fkden: Cross-validation MLE Fitting of Kernel Density Estimator, With Variety of Kernels

Description

Maximum (cross-validation) likelihood estimation for fitting kernel density estimator for a variety of possible kernels, by treating it as a mixture model.

Usage

fkden(x, linit = NULL, bwinit = NULL, kernel = "gaussian",
  extracentres = NULL, add.jitter = FALSE, factor = 0.1, amount = NULL,
  std.err = TRUE, method = "BFGS", control = list(maxit = 10000),
  finitelik = TRUE, ...)
lkden(x, lambda = NULL, bw = NULL, kernel = "gaussian",
  extracentres = NULL, log = TRUE)
nlkden(lambda, x, bw = NULL, kernel = "gaussian", extracentres = NULL,
  finitelik = FALSE)

Arguments

vector of sample data

linit

initial value for bandwidth (as kernel half-width) or NULL

bwinit

initial value for bandwidth (as kernel standard deviations) or NULL

kernel

kernel name (default = "gaussian")

extracentres

extra kernel centres used in KDE, but likelihood contribution not evaluated, or NULL

add.jitter

logical, whether jitter is needed for rounded kernel centres

factor

see jitter

amount

see jitter

std.err

logical, should standard errors be calculated

method

optimisation method (see optim)

control

optimisation control list (see optim)

finitelik

logical, should log-likelihood return finite value for invalid parameters

...

optional inputs passed to optim

lambda

bandwidth for kernel (as half-width of kernel) or NULL

bandwidth for kernel (as standard deviations of kernel) or NULL

log

logical, if TRUE then log-likelihood rather than likelihood is output

Value

Log-likelihood is given by lkden and it's wrappers for negative log-likelihood from nlkden. Fitting function fkden returns a simple list with the following elements

`call`:	`optim` call
`x`:	(jittered) data vector `x`
`kerncentres`:	actual kernel centres used `x`
`init`:	`linit` for lambda
`optim`:	complete `optim` output
`mle`:	vector of MLE of bandwidth
`cov`:	variance of MLE of bandwidth
`se`:	standard error of MLE of bandwidth
`nllh`:	minimum negative cross-validation log-likelihood
`n`:	total sample size
`lambda`:	MLE of lambda (kernel half-width)
`bw`:	MLE of bw (kernel standard deviations)
`kernel`:	kernel name

Warning

Two important practical issues arise with MLE for the kernel bandwidth: 1) Cross-validation likelihood is needed for the KDE bandwidth parameter as the usual likelihood degenerates, so that the MLE $\hat{λ} \to 0$ as $n \to \infty$ , thus giving a negative bias towards a small bandwidth. Leave one out cross-validation essentially ensures that some smoothing between the kernel centres is required (i.e. a non-zero bandwidth), otherwise the resultant density estimates would always be zero if the bandwidth was zero.

This problem occassionally rears its ugly head for data which has been heavily rounded, as even when using cross-validation the density can be non-zero even if the bandwidth is zero. To overcome this issue an option to add a small jitter should be added to the data (x only) has been included in the fitting inputs, using the jitter function, to remove the ties. The default options red in the jitter are specified above, but the user can override these. Notice the default scaling factor=0.1, which is a tenth of the default value in the jitter function itself.

A warning message is given if the data appear to be rounded (i.e. more than 5 data rounding is the likely culprit. Only use the jittering when the MLE of the bandwidth is far too small.

2) For heavy tailed populations the bandwidth is positively biased, giving oversmoothing (see example). The bias is due to the distance between the upper (or lower) order statistics not necessarily decaying to zero as the sample size tends to infinity. Essentially, as the distance between the two largest (or smallest) sample datapoints does not decay to zero, some smoothing between them is required (i.e. bandwidth cannot be zero). One solution to this problem is to trim the data at a suitable threshold to remove the problematic tail from the inference for the bandwidth, using either the fkdengpd function for a single heavy tail or the fgkg function if both tails are heavy. See MacDonald et al (2013).

Acknowledgments

See Acknowledgments in fnormgpd, type help fnormgpd. Based on code by Anna MacDonald produced for MATLAB.

Details

The kernel density estimator (KDE) with one of possible kernels is fitted to the entire dataset using maximum (cross-validation) likelihood estimation. The estimated bandwidth, variance and standard error are automatically output.

The alternate bandwidth definitions are discussed in the kernels, with the lambda used here but bw also output. 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.

Missing values (NA and NaN) are assumed to be invalid data so are ignored.

Cross-validation likelihood is used for kernel density component, obtained by leaving each point out in turn and evaluating the KDE at the point left out: $L (λ) \prod_{i = 1}^{n} {\hat{f}}_{- i} (x_{i})$ where ${\hat{f}}_{- i} (x_{i}) = \frac{1}{(n - 1) λ} \sum_{j = 1 : j \neq i}^{n} K (\frac{x_{i} - x_{j}}{λ})$ is the KDE obtained when the $i$ th datapoint is dropped out and then evaluated at that dropped datapoint at $x_{i}$ .

