fda.usc (version 1.2.3)

fregre.pls: Functional Penalized PLS regression with scalar response

Description

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 $<.,.>$ denotes the inner product on $L_2$ and $\epsilon$ are random errors with mean zero , finite variance $\sigma^2$ and $E[X(t)\epsilon]=0$.

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-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.

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 ${\nu_k}_k=1:\infty$ the functional PLS components and $X_i(t)=\sum{k=1:\infty} \gamma_{ik} \nu_k$ and $\beta(t)=\sum{k=1:\infty} \beta_k \nu_k$. 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 $P!=0$. 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. 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/

See Also

See Also as: P.penalty and fregre.pls.cv. Alternative method: fregre.pc.

Examples

## Not run: 
# data(tecator)
# x<-tecator$absorp.fdata
# y<-tecator$y$Fat
# res=fregre.pc(x,y,c(1:8))
# summary(res)
# res2=fregre.pls(x,y,c(1:8),lambda=10)
# summary(res2)
# ## End(Not run)