density {stats} | R Documentation

## Kernel Density Estimation

### Description

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

### Usage

    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)

### 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 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.

### Details

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)`.

### Value

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.
`data.name` | the deparsed name of the `x` argument.
`has.na` | logical, for compatibility (always `FALSE`).

### 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 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. (1999) Modern Applied Statistics with S-PLUS. New York: Springer.

### See Also

`bw.nrd`, `plot.density`, `hist`.

### Examples

    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 = "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))

    (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, 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)

    (RKs <- cbind(sapply(kernels, function(k)density(kern = k, give.Rkern = TRUE))))
    100*round(RKs["epanechnikov",]/RKs, 4) ## Efficiencies

    if(interactive()) {
    data(precip)
    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)
    }

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