Nothing
Using the wishmom Package"
In wishmom: Compute Moments Related to Beta-Wishart and Inverse Beta-Wishart Distributions
knitr::opts_chunk$set(
collapse = TRUE,
comment = "#>"
)
Introduction
The wishmom package provides functions to compute the expectation of matrix-valued functions of $\beta$-Wishart and inverse $\beta$-Wishart distributions ($\beta=1$: Real Wishart, $\beta=2$: Complex Wishart, $\beta=4$: Quaternion Wishart, $\beta=8$: Octonion Wishart). The main functions in this package are wishmom() and iwishmom(), which handle the numerical computation of moments of $\beta$-Wishart and inverse $\beta$-Wishart distributions, respectively. The corresponding functions for generating analytical expressions of moments
of $\beta$-Wishart and inverse $\beta$-Wishart distributions are wishmom_sym() and iwishmom_sym(). These programs are developed based on the results in Letac and Massam (2004)
and Hillier and Kan (2024).
You can install the package and load it using:
install.packages("wishmom")
library("wishmom")
Mathematical Background
$\beta$-Wishart Distribution
The $\beta$-Wishart distribution is a fundamental distribution in multivariate statistics. When $\beta=1,\;2,\;4,\;8$, it is the real Wishart, complex Wishart, quaternion Wishart, and octonion Wishart, respectively. The density function of $W \sim W_m^\beta(n,\Sigma)$, i,e., a $\beta$-Wishart distribution
with $n$ degrees of freedom and an $m \times m$ covariance matrix $\Sigma$, is given by (when $n > m-1$) (see Díaz-García and Gutiérrez-Jáimez (2011, Corollary 1))
[
f(W) = \frac{\left(\frac{\beta}{2}\right)^\frac{mn\beta}{2}}{\Gamma_m^{(\beta)}\left(\frac{n \beta}{2}\right)
|\Sigma|^\frac{n \beta}{2}}|W|^{\frac{(n-m+1)\beta}{2}-1}
\mbox{etr}\left(-\frac{\beta \Sigma^{-1}W}{2}\right),
]
where
[
\Gamma_m^{(\beta)}(a) = \pi^\frac{m(m-1)\beta}{4}\prod_{i=1}^m \Gamma\left(a-\frac{(i-1)\beta}{2}\right).
]
Note that we do not require $n$ to be an integer but
the definition of the density of $W$ requires $\beta = 1,\;2,\;4$ or $8$. However, if our interest is only on the functions of eigenvalues of $W$, we can
generalize this to any real $\beta>0$. Therefore, for
any symmetric functions (say power-sum) of the eigenvalues
of $W$, they can be well defined even when $\beta$ is not equal to $1,\;
2,\;4$ or $8$. For $W \sim W_m^\beta(n,\Sigma)$, the joint density of its eigenvalues $\lambda_1 \geq \cdots \geq \lambda_m$ is given by (see Dresnky, Edelman, Genoar, Kan, and Koev (2021))
[
f(\lambda_1,\ldots,\lambda_m) = \frac{| \Sigma|^{-\frac{n\beta}{2}}}{\mathcal{K}m^
{(\beta)}\left(\frac{n\beta}{2}\right)}
|\Lambda|^{\frac{(n-m+1)\beta}{2}-1}
{}_0^{}F_0^{(\beta)}\left(-\frac{\beta}{2}\Lambda,\Sigma^{-1}\right)\prod{1 \leq i < j \leq m}(\lambda_i-\lambda_j)^{\beta},
]
where $\Lambda = \mbox{Diag}(\lambda_1,\ldots,\lambda_m)$,
[
\mathcal{K}_m^{(\beta)}(a) =
\frac{\left(\frac{2}{\beta}\right)^{ma}}
{\pi^{\frac{m(m-1)\beta}{2}}}
\frac{\Gamma_m^{(\beta)}\left(\frac{m\beta}{2}\right)\Gamma_m^{(\beta)}(a)}{\left[\Gamma\left(\frac{\beta}{2}\right)\right]^m},
]
[
{}0^{}F_0^{(\beta)}(A,B)
= \sum{k=0}^\infty \sum_{\kappa \vdash k}
\frac{C_\kappa^{(\beta)}(A)C_\kappa^{(\beta)}(B)}
{k!C_\kappa^{(\beta)}(I_m)},
]
and $C_\kappa^{(\beta)}(X)$ is the Jack function of
the eigenvalues of $X$.
