SSJ
V. 2.6.

## 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 KernelDensityGenextends 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.

 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 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, toString`

Methods inherited from class java.lang.Object
`equals, getClass, hashCode, notify, notifyAll, 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. 2.6.

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