BUILT-IN MATHEMATICA SYMBOL
SmoothKernelDistribution
SmoothKernelDistribution[{{x1, y1, ...}, {x2, y2, ...}, ...}]
represents a multivariate smooth kernel distribution based on the data values 
.
SmoothKernelDistribution[..., bw, ker]
represents a smooth kernel distribution with bandwidth bw and smoothing kernel ker.
- SmoothKernelDistribution returns a DataDistribution object that can be used like any other probability distribution.
- The probability density function for SmoothKernelDistribution for a value
is given by a linearly interpolated version of 
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 deviations |
| {"Adaptive",h,s} | adaptive with initial bandwidth h and sensitivity  |
| 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" | use the method of least-squares cross-validation |
| "Oversmooth" | 1.08 times wider than the standard Gaussian |
| "Scott" | use Scott's rule to determine bandwidth |
| "SheatherJones" | use the Sheather-Jones plugin estimator |
| "Silverman" | use Silverman's rule to determine bandwidth |
| "StandardDeviation" | use 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 SmoothKernelDistribution to generate a true density estimate, the function fn should be a valid 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:
-
- SmoothKernelDistribution can be used with such functions as Mean, CDF, and RandomVariate.
Create an interpolated version of a kernel density estimate for some univariate data:
Use the resulting distribution to perform analysis, including visualizing distribution functions:
| Out[3]= |  |
Compute moments and quantiles:
| Out[4]= |  |
| Out[5]= |  |
Create an interpolated version of a kernel density estimate of some bivariate data:
Visualize the estimated PDF and CDF:
| Out[3]= |  |
Compute covariance and general moments:
Out[4]//MatrixForm= |
| |  |
| Out[5]= |  |
New in 8
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