Instead of using $\beta$, we use $\alpha = 2/\beta$
in our programs to describe the type of Wishart distribution.
Therefore, $\alpha=2$ is for real Wishart, $\alpha=1$ is for complex Wishart, and $\alpha =1/2$ is for quaternion Wishart.
Moments of Matrix-valued Functions of $\beta$-Wishart and Inverse $\beta$-Wishart Distributions
Let $\lambda = (\lambda_1,\ldots, \lambda_k)$ be an integer partition of a positive integer $k$, where $|\lambda|= \lambda_1+\ldots+\lambda_k=k$, with
$\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_k \geq 0$.
The power-sum symmetric function $p_\lambda(W)$
of $W$ corresponding to a partition $\lambda$ is defined as
[
p_{\lambda}(W) = \prod_{i=1}^{\ell(\lambda)}p_{\lambda_i}(W),
]
where $\ell(\lambda)$ is the number of non-zero parts of $\lambda$,
and $p_i(W) = \mbox{tr}(W^i)$. We are interested
in computing
[
\mathbb{E}[W^rp_{\lambda}(W)]\;\;\;\;\mbox{and}
\;\;\;\;
\mathbb{E}[W^{-r}p_{\lambda}(W^{-1})],
]
where $W \sim W^{\beta}_m(n,\Sigma)$.
The method that we use is based on a generalization of the recurrence relations given in Hillier and Kan (2024)
for which the cases of $\beta = 1$ and $2$ were developed.
Specifically, we have the following recurrence relations:
\begin{align}
\mathbb{E}[W^{r+1}p_{\lambda}(W)] & = \left[n+\left(\frac{2}{\beta}-1\right)r\right]\Sigma \mathbb{E}[W^{r} p_{\lambda}(W)]+
\sum_{j=1}^{r} \Sigma \mathbb{E}[W^{r-j}p_j(W)p_{\lambda}(W)] \nonumber \
& \;\;\;\;{}+\frac{2}{\beta}\sum_{i=1}^{\ell(\lambda)}\lambda_i \Sigma \mathbb{E}\left[W^{r+\lambda_i}p_{\lambda_{(i)}}(W)\right], \
\Sigma^{-1}\mathbb{E}[W^{-r}p_{\lambda}(W^{-1})] & = \left[\tilde{n}-\left(\frac{2}{\beta}-1\right)r\right]\mathbb{E}[W^{-(r+1)}p_{\lambda}(W^{-1})]
-\sum_{j=1}^{r} \mathbb{E}[W^{-r-1+j}p_j(W^{-1})p_{\lambda}(W^{-1})] \nonumber \
& \;\;\;\;{}-\frac{2}{\beta} \sum_{i=1}^{\ell(\lambda)}\lambda_i \mathbb{E}[W^{-r-1-\lambda_i}p_{\lambda_{(i)}}(W^{-1})],
\end{align}
where $\tilde{n}= n-m+1-(2/\beta)$ and $\lambda_{(i)}$ is
$\lambda$ with its $i$-th element removed.
Together with the boundary conditions $\mathbb{E}[W] = n\Sigma$
and $\mathbb{E}[W^{-1}] = \Sigma^{-1}/\tilde{n}$, we can obtain
$\mathbb{E}[E^rp_{\lambda}(W)]$ and $\mathbb{E}[W^{-r}p_{\lambda}(W^{-1})]$. Note
that $\mathbb{E}[W^{-r}p_{\lambda}(W^{-1})]$
exists if and only if $\tilde{n}>2(r+|\lambda|)$.
Let $k=r+|\lambda|$. Hillier and Kan (2024) show that
\begin{align}
\mathbb{E}[W^{r}p_{\lambda}(W)] & = \sum_{i=1}^k\left[\sum_{\rho \vdash k-i} c_{\lambda,\rho}p_{\rho}(\Sigma)\right]\Sigma^i, \
\mathbb{E}[W^{-r}p_{\lambda}(W^{-1})] & = \sum_{i=1}^k\left[\sum_{\rho \vdash k-i}\tilde{c}{\lambda,\rho}p{\rho}(\Sigma^{-1})\right]\Sigma^{-i},
\end{align}
where $c_{\lambda,\rho}$ and $\tilde{c}{\lambda,\rho}$
are constants that depend on $n$ and $\tilde{n}$, respectively, but they do not depend on $\Sigma$.
