Wishart and Inverse Wishart Distributions
The Wishart distribution is the distribution of the covariance matrix of samples drawn from independent multinormal random vectors. It is a generalization of distribution to multiple dimensions. The distribution appears naturally in multivariate statistics such as regression, covariance, etc.
Generate a random positive definitive matrix to use as parameters for the Wishart distribution.
![Click for copyable input](assets.en/wishart-and-inverse-wishart-distributions/In_24.png)
\[CapitalSigma] = DiagonalMatrix[RandomReal[10, 5]];
Matrices from the Wishart distribution are symmetric and positive definite. »
![Click for copyable input](assets.en/wishart-and-inverse-wishart-distributions/In_25.png)
dist = WishartMatrixDistribution[30, \[CapitalSigma]];
mat = RandomVariate[dist];
![Click for copyable input](assets.en/wishart-and-inverse-wishart-distributions/In_26.png)
SymmetricMatrixQ[mat] && PositiveDefiniteMatrixQ[mat]
![](assets.en/wishart-and-inverse-wishart-distributions/O_12.png)
Inverse Wishart distribution is the distribution of the inverse matrices from the Wishart distribution. »
![Click for copyable input](assets.en/wishart-and-inverse-wishart-distributions/In_27.png)
invdist =
InverseWishartMatrixDistribution[30, Inverse[\[CapitalSigma]]];
invmat = RandomVariate[invdist];
Matrices from the inverse Wishart distribution are symmetric and positive definite.
![Click for copyable input](assets.en/wishart-and-inverse-wishart-distributions/In_28.png)
SymmetricMatrixQ[invmat] && PositiveDefiniteMatrixQ[invmat]
![](assets.en/wishart-and-inverse-wishart-distributions/O_13.png)
Compare the distribution of eigenvalues for matrices from the Wishart and inverse Wishart distributions.
![Click for copyable input](assets.en/wishart-and-inverse-wishart-distributions/In_29.png)
eigs = Flatten[
RandomVariate[
MatrixPropertyDistribution[Eigenvalues[x], x \[Distributed] dist],
10^4]];
inveigs =
Flatten[RandomVariate[
MatrixPropertyDistribution[Eigenvalues[x]^-1,
x \[Distributed] invdist], 10^4]];
![](assets.en/wishart-and-inverse-wishart-distributions/O_14.png)
For any nonzero vector and Wishart matrix
with scale matrix
, the statistic
is
distributed.
![Click for copyable input](assets.en/wishart-and-inverse-wishart-distributions/In_31.png)
y = #/Sqrt[#.\[CapitalSigma].#] &[RandomReal[1, 5]];
data = RandomVariate[
MatrixPropertyDistribution[y.w.y,
w \[Distributed] WishartMatrixDistribution[30, \[CapitalSigma]]],
10^4];
![Click for copyable input](assets.en/wishart-and-inverse-wishart-distributions/In_32.png)
Show[Histogram[data, Automatic, PDF, PlotTheme -> "Detailed"],
Plot[PDF[ChiSquareDistribution[30], x], {x, 0, 80}],
ImageSize -> Medium]
![](assets.en/wishart-and-inverse-wishart-distributions/O_15.png)