density
Kernel Density Estimation
The (S3) generic function density
computes kernel density
estimates. Its default method does so with the given kernel and
bandwidth for univariate observations.
 Keywords
 distribution, smooth
Usage
density(x, ...)
"density"(x, bw = "nrd0", adjust = 1, kernel = c("gaussian", "epanechnikov", "rectangular", "triangular", "biweight", "cosine", "optcosine"), weights = NULL, 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 SPLUS.)
bw
can also be a character string giving a rule to choose the bandwidth. Seebw.nrd
. The default,"nrd0"
, has remained the default for historical and compatibility reasons, rather than as a general recommendation, where e.g.,"SJ"
would rather fit, see also Venables and Ripley (2002).The specified (or computed) value of
bw
is multiplied byadjust
.  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 MSEefficient. However,"cosine"
is the version used by S.  weights
 numeric vector of nonnegative observation weights,
hence of same length as
x
. The defaultNULL
is equivalent toweights = rep(1/nx, nx)
wherenx
is the length of (the finite entries of)x[]
.  width
 this exists for compatibility with S; if given, and
bw
is not, will setbw
towidth
if this is a character string, or to a kerneldependent multiple ofwidth
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 a power of 2 during the calculations (asfft
is used) and the final result is interpolated byapprox
. So it almost always makes sense to specifyn
as a power of two.  from,to
 the left and rightmost points of the grid at which the
density is to be estimated; the defaults are
cut * bw
outside ofrange(x)
.  cut
 by default, the values of
from
andto
arecut
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 fromx
. IfFALSE
any missing values cause an error.  ...
 further arguments for (nondefault) methods.
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
$sig^2 (K) = int(t^2 K(t) dt)$
which is always $= 1$ for our kernels (and hence the bandwidth
bw
is the standard deviation of the kernel) and
$R(K) = int(K^2(t) dt)$.
MSEequivalent 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 subdensity on
(Inf, +Inf)
.
Value

If
 x
 the
n
coordinates of the points where the density is estimated.  y
 the estimated density values. These will be nonnegative, but can be zero.
 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
). The
give.Rkern
is true, the number $R(K)$, otherwise
an object with class "density"
whose
underlying structure is a list containing the following components.
print
method reports summary
values on the
x
and y
components.
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 databased bandwidth selection method for kernel density estimation. J. Roy. Statist. Soc. B, 683690.
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
Examples
library(stats)
require(graphics)
plot(density(c(20, rep(0,98), 20)), xlim = c(4, 4)) # IQR = 0
# The Old Faithful geyser data
d < density(faithful$eruptions, bw = "sj")
d
plot(d)
plot(d, type = "n")
polygon(d, col = "wheat")
## Missing values:
x < xx < faithful$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))
## Weighted observations:
fe < sort(faithful$eruptions) # has quite a few nonunique values
## use 'counts / n' as weights:
dw < density(unique(fe), weights = table(fe)/length(fe), bw = d$bw)
utils::str(dw) ## smaller n: only 126, but identical estimate:
stopifnot(all.equal(d[1:3], dw[1:3]))
## simulation from a density() fit:
# a kernel density fit is an equallyweighted mixture.
fit < density(xx)
N < 1e6
x.new < rnorm(N, sample(xx, size = N, replace = TRUE), fit$bw)
plot(fit)
lines(density(x.new), col = "blue")
(kernels < eval(formals(density.default)$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, kernel = kernels[i]), col = i)
legend(1.5,.4, legend = kernels, col = seq(kernels),
lty = 1, cex = .8, y.intersp = 1)
## show the kernels in the S parametrization
plot(density(0, from = 1.2, to = 1.2, width = 2, kernel = "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, kernel = kernels[i]), col = i)
legend(0.6, 1.0, legend = kernels, col = seq(kernels), lty = 1)
## Semiadvanced theoretic from here on 
(RKs < cbind(sapply(kernels,
function(k) density(kernel = k, give.Rkern = TRUE))))
100*round(RKs["epanechnikov",]/RKs, 4) ## Efficiencies
bw < bw.SJ(precip) ## sensible automatic choice
plot(density(precip, bw = bw),
main = "same sd bandwidths, 7 different kernels")
for(i in 2:length(kernels))
lines(density(precip, bw = bw, kernel = kernels[i]), col = i)
## Bandwidth Adjustment for "Exactly Equivalent Kernels"
h.f < sapply(kernels, function(k)density(kernel = 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),
main = "equivalent bandwidths, 7 different kernels")
for(i in 2:length(kernels))
lines(density(precip, bw = bw, adjust = h.f[i], kernel = kernels[i]),
col = i)
legend(55, 0.035, legend = kernels, col = seq(kernels), lty = 1)