bw.circular: Bandwidth Selectors for Kernel Density Estimation for...
In circular: Circular Statistics
bandwidth R Documentation
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)
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.
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.
Value
A bandwidth on a scale suitable for the bw argument
of density.circular.
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(s)
Claudio Agostinelli and Eduardo Garcia–Portugues
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)
circular documentation built on Sept. 11, 2024, 8:21 p.m.
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| bandwidth | R Documentation |
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)
Arguments
x |
the data from which the bandwidth is to be computed. The object is coerced to class |
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 |
kernel |
a character string giving the smoothing kernel to be used. This must be one of |
K |
number of terms to be used in approximating the wrappednormal density. See |
min.k |
minimum number of terms used in approximating the
wrappednormal density. See |
kappa.est |
a numerical value or one available method. |
kappa.bias |
logical. If |
P |
integer, the maximum order of the sample trigonometric
moments used in the estimation of |
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.
Value
A bandwidth on a scale suitable for the bw argument
of density.circular.
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(s)
Claudio Agostinelli and Eduardo Garcia–Portugues
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)
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