KernelMixtureDistribution
KernelMixtureDistribution[{x1,x2,…}]
represents a kernel mixture distribution based on the data values xi.
KernelMixtureDistribution[{{x1,y1,…},{x2,y2,…},…}]
represents a multivariate kernel mixture distribution based on data values {xi,yi,…}.
KernelMixtureDistribution[…,bw]
represents a kernel mixture distribution with bandwidth bw.
KernelMixtureDistribution[…,bw,ker]
represents a kernel mixture distribution with bandwidth bw and smoothing kernel ker.
Details and Options
- KernelMixtureDistribution returns a DataDistribution object that can be used like any other probability distribution.
- The probability density function for KernelMixtureDistribution for a value
is given by 
for a smoothing kernel 
and bandwidth parameter 
. - The following bandwidth specifications bw can be given:
-
h bandwidth to use {"Standardized",h} bandwidth in units of standard deviation {"Adaptive",h,s} adaptive bandwidth with initial bandwidth h and sensitivity s Automatic automatically computed bandwidth "name" use a named bandwidth selection method {bwx,bwy,…} separate bandwidth specifications for x, y, etc. - For multivariate densities, h can be a positive definite symmetric matrix.
- For adaptive bandwidths, the sensitivity s must be a real number between 0 and 1 or Automatic. If Automatic is used, s is set to
, where 
is the dimensionality of the data. - Possible named bandwidth selection methods include:
-
"LeastSquaresCrossValidation" uses the method of least-squares cross-validation "Oversmooth" 1.08 times wider than the standard Gaussian "Scott" uses Scott's rule to determine bandwidth "SheatherJones" uses the Sheather–Jones plugin estimator "Silverman" uses Silverman's rule to determine bandwidth "StandardDeviation" uses the standard deviation as bandwidth "StandardGaussian" optimal bandwidth for standard normal data - By default, the "Silverman" method is used.
- For automatic bandwidth computation, constant arrays are assumed to have unit variance.
- The following kernel specifications ker can be given:
-
"Biweight" 
"Cosine" 
"Epanechnikov" 
"Gaussian" 
"Rectangular" 
"SemiCircle" 
"Triangular" 
"Triweight" 
func 
- In order for KernelMixtureDistribution to generate a true density estimate, the function fn should be a valid univariate probability density function.
- By default, the "Gaussian" kernel is used.
- For multivariate densities, the kernel function ker can be specified as product and radial types using {"Product",ker} and {"Radial",ker}, respectively. Product-type kernels are used if no type is specified.
- The precision used for density estimation is the minimum precision given in the bw and data.
- The following options can be given:
-
MaxMixtureKernels Automatic max number of kernels to use - KernelMixtureDistribution can be used with such functions as Mean, CDF, and RandomVariate.
Examples
open all close allBasic Examples (3)
Create a kernel density estimate of univariate data:
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Use the resulting distribution to perform analysis, including visualizing distribution functions:
Compute moments and quantiles:
Create a kernel density estimate of some bivariate data:
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Visualize the estimated PDF and CDF:
Compute covariance and general moments:
Create symbolic representations of kernel density estimates:
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