BUILT-IN MATHEMATICA SYMBOL

# KernelMixtureDistribution

KernelMixtureDistribution[{x1, x2, ...}]
represents a kernel mixture distribution based on the data values .

KernelMixtureDistribution[{{x1, y1, ...}, {x2, y2, ...}, ...}]
represents a multivariate kernel mixture distribution based on data values .

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 OptionsDetails 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 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 kernel is used.
• For multivariate densities, the kernel function ker can be specified as product and radial types using and 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.

## ExamplesExamplesopen allclose all

### Basic Examples (3)Basic Examples (3)

Create a kernel density estimate of univariate data:

Use the resulting distribution to perform analysis, including visualizing distribution functions:

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Compute moments and quantiles:

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Create a kernel density estimate of some bivariate data:

Visualize the estimated PDF and CDF:

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Compute covariance and general moments:

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Create symbolic representations of kernel density estimates:

Investigate symbolic properties:

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