The deviance score

Returns the deviance of a fitted model object by GCV score.

Usage

dev.S(
  y,
  S,
  obs,
  family = gaussian(),
  off,
  offdf,
  criteria = "GCV",
  W = diag(1, ncol = ncol(S), nrow = nrow(S)),
  trim = 0,
  draw = FALSE,
  ...
)

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.

obs

observed response.

family

a description of the error distribution and link function to be used in the model. This can be a character string naming a family function, a family function or the result of a call to a family function. (See family for details of family functions.)

off

off

offdf

off, degrees of freedom

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.

...

Further arguments passed to or from other methods.

Value

Returns GCV score calculated for input parameters.

Details

Up to a constant, minus twice the maximized log-likelihood. Where sensible, the constant is chosen so that a saturated model has deviance zero.

$$GCV(h)=p(h) \Xi(n^{-1}h^{-1})$$

Where
$$p(h)=\frac{1}{n} \sum_{i=1}^{n}{\Big(y_i-r_{i}(x_i)\Big)^{2}w(x_i)}$$
and 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. https://www.jstatsoft.org/v51/i04/

See also

See Also as GCV.S.
Alternative method: CV.S

Author

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

Examples

data(phoneme)
mlearn<-phoneme$learn
np<-ncol(mlearn)
tt<-mlearn[["argvals"]]
S1 <- S.NW(tt,2.5)
gcv1 <- dev.S(mlearn$data[1,],obs=(sample(150)), 
S1,off=rep(1,150),offdf=3)
gcv2 <- dev.S(mlearn$data[1,],obs=sort(sample(150)), 
S1,off=rep(1,150),offdf=3)