Functional penalized PLS regression with scalar response using selection of number of PLS components

Functional Regression with scalar response using selection of number of penalized principal componentes 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 PPLS components.

Usage

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

Arguments

fdataobj: fdata class object.
y: 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.
P: The vector of coefficients to define the penalty matrix object. For example, if P=c(0,0,1), penalized regression is computed penalizing the second derivative (curvature).
criteria: Type of cross-validation (CV) or Model Selection Criteria (MSC) applied. Possible values are "CV", "AIC", "AICc", "SIC", "SICc", "HQIC".
...: Further arguments passed to fregre.pls.

Value

Return:

fregre.pls: Fitted regression object by the best (pls.opt) components.
pls.opt: Index of PLS components selected.
MSC.min: Minimum Model Selection Criteria (MSC) value for the (pls.opt) components.
MSC: Minimum Model Selection Criteria (MSC) value for kmax components.

Details

The algorithm selects the best principal components pls.opt from the first kmax PLS and (optionally) the best penalized parameter lambda.opt from a sequence of non-negative numbers lambda.

The method selects the best principal components with minimum MSC criteria by stepwise regression using fregre.pls in each step.
The process (point 1) is repeated for each lambda value.
The method selects the principal components (pls.opt=pls.order[1:k.min]) and (optionally) the lambda parameter with minimum MSC criteria.

Finally, is computing functional PLS regression between functional explanatory variable \(X(t)\) and scalar response \(Y\) using the best selection of PLS pls.opt and ridge parameter rn.opt. 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”
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\(=\lambda\).

Note

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.

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/

Author

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

Examples

if (FALSE) { # \dontrun{
data(tecator)
x<-tecator$absorp.fdata[1:129]
y<-tecator$y$Fat[1:129]
# no penalization
pls1<- fregre.pls.cv(x,y,8)
# 2nd derivative penalization
pls2<-fregre.pls.cv(x,y,8,lambda=0:5,P=c(0,0,1))
} # }