Functional Regression with scalar response using selection of number of (penalized) principal components PC through cross-validation. The algorithm selects the PC 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 principal components.
fregre.pc.cv(
fdataobj,
y,
kmax = 8,
lambda = 0,
P = c(0, 0, 1),
criteria = "SIC",
weights = rep(1, len = n),
...
)Return:
fregre.pc Fitted regression object by the best (pc.opt) components.
pc.opt Index of PC components selected.
MSC.min Minimum Model Selection Criteria (MSC) value for the (pc.opt components.
MSC Minimum Model Selection Criteria (MSC) value for kmax components.
fdata class object.
Scalar response with length n.
The number of components to include in the model.
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.
The vector of coefficients to define the penalty matrix object. For
example, if P=c(1,0,0), ridge regresion is computed and if
P=c(0,0,1), penalized regression is computed penalizing the second
derivative (curvature).
Type of cross-validation (CV) or Model Selection Criteria (MSC) applied. Possible values are "CV", "AIC", "AICc", "SIC", "SICc", "HQIC".
weights
Further arguments passed to fregre.pc or
fregre.pls
Manuel Febrero-Bande, Manuel Oviedo de la Fuente manuel.oviedo@usc.es
The algorithm selects the best principal components pc.opt from the first kmax PC and (optionally) the best penalized parameter lambda.opt from a sequence of non-negative
numbers lambda.
If kmax is a integer (by default and recomended) the procedure is as follows (see example 1):
Calculate the best principal component (pc.order[1]) between kmax by
fregre.pc.
Calculate the second-best principal component (pc.order [2]) between the (kmax-1) by
fregre.pc and calculate the criteria value of the two
principal components.
The process (point 1 and 2) is repeated until kmax principal component (pc.order[kmax]).
The proces (point 1, 2 and 3) is repeated for each lambda value.
The method selects the principal components (pc.opt=pc.order[1:k.min]) and (optionally) the lambda
parameter with minimum MSC criteria.
If kmax is a sequence of integer the procedure is as follows (see example 2):
The method selects the best principal components with minimum MSC criteria by
stepwise regression using fregre.pc in each step.
The process (point 1) is repeated for each lambda value.
The method selects the principal components (pc.opt=pc.order[1:k.min]) and (optionally) the lambda
parameter with minimum MSC criteria.
Finally, is computing functional PC regression between functional explanatory variable \(X(t)\) and scalar
response \(Y\) using the best selection of PC pc.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\(=\lambda\).
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:fregre.pc .
if (FALSE) {
data(tecator)
x<-tecator$absorp.fdata[1:129]
y<-tecator$y$Fat[1:129]
# no penalization
res.pc1=fregre.pc.cv(x,y,8)
# 2nd derivative penalization
res.pc2=fregre.pc.cv(x,y,8,lambda=TRUE,P=c(0,0,1))
# Ridge regression
res.pc3=fregre.pc.cv(x,y,1:8,lambda=TRUE,P=1)
}
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