circular (version 0.5-1)

bandwidth: Bandwidth Selectors for Kernel Density Estimation for Circular Data

Description

Bandwidth selectors for circular kernels in density.circular.

Usage

bw.cv.mse.circular(x, lower=NULL, upper=NULL, tol = 1e-4,
  kernel = c("vonmises", "wrappednormal"), K = NULL, min.k = 10)
bw.cv.ml.circular(x, lower=NULL, upper=NULL, tol = 1e-4,
  kernel = c("vonmises", "wrappednormal"), K = NULL, min.k = 10)
bw.nrd.circular(x, lower=NULL, upper=NULL,
  kappa.est=c("ML","trigmoments"), kappa.bias=FALSE, P=3)

Value

A bandwidth on a scale suitable for the bw argument of density.circular.

Arguments

x

the data from which the bandwidth is to be computed. The object is coerced to class circular.

lower, upper

range over which to minimize for cross validatory bandwidths. The default is almost always satisfactory, although it is recommended experiment a little with different ranges. A warning message indicates if the resulting bandwidth is too near to the endpoints of the interval search.

tol

for cross validatory bandwidths, the convergence tolerance for optimize.

kernel

a character string giving the smoothing kernel to be used. This must be one of "vonmises" or "wrappednormal".

K

number of terms to be used in approximating the wrappednormal density. See dwrappednormal.

min.k

minimum number of terms used in approximating the wrappednormal density. See dwrappednormal.

kappa.est

a numerical value or one available method.

kappa.bias

logical. If TRUE, when kappa.est=="ML" a bias correction in the estimation of kappa is applied.

P

integer, the maximum order of the sample trigonometric moments used in the estimation of kappa when kappa.est=="trigmoments", see Details.

Warning

Plug-in bandwidth selector bw.nrd.circular assumes that the underlying population is von Mises. If this is not true, it might lead to serious misestimations of the circular bandwidth. Example 2 below shows how this behaviour can appear with multimodality populations. In those cases, the use of kappa.est="trigmoments" could be of help.

Author

Claudio Agostinelli and Eduardo Garcia--Portugues

Details

bw.cv.mse.circular and bw.cv.ml.circular implement cross validatory bandwidths minimizing squared--error loss and Kullback--Leibler loss, respectively. This is done by minimizing the second and third equations in section 5 of Hall, Watson and Cabrera (1987). Kullback--Leibler loss is equivalent to maximize the cross validation log--likelihood with respect to the bandwidth parameter.

bw.nrd.circular implements a rule-of-thumb for choosing the bandwidth of a von Mises kernel density estimator with underlying population von Mises. It was proposed by Taylor (2008, equation (7)) and is the circular analogue of the usual rule of thumb used for the normal distribution. The only remarkable difference between them is that Taylor's bandwidth supposes a von Mises population for the derivation of AMISE, while normal rule of thumb only introduces distribution assumption to compute the density curvature. Estimation of the spread is done by maximum likelihood. The "trigmoments" method for the estimation of kappa is implemented as follows. Let \(\mu_p\) be the p-th sample trigonometric moment. Let \(k_p\) be the estimates of kappa using the p-th sample trigonometric moment, as solution (using uniroot function) of the equation \(A_p(k) = \frac{1}{n} \sum_{i=1}^n \cos(p x_i - \mu_p)\). We let kappa equal to \(max(k_1, k_2, \cdots, k_P)\), see Taylor (2008) for further details.

Note that circular bandwidth has a different scale from linear bandwidth (see Hall, Watson and Cabrera (1987)). The behaviour of the circular bandwidth is the inverse of the linear: large values overestimate the density, whereas small values underestimate.

References

P. Hall and G.S. Watson and J. Cabrera (1987). Kernel Density Estimation with Spherical Data, Biometrika, 74, 4, 751--762.

C.C Taylor (2008). Automatic bandwidth selection for circular density estimation. Computational Statistics and Data Analysis, 52, 7, 3493--3500.

