fda.usc (version 1.1.0)

GCV.S: The generalized cross-validation (GCV) score.

Description

The generalized cross-validation (GCV) score.

Usage

GCV.S(y,S,criteria="GCV",W=NULL,trim=0,draw=FALSE,metric=metric.lp,...)

Arguments

y

Matrix of set cases with dimension (n x m), where n is the number of curves and m are the points observed in each curve.

S

Smoothing matrix, see S.NW, S.LLR or $S.KNN$.

criteria

The penalizing function. By default "Rice" criteria. Possible values are "GCV", "AIC", "FPE", "Shibata", "Rice".

W

Matrix of weights.

trim

The alpha of the trimming.

draw

=TRUE, draw the curves, the sample median and trimmed mean.

metric

Metric function, by default metric.lp.

...

Further arguments passed to or from other methods.

Value

• res
  Returns GCV score calculated for input parameters.

eqn

$S.KNN$

code

S.LLR

Details

$$GCV(h)=p(h) \Xi(n^{-1}h^{-1})$$ Where A.-If trim=0: $$p(h)={\left\|\sqrt(W)\left(y_i-\hat{y}_{i}\right)\right\|}$$ B.-If trim>0: $$p(h)=\frac{1}{l} \sum_{i=1}^{l}{\Big(y_i-r_{i}(x_i)\Big)^{2}w(x_i)}$$ l: index of (1-trim) curves with less error. where $h$ is the bandwidth parameter, $w$ the weights and the penalty function $\Xi$ can be selected from the following criteria:

• Generalized Cross-validation (GCV): $$\Xi_{GCV}(n^{-1}h^{-1})=(1-n^{-1}S_{ii})^{-2}$$

Akaike's Information Criterion (AIC): $$\Xi_{AIC}(n^{-1}h^{-1})=exp(2n^{-1}S_{ii})$$ Finite Prediction Error (FPE): $$\Xi_{FPE}(n^{-1}h^{-1})=\frac{(1+n^{-1}S_{ii})}{(1-n^{-1}S_{ii})}$$ Shibata's model selector (Shibata): $$\Xi_{Shibata}(n^{-1}h^{-1})=(1+2n^{-1}S_{ii})$$ Rice's bandwidth selector (Rice): $$\Xi_{Rice}(n^{-1}h^{-1})=(1-2n^{-1}S_{ii})^{-1}$$

References

Wasserman, L. All of Nonparametric Statistics. Springer Texts in Statistics, 2006. Hardle, W. Applied Nonparametric Regression. Cambridge University Press, 1994. 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

See Also as min.np. Alternative method: CV.S

Examples

data(phoneme)
mlearn<-phoneme$learn
tt<-1:ncol(mlearn)
S1 <- S.NW(tt,2.5)
S2 <- S.LLR(tt,2.5)
S3 <- S.KNN(tt,2.5)
gcv1 <- GCV.S(mlearn, S1)
gcv2 <- GCV.S(mlearn, S2)
gcv3 <- GCV.S(mlearn, S3)
gcv4 <- GCV.S(mlearn, S1,criteria="AIC")
gcv5 <- GCV.S(mlearn, S1,trim=0.01,draw=TRUE)
gcv1; gcv2; gcv3; gcv4; gcv5