density {stats}R Documentation

Kernel Density Estimation


The function density computes kernel density estimates with the given kernel and bandwidth.


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)


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 string giving a rule to choose the bandwidth. See bw.nrd.
The specified (or computed) value of bw is multiplied by adjust.
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 "optcosine", with default "gaussian", and may be abbreviated to a unique prefix (single letter).
"cosine" is smoother than "optcosine", which is the usual “cosine” kernel in the literature and almost MSE-efficient. However, "cosine" is the version used by S.
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.


The algorithm used in density 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 sig^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 sig(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).


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. the deparsed name of the x argument. logical, for compatibility (always FALSE).


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. (2002) Modern Applied Statistics with S. New York: Springer.

See Also

bw.nrd, plot.density, hist.


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

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

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)$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, = 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()) {
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)

[Package stats version 2.1.0 Index]