InverseWishartMatrixDistribution

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`InverseWishartMatrixDistribution`

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InverseWishartMatrixDistribution[ν,Σ]

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

Details

The probability density for a symmetric matrix in an inverse Wishart matrix distribution is proportional to , where is the size of matrix Σ.
For a matrix distributed as InverseWishartMatrixDistribution[ν,Σ], the inverse is distributed as WishartMatrixDistribution[ν,Σ^-1].
The covariance matrix can be any positive definite symmetric matrix of dimensions and ν can be any real number greater than .
InverseWishartMatrixDistribution can be used with such functions as MatrixPropertyDistribution, EstimatedDistribution, and RandomVariate.

Examples

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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/0ce0v1uwsetz3mjigjgyi-lcf25m

Out[1]=1

Check that it is positive definite:

In[2]:=2

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https://wolfram.com/xid/0ce0v1uwsetz3mjigjgyi-436hr

Out[2]=2

Sample eigenvalues of an inverse Wishart random matrix using MatrixPropertyDistribution:

In[1]:=1

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https://wolfram.com/xid/0ce0v1uwsetz3mjigjgyi-c7w4o4

In[2]:=2

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https://wolfram.com/xid/0ce0v1uwsetz3mjigjgyi-eq63i5

Out[2]=2

Mean and variance:

In[1]:=1

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https://wolfram.com/xid/0ce0v1uwsetz3mjigjgyi-sxr

In[2]:=2

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https://wolfram.com/xid/0ce0v1uwsetz3mjigjgyi-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/0ce0v1uwsetz3mjigjgyi-d13sc9

Out[1]=1

Generate a set of pseudorandom matrices:

In[1]:=1

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https://wolfram.com/xid/0ce0v1uwsetz3mjigjgyi-y6wkjj

Out[1]=1

Sample at extended precision:

In[1]:=1

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https://wolfram.com/xid/0ce0v1uwsetz3mjigjgyi-exxjnv

Out[1]=1

Compute statistical properties numerically:

In[1]:=1

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https://wolfram.com/xid/0ce0v1uwsetz3mjigjgyi-6gedk9

Numerically approximate expectation of the largest matrix eigenvalue :

In[2]:=2

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https://wolfram.com/xid/0ce0v1uwsetz3mjigjgyi-3m1a2c

Out[2]=2

Distribution parameters estimation:

In[1]:=1

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https://wolfram.com/xid/0ce0v1uwsetz3mjigjgyi-ndcda

Estimate the distribution parameters from sample data:

In[2]:=2

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https://wolfram.com/xid/0ce0v1uwsetz3mjigjgyi-epi747

Out[2]=2

Compare LogLikelihood for both distributions:

In[3]:=3

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https://wolfram.com/xid/0ce0v1uwsetz3mjigjgyi-2slvjw

Out[3]=3

Skewness:

In[1]:=1

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https://wolfram.com/xid/0ce0v1uwsetz3mjigjgyi-h521li

Properties & Relations (3)Properties of the function, and connections to other functions

, where and are independent Gaussian vector and Wishart matrix follows HotellingTSquareDistribution:

In[1]:=1

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https://wolfram.com/xid/0ce0v1uwsetz3mjigjgyi-bsax8f

Use MatrixPropertyDistribution to sample expressions :

In[4]:=4

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https://wolfram.com/xid/0ce0v1uwsetz3mjigjgyi-ck8vqy

In[5]:=5

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https://wolfram.com/xid/0ce0v1uwsetz3mjigjgyi-bcpdu4

Out[5]=5

In[6]:=6

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https://wolfram.com/xid/0ce0v1uwsetz3mjigjgyi-lidu3v

Out[6]=6

Any diagonal element of inverse Wishart random matrix follows scaled inverse χ² distribution:

In[1]:=1

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https://wolfram.com/xid/0ce0v1uwsetz3mjigjgyi-uslpw

In[2]:=2

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https://wolfram.com/xid/0ce0v1uwsetz3mjigjgyi-q5ery4

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0ce0v1uwsetz3mjigjgyi-bgvtqd

Out[3]=3

Diagonal elements are not independent:

In[4]:=4

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https://wolfram.com/xid/0ce0v1uwsetz3mjigjgyi-ipjqm3

Out[4]=4

In[5]:=5

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https://wolfram.com/xid/0ce0v1uwsetz3mjigjgyi-l2xbzm

Out[5]=5

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

In[1]:=1

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https://wolfram.com/xid/0ce0v1uwsetz3mjigjgyi-dus3vi

In[2]:=2

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https://wolfram.com/xid/0ce0v1uwsetz3mjigjgyi-b4tu5a

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0ce0v1uwsetz3mjigjgyi-fmhnqz

Out[3]=3

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