Functional Penalized PLS regression with scalar response

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 $X_i(t)=\sum_{k=1}^{\infty}\gamma_{ik}\nu_k$.

Usage

fregre.pls(fdataobj, y = NULL, l = NULL, lambda = 0, P = c(0, 0, 1), ...)

Arguments

fdataobj: fdata class object.
y: Scalar response with length n.
l: Index of components to include in the model.
lambda: Amount of penalization. Default value is 0, i.e. no penalization is used.
P: 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.

Value

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.
residuals: y minus 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.

Details

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), $$\hat{y}=\nu_k^{\top}(\nu_k\top \nu_k+\lambda \nu_k^{\top} \textbf{P}\nu_k)^{-1}\nu_k^{\top}y$$

References

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. 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/

Author

Manuel Febrero-Bande, Manuel Oviedo de la Fuente manuel.oviedo@udc.es

Examples

if (FALSE) { # \dontrun{
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)
} # }