umontreal.iro.lecuyer.randvar

Class KernelDensityGen

java.lang.Object

umontreal.iro.lecuyer.randvar.RandomVariateGen

umontreal.iro.lecuyer.randvar.KernelDensityGen

Direct Known Subclasses:

KernelDensityVarCorrectGen

public class KernelDensityGen
extends RandomVariateGen

This class implements random variate generators for distributions obtained via kernel density estimation methods from a set of n individual observations x₁,..., x_n. The basic idea is to center a copy of the same symmetric density at each observation and take an equally weighted mixture of the n copies as an estimator of the density from which the observations come. The resulting kernel density has the general form

f_n(x) = (1/nh)∑_i=1ⁿk((x - x_i)/h),

where k is a fixed pre-selected density called the kernel and h is a positive constant called the bandwidth or smoothing factor. A difficult practical issue is the selection of k and h. Several approaches have been proposed for that; see, e.g.,.

The constructor of a generator from a kernel density requires a random stream s, the n observations in the form of an empirical distribution, a random variate generator for the kernel density k, and the value of the bandwidth h. The random variates are then generated as follows: select an observation x_I at random, by inversion, using stream s, then generate random variate Y with the generator provided for the density k, and return x_I + hY.

A simple formula for the bandwidth, suggested in, is h = α_kh₀, where

h₀ = 1.36374min(s_n, q/1.34)n^-1/5,

s_n and q are the empirical standard deviation and the interquartile range of the n observations, and α_k is a constant that depends on the type of kernel k. It is defined by

α_k = (σ_k^-4∫_-∞^∞k(x)dx)^1/5

where σ_k is the standard deviation of the density k. The static method getBaseBandwidth permits one to compute h₀ for a given empirical distribution.

Table: Some suggested kernels

name	constructor	αk	σk2	efficiency
Epanechnikov	BetaSymmetricalDist(2, -1, 1)	1.7188	1/5	1.000
triangular	TriangularDist(-1, 1, 0)	1.8882	1/6	0.986
Gaussian	NormalDist()	0.7764	1	0.951
boxcar	UniformDist(-1, 1)	1.3510	1/3	0.930
logistic	LogisticDist()	0.4340	3.2899	0.888
Student-t(3)	StudentDist(3)	0.4802	3	0.674

Table gives the precomputed values of σ_k and α_k for selected (popular) kernels. The values are taken from. The second column gives the name of a function (in this package) that constructs the corresponding distribution. The efficiency of a kernel is defined as the ratio of its mean integrated square error over that of the Epanechnikov kernel, which has optimal efficiency and corresponds to the beta distribution with parameters (2, 2) over the interval (- 1, 1).

Constructor Summary

Constructors

Constructor and Description
KernelDensityGen(RandomStream s, EmpiricalDist dist, NormalGen kGen) This constructor uses a gaussian kernel and the default bandwidth h = αkh0 with the αk suggested in Table for the gaussian distribution.
KernelDensityGen(RandomStream s, EmpiricalDist dist, RandomVariateGen kGen, double h) Creates a new generator for a kernel density estimated from the observations given by the empirical distribution dist, using stream s to select the observations, generator kGen to generate the added noise from the kernel density, and bandwidth h.

Method Summary

Methods

Modifier and Type	Method and Description
static double	getBaseBandwidth(EmpiricalDist dist) Computes and returns the value of h0 in.
double	nextDouble() Generates a random number from the continuous distribution contained in this object.
void	setBandwidth(double h) Sets the bandwidth to h.
void	setPositiveReflection(boolean reflect) After this method is called with true, the generator will produce only positive values, by using the reflection method: replace all negative values by their absolute values.

Methods inherited from class umontreal.iro.lecuyer.randvar.RandomVariateGen

getDistribution, getStream, nextArrayOfDouble, setStream

Methods inherited from class java.lang.Object

equals, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

KernelDensityGen

public KernelDensityGen(RandomStream s,
                EmpiricalDist dist,
                RandomVariateGen kGen,
                double h)

Creates a new generator for a kernel density estimated from the observations given by the empirical distribution dist, using stream s to select the observations, generator kGen to generate the added noise from the kernel density, and bandwidth h.

KernelDensityGen

public KernelDensityGen(RandomStream s,
                EmpiricalDist dist,
                NormalGen kGen)

This constructor uses a gaussian kernel and the default bandwidth h = α_kh₀ with the α_k suggested in Table for the gaussian distribution. This kernel has an efficiency of 0.951.

Method Detail

getBaseBandwidth

public static double getBaseBandwidth(EmpiricalDist dist)

Computes and returns the value of h₀ in.

setBandwidth

public void setBandwidth(double h)

Sets the bandwidth to h.

setPositiveReflection

public void setPositiveReflection(boolean reflect)

After this method is called with true, the generator will produce only positive values, by using the reflection method: replace all negative values by their absolute values. That is, nextDouble will return | x| if x is the generated variate. The mecanism is disabled when the method is called with false.

nextDouble

public double nextDouble()

Description copied from class: RandomVariateGen

Generates a random number from the continuous distribution contained in this object. By default, this method uses inversion by calling the inverseF method of the distribution object. Alternative generating methods are provided in subclasses.

Overrides:

nextDouble in class RandomVariateGen

Returns:

the generated value