kden: Kernel Density Estimation, With Variety of Kernels
In evmix: Extreme Value Mixture Modelling, Threshold Estimation and Boundary Corrected Kernel Density Estimation

Description Usage Arguments Details Value Acknowledgments Note Author(s) References See Also Examples

Density, cumulative distribution function, quantile function and random number generation for the kernel density estimation using the kernel specified by kernel, with a constant bandwidth specified by either lambda or bw.

dkden(x, kerncentres, lambda = NULL, bw = NULL, kernel = "gaussian",
  log = FALSE)

pkden(q, kerncentres, lambda = NULL, bw = NULL, kernel = "gaussian",
  lower.tail = TRUE)

qkden(p, kerncentres, lambda = NULL, bw = NULL, kernel = "gaussian",
  lower.tail = TRUE)

rkden(n = 1, kerncentres, lambda = NULL, bw = NULL,
  kernel = "gaussian")

`x`	quantiles
`kerncentres`	kernel centres (typically sample data vector or scalar)
`lambda`	bandwidth for kernel (as half-width of kernel) or `NULL`
`bw`	bandwidth for kernel (as standard deviations of kernel) or `NULL`
`kernel`	kernel name (`default = "gaussian"`)
`log`	logical, if TRUE then log density
`q`	quantiles
`lower.tail`	logical, if FALSE then upper tail probabilities
`p`	cumulative probabilities
`n`	sample size (positive integer)

Kernel density estimation using one of many possible kernels with a constant bandwidth.

The alternate bandwidth definitions are discussed in the kernels, with the lambda as the default. The bw specification is the same as used in the density function.

The possible kernels are also defined in kernels help documentation with the "gaussian" as the default choice.

The density function dkden produces exactly the same density estimate as density when a sequence of x values are provided, see examples. The latter function is far more efficient in this situation as it takes advantage of the computational savings from doing the kernel smoothing in the spectral domain (using the FFT), where the convolution becomes a multiplication. So even after accounting for applying the (Fast) Fourier Transform (FFT) and its inverse it is much more efficient especially for a large sample size or large number of evaluation points.

However, this KDE function applies the less efficient convolution using the standard definition:

\hat{f}_(x) = \frac{1}{n} ∑_{j=1}^{n} K(\frac{x - x_j}{λ})

where K(.) is the density function for the standard kernel. Thus are no restriction on the values x can take. For example, in the "gaussian" kernel case for a particular x the density is evaluated as mean(dnorm(x, kerncentres, lambda)) for the density and mean(pnorm(x, kerncentres, lambda)) for cumulative distribution function which is slower than the FFT but is more adaptable.

An inversion sampler is used for random number generation which also rather inefficient, as it can be carried out more efficiently using a mixture representation.

The quantile function is rather complicated as there is no closed form solution, so is obtained by numerical approximation of the inverse cumulative distribution function P(X ≤ q) = p to find q. The quantile function qkden evaluates the KDE cumulative distribution function over the range from c(max(kerncentre) - lambda, max(kerncentre) + lambda), or c(max(kerncentre) - 5*lambda, max(kerncentre) + 5*lambda) for normal kernel. Outside of this range the quantiles are set to -Inf for lower tail and Inf for upper tail. A sequence of values of length fifty times the number of kernels (with minimum of 1000) is first calculated. Spline based interpolation using splinefun, with default monoH.FC method, is then used to approximate the quantile function. This is a similar approach to that taken by Matt Wand in the qkde in the ks package.

If no bandwidth is provided lambda=NULL and bw=NULL then the normal reference rule is used, using the bw.nrd0 function, which is consistent with the density function. At least two kernel centres must be provided as the variance needs to be estimated.

dkden gives the density, pkden gives the cumulative distribution function, qkden gives the quantile function and rkden gives a random sample.

Based on code by Anna MacDonald produced for MATLAB.

Unlike most of the other extreme value mixture model functions the kden functions have not been vectorised as this is not appropriate. The main inputs (x, p or q) must be either a scalar or a vector, which also define the output length.

The kernel centres kerncentres can either be a single datapoint or a vector of data. The kernel centres (kerncentres) and locations to evaluate density (x) and cumulative distribution function (q) would usually be different.

Default values are provided for all inputs, except for the fundamentals kerncentres, x, q and p. The default sample size for rkden is 1.

Missing (NA) and Not-a-Number (NaN) values in x, p and q are passed through as is and infinite values are set to NA. None of these are not permitted for the parameters.

Error checking of the inputs (e.g. invalid probabilities) is carried out and will either stop or give warning message as appropriate.

