fregre.pls {fda.usc}

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 \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.

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.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.

Author(s)

Manuel Febrero-Bande, Manuel Oviedo de la Fuente [e-mail]

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/

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$Fat
res <- fregre.pls(x,y,c(1:4))
summary(res)
res1 <- fregre.pls(x,y,l=1:4,lambda=100,P=c(1))
res4 <- fregre.pls(x,y,l=1:4,lambda=1,P=c(0,0,1))
summary(res4)#' plot(res$beta.est)
lines(res1$beta.est,col=4)
lines(res4$beta.est,col=2)

## End(Not run)