In addition, we have
\begin{align}
\mathbb{E}[p{\lambda}(W)] & = \sum_{\kappa \vdash k} h_{\kappa}p_{\kappa}(\Sigma), \
\mathbb{E}[p_{\lambda}(W^{-1})] & = \sum_{\kappa \vdash k} \tilde{h}{\kappa}p{\kappa}(\Sigma^{-1}),
\end{align}
where $h_{\kappa}$ and $\tilde{h}_{\kappa}$ are constants
that depend on $n$ and $\tilde{n}$, repsectively, but they do
not depend on $\Sigma$.
Using the recurrence relations, Hillier and Kan (2024)
develop efficient algorithms for computing the constants
$c_{\lambda,\rho}$, $\tilde{c}{\lambda,\rho}$, $h{\kappa}$
and $\tilde{h}_{\kappa}$.
Main Functions in the Package
There are four main functions in this package:
wishmom(), iwishmom(), wishmom_sym(), and iwishmom_sym(). The first two are used to numerically
compute $\mathbb{E}[W^rp_{\lambda}(W)]$
and $\mathbb{E}[W^{-r}p_{\lambda}(W^{-1})]$, respectively.
The last two are used to generate an analytical expression
of $\mathbb{E}[W^rp_{\lambda}(W)]$ and
$\mathbb{E}[W^{-r}p_{\lambda}(W^{-1})]$, respectively.
Moments of $\beta$-Wishart:
wishmom()
The function wishmom() computes
$\mathbb{E}\left[\prod_{j=1}^r \mbox{tr}(W^j)^{f_j}W^{iw}\right]$
numerically, where $W \sim W_m^\beta(n, \Sigma)$. When $iw=0$,
it computes $\mathbb{E}[\prod_{j=1}^r \mbox{tr}(W^j)^{f_j}]$.
Arguments
n: degrees of freedom of the $\beta$-Wishart distributionS: covariance matrix of the $\beta$-Wishart distributionf: a vector of nonnegative integers $f_j$ that represents the power for $\mbox{tr}(W^j)$, $j=1,\ldots, r$iw: Power of $W$alpha: The type of Wishart distribution ($\alpha=2/\beta$):1/2: Quaternion Wishart1: Complex Wishart2: Real Wishart (default)
Output
When $iw=0$, it returns $\mathbb{E}[\prod_{j=1}^r \mbox{tr}(W^j)^{f_j}]$.
When $iw \neq 0$, it returns $\mathbb{E}[\prod_{j=1}^r \mbox{tr}(W^j)^{f_j}W^{iw}]$.
Examples
# Example 1: For E[tr(W)^4] with W ~ W_m^1(n,S), where n and S are defined below:
n <- 20
S <- matrix(c(25, 49,
49, 109), nrow=2, ncol=2)
wishmom(n, S, 4) # iw = 0, for real Wishart distribution
#> [1] 8.705084e+13
# Example 2: For E[tr(W)^2*tr(W^3)*W^2] with W ~ W_m^1(n,S), where n and S, are defined below:
n <- 20
S <- matrix(c(25, 49,
49, 109), nrow=2, ncol=2)
wishmom(n, S, c(2, 0, 1), 2, 2) # iw = 2, for real Wishart distribution
#> [,1] [,2]
#> [1,] 9.039462e+23 1.956948e+24
#> [2,] 1.956948e+24 4.258714e+24
# Example 3: For E[tr(W)^2*tr(W^3)] with W ~ W_m^2(n,S), where n and S are defined below:
n <- 20
S <- matrix(c(25, 49 + 2i,
49 - 2i, 109), nrow=2, ncol=2)
wishmom(n, S, c(2, 0, 1), 0, 1) # iw = 0, for complex Wishart distribution
#> [1] 2.078126e+17
# Example 4: For E[tr(W)*tr(W^2)^2*tr(W^3)^2*W] with W ~ W_m^2(n,S), where n, S, are defined below:
n <- 20
S <- matrix(c(25, 49 + 2i,
49 - 2i, 109), nrow=2, ncol=2)
wishmom(n, S, c(1, 2, 2), 1, 1) # iw = 1, for complex Wishart distribution
#> [,1] [,2]
#> [1,] 3.418999e+41+5.014362e+20i 6.943130e+41-2.833930e+40i
#> [2,] 6.943130e+41+2.833930e+40i 1.532151e+42-2.882805e+22i
wishmom_sym()
The function wishmom_sym() generates an analytical expression of
$\mathbb{E}\left[\prod_{j=1}^r \mbox{tr}(W^j)^{f_j}W^{iw}\right]$,
where $W \sim W_m^\beta(n, \Sigma)$. When $iw=0$,
it generates an analytical expression of $\mathbb{E}[\prod_{j=1}^r \mbox{tr}(W^j)^{f_j}]$.
