fregre.pls

From fda.usc v2.0.1 by Manuel Oviedo de la Fuente

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

Keywords: regression

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.

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

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.
residualsy-fitted values.
H Hat matrix.
df 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.

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. http://dx.doi.org/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. http://www.jstatsoft.org/v51/i04/

Examples

# NOT RUN {
data(tecator)
x<-tecator$absorp.fdata
y<-tecator$y$Fat
res=fregre.pls(x,y,c(1:8),lambda=10)
summary(res)
# }

Documentation reproduced from package fda.usc, version 2.0.1, License: GPL-2

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