\HeaderA{density}{Kernel Density Estimation}{density}
\methaliasA{density.default}{density}{density.default}
\aliasA{print.density}{density}{print.density}
\keyword{distribution}{density}
\keyword{smooth}{density}
\begin{Description}\relax
The (S3) generic function \code{density} computes kernel density
estimates.  Its default method does so with the given kernel and
bandwidth for univariate observations.
\end{Description}
\begin{Usage}
\begin{verbatim}
density(x, ...)
## Default S3 method:
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, ...)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{x}] the data from which the estimate is to be computed.
\item[\code{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.)

\code{bw} can also be a character string giving a rule to choose the
bandwidth.  See \code{\LinkA{bw.nrd}{bw.nrd}}.

The specified (or computed) value of \code{bw} is multiplied by
\code{adjust}.

\item[\code{adjust}] the bandwidth used is actually \code{adjust*bw}.
This makes it easy to specify values like \dQuote{half the default}
bandwidth.
\item[\code{kernel, window}] a character string giving the smoothing kernel
to be used. This must be one of \code{"gaussian"},
\code{"rectangular"}, \code{"triangular"}, \code{"epanechnikov"},
\code{"biweight"}, \code{"cosine"} or \code{"optcosine"}, with default
\code{"gaussian"}, and may be abbreviated to a unique prefix (single
letter).

\code{"cosine"} is smoother than \code{"optcosine"}, which is the
usual \dQuote{cosine} kernel in the literature and almost MSE-efficient.
However, \code{"cosine"} is the version used by S.

\item[\code{weights}] numeric vector of non-negative observation weights,
hence of same length as \code{x}. The default \code{NULL} is
equivalent to \code{weights = rep(1/nx, nx)} where \code{nx} is the
length of (the finite entries of) \code{x[]}.
\item[\code{width}] this exists for compatibility with S; if given, and
\code{bw} is not, will set \code{bw} to \code{width} if this is a
character string, or to a kernel-dependent multiple of \code{width}
if this is numeric.
\item[\code{give.Rkern}] logical; if true, \emph{no} density is estimated, and
the \dQuote{canonical bandwidth} of the chosen \code{kernel} is returned
instead.
\item[\code{n}] the number of equally spaced points at which the density
is to be estimated.  When \code{n > 512}, it is rounded up to the next
power of 2 for efficiency reasons (\code{\LinkA{fft}{fft}}).
\item[\code{from,to}] the left and right-most points of the grid at which the
density is to be estimated.
\item[\code{cut}] by default, the values of \code{left} and \code{right} are
\code{cut} bandwidths beyond the extremes of the data. This allows the
estimated density to drop to approximately zero at the extremes.
\item[\code{na.rm}] logical; if \code{TRUE}, missing values are removed
from \code{x}. If \code{FALSE} any missing values cause an error.
\item[\code{...}] further arguments for (non-default) methods.
\end{ldescription}
\end{Arguments}
\begin{Details}\relax
The algorithm used in \code{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
\eqn{\sigma^2_K = \int t^2 K(t) dt}{sig^2 (K) = int(t^2 K(t) dt)}
which is always \eqn{= 1}{} for our kernels (and hence the bandwidth
\code{bw} is the standard deviation of the kernel) and
\eqn{R(K) = \int K^2(t) dt}{R(K) = int(K^2(t) dt)}.\\
MSE-equivalent bandwidths (for different kernels) are proportional to
\eqn{\sigma_K R(K)}{sig(K) R(K)} which is scale invariant and for our
kernels equal to \eqn{R(K)}{}.  This value is returned when
\code{give.Rkern = TRUE}.  See the examples for using exact equivalent
bandwidths.

Infinite values in \code{x} are assumed to correspond to a point mass at
\code{+/-Inf} and the density estimate is of the sub-density on
\code{(-Inf, +Inf)}.
\end{Details}
\begin{Value}
If \code{give.Rkern} is true, the number \eqn{R(K)}{}, otherwise
an object with class \code{"density"} whose
underlying structure is a list containing the following components.
\begin{ldescription}
\item[\code{x}] the \code{n} coordinates of the points where the density is
estimated.
\item[\code{y}] the estimated density values.
\item[\code{bw}] the bandwidth used.
\item[\code{n}] the sample size after elimination of missing values.
\item[\code{call}] the call which produced the result.
\item[\code{data.name}] the deparsed name of the \code{x} argument.
\item[\code{has.na}] logical, for compatibility (always \code{FALSE}).
\end{ldescription}
\end{Value}
\begin{References}\relax
Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988)
\emph{The New S Language}.
Wadsworth \& Brooks/Cole (for S version).

Scott, D. W. (1992)
\emph{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.
\emph{J. Roy. Statist. Soc.} \bold{B}, 683--690.

Silverman, B. W. (1986)
\emph{Density Estimation}.
London: Chapman and Hall.

Venables, W. N. and Ripley, B. D. (2002)
\emph{Modern Applied Statistics with S}.
New York: Springer.
\end{References}
\begin{SeeAlso}\relax
\code{\LinkA{bw.nrd}{bw.nrd}},
\code{\LinkA{plot.density}{plot.density}}, \code{\LinkA{hist}{hist}}.
\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
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$eruptions) # has quite a few non-unique values
## use 'counts / n' as weights:
dw <- density(unique(fe), weights = table(fe)/length(fe), bw = d$bw)
str(dw) ## smaller n: only 126, but identical estimate:
stopifnot(all.equal(d[1:3], dw[1:3]))

(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)

##-------- Semi-advanced theoretic from here on ---------------------------

(RKs <- cbind(sapply(kernels, function(k)density(kern = 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, kern =  kernels[i]), 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),
     main = "equivalent bandwidths, 7 different kernels")
for(i in 2:length(kernels))
   lines(density(precip, bw = bw, adjust = h.f[i], kern = kernels[i]),
         col = i)
legend(55, 0.035, legend = kernels, col = seq(kernels), lty = 1)
\end{ExampleCode}
\end{Examples}

