SCBmeanfd (version 1.2.2)

cv.score: Leave-One-Curve-out Cross-Validation Score

Description

Compute the cross-validation score of Rice and Silverman (1991) for the local polynomial estimation of a mean function.

Usage

cv.score(bandwidth, x, y, degree = 1, gridsize = length(x))

Arguments

bandwidth

kernel bandwidth.

x

observation points. Missing values are not accepted.

y

matrix or data frame with functional observations (= curves) stored in rows. The number of columns of y must match the length of x. Missing values are not accepted.

degree

degree of the local polynomial fit.

gridsize

size of evaluation grid for the smoothed data.

Value

Details

The cross-validation score is obtained by leaving in turn each curve out and computing the prediction error of the local polynomial smoother based on all other curves. For a bandwith value $h$, this score is $$ CV(h) = \frac{1}{np} \sum_{i=1}^n \sum_{j=1}^p \left( Y_{ij} - \hat{\mu}^{-(i)}(x_j;h) \right)^2, $$ where $Y[ij]$ is the measurement of the $i$-th curve at location $x[j]$ for $i=1,\ldots,n$ and $j=1,\ldots,p$, and $\mu^{-i}(x[j];h)$ is the local polynomial estimator with bandwidth $h$ based on all curves except the $i$-th.

If the x values are not equally spaced, the data are first smoothed and evaluated on a grid of length gridsize spanning the range of x. The smoothed data are then interpolated back to x.

References

Rice, J. A. and Silverman, B. W. (1991). Estimating the mean and covariance structure nonparametrically when the data are curves. Journal of the Royal Statistical Society. Series B (Methodological), 53, 233--243.

See Also

cv.select

Examples

## Artificial example 
x <- seq(0, 1, len = 100)
mu <- x + .2 * sin(2 * pi * x)
y <- matrix(mu + rnorm(2000, sd = .25), 20, 100, byrow = TRUE)
h <- c(.005, .01, .02, .05, .1, .15)
cv <- numeric()
for (i in 1:length(h)) cv[i] <- cv.score(h[i], x, y, 1)
plot(h, cv, type = "l")

## Plasma citrate data
## Compare cross-validation scores and bandwidths  
## for local linear and local quadratic smoothing
## Not run: 
# data(plasma)
# time <- 8:21
# h1 <- seq(.5, 1.3, .05)
# h2 <- seq(.75, 2, .05)
# cv1 <- sapply(h1, cv.score, x = time, y = plasma, degree = 1)
# cv2 <- sapply(h2, cv.score, x = time, y = plasma, degree = 2)
# plot(h1, cv1, type = "l", xlim = range(c(h1,h2)), ylim = range(c(cv1, cv2)), 
#   xlab = "Bandwidth (hour)", ylab = "CV score", 
#   main = "Cross validation for local polynomial estimation")
# lines(h2, cv2, col = 2)
# legend("topleft", legend = c("Linear", "Quadratic"), lty = 1, 
#   col = 1:2, cex = .9)
# 
# ## Note: using local linear (resp. quadratic) smoothing 
# ## with a bandwidth smaller than .5 (resp. .75) can result 
# ## in non-definiteness or numerical instability of the estimator. 
# ## End(Not run)