# Simulation of the fractional noncentral Wishart distribution

Posted on December 9, 2017 by Stéphane Laurent
Tags: R, maths, statistics

It is well known how to simulate the noncentral Wishart distribution when the number of degrees of freedom $$\nu$$ and the dimension $$d$$ satisfy $$\nu > 2d-1$$, or when $$\nu \geq d$$ is an integer. In their paper Exact and high-order discretization schemes for Wishart processes and their affine extensions, Ahdida & Alfonsi provide a method that allows to simulate the Wishart process of dimension $$d$$ for any number of degrees of freedom $$\nu \geq d-1$$ and without restrictions on the other parameters. This method allows to simulate the noncentral Wishart distribution, in the way we will expose now.

# Two properties of the noncentral Wishart distribution

We will need the two following properties.

Recall that the characteristic function of the noncentral Wishart distribution $$\mathcal{W}(\nu, \Sigma, \Theta)$$ at $$Z$$ is $\phi_{\nu,\Sigma,\Theta}(Z) = \frac{\exp\Bigl(i\textrm{tr}\bigl({(I_d - 2i Z\Sigma)}^{-1}Z\Theta\bigr)\Bigr)}{{\det(I_d - 2i Z\Sigma)}^{\frac{\nu}{2}}}.$

$$\bullet$$ First property of the Wishart distribution. Using the characteristic function, it is easy to check that $$A W A' \sim \mathcal{W}(\nu, A\Sigma A', A\Theta A')$$ when $$W \sim \mathcal{W}(\nu, \Sigma, \Theta)$$.

$$\bullet$$ Second property of the Wishart distribution. Using the characteristic function, it is not hard to check that if $$W_1$$ and $$W_2$$ are two random matrices such that $$W_1 \sim \mathcal{W}(\nu, \Sigma_1, \Theta)$$ and $$(W_2 \mid W_1) \sim \mathcal{W}(\nu, \Sigma_2, W_1)$$, then $$W_2 \sim \mathcal{W}(\nu, \Sigma_1+\Sigma_2, \Theta)$$. This result is proved in A&A’s paper only for the covariance matrices $$J_d^i$$ we will see later, by means of another method.

Let’s prove this result with the characteristic function. The conditional characteristic function of $$W_2$$ given $$W_1$$ at $$Z$$ is $\frac{\exp\Bigl(i\textrm{tr}\bigl({(I_d - 2i Z\Sigma_2)}^{-1}ZW_1\bigr)\Bigr)}{{\det(I_d - 2i Z\Sigma_2)}^{\frac{\nu}{2}}}.$ The characteristic function of $$W_2$$ is obtained by taking the expectation of this expression, and doing so we get $\frac{\phi_{\nu,\Sigma_1,\Theta}\bigl({(I_d - 2i Z\Sigma_2)}^{-1}Z\bigr)}{{\det(I_d - 2i Z\Sigma_2)}^{\frac{\nu}{2}}} = \frac{\exp\left(i\textrm{tr}\Bigl({\bigr(I_d - 2i {(I_d - 2i Z\Sigma_2)}^{-1}Z\Sigma_1\bigl)}^{-1}{(I_d - 2i Z\Sigma_2)}^{-1}Z\Theta\Bigr)\right)}{{\det(I_d - 2i Z\Sigma_2)}^{\frac{\nu}{2}} {\det(I_d - 2i {\bigl(I_d - 2i Z\Sigma_2)}^{-1}Z\Sigma_1\bigr)}^{\frac{\nu}{2}}}.$

It is easy to check that the denominator is $${\det\bigl(I_d - 2iZ(\Sigma_1+\Sigma_2)\bigr)}^{\frac{\nu}{2}}$$. The expression inside $$\textrm{tr}(\ldots)$$ at the numerator is ${\Bigl((I_d - 2i Z\Sigma_2)\bigr(I_d - 2i {(I_d - 2i Z\Sigma_2)}^{-1}Z\Sigma_1\bigl)\Bigr)}^{-1}Z\Theta = {\bigr(I_d - 2iZ(\Sigma_1+\Sigma_2)\bigl)}^{-1}Z\Theta.$

Thus we find that the the characteristic function of $$W_2$$ is $$\phi_{\nu, \Sigma_1+\Sigma_2,\Theta}$$, that is to say $$W_2 \sim \mathcal{W}(\nu, \Sigma_1+\Sigma_2, \Theta)$$.

# A&A’s simulation method

A&A’s simulation method has three steps:

• it firstly gives an algorithm to simulate $$\mathcal{W}(\nu, J_d^1, \Theta)$$, denoting by $$J_d^i$$ the $$d \times d$$ covariance matrix whose all entries are equal to zero except the $$(i,i)$$-entry which is equal to one;

• using the first step and the two properties of the Wishart distribution that we have seen, it provides a way to simulate $$\mathcal{W}(\nu, I_d^n, \Theta)$$ where $$I_d^n = J_d^1 + \ldots + J_d^n$$;

• using the second step and the first property of the Wishart distribution that we have seen, it provides a way to simulate $$\mathcal{W}(\nu, \Sigma, \Theta)$$ for any covariance matrix $$\Sigma$$.

