fda.usc (version 2.0.2)

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) $$Y=⟨\tilde{X},\beta ⟩+ϵ={\int }_T\tilde{X}(t)\beta (t)dt+ϵ$$ where $⟨\cdot ,\cdot ⟩$ denotes the inner product on $L_2$ and $ϵ$ are random errors with mean zero , finite variance ${\sigma }^2$ and $E[\tilde{X}(t)ϵ]=0$.
${\left\{{\nu }_k\right\}}_{k=1}^{\mathrm{\infty }}$ orthonormal basis of PLS to represent the functional data as $X_i(t)=\sum _{k=1}^{\mathrm{\infty }}{\gamma }_{ik}{\nu }_k$.

Usage

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

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.

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.

Author

Manuel Febrero-Bande, Manuel Oviedo de la Fuente

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 ${\left\{{\tilde{\nu }}_k\right\}}_{k=1}^{\mathrm{\infty }}$ the functional PLS components and ${\tilde{X}}_i(t)=\sum _{k=1}^{\mathrm{\infty }}{\tilde{\gamma }}_{ik}{\tilde{\nu }}_k$ and $\beta (t)=\sum _{k=1}^{\mathrm{\infty }}{\tilde{\beta }}_k{\tilde{\nu }}_k$. The functional linear model is estimated by: $$\hat{y}=⟨X,\hat{\beta }⟩\approx \sum _{k=1}^{k_n}{\tilde{\gamma }}_k{\tilde{\beta }}_k$$
The response can be fitted by:

• $\lambda =0$, no penalization, $$\hat{y}={\nu }_k^{\mathrm{\top }}({\nu }_k^{\mathrm{\top }}{\nu }_k{)}^{-1}{\nu }_k^{\mathrm{\top }}y$$

  • Penalized regression, $\lambda >0$ and $P\ne 0$. For example, $P=c(0,0,1)$ penalizes the second derivative (curvature) by P=P.penalty(fdataobj["argvals"],P), $$\hat{y}={\nu }_k^{\mathrm{\top }}({\nu }_k\mathrm{\top }{\nu }_k+\lambda {\nu }_k^{\mathrm{\top }}\mathbf{\text{P}}{\nu }_k{)}^{-1}{\nu }_k^{\mathrm{\top }}y$$

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

if (FALSE) {
data(tecator)
x<-tecator$absorp.fdata
y<-tecator$y$Fat
res=fregre.pls(x,y,c(1:8),lambda=10)
summary(res)
}