Yang Hu and Carl Scarrott carl.scarrott@canterbury.ac.nz.

http://en.wikipedia.org/wiki/Kernel_density_estimation

http://en.wikipedia.org/wiki/Cross-validation_(statistics)

Scarrott, C.J. and MacDonald, A. (2012). A review of extreme value threshold estimation and uncertainty quantification. REVSTAT - Statistical Journal 10(1), 33-59. Available from http://www.ine.pt/revstat/pdf/rs120102.pdf

Hu Y. and Scarrott, C.J. (2018). evmix: An R Package for Extreme Value Mixture Modeling, Threshold Estimation and Boundary Corrected Kernel Density Estimation. Journal of Statistical Software 84(5), 1-27. doi: 10.18637/jss.v084.i05.

Bowman, A.W. (1984). An alternative method of cross-validation for the smoothing of density estimates. Biometrika 71(2), 353-360.

Duin, R.P.W. (1976). On the choice of smoothing parameters for Parzen estimators of probability density functions. IEEE Transactions on Computers C25(11), 1175-1179.

MacDonald, A., Scarrott, C.J., Lee, D., Darlow, B., Reale, M. and Russell, G. (2011). A flexible extreme value mixture model. Computational Statistics and Data Analysis 55(6), 2137-2157.

Wand, M. and Jones, M.C. (1995). Kernel Smoothing. Chapman && Hall.

kernels, kfun, density, bw.nrd0 and dkde in ks package.

Other kden: bckden, fbckden, fgkgcon, fgkg, fkdengpdcon, fkdengpd, fkden, kdengpdcon, kdengpd

Other kdengpd: bckdengpd, fbckdengpd, fgkg, fkdengpdcon, fkdengpd, fkden, gkg, kdengpdcon, kdengpd

Other gkg: fgkgcon, fgkg, fkdengpd, gkgcon, gkg, kdengpd

Other bckden: bckdengpdcon, bckdengpd, bckden, fbckdengpdcon, fbckdengpd, fbckden, fkden

Other bckdengpd: bckdengpdcon, bckdengpd, bckden, fbckdengpdcon, fbckdengpd, fbckden, fkdengpd, gkg, kdengpd

Other fkden: fkden

## Not run: 
set.seed(1)
par(mfrow = c(2, 2))

nk=50
x = rnorm(nk)
xx = seq(-5, 5, 0.01)
plot(xx, dnorm(xx))
rug(x)
for (i in 1:nk) lines(xx, dnorm(xx, x[i], sd = bw.nrd0(x))*0.05)
lines(xx, dkden(xx, x), lwd = 2, col = "red")
lines(density(x), lty = 2, lwd = 2, col = "green")
legend("topright", c("True Density", "KDE Using evmix", "KDE Using density function"),
lty = c(1, 1, 2), lwd = c(1, 2, 2), col = c("black", "red", "green"))

# Estimate bandwidth using cross-validation likelihood
x = rnorm(nk)
fit = fkden(x)
hist(x, nk/5, freq = FALSE, xlim = c(-5, 5), ylim = c(0, 0.6)) 
rug(x)
for (i in 1:nk) lines(xx, dnorm(xx, x[i], sd = fit$bw)*0.05)
lines(xx,dnorm(xx), col = "black")
lines(xx, dkden(xx, x, lambda = fit$lambda), lwd = 2, col = "red")
lines(density(x), lty = 2, lwd = 2, col = "green")
lines(density(x, bw = fit$bw), lwd = 2, lty = 2,  col = "blue")
legend("topright", c("True Density", "KDE fitted evmix",
"KDE Using density, default bandwidth", "KDE Using density, c-v likelihood bandwidth"),
lty = c(1, 1, 2, 2), lwd = c(1, 2, 2, 2), col = c("black", "red", "green", "blue"))

plot(xx, pnorm(xx), type = "l")
rug(x)
lines(xx, pkden(xx, x), lwd = 2, col = "red")
lines(xx, pkden(xx, x, lambda = fit$lambda), lwd = 2, col = "green")
# green and blue (quantile) function should be same
p = seq(0, 1, 0.001)
lines(qkden(p, x, lambda = fit$lambda), p, lwd = 2, lty = 2, col = "blue") 
legend("topleft", c("True Density", "KDE using evmix, normal reference rule",
"KDE using evmix, c-v likelihood","KDE quantile function, c-v likelihood"),
lty = c(1, 1, 1, 2), lwd = c(1, 2, 2, 2), col = c("black", "red", "green", "blue"))

xnew = rkden(10000, x, lambda = fit$lambda)
hist(xnew, breaks = 100, freq = FALSE, xlim = c(-5, 5))
rug(xnew)
lines(xx,dnorm(xx), col = "black")
lines(xx, dkden(xx, x), lwd = 2, col = "red")
legend("topright", c("True Density", "KDE Using evmix"),
lty = c(1, 2), lwd = c(1, 2), col = c("black", "red"))

## End(Not run)