| Title: | Univariate Kernel Density Estimation |
|---|---|
| Description: | Provides an efficient implementation of univariate local polynomial kernel density estimators that can handle bounded and discrete data. See Geenens (2014) <arXiv:1303.4121>, Geenens and Wang (2018) <arXiv:1602.04862>, Nagler (2018a) <arXiv:1704.07457>, Nagler (2018b) <arXiv:1705.05431>. |
| Authors: | Thomas Nagler [aut, cre], Thibault Vatter [aut] |
| Maintainer: | Thomas Nagler <[email protected]> |
| License: | MIT + file LICENSE |
| Version: | 1.0.6 |
| Built: | 2024-09-11 04:52:16 UTC |
| Source: | https://github.com/tnagler/kde1d |
Provides an efficient implementation of univariate local polynomial kernel density estimators that can handle bounded and discrete data. The implementation utilizes spline interpolation to reduce memory usage and computational demand for large data sets.
Geenens, G. (2014). Probit transformation for kernel density estimation on the unit interval. Journal of the American Statistical Association, 109:505, 346-358, arXiv:1303.4121
Geenens, G., Wang, C. (2018). Local-likelihood transformation kernel density estimation for positive random variables. Journal of Computational and Graphical Statistics, to appear, arXiv:1602.04862
Nagler, T. (2018a). A generic approach to nonparametric function estimation with mixed data. Statistics & Probability Letters, 137:326–330, arXiv:1704.07457
Nagler, T. (2018b). Asymptotic analysis of the jittering kernel density estimator. Mathematical Methods of Statistics, in press, arXiv:1705.05431
Density, distribution function, quantile function and random generation for a 'kde1d' kernel density estimate.
dkde1d(x, obj) pkde1d(q, obj) qkde1d(p, obj) rkde1d(n, obj, quasi = FALSE)dkde1d(x, obj) pkde1d(q, obj) qkde1d(p, obj) rkde1d(n, obj, quasi = FALSE)
x |
vector of density evaluation points. |
obj |
a |
q |
vector of quantiles. |
p |
vector of probabilities. |
n |
integer; number of observations. |
quasi |
logical; the default ( |
dkde1d() gives the density, pkde1d() gives
the distribution function, qkde1d() gives the quantile function,
and rkde1d() generates random deviates.
The length of the result is determined by n for rkde1d(), and
is the length of the numerical argument for the other functions.
The density, distribution function or quantile functions estimates
evaluated respectively at x, q, or p, or a sample of n random
deviates from the estimated kernel density.
set.seed(0) # for reproducibility x <- rnorm(100) # simulate some data fit <- kde1d(x) # estimate density dkde1d(0, fit) # evaluate density estimate (close to dnorm(0)) pkde1d(0, fit) # evaluate corresponding cdf (close to pnorm(0)) qkde1d(0.5, fit) # quantile function (close to qnorm(0)) hist(rkde1d(100, fit)) # simulateset.seed(0) # for reproducibility x <- rnorm(100) # simulate some data fit <- kde1d(x) # estimate density dkde1d(0, fit) # evaluate density estimate (close to dnorm(0)) pkde1d(0, fit) # evaluate corresponding cdf (close to pnorm(0)) qkde1d(0.5, fit) # quantile function (close to qnorm(0)) hist(rkde1d(100, fit)) # simulate
Converts ordered variables to numeric and Adds deterministic uniform noise. See Details.
equi_jitter(x)equi_jitter(x)
x |
observations; the function does nothing if |
Jittering makes discrete variables continuous by adding noise. This simple trick allows to consistently estimate densities with tools designed for the continuous case (see, Nagler, 2018a/b). The drawback is that estimates are random and the noise may deteriorate the estimate by chance.
Here, we add a form of deterministic noise that makes estimators well
behaved. Tied occurences of a factor level are spread out uniformly
(i.e., equidistantly) on the interval . This is similar to
adding random noise that is uniformly distributed, conditional on the
observed outcome. Integrating over the outcome, one can check that the
unconditional noise distribution is also uniform on .
Asymptotically, the deterministic jittering variant is equivalent to the random one.
Nagler, T. (2018a). A generic approach to nonparametric function estimation with mixed data. Statistics & Probability Letters, 137:326–330, arXiv:1704.07457
Nagler, T. (2018b). Asymptotic analysis of the jittering kernel density estimator. Mathematical Methods of Statistics, in press, arXiv:1705.05431
x <- as.factor(rbinom(10, 1, 0.5)) equi_jitter(x)x <- as.factor(rbinom(10, 1, 0.5)) equi_jitter(x)
The estimators can handle data with bounded, unbounded, and discrete support, see Details.
kde1d( x, xmin = NaN, xmax = NaN, mult = 1, bw = NA, deg = 2, weights = numeric(0) )kde1d( x, xmin = NaN, xmax = NaN, mult = 1, bw = NA, deg = 2, weights = numeric(0) )
x |
vector (or one-column matrix/data frame) of observations; can be
|
xmin |
lower bound for the support of the density (only for continuous
data); |
xmax |
upper bound for the support of the density (only for continuous
data); |
mult |
positive bandwidth multiplier; the actual bandwidth used is
|
bw |
bandwidth parameter; has to be a positive number or |
deg |
degree of the polynomial; either |
weights |
optional vector of weights for individual observations. |
A gaussian kernel is used in all cases. If xmin or xmax are
finite, the density estimate will be 0 outside of . A
log-transform is used if there is only one boundary (see, Geenens and Wang,
2018); a probit transform is used if there are two (see, Geenens, 2014).
