SSJ
V. 1.2.5.

umontreal.iro.lecuyer.randvar
Class KernelDensityGen

java.lang.Object
  extended by umontreal.iro.lecuyer.randvar.RandomVariateGen
      extended by 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,..., xn. 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

fn(x) = (1/nh)∑i=1nk((x - xi)/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 xI at random, by inversion, using stream s, then generate random variate Y with the generator provided for the density k, and return xI + hY.

A simple formula for the bandwidth, suggested in, is h = αkh0, where

h0 = 1.36374min(sn, q/1.34)n-1/5,

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

αk = (σk-4-∞k(x)dx)1/5

where σk 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 BetaSymmetricDist (2.0, -1.0, 1.0) 1.7188 1/5 1.000
triangular TriangularDist (-1.0, 1.0, 0.0) 1.8882 1/6 0.986
Gaussian NormalDist() 0.7764 1 0.951
boxcar UniformDist (-1.0, 1.0) 1.3510 1/3 0.930
logistic LogisticDist() 0.4340 3.2899 0.888
Student-t(3) StudentDist (3.0) 0.4802 3 0.674

Table  gives the precomputed values of σk and αk 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
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
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 = αkh0 with the αk 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

SSJ
V. 1.2.5.

To submit a bug or ask questions, send an e-mail to Pierre L'Ecuyer.