$$\bullet$$ Simulation of $$\mathcal{W}(\nu, J_d^1, \Theta)$$. This algorithm runs as follows. Let $$(L,M,P)$$ be an extended Cholesky decomposition of $$\Theta_{2:d,2:d}$$. Set $$Q = \begin{pmatrix} 1 & 0 \\ 0 & P \end{pmatrix}$$ and $$\widetilde{\Theta} = Q\Theta Q'$$, then set $$u = L^{-1}\widetilde{\Theta}_{1, 2:(r+1)}'$$ and $$v = \widetilde{\Theta}_{1,1} - \sum_{i=1}^r u_i^2$$. Take $$Z_1, \ldots, Z_r \sim_{\text{iid}} \mathcal{N}(0,1)$$ and set $$G_i = u_i + Z_i$$. Finally, take $$X \sim \chi^2_{\nu-r, v}$$ (noncentral chi-squared distribution) independent of the $$Z_i$$, and set $W = Q' \begin{pmatrix} 1 & 0 & 0 \\ 0 & L & 0 \\ 0 & M & I_{d-r-1} \end{pmatrix} \begin{pmatrix} X + \sum_{i=1}^r G_i^2 & G' & 0 \\ G & I_r & 0 \\ 0 & 0 & 0 \end{pmatrix} \begin{pmatrix} 1 & 0 & 0 \\ 0 & L' & M' \\ 0 & 0 & I_{d-r-1} \end{pmatrix} Q.$ Then A&A have shown that $$W \sim \mathcal{W}(\nu, J_d^1, \Theta)$$.

$$\bullet$$ Simulation of $$\mathcal{W}(\nu, I_d^n, \Theta)$$. Let $$P$$ be the permutation matrix exchanging rows $$1$$ and $$2$$. Using the previous algorithm, simulate $$W_1 \sim \mathcal{W}(\nu, J_d^1, \Theta)$$. By the first property of $$\mathcal{W}$$ we have seen, $$P W_1 P \sim \mathcal{W}(\nu, J_d^2, P\Theta P)$$. Then, still using the previous algorithm, simulate $$(W_2 \mid W_1) \sim \mathcal{W}(\nu, J_d^1, P W_1 P)$$. By the second property of $$\mathcal{W}$$ we have seen, $$W_2 \sim \mathcal{W}(\nu, I_d^2, P \Theta P)$$. And by the first property, $$P W_2 P \sim \mathcal{W}(\nu, I_d^2, \Theta)$$. Continuing so on, we can simulate $$\mathcal{W}(\nu, I_d^n, \Theta)$$ for any $$n \leq d$$.

$$\bullet$$ Simulation of $$\mathcal{W}(\nu, \Sigma, \Theta)$$. Finally, given any covariance matrix $$\Sigma$$ of rank $$n$$, take the $$\widetilde{C}$$ matrix of an extended Cholesky decomposition of $$\Sigma$$ with permutation matrix $$P$$, and set $$A = P'\widetilde{C}$$. Simulate $$Y \sim \mathcal{W}\bigl(\nu, I_d^n, A^{-1}\Theta{(A^{-1})}'\bigr)$$ with the previous algorithm and finally set $$W = AYA'$$, so that $$W \sim \mathcal{W}(\nu, \Sigma, \Theta)$$ by the first property and by the property of the Cholesky decomposition.

# Checking

The algorithm is implemented in my package matrixsampling. Let’s try it.

library(matrixsampling)
p <- 6
nu <- 6.3
Sigma <- toeplitz(p:1)
Theta <- matrix(1, p, p)
nsims <- 100000
W <- rwishart(nsims, nu, Sigma, Theta)

As expected, the average simulated matrix is close to the theoretical mean $$\nu \Sigma + \Theta$$:

round((nu*Sigma + Theta) - apply(W, 1:2, mean), 2)
##       [,1]  [,2]  [,3]  [,4]  [,5]  [,6]
## [1,] -0.05 -0.04  0.00  0.00  0.02  0.04
## [2,] -0.04 -0.06 -0.03 -0.03  0.00  0.01
## [3,]  0.00 -0.03 -0.04 -0.06 -0.05 -0.05
## [4,]  0.00 -0.03 -0.06 -0.07 -0.04 -0.07
## [5,]  0.02  0.00 -0.05 -0.04 -0.03 -0.05
## [6,]  0.04  0.01 -0.05 -0.07 -0.05 -0.09

Let’s compare the theoretical characteristic function to its approximation obtained from the simulations:

z <- seq(0.001, 0.004, length.out = 20)
Z <- sapply(z, function(z){
z*diag(p) + matrix(z, p, p)
}, simplify=FALSE)
tr <- function(A) sum(diag(A))
Phi_theoretical <- sapply(Z, function(Z){
complexplus::Det(diag(p) - 2*1i*Z%*%Sigma)^(-nu/2) *
exp(1i*tr(solve(diag(p) - 2*1i*Z%*%Sigma) %*% Z %*% Theta))
})
Phi_sims <- sapply(Z, function(Z){
mean(apply(W, 3, function(W){
exp(1i*tr(Z%*%W))
}))
})
layout(t(1:2))
plot(z, Re(Phi_theoretical), type="o", pch=19)
lines(z, Re(Phi_sims), type="o", pch=19, col="red")
plot(z, Im(Phi_theoretical), type="o", pch=19)
lines(z, Im(Phi_sims), type="o", pch=19, col="red") 