circular (version 0.3-1)

density: Kernel Density Estimation

Description

The function density computes kernel density estimates with the given kernel and bandwidth. The density function from the base is replace by a new method. density.default is copied from the original function density. The behavior would be the same for objects which are not from class circular (where the function density.circular is called).

Usage

density(x, ...)
## S3 method for class 'default':
density(x, bw = "nrd0", adjust = 1, kernel = c("gaussian",
    "epanechnikov", "rectangular", "triangular", "biweight",
    "cosine", "optcosine"), window = kernel, width, give.Rkern = FALSE,
    n = 512, from, to, cut = 3, na.rm = FALSE, ...)

Arguments

x

the data from which the estimate is to be computed.

bw

the smoothing bandwidth to be used. The kernels are scaled such that this is the standard deviation of the smoothing kernel. (Note this differs from the reference books cited below, and from S-PLUS.) bw can also be a character

adjust

the bandwidth used is actually adjust*bw. This makes it easy to specify values like half the default bandwidth.

kernel, window

a character string giving the smoothing kernel to be used. This must be one of "gaussian", "rectangular", "triangular", "epanechnikov", "biweight", "cosine" or

width

this exists for compatibility with S; if given, and bw is not, will set bw to width if this is a character string, or to a kernel-dependent multiple of width if this is numeric.

give.Rkern

logical; if true, no density is estimated, and the canonical bandwidth of the chosen kernel is returned instead.

n

the number of equally spaced points at which the density is to be estimated. When n > 512, it is rounded up to the next power of 2 for efficiency reasons (fft).

from,to

the left and right-most points of the grid at which the density is to be estimated.

cut

by default, the values of left and right are cut bandwidths beyond the extremes of the data. This allows the estimated density to drop to approximately zero at the extremes.

na.rm

logical; if TRUE, missing values are removed from x. If FALSE any missing values cause an error.

...

further arguments passed to or from other methods.

Value

• If give.Rkern is true, the number $R(K)$, otherwise an object with class "density" whose underlying structure is a list containing the following components.
• x
  the n coordinates of the points where the density is estimated.
• y
  the estimated density values.
• bw
  the bandwidth used.
• N
  the sample size after elimination of missing values.
• call
  the call which produced the result.
• data.name
  the deparsed name of the x argument.
• has.na
  logical, for compatibility (always FALSE).

Details

The algorithm used in density.default disperses the mass of the empirical distribution function over a regular grid of at least 512 points and then uses the fast Fourier transform to convolve this approximation with a discretized version of the kernel and then uses linear approximation to evaluate the density at the specified points. The statistical properties of a kernel are determined by $\sigma^2_K = \int t^2 K(t) dt$ which is always $= 1$ for our kernels (and hence the bandwidth bw is the standard deviation of the kernel) and $R(K) = \int K^2(t) dt$. MSE-equivalent bandwidths (for different kernels) are proportional to $\sigma_K R(K)$ which is scale invariant and for our kernels equal to $R(K)$. This value is returned when give.Rkern = TRUE. See the examples for using exact equivalent bandwidths. Infinite values in x are assumed to correspond to a point mass at +/-Inf and the density estimate is of the sub-density on (-Inf, +Inf).

References

Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole (for S version). Scott, D. W. (1992) Multivariate Density Estimation. Theory, Practice and Visualization. New York: Wiley. Sheather, S. J. and Jones M. C. (1991) A reliable data-based bandwidth selection method for kernel density estimation. J. Roy. Statist. Soc. B, 683--690. Silverman, B. W. (1986) Density Estimation. London: Chapman and Hall. Venables, W. N. and Ripley, B. D. (1999) Modern Applied Statistics with S-PLUS. New York: Springer.

See Also

density, density.circular, bw.nrd, plot.density, hist.

Examples

plot(density(c(-20,rep(0,98),20)), xlim = c(-4,4))# IQR = 0

# The Old Faithful geyser data
data(faithful)
d <- density(faithful$eruptions, bw = "sj")
d
plot(d)

plot(d, type = "n")
polygon(d, col = "wheat")

## Missing values:
x <- xx <- faithful$eruptions
x[i.out <- sample(length(x), 10)] <- NA
doR <- density(x, bw = 0.15, na.rm = TRUE)
lines(doR, col = "blue")
points(xx[i.out], rep(0.01, 10))


(kernels <- eval(formals(density.default)$kernel))

## show the kernels in the R parametrization
plot (density(0, bw = 1), xlab = "",
      main="R's density() kernels with bw = 1")
for(i in 2:length(kernels))
   lines(density(0, bw = 1, kern =  kernels[i]), col = i)
legend(1.5,.4, legend = kernels, col = seq(kernels),
       lty = 1, cex = .8, y.int = 1)

## show the kernels in the S parametrization
plot(density(0, from=-1.2, to=1.2, width=2, kern="gaussian"), type="l",
     ylim = c(0, 1), xlab="", main="R's density() kernels with width = 1")
for(i in 2:length(kernels))
   lines(density(0, width=2, kern =  kernels[i]), col = i)
legend(0.6, 1.0, legend = kernels, col = seq(kernels), lty = 1)

(RKs <- cbind(sapply(kernels, function(k)density(kern = k, give.Rkern = TRUE))))
100*round(RKs["epanechnikov",]/RKs, 4) ## Efficiencies

if(interactive()) {
data(precip)
bw <- bw.SJ(precip) ## sensible automatic choice
plot(density(precip, bw = bw, n = 2^13),
     main = "same sd bandwidths, 7 different kernels")
for(i in 2:length(kernels))
   lines(density(precip, bw = bw, kern =  kernels[i], n = 2^13), col = i)

## Bandwidth Adjustment for "Exactly Equivalent Kernels"
h.f <- sapply(kernels, function(k)density(kern = k, give.Rkern = TRUE))
(h.f <- (h.f["gaussian"] / h.f)^ .2)
## -> 1, 1.01, .995, 1.007,... close to 1 => adjustment barely visible..

plot(density(precip, bw = bw, n = 2^13),
     main = "equivalent bandwidths, 7 different kernels")
for(i in 2:length(kernels))
   lines(density(precip, bw = bw, adjust = h.f[i], kern =  kernels[i],
         n = 2^13), col = i)
legend(55, 0.035, legend = kernels, col = seq(kernels), lty = 1)
}