See Also

density.circular

Examples

set.seed(12345)

## Example 1: von Mises ##
theta1 <- rvonmises(n=150,mu=circular(pi),kappa=2)

bw.nrd1 <- bw.nrd.circular(theta1)
bw.cv.mse1 <- bw.cv.mse.circular(theta1)
bw.cv.ml1 <- bw.cv.ml.circular(theta1)

## Linear plot
plot(function(x) dvonmises(circular(x), mu=circular(pi), kappa=2),
type="l", lwd=2, col=1, main="von Mises", xlab=expression(theta),
ylab="Density", from=0, to=2*pi)
plot(approxfun(density.circular(x=theta1, bw=bw.nrd1)), col=2, from=0, to=2*pi, add=TRUE)
plot(approxfun(density.circular(x=theta1, bw=bw.cv.mse1)), col=3,
from=0, to=2*pi, add=TRUE)
plot(approxfun(density.circular(x=theta1, bw=bw.cv.ml1)), col=4, from=0,
to=2*pi, add=TRUE)
legend("topright", legend=c("True", "Taylor", "LSCV", "MLCV"), col=1:4, lwd=2)
rug(theta1)

## Circular plot
dvonmises1 <- function(x) dvonmises(circular(x), mu=circular(pi), kappa=2)
curve.circular(dvonmises1, lwd=2, col=1, main="von Mises", xlim=c(-1.5,
1.5), ylim=c(-1.5,1.5))
lines(density.circular(x=theta1, bw=bw.nrd1), col=2)
lines(density.circular(x=theta1, bw=bw.cv.mse1), col=3)
lines(density.circular(x=theta1, bw=bw.cv.ml1), col=4)
legend("topright", legend=c("True", "Taylor", "LSCV", "MLCV"), col=1:4, lwd=2)
points(theta1)

## Example 2: mixture of von Mises ##

theta2 <- rmixedvonmises(n=150, mu1=circular(pi/2),
mu2=circular(3*pi/2), kappa1=5, kappa2=5,p=0.5)

bw.nrd2 <- bw.nrd.circular(theta2)
bw.cv.mse2 <- bw.cv.mse.circular(theta2)
bw.cv.ml2 <- bw.cv.ml.circular(theta2)

## Linear plot
plot(function(x) dmixedvonmises(circular(x), mu1=circular(pi/2),
mu2=circular(3*pi/2), kappa1=5, kappa2=5, p=0.5), type="l", lwd=2,
col=1, main="mixture of von Mises", xlab=expression(theta),
ylab="Density", from=0, to=2*pi)
lines(density.circular(x=theta2, bw=bw.nrd2), plot.type='line', col=2)
lines(density.circular(x=theta2, bw=bw.cv.mse2), plot.type='line',
col=3)
lines(density.circular(x=theta2, bw=bw.cv.ml2), plot.type='line', col=4)
rug(theta2)
legend("topright", legend=c("True", "Taylor", "LSCV", "MLCV"), col=1:4, lwd=2)

## Circular plot
dmixedvonmises1 <- function(x) dmixedvonmises(circular(x), mu1=circular(pi/2),
mu2=circular(3*pi/2), kappa1=5, kappa2=5, p=0.5)
curve.circular(dmixedvonmises1, join=TRUE,
xlim=c(-1.5, 1.5), ylim=c(-1.5, 1.5), lwd=2, col=1, main="mixture of von
Mises")
lines(density.circular(x=theta2, bw=bw.nrd2), col=2)
lines(density.circular(x=theta2, bw=bw.cv.mse2), col=3)
lines(density.circular(x=theta2, bw=bw.cv.ml2), col=4)
points(theta2)
legend("topright", legend=c("True", "Taylor", "LSCV", "MLCV"), col=1:4, lwd=2)

## Example 3: mixture of von Mises and Wrapped Cauchy ##

rmixture <- function(n){
  x <- circular(sapply(runif(n), function(u) ifelse(u>0.5,
  rvonmises(n=1, mu=circular(pi),kappa=10),
  rwrappedcauchy(n=1,mu=circular(pi/2),rho=0.75))))
  return(x)
}

theta3 <- rmixture(n=150)

bw.nrd3 <- bw.nrd.circular(theta3)
bw.cv.mse3 <- bw.cv.mse.circular(theta3, lower=0.1, upper=100)
bw.cv.ml3 <- bw.cv.ml.circular(theta3, lower=0.1, upper=100)

dmixture <- function(x) (dvonmises(x, mu=circular(pi),
kappa=10)+dwrappedcauchy(x, mu=circular(pi/2), rho=0.75))/2
curve.circular(dmixture, join=TRUE, xlim=c(-1.5, 1.5), ylim=c(-1.5,
1.5), lwd=2, col=1, main="mixture of von Mises and Wrapped Normal")
lines(density.circular(x=theta3, bw=bw.nrd3), col=2)
lines(density.circular(x=theta3, bw=bw.cv.mse3), col=3)
lines(density.circular(x=theta3, bw=bw.cv.ml3), col=4)
legend("topright", legend=c("True", "Taylor", "LSCV", "MLCV"), col=1:4, lwd=2)
points(theta3)