Normally for likelihood estimation of the bandwidth the kernel centres and the data where the likelihood is evaluated are the same. However, when using KDE for extreme value mixture modelling the likelihood only those data in the bulk of the distribution should contribute to the likelihood, but all the data (including those beyond the threshold) should contribute to the density estimate. The extracentres option allows the use to specify extra kernel centres used in estimating the density, but not evaluated in the likelihood. Suppose the first nb data are below the threshold, followed by nu exceedances of the threshold, so $i = 1, \dots, n b, n b + 1, \dots, n b + n u$ . The cross-validation likelihood using the extra kernel centres is then: $L (λ) \prod_{i = 1}^{n b} {\hat{f}}_{- i} (x_{i})$ where ${\hat{f}}_{- i} (x_{i}) = \frac{1}{(n b + n u - 1) λ} \sum_{j = 1 : j \neq i}^{n b + n u} K (\frac{x_{i} - x_{j}}{λ})$ which shows that the complete set of data is used in evaluating the KDE, but only those below the threshold contribute to the cross-validation likelihood. The default is to use the existing data, so extracentres=NULL.

The following functions are provided:

fkden - maximum (cross-validation) likelihood fitting with all the above options;
lkden - cross-validation log-likelihood;
nlkden - negative cross-validation log-likelihood;

The log-likelihood functions are provided for wider usage, e.g. constructing profile likelihood functions.

The log-likelihood and negative log-likelihood are also provided for wider usage, e.g. constructing your own extreme value mixture models or profile likelihood functions. The parameter lambda must be specified in the negative log-likelihood nlkden.

Log-likelihood calculations are carried out in lkden, which takes bandwidths as inputs in the same form as distribution functions. The negative log-likelihood is a wrapper for lkden, designed towards making it useable for optimisation (e.g. lambda given as first input).

Defaults values for the bandwidth linit and lambda are given in the fitting fkden and cross-validation likelihood functions lkden. The bandwidth linit must be specified in the negative log-likelihood function nlkden.

Missing values (NA and NaN) are assumed to be invalid data so are ignored, which is inconsistent with the evd library which assumes the missing values are below the threshold.

The function lkden carries out the calculations for the log-likelihood directly, which can be exponentiated to give actual likelihood using (log=FALSE).

The default optimisation algorithm is "BFGS", which requires a finite negative log-likelihood function evaluation finitelik=TRUE. For invalid parameters, a zero likelihood is replaced with exp(-1e6). The "BFGS" optimisation algorithms require finite values for likelihood, so any user input for finitelik will be overridden and set to finitelik=TRUE if either of these optimisation methods is chosen.

It will display a warning for non-zero convergence result comes from optim function call or for common indicators of lack of convergence (e.g. estimated bandwidth equal to initial value).

If the hessian is of reduced rank then the variance covariance (from inverse hessian) and standard error of parameters cannot be calculated, then by default std.err=TRUE and the function will stop. If you want the parameter estimates even if the hessian is of reduced rank (e.g. in a simulation study) then set std.err=FALSE.

References

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.

MacDonald, A., C. J. Scarrott, and D. S. Lee (2011). Boundary correction, consistency and robustness of kernel densities using extreme value theory. Submitted. Available from: http://www.math.canterbury.ac.nz/~c.scarrott.

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

Examples

Run this code

# NOT RUN {
set.seed(1)
par(mfrow = c(1, 1))

nk=50
x = rnorm(nk)
xx = seq(-5, 5, 0.01)
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$lambda)*0.05)
lines(xx,dnorm(xx), col = "black")
lines(xx, dkden(xx, x, lambda = fit$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"))

par(mfrow = c(2, 1))

# bandwidth is biased towards oversmoothing for heavy tails
nk=100
x = rt(nk, df = 2)
xx = seq(-8, 8, 0.01)
fit = fkden(x)
hist(x, seq(floor(min(x)), ceiling(max(x)), 0.5), freq = FALSE, xlim = c(-8, 10)) 
rug(x)
for (i in 1:nk) lines(xx, dnorm(xx, x[i], sd = fit$lambda)*0.05)
lines(xx,dt(xx , df = 2), col = "black")
lines(xx, dkden(xx, x, lambda = fit$lambda), lwd = 2, col = "red")
legend("topright", c("True Density", "KDE fitted evmix, c-v likelihood bandwidth"),
lty = c(1, 1), lwd = c(1, 2), col = c("black", "red"))

# remove heavy tails from cv-likelihood evaluation, but still include them in KDE within likelihood
# often gives better bandwidth (see MacDonald et al (2011) for justification)
nk=100
x = rt(nk, df = 2)
xx = seq(-8, 8, 0.01)
fit2 = fkden(x[(x > -4) & (x < 4)], extracentres = x[(x <= -4) | (x >= 4)])
hist(x, seq(floor(min(x)), ceiling(max(x)), 0.5), freq = FALSE, xlim = c(-8, 10)) 
rug(x)
for (i in 1:nk) lines(xx, dnorm(xx, x[i], sd = fit2$lambda)*0.05)
lines(xx,dt(xx , df = 2), col = "black")
lines(xx, dkden(xx, x, lambda = fit2$lambda), lwd = 2, col = "red")
lines(xx, dkden(xx, x, lambda = fit$lambda), lwd = 2, col = "blue")
legend("topright", c("True Density", "KDE fitted evmix, tails removed",
"KDE fitted evmix, tails included"),
lty = c(1, 1, 1), lwd = c(1, 2, 2), col = c("black", "red", "blue"))
# }
# NOT RUN {
# }

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