Wishart Bartlett Construction

The Wishart distribution is a family of probability distributions defined over symmetric, nonnegative-definite matrices. It represents the distribution of the sample covariance matrix when drawing independent samples from a multivariate normal distribution. The Bartlett decomposition provides an efficient and insightful way to construct a random Wishart matrix from independent, scalar random variables without directly simulating high-dimensional vectors and forming their sample covariance.

Worked Example

Let us trace the Bartlett construction for a 2-dimensional Wishart matrix W with scale matrix L (where Σ = L L^T), degrees of freedom ν = 3, and scalar primitives q₁ = 1, q₂ = 1, and z = 1.

We use the Correlated scale where:

L = [[1, 0], [0.5, √0.75]] Σ = L L^T = [[1, 0.5], [0.5, 1]]

The Bartlett factor A is constructed as a lower triangular matrix:

A = [[√q1, 0], [z, √q2]] = [[1, 0], [1, 1]]

We then compute W = (L A) (L A)^T. First, we find L A:

LA = [[1*1 + 0*1, 1*0 + 0*1], [0.5*1 + √0.75*1, 0.5*0 + √0.75*1]] = [[1, 0], [1.366, 0.866]]

Multiplying L A by its transpose yields W:

W = LA LA^T = [[1, 1.366], [1.366, 2.616]]

The expected trace is ν * tr(Σ) = 3 * (1 + 1) = 6.0, and the observed determinant is det(Σ) * q₁ * q₂ = 0.75 * 1 * 1 = 0.75.

Experiment

We can observe how these parameters independently affect the matrix structure and geometry by interacting with the module.

Change the Scale to "Anisotropic". The ellipses stretch along the vertical axis, and the expected trace jumps from 6.0 (for Correlated) to 15.0, reflecting Σ = [[1, 0], [0, 4]].
Increase the Degrees to 5. Notice how the expected trace and determinant increase proportionally, but the matrix W remains unchanged since the primitive values q₁, q₂, and z are fixed.
Set Normal value to 0. This removes the off-diagonal correlation introduced by z, which changes the covariance ellipse of W to perfectly align with the axes when the scale is "Identity".

Limitations

This construction relies on continuous probability distributions (chi-square and standard normal). The visualization uses specific fixed numeric values for q₁, q₂, and z to clearly demonstrate the algebraic process of forming the Wishart matrix.

However, this deterministic interaction replaces the true random sampling step. A real Wishart variate requires sampling q₁ ~ χ²(ν), q₂ ~ χ²(ν-1), and z ~ N(0,1). Additionally, the module restricts degrees of freedom ν ≥ 2 for the 2D case to ensure that the constructed matrix is almost surely strictly positive definite. If we allowed ν < p (where p is the dimension), the matrix would be positive semi-definite and singular, causing det(W) to be 0.

References

The Bartlett decomposition of the Wishart distribution provides a fundamental method for efficient generation of random covariance matrices.

Read the primary reference on matrix decompositions and Wishart sampling: M. S. Srivastava (1965) On the distribution of a multiple correlation matrix for more insights into random matrix theory.

Wishart Bartlett Construction