density {base} | R Documentation |
The function density
computes kernel density estimates
with the given kernel and bandwidth.
The generic functions plot
and print
have
methods for density objects.
density(x, bw, 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) print(dobj) plot(dobj, main = NULL, xlab = NULL, ylab = "Density", type = "l", zero.line = TRUE, ...)
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.
It defaults to 0.9 times the
minimum of the standard deviation and the interquartile range divided by
1.34 times the sample size to the negative one-fifth power
(= Silverman's ``rule of thumb'') unless the quartiles
coincide where bw > 0 will be guaranteed.
The specified (or default) 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).
|
width |
this exists for compatibility with S; if given, and
bw is not, will set bw = width/4 . |
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. |
dobj |
a ``density'' object. |
main, xlab, ylab, type |
plotting parameters with useful defaults. |
... |
further plotting parameters. |
zero.line |
logical; if TRUE , add a base line at y = 0 |
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)
.
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). |
Silverman, B. W. (1986) Density Estimation. London: Chapman and Hall.
Venables, W. N. and B. D. Ripley (1994, 7, 9) Modern Applied Statistics with S-PLUS. New York: Springer.
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, 683690.
hist
.
plot(density(c(-20,rep(0,98),20)), xlim = c(-4,4))# IQR = 0 # The Old Faithful geyser data data(faithful) d <- density(faithful$eruptions, bw = 0.15) d plot(d) plot(d, type = "n") polygon(d, col = "wheat") ## Missing values: x <- xx <- faithful$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(.01,10)) (kernels <- eval(formals(density)$kernel)) plot (density(0,bw = 1)) for(i in 2:length(kernels)) lines(density(0,bw = 1, kern = kernels[i]), col = i) mtext(side = 3, "R's density() kernels with bw = 1") legend(1.5,.4, leg = kernels, col = seq(kernels),lty = 1, cex = .8, y.int = 1) (RKs <- cbind(sapply(kernels, function(k)density(kern = k, give.Rkern = TRUE)))) 100*round(RKs["epanechnikov",]/RKs, 4) ## Efficiencies data(precip) plot(density(precip, n = 2^13)) for(i in 2:length(kernels)) lines(density(precip, kern = kernels[i], n = 2^13), col = i) mtext(side = 3, "same scale bandwidths, 7 different kernels") ## 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, n = 2^13)) for(i in 2:length(kernels)) lines(density(precip, adjust = h.f[i], kern = kernels[i], n = 2^13), col = i) mtext(side = 3, "equivalent bandwidths, 7 different kernels") legend(55,.035, leg = kernels, col = seq(kernels), lty = 1)