# fregre.pls: Functional Penalized PLS regression with scalar response In fda.usc: Functional Data Analysis and Utilities for Statistical Computing

## Description

Computes functional linear regression between functional explanatory variable X(t) and scalar response Y using penalized Partial Least Squares (PLS)

Y=<\tilde{X},β>+ε

where <.,.> denotes the inner product on L_2 and ε are random errors with mean zero , finite variance σ^2 and E[X(t)ε]=0.
ν_1,...,ν_∞ orthonormal basis of PLS to represent the functional data as X(t)=∑_(k=1:∞) γ_k ν_k.

## Usage

 1 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 {ν_k}_k=1:∞ the functional PLS components and X_i(t)=∑{k=1:∞} γ_{ik} ν_k and β(t)=∑{k=1:∞} β_k ν_k. The functional linear model is estimated by:

y.est=< X,β.est > \approx ∑{k=1:k_n} γ_k β_k

The response can be fitted by:

• λ=0, no penalization,

y.est= ν'(ν'ν)^{-1}ν'y

• Penalized regression, λ>0 and P!=0. For example, P=c(0,0,1) penalizes the second derivative (curvature) by P=P.penalty(fdataobj["argvals"],P),

y.est=ν'(ν'ν+λ v'Pv)^{-1}ν'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.

## Author(s)

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

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

## Examples

 1 2 3 4 5 6 7 8 ## Not run: data(tecator) x<-tecator$absorp.fdata y<-tecator$y\$Fat res=fregre.pls(x,y,c(1:8),lambda=10) summary(res) ## End(Not run) 

### Example output Loading required package: fda

Attaching package: ‘fda’

The following object is masked from ‘package:graphics’:

matplot

This is mgcv 1.8-33. For overview type 'help("mgcv-package")'.
----------------------------------------------------------------------------------
Functional Data Analysis and Utilities for Statistical Computing
fda.usc version 2.0.2 (built on 2020-02-17) is now loaded
fda.usc is running sequentially usign foreach package
Please, execute ops.fda.usc() once to run in local parallel mode
Deprecated functions: min.basis, min.np, anova.hetero, anova.onefactor, anova.RPm
New functions: optim.basis, optim.np, fanova.hetero, fanova.onefactor, fanova.RPm
----------------------------------------------------------------------------------

Warning message:
In if (class(Minv) != "try-error") { :
the condition has length > 1 and only the first element will be used
*** Summary Functional Regression with Partial Least Squares***

-Call: fregre.pls(fdataobj = x, y = y, l = c(1:8), lambda = 10)

Estimate Std. Error   t value      Pr(>|t|)
(Intercept) 18.1423256  0.1899909 95.490516 1.225261e-171
PLS1        -0.8147606  0.1206244 -6.754528  1.456586e-10
PLS2         8.6253070  0.3260360 26.455075  5.914682e-68
PLS3         5.7642481  0.2327310 24.767859  1.738403e-63
PLS4         6.3763077  0.4174746 15.273523  1.194715e-35
PLS5        19.7023847  1.8253711 10.793632  8.690502e-22
PLS6         9.6759513  2.8464063  3.399357  8.119417e-04
PLS7        24.0614833  3.6205276  6.645850  2.680142e-10
PLS8        53.6760792  2.3935314 22.425475  5.023804e-57

-R squared:  0.9542937
-Residual variance:  7.760754 on  204.5715  degrees of freedom
-Names of possible atypical curves: 43 44
-Names of possible influence curves: 34 35 86 89 140 180 181


fda.usc documentation built on Feb. 18, 2020, 1:07 a.m.