Discrete variables are handled via jittering (see, Nagler, 2018a, 2018b).
A specific form of deterministic jittering is used, see equi_jitter().
An object of class kde1d.
Geenens, G. (2014). Probit transformation for kernel density estimation on the unit interval. Journal of the American Statistical Association, 109:505, 346-358, arXiv:1303.4121
Geenens, G., Wang, C. (2018). Local-likelihood transformation kernel density estimation for positive random variables. Journal of Computational and Graphical Statistics, to appear, arXiv:1602.04862
Nagler, T. (2018a). A generic approach to nonparametric function estimation with mixed data. Statistics & Probability Letters, 137:326–330, arXiv:1704.07457
Nagler, T. (2018b). Asymptotic analysis of the jittering kernel density estimator. Mathematical Methods of Statistics, in press, arXiv:1705.05431
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.
dkde1d(), pkde1d(), qkde1d(), rkde1d(),
plot.kde1d(), lines.kde1d()
## unbounded data x <- rnorm(500) # simulate data fit <- kde1d(x) # estimate density dkde1d(0, fit) # evaluate density estimate summary(fit) # information about the estimate plot(fit) # plot the density estimate curve(dnorm(x), add = TRUE, # add true density col = "red" ) ## bounded data, log-linear x <- rgamma(500, shape = 1) # simulate data fit <- kde1d(x, xmin = 0, deg = 1) # estimate density dkde1d(seq(0, 5, by = 1), fit) # evaluate density estimate summary(fit) # information about the estimate plot(fit) # plot the density estimate curve(dgamma(x, shape = 1), # add true density add = TRUE, col = "red", from = 1e-3 ) ## discrete data x <- rbinom(500, size = 5, prob = 0.5) # simulate data x <- ordered(x, levels = 0:5) # declare as ordered fit <- kde1d(x) # estimate density dkde1d(sort(unique(x)), fit) # evaluate density estimate summary(fit) # information about the estimate plot(fit) # plot the density estimate points(ordered(0:5, 0:5), # add true density dbinom(0:5, 5, 0.5), col = "red" ) ## weighted estimate x <- rnorm(100) # simulate data weights <- rexp(100) # weights as in Bayesian bootstrap fit <- kde1d(x, weights = weights) # weighted fit plot(fit) # compare with unweighted fit lines(kde1d(x), col = 2)## unbounded data x <- rnorm(500) # simulate data fit <- kde1d(x) # estimate density dkde1d(0, fit) # evaluate density estimate summary(fit) # information about the estimate plot(fit) # plot the density estimate curve(dnorm(x), add = TRUE, # add true density col = "red" ) ## bounded data, log-linear x <- rgamma(500, shape = 1) # simulate data fit <- kde1d(x, xmin = 0, deg = 1) # estimate density dkde1d(seq(0, 5, by = 1), fit) # evaluate density estimate summary(fit) # information about the estimate plot(fit) # plot the density estimate curve(dgamma(x, shape = 1), # add true density add = TRUE, col = "red", from = 1e-3 ) ## discrete data x <- rbinom(500, size = 5, prob = 0.5) # simulate data x <- ordered(x, levels = 0:5) # declare as ordered fit <- kde1d(x) # estimate density dkde1d(sort(unique(x)), fit) # evaluate density estimate summary(fit) # information about the estimate plot(fit) # plot the density estimate points(ordered(0:5, 0:5), # add true density dbinom(0:5, 5, 0.5), col = "red" ) ## weighted estimate x <- rnorm(100) # simulate data weights <- rexp(100) # weights as in Bayesian bootstrap fit <- kde1d(x, weights = weights) # weighted fit plot(fit) # compare with unweighted fit lines(kde1d(x), col = 2)
Plotting kde1d objects
## S3 method for class 'kde1d' plot(x, ...) ## S3 method for class 'kde1d' lines(x, ...)## S3 method for class 'kde1d' plot(x, ...) ## S3 method for class 'kde1d' lines(x, ...)
x |
|
... |
further arguments passed to |
## continuous data x <- rbeta(100, shape1 = 0.3, shape2 = 0.4) # simulate data fit <- kde1d(x) # unbounded estimate plot(fit, ylim = c(0, 4)) # plot estimate curve(dbeta(x, 0.3, 0.4), # add true density col = "red", add = TRUE ) fit_bounded <- kde1d(x, xmin = 0, xmax = 1) # bounded estimate lines(fit_bounded, col = "green") ## discrete data x <- rpois(100, 3) # simulate data x <- ordered(x, levels = 0:20) # declare variable as ordered fit <- kde1d(x) # estimate density plot(fit, ylim = c(0, 0.25)) # plot density estimate points(ordered(0:20, 0:20), # add true density values dpois(0:20, 3), col = "red" )## continuous data x <- rbeta(100, shape1 = 0.3, shape2 = 0.4) # simulate data fit <- kde1d(x) # unbounded estimate plot(fit, ylim = c(0, 4)) # plot estimate curve(dbeta(x, 0.3, 0.4), # add true density col = "red", add = TRUE ) fit_bounded <- kde1d(x, xmin = 0, xmax = 1) # bounded estimate lines(fit_bounded, col = "green") ## discrete data x <- rpois(100, 3) # simulate data x <- ordered(x, levels = 0:20) # declare variable as ordered fit <- kde1d(x) # estimate density plot(fit, ylim = c(0, 0.25)) # plot density estimate points(ordered(0:20, 0:20), # add true density values dpois(0:20, 3), col = "red" )