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WishartMatrixDistribution
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WishartMatrixDistribution

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represents a Wishart matrix distribution with ν degrees of freedom and covariance matrix Σ.

Details

  • WishartMatrixDistribution is the distribution of the sample covariance from ν independent realizations of a multivariate Gaussian distribution with covariance matrix Σ when the degrees of freedom parameter ν is an integer.
  • WishartMatrixDistribution is also known as Wishart–Laguerre ensemble.
  • The probability density for a symmetric matrix in a Wishart matrix distribution is proportional to , where is the size of matrix Σ.
  • The covariance matrix can be any positive definite symmetric matrix of dimensions and ν can be any real number greater than .
  • WishartMatrixDistribution can be used with such functions as MatrixPropertyDistribution, EstimatedDistribution, and RandomVariate.

Examples

open allclose all

Basic Examples  (3)Summary of the most common use cases

Generate a pseudorandom matrix:

In[1]:=1
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https://wolfram.com/xid/0dblm6mpdlz4pnsn7u-lcf25m
Out[1]=1

Check that it is symmetric and positive definite:

In[2]:=2
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https://wolfram.com/xid/0dblm6mpdlz4pnsn7u-436hr
Out[2]=2

Sample eigenvalues of a Wishart random matrix using MatrixPropertyDistribution:

In[1]:=1
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https://wolfram.com/xid/0dblm6mpdlz4pnsn7u-4dxs2

Estimate joint distribution of eigenvalues:

In[2]:=2
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https://wolfram.com/xid/0dblm6mpdlz4pnsn7u-eq63i5
Out[2]=2

Mean and variance:

In[1]:=1
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https://wolfram.com/xid/0dblm6mpdlz4pnsn7u-sxr
In[2]:=2
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https://wolfram.com/xid/0dblm6mpdlz4pnsn7u-g5k

Scope  (6)Survey of the scope of standard use cases

Generate a single pseudorandom matrix:

In[1]:=1
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https://wolfram.com/xid/0dblm6mpdlz4pnsn7u-d13sc9
Out[1]=1

Generate a set of pseudorandom matrices:

In[1]:=1
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https://wolfram.com/xid/0dblm6mpdlz4pnsn7u-y6wkjj
Out[1]=1

Sample at extended precision:

In[1]:=1
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https://wolfram.com/xid/0dblm6mpdlz4pnsn7u-exxjnv
Out[1]=1

Compute statistical properties numerically:

In[1]:=1
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https://wolfram.com/xid/0dblm6mpdlz4pnsn7u-6gedk9

Numerically approximate expectation of the largest matrix eigenvalue :

In[2]:=2
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https://wolfram.com/xid/0dblm6mpdlz4pnsn7u-3m1a2c
Out[2]=2

Distribution parameters estimation:

In[1]:=1
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https://wolfram.com/xid/0dblm6mpdlz4pnsn7u-0fi0u2

Estimate the distribution parameters from sample data:

In[2]:=2
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https://wolfram.com/xid/0dblm6mpdlz4pnsn7u-epi747
Out[2]=2

Compare LogLikelihood for both distributions:

In[3]:=3
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https://wolfram.com/xid/0dblm6mpdlz4pnsn7u-2slvjw
Out[3]=3

Skewness and kurtosis:

In[1]:=1
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https://wolfram.com/xid/0dblm6mpdlz4pnsn7u-h521li
In[2]:=2
✖
https://wolfram.com/xid/0dblm6mpdlz4pnsn7u-6eko8

Applications  (2)Sample problems that can be solved with this function

When n and p (the dimension of the covariance matrix Σ) are both large, the scaled largest eigenvalue of a matrix from a Wishart ensemble with identity covariance is approximately distributed as a Tracy–Widom distribution:

In[1]:=1
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https://wolfram.com/xid/0dblm6mpdlz4pnsn7u-hkqwkr

Sample the scaled largest eigenvalue:

In[2]:=2
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https://wolfram.com/xid/0dblm6mpdlz4pnsn7u-ez2ac6

Check goodness of fit with TracyWidomDistribution:

In[3]:=3
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https://wolfram.com/xid/0dblm6mpdlz4pnsn7u-gd9lj
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0dblm6mpdlz4pnsn7u-deemk4
Out[4]=4

Algebraically independent components of a symmetric Wishart matrix have a known PDF:

In[1]:=1
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https://wolfram.com/xid/0dblm6mpdlz4pnsn7u-bbmko0
In[2]:=2
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https://wolfram.com/xid/0dblm6mpdlz4pnsn7u-qkizda
In[3]:=3
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https://wolfram.com/xid/0dblm6mpdlz4pnsn7u-fmq8va

Build the distribution of independent components of a Wishart matrix:

In[4]:=4
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https://wolfram.com/xid/0dblm6mpdlz4pnsn7u-e05of2
In[5]:=5
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https://wolfram.com/xid/0dblm6mpdlz4pnsn7u-b3zfyf
Out[5]=5

Find the joint distribution of a diagonal element:

In[6]:=6
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https://wolfram.com/xid/0dblm6mpdlz4pnsn7u-gl2gqn
Out[6]=6

Use MatrixPropertyDistribution to sample diagonal elements of Wishart matrices:

