density | R Documentation |
The (S3) generic function density
computes kernel density
estimates. Its default method does so with the given kernel and
bandwidth for univariate observations.
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, ...)
x |
the data from which the estimate is to be computed. For the default method a numeric vector: long vectors are not supported. |
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.)
The specified (or computed) value of |
adjust |
the bandwidth used is actually |
kernel, window |
a character string giving the smoothing kernel
to be used. This must partially match one of
|
weights |
numeric vector of non-negative observation weights,
hence of same length as |
width |
this exists for compatibility with S; if given, and
|
give.Rkern |
logical; if true, no density is estimated, and
the ‘canonical bandwidth’ of the chosen |
n |
the number of equally spaced points at which the density is
to be estimated. When |
from,to |
the left and right-most points of the grid at which the
density is to be estimated; the defaults are |
cut |
by default, the values of |
na.rm |
logical; if |
... |
further arguments for (non-default) methods. |
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).
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 |
y |
the estimated density values. These will be non-negative, 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 |
has.na |
logical, for compatibility (always |
The print
method reports summary
values on the
x
and y
components.
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. Journal of the Royal Statistical Society series B, 53, 683–690. \Sexpr[results=rd,stage=build]{tools:::Rd_expr_doi("10.1111/j.2517-6161.1991.tb01857.x")}. https://www.jstor.org/stable/2345597.
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.
bw.nrd
,
plot.density
, hist
.
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$eruptions) # has quite a few non-unique values ## use 'counts / n' as weights: dw <- density(unique(fe), weights = table(fe)/length(fe), bw = d$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 equally-weighted 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) ##-------- Semi-advanced 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)
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