Arguments
f: a vector of nonnegative integers $f_j$ that represents the power for $\mbox{tr}(W^j)$, $j=1,\ldots, r$iw: Power of $W$alpha: The type of Wishart distribution ($\alpha=2/\beta$):1/2: Quaternion Wishart1: Complex Wishart2: Real Wishart (default)latex: The type of outputTRUE: LaTeX expressionFALSE: Dataframe (default)
Output
When $iw = 0$, the output is an analytical expression of $\mathbb{E}[\prod_{j=1}^r \mbox{tr}(W^j)^{f_j}]$.
When $iw \neq 0$, the output is an analytical expression of $\mathbb{E}[\prod_{j=1}^r \mbox{tr}(W^j)^{f_j}W^{iw}]$.
If latex = FALSE (default), the output is a data frame that stores the coefficients for the analytical expression.
If latex = TRUE, the output is a $\LaTeX$ formatted string of the result in terms of $n$ and $\Sigma$.
Examples
# Example 1: For E[tr(W)^4] with W ~ W_m^1(n,Sigma), represented as a dataframe:
wishmom_sym(4) # iw = 0, for real Wishart distribution
#> kappa h_kappa
#> 1 4 48
#> 2 3,1 32n
#> 3 2,2 12n
#> 4 2,1,1 12n^2
#> 5 1,1,1,1 n^3
# Example 2: For E[tr(W)*tr(W^2)W] with W ~ W_m^1(n,Sigma), represented as a dataframe:
wishmom_sym(c(1,1), 1) # iw = 1, for real Wishart distribution
#> i rho c
#> 1 1 3 4n^2+4n
#> 2 1 2,1 n^3+n^2+4n
#> 3 1 1,1,1 n^2
#> 4 2 2 2n^2+2n+8
#> 5 2 1,1 6n
#> 6 3 1 4n^2+4n+16
#> 7 4 <NA> 24n+24
# Example 3: For E[tr(W)^4] with W ~ W_m^2(n,Sigma), represented as a LaTeX string:
writeLines(wishmom_sym(4, 0, 1, latex=TRUE)) # iw = 0, for complex Wishart distribution
#> (6)p_{(4)}(\Sigma)
#> +(8n)p_{(3,1)}(\Sigma)
#> +(3n)p_{(2,2)}(\Sigma)
#> +(6n^2)p_{(2,1,1)}(\Sigma)
#> +(n^3)p_{(1,1,1,1)}(\Sigma)
# Example 4: For E[tr(W)*tr(W^2)W] with W ~ W_m^2(n,Sigma), represented as a LaTeX string:
writeLines(wishmom_sym(c(1, 1), 1, 1, latex=TRUE)) # iw = 1, for complex Wishart distribution
#> [(2n^2)p_{(3)}(\Sigma)+(n^3+2n)p_{(2, 1)}(\Sigma)+(n^2)p_{(1, 1, 1)}(\Sigma)]\Sigma
#> +[(n^2+2)p_{(2)}(\Sigma)+(3n)p_{(1, 1)}(\Sigma)]\Sigma^2
#> +(2n^2+4)p_{(1)}(\Sigma)\Sigma^3
#> +(6n)\Sigma^4
Moments of Inverse $\beta$-Wishart:
iwishmom()
The function iwishmom() computes
$\mathbb{E}\left[\prod_{j=1}^r \mbox{tr}(W^{-j})^{f_j}W^{-iw}\right]$ numerically,
where $W \sim W_m^\beta(n, \Sigma)$. When $iw=0$,
it computes $\mathbb{E}[\prod_{j=1}^r \mbox{tr}(W^{-j})^{f_j}]$.
