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"Multinormal" (Machine Learning Method)

  • Method for LearnDistribution.
  • Models the probability density using a multivariate normal (Gaussian) distribution.

Details & Suboptions

  • "Multinormal" models the probability density of a numeric space using a multivariate normal distribution as in MultinormalDistribution.
  • The probability density for vector is proportional to , where and are learned parameters. If n is the size of the input numeric vector, is an n×n symmetric positive definite matrix called covariance, and is a size-n vector.
  • The following options can be given:
  • "CovarianceType" "Full"type of constraint on the covariance matrix
    "IntrinsicDimension" Automaticeffective dimensionality of the data to assume
  • Possible settings for "CovarianceType" include:
  • "Diagonal"only diagonal elements are learned (the others are set to 0)
    "Full"all n×n elements are learned
    "Spherical"only diagonal elements are learned and are set to be equal
  • Possible settings for "IntrinsicDimension" include:
  • Automatictry several possible dimensions
    "Heuristic"use a heuristic based on the data
    kuse the specified dimension
  • When "CovarianceType""Full" and "IntrinsicDimension"k, with k<n, a linear dimensionality reduction is performed on the data. A full k×k covariance matrix is used to model data in the reduced space (which can be interpreted as the "signal" part), while a spherical covariance matrix is used to model the n-k remaining dimensions (which can be interpreted as the "noise" part).
  • The value of "IntrinsicDimension" is ignored when "CovarianceType""Diagonal" or "CovarianceType""Spherical".
  • 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)Summary of the most common use cases

Train a "Multinormal" distribution on a numeric dataset:

In[1]:=1
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https://wolfram.com/xid/0bqb5mj7q502-6wc05d
Out[1]=1

Look at the distribution Information:

In[2]:=2
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https://wolfram.com/xid/0bqb5mj7q502-zci6sv
Out[2]=2

Obtain options information:

In[3]:=3
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https://wolfram.com/xid/0bqb5mj7q502-d4gaww
Out[3]=3

Obtain an option value directly:

In[4]:=4
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https://wolfram.com/xid/0bqb5mj7q502-tm1l4s
Out[4]=4

Compute the probability density for a new example:

In[5]:=5
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https://wolfram.com/xid/0bqb5mj7q502-37we6q
Out[5]=5

Plot the PDF along with the training data:

In[6]:=6
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https://wolfram.com/xid/0bqb5mj7q502-jfmxrj
Out[6]=6

Generate and visualize new samples:

In[7]:=7
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https://wolfram.com/xid/0bqb5mj7q502-trbtxb
Out[7]=7

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

In[1]:=1
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https://wolfram.com/xid/0bqb5mj7q502-g4novm
In[2]:=2
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https://wolfram.com/xid/0bqb5mj7q502-4ewt2p
Out[2]=2

Plot the PDF along with the training data:

In[3]:=3
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https://wolfram.com/xid/0bqb5mj7q502-i4ttni
Out[3]=3

Use SynthesizeMissingValues to impute missing values using the learned distribution:

In[4]:=4
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https://wolfram.com/xid/0bqb5mj7q502-og5slp
Out[4]=4
In[5]:=5
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https://wolfram.com/xid/0bqb5mj7q502-feqqk1
Out[5]=5

Train a "Multinormal" distribution on a nominal dataset:

In[1]:=1
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https://wolfram.com/xid/0bqb5mj7q502-36gxxt
Out[1]=1

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

In[2]:=2
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https://wolfram.com/xid/0bqb5mj7q502-wbm3in
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0bqb5mj7q502-jokluj
Out[3]=3

Use ComputeUncertainty to obtain the uncertainty on the result:

In[4]:=4
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https://wolfram.com/xid/0bqb5mj7q502-29ml4l
Out[4]=4

Increase MaxIterations to improve the estimation precision:

In[5]:=5
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https://wolfram.com/xid/0bqb5mj7q502-5lufi6
Out[5]=5

Options  (2)Common values & functionality for each option

"CovarianceType"  (1)

Train a "Multinormal" distribution with a "Full" covariance:

In[1]:=1
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https://wolfram.com/xid/0bqb5mj7q502-ol3902
In[2]:=2
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https://wolfram.com/xid/0bqb5mj7q502-1xytp2
Out[2]=2

Evaluate the PDF of the distribution at a specific point:

In[3]:=3
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https://wolfram.com/xid/0bqb5mj7q502-v8vvn8
Out[3]=3

Visualize the PDF obtained after training a multinormal with covariance types "Full", "Diagonal" and "Spherical":

In[4]:=4
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https://wolfram.com/xid/0bqb5mj7q502-3oji9z
In[5]:=5
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https://wolfram.com/xid/0bqb5mj7q502-6gqzwl
Out[5]=5

"IntrinsicDimension"  (1)

Train a "Multinormal" distribution with a "Full" covariance and an intrinsic dimension of 1:

In[1]:=1
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https://wolfram.com/xid/0bqb5mj7q502-e34hxq
In[2]:=2
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https://wolfram.com/xid/0bqb5mj7q502-vf8dac
Out[2]=2

Visualize samples generated from the distribution:

In[3]:=3
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https://wolfram.com/xid/0bqb5mj7q502-bhak2v
Out[3]=3

Compare with samples generated with an intrinsic dimension of 3:

In[4]:=4
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https://wolfram.com/xid/0bqb5mj7q502-9kjfcf
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0bqb5mj7q502-gb5sg7
Out[5]=5