`scipy.stats._multivariate:invwishart_gen`

source: /scipy/stats/_multivariate.py :3295

Signature

class invwishart_gen ( seed = None )

Members

Summary

An inverse Wishart random variable.

Extended Summary

The df keyword specifies the degrees of freedom. The scale keyword specifies the scale matrix, which must be symmetric and positive definite. In this context, the scale matrix is often interpreted in terms of a multivariate normal covariance matrix.

Parameters

%(_doc_default_callparams)s
%(_doc_random_state)s

Methods

pdf(x, df, scale): Probability density function.
logpdf(x, df, scale): Log of the probability density function.
rvs(df, scale, size=1, random_state=None): Draw random samples from an inverse Wishart distribution.
entropy(df, scale): Differential entropy of the distribution.

Raises

: scipy.linalg.LinAlgError: If the scale matrix scale is not positive definite.

Notes

%(_doc_callparams_note)s

The scale matrix scale must be a symmetric positive definite matrix. Singular matrices, including the symmetric positive semi-definite case, are not supported. Symmetry is not checked; only the lower triangular portion is used.

The inverse Wishart distribution is often denoted

W_{p}^{- 1} (ν, Ψ)

where $ν$ is the degrees of freedom and $Ψ$ is the $p \times p$ scale matrix.

The probability density function for invwishart has support over positive definite matrices $S$ ; if $S \sim W_{p}^{- 1} (ν, Σ)$ , then its PDF is given by:

f (S) = \frac{∣Σ ∣ ^{\frac{ν}{2}}}{2 ^{\frac{ν p}{2}} ∣ S ∣ ^{\frac{ν + p + 1}{2}} Γ _{p} ( \frac{ν}{2} )} exp (- t r (Σ S^{- 1}) /2)

If $S \sim W_{p}^{- 1} (ν, Ψ)$ (inverse Wishart) then $S^{- 1} \sim W_{p} (ν, Ψ^{- 1})$ (Wishart).

If the scale matrix is 1-dimensional and equal to one, then the inverse Wishart distribution $W_{1} (ν, 1)$ collapses to the inverse Gamma distribution with parameters shape = $\frac{ν}{2}$ and scale = $\frac{1}{2}$ .

Instead of inverting a randomly generated Wishart matrix as described in [2], here the algorithm in [4] is used to directly generate a random inverse-Wishart matrix without inversion.

Examples

import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import invwishart, invgamma
x = np.linspace(0.01, 1, 100)
iw = invwishart.pdf(x, df=6, scale=1)

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iw[:3]

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ig = invgamma.pdf(x, 6/2., scale=1./2)

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ig[:3]
plt.plot(x, iw)

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plt.show()

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The input quantiles can be any shape of array, as long as the last axis labels the components. Alternatively, the object may be called (as a function) to fix the degrees of freedom and scale parameters, returning a "frozen" inverse Wishart random variable:

rv = invwishart(df=1, scale=1)

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Aliases

scipy.stats._multivariate.invwishart_gen