SCBmeanfd (version 1.1)

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)

Arguments

bandwidth

the kernel bandwidth smoothing parameter.

x

numeric vector of x data. This observation grid must be uniform and 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 local polynomial used.

Value

Details

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

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)