Computes functional linear regression between functional explanatory variable \(X(t)\) and scalar response \(Y\) using penalized Partial
Least Squares (PLS) $$Y=\big<\tilde{X},\beta\big>+\epsilon=\int_{T}{\tilde{X}(t)\beta(t)dt+\epsilon}$$ where \( \big< \cdot , \cdot \big>\) denotes the inner product on
\(L_2\) and \(\epsilon\) are random errors with mean zero , finite variance \(\sigma^2\) and \(E[\tilde{X}(t)\epsilon]=0\).
\(\left\{\nu_k\right\}_{k=1}^{\infty}\) orthonormal basis of PLS to represent the functional data as \(X_i(t)=\sum_{k=1}^{\infty}\gamma_{ik}\nu_k\).
fregre.pls(fdataobj, y = NULL, l = NULL, lambda = 0, P = c(0, 0, 1), ...)Return:
call The matched call of fregre.pls function.
beta.est Beta coefficient estimated of class fdata.
coefficients A named vector of coefficients.
fitted.values Estimated scalar response.
residualsy-fitted values.
H Hat matrix.
df.residual The residual degrees of freedom.
r2 Coefficient of determination.
GCV GCV criterion.
sr2 Residual variance.
l Index of components to include in the model.
lambda Amount of shrinkage.
fdata.comp Fitted object in fdata2pls function.
lm Fitted object in lm function
fdataobj Functional explanatory data.
y Scalar response.
fdata class object.
Scalar response with length n.
Index of components to include in the model.
Amount of penalization. Default value is 0, i.e. no penalization is used.
If P is a vector: P are coefficients to define the
penalty matrix object. By default P=c(0,0,1) penalize the second
derivative (curvature) or acceleration. If P is a matrix: P is the
penalty matrix object.
Further arguments passed to or from other methods.
Manuel Febrero-Bande, Manuel Oviedo de la Fuente manuel.oviedo@udc.es
Functional (FPLS) algorithm maximizes the covariance between \(X(t)\) and the scalar response \(Y\) via the partial least squares (PLS) components.
The functional penalized PLS are calculated in fdata2pls by alternative formulation of the NIPALS algorithm proposed by Kraemer and
Sugiyama (2011).
Let \(\left\{\tilde{\nu}_k\right\}_{k=1}^{\infty}\) the functional PLS components and \(\tilde{X}_i(t)=\sum_{k=1}^{\infty}\tilde{\gamma}_{ik}\tilde{\nu}_k\) and \(\beta(t)=\sum_{k=1}^{\infty}\tilde{\beta}_k\tilde{\nu}_k\). The functional linear model is estimated by: $$\hat{y}=\big< X,\hat{\beta} \big> \approx \sum_{k=1}^{k_n}\tilde{\gamma}_{k}\tilde{\beta}_k $$
The response can be fitted by:
\(\lambda=0\), no penalization, $$\hat{y}=\nu_k^{\top}(\nu_k^{\top}\nu_k)^{-1}\nu_k^{\top}y$$
Penalized regression, \(\lambda>0\) and \(P\neq0\). For example, \(P=c(0,0,1)\) penalizes the
second derivative (curvature) by P=P.penalty(fdataobj["argvals"],P),
$$\hat{y}=\nu_k^{\top}(\nu_k\top \nu_k+\lambda \nu_k^{\top} \textbf{P}\nu_k)^{-1}\nu_k^{\top}y$$
Preda C. and Saporta G. PLS regression on a stochastic process. Comput. Statist. Data Anal. 48 (2005): 149-158.
N. Kraemer, A.-L. Boulsteix, and G. Tutz (2008). Penalized Partial Least Squares with Applications to B-Spline Transformations and Functional Data. Chemometrics and Intelligent Laboratory Systems, 94, 60 - 69. tools:::Rd_expr_doi("10.1016/j.chemolab.2008.06.009")
Martens, H., Naes, T. (1989) Multivariate calibration. Chichester: Wiley.
Kraemer, N., Sugiyama M. (2011). The Degrees of Freedom of Partial Least Squares Regression. Journal of the American Statistical Association. Volume 106, 697-705.
Febrero-Bande, M., Oviedo de la Fuente, M. (2012). Statistical Computing in Functional Data Analysis: The R Package fda.usc. Journal of Statistical Software, 51(4), 1-28. https://www.jstatsoft.org/v51/i04/
See Also as: P.penalty and
fregre.pls.cv.
Alternative method: fregre.pc.
if (FALSE) {
data(tecator)
x <- tecator$absorp.fdata
y <- tecator$y$Fat
res <- fregre.pls(x,y,c(1:4))
summary(res)
res1 <- fregre.pls(x,y,l=1:4,lambda=100,P=c(1))
res4 <- fregre.pls(x,y,l=1:4,lambda=1,P=c(0,0,1))
summary(res4)#' plot(res$beta.est)
lines(res1$beta.est,col=4)
lines(res4$beta.est,col=2)
}
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