Arguments
n: degrees of freedom of the $\beta$-Wishart distributionS: covariance matrix of the $\beta$-Wishart distributionf: a vector of nonnegative integers $f_j$ that represents the power for $\mbox{tr}(W^{-j})$, $j=1,\ldots, r$iw: Power of $W^{-1}$alpha: The type of Wishart distribution ($\alpha=2/\beta$):1/2: Quaternion Wishart1: Complex Wishart2: Real Wishart (default)
Output
When $iw=0$, it returns $\mathbb{E}[\prod_{j=1}^r \mbox{tr}(W^{-j})^{f_j}]$.
When $iw \neq 0$, it returns $\mathbb{E}[\prod_{j=1}^r \mbox{tr}(W^{-j})^{f_j}W^{-iw}]$.
Examples
# Example 1: For E[tr(W^{-1})^2] with W ~ W_m^1(n,S), where n and S are defined below:
n <- 20
S <- matrix(c(25, 49,
49, 109), nrow=2, ncol=2)
iwishmom(n, S, 2) # iw = 0, for real Wishart distribution
#> [1] 0.0006680892
# Example 2: For E[tr(W^{-1})^2*tr(W^{-3})W^{-2}] with W ~ W_m^1(n,S), where n and S are defined below:
n <- 20
S <- matrix(c(25, 49,
49, 109), nrow=2, ncol=2)
iwishmom(n, S, c(2, 0, 1), 2, 2) # iw = 2, for real Wishart distribution
#> [,1] [,2]
#> [1,] 1.328434e-10 -6.101692e-11
#> [2,] -6.101692e-11 2.824292e-11
# Example 3: For E[tr(W^{-1})^2*tr(W^{-3})] with W ~ W_m^2(n,S), where n and S are defined below:
n <- 20
S <- matrix(c(25, 49 + 2i,
49 - 2i, 109), nrow=2, ncol=2)
iwishmom(n, S, c(2, 0, 1), 0, 1) # iw = 0, for complex Wishart distribution
#> [1] 1.17985e-08
# Example 4: For E[tr(W^{-1})*tr(W^{-2})^2*tr(W^{-3})^2*W^{-1}] with W ~ W_m^2(n,S), where n and S are defined below:
n <- 30
S <- matrix(c(25, 49 + 2i,
49 -2i, 109), nrow=2, ncol=2)
iwishmom(n, S, c(1, 2, 2), 1, 1) # iw = 1, for complex Wishart distribution
#> [,1] [,2]
#> [1,] 1.348928e-21+0.000000e+00i -6.116211e-22+2.496413e-23i
#> [2,] -6.116211e-22-2.496413e-23i 3.004350e-22+0.000000e+00i
iwishmom_sym()
The function iwishmom_sym() generates an analytical expression of $\mathbb{E}\left[\prod_{j=1}^r \mbox{tr}(W^{-j})^{f_j}W^{-iw}\right]$,
where $W \sim W_m^\beta(n, \Sigma)$. When $iw=0$,
it generates an analytical expression of $\mathbb{E}[\prod_{j=1}^r \mbox{tr}(W^{-j})^{f_j}]$.
Arguments
f: a vector of nonnegative integers $f_j$ that represents the power for $\mbox{tr}(W^{-j})$, $j=1,\ldots, r$iw: Power of $W^{-1}$alpha: The type of Wishart distribution ($\alpha=2/\beta$):1/2: Quaternion Wishart1: Complex Wishart2: Real Wishart (default)latex: The type of outputTRUE: LaTeX expressionFALSE: Dataframe (default)
Output
When $iw = 0$, the output is an analytical expression of $\mathbb{E}[\prod_{j=1}^r \mbox{tr}(W^{-j})^{f_j}]$.
When $iw \neq 0$, the output is an analytical expression of $\mathbb{E}[\prod_{j=1}^r \mbox{tr}(W^{-j})^{f_j}W^{-iw}]$.
If latex = FALSE (default), the output is a data frame that stores the coefficients for the analytical expression. If latex = TRUE, the output is a $\LaTeX$ formatted string of the result in terms of $\tilde{n}$ and $\Sigma$, where $\tilde{n} = n-m+1-\alpha$ and $m$ is the dimension of the $\beta$-Wishart
distribution.
Examples
# Example 1: For E[tr(W^{-1})^4] with W ~ W_m^1(n,Sigma), represented as a dataframe:
iwishmom_sym(4) # iw = 0, for real Wishart distribution
#> $dataframe
#> kappa h_kappa_numerator
#> 1 4 240n1-288
#> 2 3,1 64n1^2-256n1
#> 3 2,2 12n1^2-60n1+216
#> 4 2,1,1 12n1^3-72n1^2+36n1+72
#> 5 1,1,1,1 n1^4-7n1^3+n1^2+35n1-6
#>
#> $denominator
#> [1] "n1^8-7n1^7-11n1^6+107n1^5+34n1^4-388n1^3-24n1^2+288n1"
# Example 2: For E[tr(W^{-1})*tr(W^{-2})W^{-1}] with W ~ W_m^1(n,Sigma), represented as a dataframe:
iwishmom_sym(c(1,1), 1) # iw = 1, for real Wishart distribution
#> $dataframe
#> i rho c_numerator
#> 1 1 3 4n1^3-16n1^2-20n1+24
#> 2 1 2,1 n1^4-6n1^3+3n1^2+6n1
#> 3 1 1,1,1 n1^3-6n1^2+3n1+6
#> 4 2 2 2n1^3-10n1^2+36n1
#> 5 2 1,1 10n1^2-42n1+36
#> 6 3 1 4n1^3-16n1^2+60n1-72
#> 7 4 <NA> 40n1^2-48n1
#>
#> $denominator
#> [1] "n1^8-7n1^7-11n1^6+107n1^5+34n1^4-388n1^3-24n1^2+288n1"
# Example 3: For E[tr(W^{-1})^4] with W ~ W_m^2(n,Sigma), represented as a LaTeX string:
writeLines(iwishmom_sym(4, 0, 1, latex=TRUE)) # iw = 0, for complex Wishart distribution
#> [(30\tilde{n})p_{(4)}(\Sigma^{-1})
#> +(16\tilde{n}^2-24)p_{(3,1)}(\Sigma^{-1})
#> +(3\tilde{n}^2+18)p_{(2,2)}(\Sigma^{-1})
#> +(6\tilde{n}^3-24\tilde{n})p_{(2,1,1)}(\Sigma^{-1})
#> +(\tilde{n}^4-8\tilde{n}^2+6)p_{(1,1,1,1)}(\Sigma^{-1})]
#> /(\tilde{n}^8-14\tilde{n}^6+49\tilde{n}^4-36\tilde{n}^2)
# Example 4: For E[tr(W^{-1})*tr(W^{-2})W^{-1}] with W ~ W_m^2(n,Sigma), represented as a LaTeX string:
writeLines(iwishmom_sym(c(1, 1), 1, 1, latex=TRUE)) # iw = 1, for complex Wishart distribution
#> [[(2\tilde{n}^3-8\tilde{n})p_{(3)}(\Sigma^{-1})+(\tilde{n}^4-4\tilde{n}^2)p_{(2,1)}(\Sigma^{-1})+(\tilde{n}^3-4\tilde{n})p_{(1,1,1)}(\Sigma^{-1})]\Sigma^{-1}
#> +[(\tilde{n}^3+6\tilde{n})p_{(2)}(\Sigma^{-1})+(5\tilde{n}^2)p_{(1,1)}(\Sigma^{-1})]\Sigma^{-2}
#> +(2\tilde{n}^3+12\tilde{n})p_{(1)}(\Sigma^{-1})\Sigma^{-3}
#> +(10\tilde{n}^2)\Sigma^{-4}]
#> /(\tilde{n}^8-14\tilde{n}^6+49\tilde{n}^4-36\tilde{n}^2)
Auxiliary Functions
Below is a list of auxiliary functions that are called by
wishmom, iwishmom, wishmom_sym, and iwishmom_sym.
ip_desc()
The function ip_desc() generates all integer partitions of a given integer k in a reverse lexicographical order.
Arguments
k: A positive integer to be partitioned
Output
A matrix where each row represents an integer partition of k, listed in a reverse lexicographical order.
Examples
# Example 1: List of integer partitions of 3
ip_desc(3)
#> [,1] [,2] [,3]
#> [1,] 3 0 0
#> [2,] 2 1 0
#> [3,] 1 1 1
# Example 2: List of integer partitions of 5
ip_desc(5)
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 5 0 0 0 0
#> [2,] 4 1 0 0 0
#> [3,] 3 2 0 0 0
#> [4,] 3 1 1 0 0
#> [5,] 2 2 1 0 0
#> [6,] 2 1 1 1 0
#> [7,] 1 1 1 1 1
dkmap()
The function dkmap() computes the mapping matrix $D_k$
discussed in Appendix B of Hillier and Kan (2024), modified
for the general $\beta$-Wishart case. The returned matrix
is $D_k$ but with $n$ in the diagonal elements removed.
