, pp. 227-250.

Rosipal, R., Trejo, L.J., 2001. Kernel Partial Least Squares Regression in Reproducing Kernel Hilbert Space. Journal of Machine Learning Research 2, 97-123.

Examples

## EXAMPLE 1

n <- 6 ; p <- 4
Xtrain <- matrix(rnorm(n * p), ncol = p)
ytrain <- rnorm(n)
Ytrain <- cbind(y1 = ytrain, y2 = 100 * ytrain)
m <- 3
Xtest <- Xtrain[1:m, , drop = FALSE] 
Ytest <- Ytrain[1:m, , drop = FALSE] ; ytest <- Ytest[1:m, 1]

nlv <- 2
fm <- dkplsr(Xtrain, Ytrain, nlv = nlv, kern = "krbf", gamma = .8)
transform(fm, Xtest)
transform(fm, Xtest, nlv = 1)
coef(fm)
coef(fm, nlv = 1)

predict(fm, Xtest)
predict(fm, Xtest, nlv = 0:nlv)$pred

pred <- predict(fm, Xtest)$pred
msep(pred, Ytest)

nlv <- 2
fm <- dkplsr(Xtrain, Ytrain, nlv = nlv, kern = "kpol", degree = 2, coef0 = 10)
predict(fm, Xtest, nlv = nlv)

## EXAMPLE 2

x <- seq(-10, 10, by = .2)
x[x == 0] <- 1e-5
n <- length(x)
zy <- sin(abs(x)) / abs(x)
y <- zy + rnorm(n, 0, .2)
plot(x, y, type = "p")
lines(x, zy, lty = 2)
X <- matrix(x, ncol = 1)

nlv <- 3
fm <- dkplsr(X, y, nlv = nlv)
pred <- predict(fm, X)$pred
plot(X, y, type = "p")
lines(X, zy, lty = 2)
lines(X, pred, col = "red")