# "KernelDensityEstimation"(Machine Learning Method)

• Method for LearnDistribution.
• Models probability density with a mixture of simple distributions.

# Details & Suboptions

• "KernelDensityEstimation" is a nonparametric method that models the probability density of a numeric space with a mixture of simple distributions (called kernels) centered around each training example, as in KernelMixtureDistribution.
• The probability density function for a vector is given by for a kernel function , kernel size and a number of training examples m.
• The following options can be given:
•  Method "Fixed" kernel size method "KernelSize" Automatic size of the kernels when Method"Fixed" "KernelType" "Gaussian" type of kernel used "NeighborsNumber" Automatic kernel size expressed as a number of neighbors
• Possible settings for "KernelType" include:
•  "Gaussian" each kernel is a Gaussian distribution "Ball" each kernel is a uniform distribution on a ball
• Possible settings for Method include:
•  "Adaptive" kernel sizes can differ from each other "Fixed" all kernels have the same size
• When "KernelType""Gaussian", each kernel is a spherical Gaussian (product of independent normal distributions ), and "KernelSize" h refers to the standard deviation of the normal distribution.
• When "KernelType""Ball", each kernel is a uniform distribution inside a sphere, and "KernelSize" refers to the radius of the sphere.
• The value of "NeighborsNumber"k is converted into kernel size(s), so that a kernel centered around a training example typically "contains" k other training examples. If "KernelType""Ball", "contains" refers to examples that are inside the ball. If "KernelType""Gaussian", "contains" refers to examples that are inside a ball of radius h where n is the dimension of the data.
• When Method"Fixed" and "NeighborsNumber"k, a unique kernel size is found such that training examples contain on average k other examples.
• When Method"Adaptive" and "NeighborsNumber"k, each training example adapts its kernel size such that it contains about k other examples.
• Because of preprocessing, the "NeighborsNumber" option is typically a more convenient way to control kernel sizes than "KernelSize". When Method"Fixed", the value of "KernelSize" supersedes the value of "NeighborsNumber".
• Information[LearnedDistribution[],"MethodOption"] can be used to extract the values of options chosen by the automation system.
• LearnDistribution[,FeatureExtractor"Minimal"] can be used to remove most preprocessing and directly access the method.

# Examples

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## Basic Examples(3)

Train a "KernelDensityEstimation" distribution on a numeric dataset:

Look at the distribution Information:

Obtain options information:

Obtain an option value directly:

Compute the probability density for a new example:

Plot the PDF along with the training data:

Generate and visualize new samples:

Train a "KernelDensityEstimation" distribution on a two-dimensional dataset:

Plot the PDF along with the training data:

Use SynthesizeMissingValues to impute missing values using the learned distribution:

Train a "KernelDensityEstimation" distribution on a nominal dataset:

Because of the necessary preprocessing, the PDF computation is not exact:

Use ComputeUncertainty to obtain the uncertainty on the result:

Increase MaxIterations to improve the estimation precision:

## Options(4)

### "KernelSize"(1)

Train a kernel mixture distribution with a kernel size of 0.2:

Evaluate the PDF of the distribution at a specific point:

Visualize the PDF obtained after training a kernel mixture distribution with various kernel sizes:

### "KernelType"(1)

Train a "KernelDensityEstimation" distribution with a "Ball" kernel:

Evaluate the PDF of the distribution at a specific point:

Visualize the PDF obtained after training a kernel mixture distribution with a "Ball" and a "Gaussian" kernel:

### Method(1)

Train a "KernelDensityEstimation" distribution with the "Adaptive" method:

Evaluate the PDF of the distribution at a specific point:

Visualize the PDF obtained after training a kernel mixture distribution with a "Ball" and a "Gaussian" kernel:

### "NeighborsNumber"(1)

Train a kernel mixture distribution with a kernel size of about 10 neighbors:

Evaluate the PDF of the distribution at a specific point:

Visualize the PDF obtained after training a kernel mixture distribution with various kernel sizes expressed as neighbors numbers: