fregre.ppc.cv: Functional penalized PC (or PLS) regression with scalar response using selection of number of PC (or PLS) components

Description

Functional Regression with scalar response using selection of number of penalized principal componentes PPC(or partial least squares components PPLS) through cross-validation. The algorithm selects the PPLS components with best estimates the response. The selection is performed by cross-validation (CV) or Model Selection Criteria (MSC). After is computing functional regression using the best selection of PPC (or PPLS) components.

Usage

fregre.ppc.cv(fdataobj, y, kmax=8,  lambda = 0, P = c(0, 0, 1), 
    criteria = "SIC", ...) 

fregre.ppls.cv(fdataobj, y, kmax=8,  lambda = 0, P = c(0, 0, 1), 
    criteria = "SIC", ...)

Arguments

fdataobj

fdata class object.

Scalar response with length n.

kmax

The number of components to include in the model.

lambda

Vector with the amounts of penalization. Default value is 0, i.e. no penalization is used. If lambda=TRUE the algorithm computes a sequence of lambda values.

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 o

criteria

Type of cross-validation (CV) or Model Selection Criteria (MSC) applied. Possible values are "CV", "AIC", "AICc", "SIC".

...

Further arguments passed to fregre.ppc or fregre.ppls

Value

Return:
pls.optIndex of PC or PLS components selected.
MSC.minMinimum Model Selection Criteria (MSC) value for the (pc.opt or pls.opt) components.
MSCMinimum Model Selection Criteria (MSC) value for kmax components.
fregre.pplsFitted regression object by the best (pc.opt or pls.opt) components.

Details

The algorithm is as follows:

Select the bests components (pc.optorpls.opt) with minimum MSC criteria by stepwise regression usingfregre.ppcorfregre.pplsin each step.
Fit the functional PPLS regression between$\tilde{X}(t)$and$Y$using the best selection of FPLS componentspls.opt.

For more details in estimation process see fregre.ppc or fregre.ppls. The criteria selection is done by cross-validation (CV) or Model Selection Criteria (MSC).

Predictive Cross-Validation:$PCV(k_n)=\frac{1}{n}\sum_{i=1}^{n}{\Big(y_i -\hat{y}_{(-i,k_n)} \Big)^2}$, criteria=``CV''
Model Selection Criteria:$MSC(k_n)=log \left[ \frac{1}{n}\sum_{i=1}^{n}{\Big(y_i-\hat{y}_i\Big)^2} \right] +p_n\frac{k_n}{n}$ $p_n=\frac{log(n)}{n}$,criteria=``SIC'' (by default) $p_n=\frac{log(n)}{n-k_n-2}$,criteria=``SICc'' $p_n=2$,criteria=``AIC'' $p_n=\frac{2n}{n-k_n-2}$,criteria=``AICc'' $p_n=\frac{2log(log(n))}{n}$,criteria=``HQIC''
The generalized minimum description length (gmdl) criteria: $gmdl(k_n)=log \left[ \frac{1}{n-k_n}\sum_{i=1}^{n}{\Big(y_i-\hat{y}_i\Big)^2} \right] +K_n log \left(\frac{(n-k_n)\sum_{i=1}^{n}\hat{y}_i^2}{{\sum_{i=1}^{n}\Big(y_i-\hat{y}_i\Big)^2} }\right)+log(n)$%{MSC(k_n)=log [ 1/(n-k_n) \sum_(i=1:n){ (y_i- < X_i , \beta_(i,k_n) > )^2} ] +p_n k_n/n }
The rho criteria:$rho(k_n)=log \left[ \frac{1}{n-k_n}\sum_{i=1}^{n}\left(\frac{y_i-\hat{y}_i}{1-H_{ii}} \right)^2\right]$

where criteria is an argument that controls the type of validation used in the selection of the smoothing parameter kmax$=k_n$ and penalized parameter lambda$=\lambda$. criteria=``CV'' is not recommended: time-consuming.

References

Preda C. and Saporta G. PLS regression on a stochastic process. Comput. Statist. Data Anal. 48 (2005): 149{-}158. 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

Run this code

data(tecator)
x<-tecator$absorp.fdata[1:129]
y<-tecator$y$Fat[1:129]
# fregre.ppls.cv(x,y,8)
# fregre.ppls.cv(x,y,8,lambda=TRUE)

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