In[7]:=7
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https://wolfram.com/xid/0dblm6mpdlz4pnsn7u-ftj7qd

Check goodness of fit:

In[8]:=8
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https://wolfram.com/xid/0dblm6mpdlz4pnsn7u-dxkkan
Out[8]=8
In[9]:=9
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https://wolfram.com/xid/0dblm6mpdlz4pnsn7u-kmkqw0
Out[9]=9

Properties & Relations  (4)Properties of the function, and connections to other functions

Use MatrixPropertyDistribution to represent the scaled eigenvalues of a Wishart random matrix with identity covariance:

In[1]:=1
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https://wolfram.com/xid/0dblm6mpdlz4pnsn7u-rxoefr

The limiting distribution of eigenvalues follows MarchenkoPasturDistribution:

In[2]:=2
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https://wolfram.com/xid/0dblm6mpdlz4pnsn7u-oww4ea
In[3]:=3
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https://wolfram.com/xid/0dblm6mpdlz4pnsn7u-dkiqlj
Out[3]=3

Compare the histogram of the eigenvalues with the PDF:

In[4]:=4
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https://wolfram.com/xid/0dblm6mpdlz4pnsn7u-3izrql
Out[4]=4

The expression n x.TemplateBox[{m}, Inverse].x, where and are, respectively, an independent Gaussian vector and Wishart matrix, follows HotellingTSquareDistribution:

In[1]:=1
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https://wolfram.com/xid/0dblm6mpdlz4pnsn7u-bsax8f

Use MatrixPropertyDistribution to sample expressions n x.TemplateBox[{m}, Inverse].x:

In[2]:=2
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https://wolfram.com/xid/0dblm6mpdlz4pnsn7u-ck8vqy
In[3]:=3
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https://wolfram.com/xid/0dblm6mpdlz4pnsn7u-bcpdu4
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0dblm6mpdlz4pnsn7u-lidu3v
Out[4]=4

Diagonal elements of a Wishart random matrix each follow a scaled χ2 distribution:

In[1]:=1
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https://wolfram.com/xid/0dblm6mpdlz4pnsn7u-uslpw

Test against applicably scaled χ2 distributions:

In[2]:=2
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https://wolfram.com/xid/0dblm6mpdlz4pnsn7u-fvpoh2
In[3]:=3
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https://wolfram.com/xid/0dblm6mpdlz4pnsn7u-q5ery4
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0dblm6mpdlz4pnsn7u-mb30uv
In[5]:=5
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https://wolfram.com/xid/0dblm6mpdlz4pnsn7u-bgvtqd
Out[5]=5

Diagonal elements are not independent:

In[6]:=6
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https://wolfram.com/xid/0dblm6mpdlz4pnsn7u-ipjqm3
Out[6]=6
In[7]:=7
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https://wolfram.com/xid/0dblm6mpdlz4pnsn7u-l2xbzm
Out[7]=7

For any nonzero vector and Wishart matrix with scale matrix , is χ2 distributed:

In[1]:=1
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https://wolfram.com/xid/0dblm6mpdlz4pnsn7u-dus3vi
In[2]:=2
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https://wolfram.com/xid/0dblm6mpdlz4pnsn7u-b4tu5a
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0dblm6mpdlz4pnsn7u-fmhnqz
Out[3]=3
Wolfram Research (2015), WishartMatrixDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/WishartMatrixDistribution.html (updated 2017).
✖
Wolfram Research (2015), WishartMatrixDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/WishartMatrixDistribution.html (updated 2017).

Text

Wolfram Research (2015), WishartMatrixDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/WishartMatrixDistribution.html (updated 2017).

✖
Wolfram Research (2015), WishartMatrixDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/WishartMatrixDistribution.html (updated 2017).

CMS

Wolfram Language. 2015. "WishartMatrixDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2017. https://reference.wolfram.com/language/ref/WishartMatrixDistribution.html.

✖
Wolfram Language. 2015. "WishartMatrixDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2017. https://reference.wolfram.com/language/ref/WishartMatrixDistribution.html.

APA

Wolfram Language. (2015). WishartMatrixDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/WishartMatrixDistribution.html

✖
Wolfram Language. (2015). WishartMatrixDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/WishartMatrixDistribution.html

BibTeX

@misc{reference.wolfram_2025_wishartmatrixdistribution, author="Wolfram Research", title="{WishartMatrixDistribution}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/WishartMatrixDistribution.html}", note=[Accessed: 29-March-2025 ]}

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@misc{reference.wolfram_2025_wishartmatrixdistribution, author="Wolfram Research", title="{WishartMatrixDistribution}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/WishartMatrixDistribution.html}", note=[Accessed: 29-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_wishartmatrixdistribution, organization={Wolfram Research}, title={WishartMatrixDistribution}, year={2017}, url={https://reference.wolfram.com/language/ref/WishartMatrixDistribution.html}, note=[Accessed: 29-March-2025 ]}

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@online{reference.wolfram_2025_wishartmatrixdistribution, organization={Wolfram Research}, title={WishartMatrixDistribution}, year={2017}, url={https://reference.wolfram.com/language/ref/WishartMatrixDistribution.html}, note=[Accessed: 29-March-2025 ]}