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:
  • hbandwidth to use
    {"Standardized",h}bandwidth in units of standard deviation
    {"Adaptive",h,s}adaptive bandwidth with initial bandwidth h and sensitivity s
    Automaticautomatically 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"
    funcf_nu in R
  • 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:
  • MaxMixtureKernelsAutomaticmax 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:

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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:

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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:

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Investigate symbolic properties:

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