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.
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", ...)
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.
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.
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
Return:
Index of PC or PLS components selected.
Minimum Model Selection Criteria (MSC) value for the (pc.opt
or pls.opt
) components.
Minimum Model Selection Criteria (MSC) value for kmax
components.
Fitted regression object by the best (pc.opt
or pls.opt
) components.
The algorithm is as follows:
Select the bests components (pc.opt
or pls.opt
) with minimum MSC criteria by stepwise regression using fregre.ppc
or fregre.ppls
in each step.
Fit the functional PPLS regression between \(\tilde{X}(t)\) and \(Y\) using the best selection of FPLS components pls.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'', Schwarz information criterion (by default).
\(p_n=\frac{log(n)}{n-k_n-2}\), criteria
=``SICc'', corrected Schwarz information criterion.
\(p_n=2\), criteria
=``AIC'', Akaike information criterion.
\(p_n=\frac{2n}{n-k_n-2}\), criteria
=``AICc'', corrected Akaike information criterion
\(p_n=\frac{2log(log(n))}{n}\), criteria
=``HQIC'', Hannan-Quinn information criterion.
The generalized minimum description length (gmdl) criteria:
\(gmdl(k_n)=log \left[ \frac{1}{n-k_n}\sum_{i=1}^{n}{\Big(y_i-\hat{y}_i\Big)^2} \right] +K_n log \left(\frac{(n-k_n)\sum_{i=1}^{n}\hat{y}_i^2}{{\sum_{i=1}^{n}\Big(y_i-\hat{y}_i\Big)^2} }\right)+log(n) \)
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.
Schwarz, G. (1978). Estimating the dimension of a model. The annals of statistics, 6(2), 461-464.
Hannan, E. J., Quinn, B. G. (1979). The determination of the order of an autoregression. Journal of the Royal Statistical Society. Series B (Methodological), 190-195.
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/
See also as: fregre.ppls
and fregre.ppc
.