Arguments
k: The order of the mapping matrix $D_k$ (a positive integer)alpha: The type of $\beta$-Wishart distribution ($\alpha =2/\beta$):1/2: Quaternion Wishart1: Complex Wishart2: Real Wishart (default)
Output
The mapping matrix $D_k$ but with $n$ removed from its diagonal.
Examples
# Example 1: Compute the mapping matrix for k = 2, real Wishart
dkmap(2)
#> [,1] [,2] [,3] [,4]
#> [1,] 2 1 1 0
#> [2,] 2 1 0 1
#> [3,] 4 0 0 0
#> [4,] 0 4 0 0
# Example 2: Compute the mapping matrix for k = 1, complex Wishart
dkmap(1, 1)
#> [,1] [,2]
#> [1,] 0 1
#> [2,] 1 0
# Example 3: Compute the mapping matrix for k = 2, quaternion Wishart
dkmap(2, 1/2)
#> [,1] [,2] [,3] [,4]
#> [1,] -1.0 1.0 1 0
#> [2,] 0.5 -0.5 0 1
#> [3,] 1.0 0.0 0 0
#> [4,] 0.0 1.0 0 0
denpoly()
The function denpoly() computes the coefficients of the denominator polynomial for the elements
$\tilde{\mathcal{H}}_k$ and $\tilde{\mathcal{C}}_k$. The function returns a vector containing the coefficients
in descending powers of $\tilde{n}$, with the last element being the coefficient of $\tilde{n}$.
Arguments
k: The order of the polynomial (a positive integer)alpha: The type of $\beta$-Wishart distribution ($\alpha =2/\beta$):1/2: Quaternion Wishart1: Complex Wishart2: Real Wishart (default)
Output
A vector containing the coefficients of the denominator
polynomial in descending powers of $\tilde{n}$ for the elements of $\mathcal{\tilde{H}}_k$ and $\tilde{\mathcal{C}}_k$.
Examples
# Example 1: Compute the denominator polynomial for k = 3 and alpha = 2
# Output corresponds to the polynomial n1^5-3n1^4-8n1^3+12n1^2+16n1,
# where n1 is \eqn{\tilde{n}}
denpoly(3)
#> [1] 1 -3 -8 12 16
# Example 2: Compute the denominator polynomial for k = 2 and alpha = 1
# Output corresponds to the polynomial n1^3-n1, where n1 is \eqn{\tilde{n}}
denpoly(2, alpha = 1)
#> [1] 1 0 -1
qk_coeff()
The function qk_coeff() computes the coefficient matrix
$\mathcal{C}_k$, which is obtained based on Corollary 1
of Hillier and Kan (2024), after a modification for the
general $\beta$-Wishart case. $\mathcal{C}_k$ is represented as a
3-dimensional array where each slice along the third dimension represents a coefficient matrix of the polynomial in descending powers of $n$.
Arguments
k: The order of the $\mathcal{C}_k$ matrixalpha: The type of Wishart distribution ($\alpha=2/\beta$):1/2: Quaternion Wishart1: Complex Wishart2: Real Wishart (default)
Output
A 3-dimensional array representing $\mathcal{C}k$, a matrix of constants that allow us to obtain $\mathbb{E}[p{\lambda}(W)W^r]$, where $r+|\lambda|=k$ and $W \sim W_m^{\beta}(n,\Sigma)$.
Examples
# Example 1:
qk_coeff(2) # For real Wishart distribution with k = 2
#> , , 1
#>
#> [,1] [,2]
#> [1,] 1 0
#> [2,] 0 1
#>
#> , , 2
#>
#> [,1] [,2]
#> [1,] 1 1
#> [2,] 2 0
# Example 2:
qk_coeff(3, 1) # For complex Wishart distribution with k = 3
#> , , 1
#>
#> [,1] [,2] [,3] [,4]
#> [1,] 1 0 0 0
#> [2,] 0 1 0 0
#> [3,] 0 0 1 0
#> [4,] 0 0 0 1
#>
#> , , 2
#>
#> [,1] [,2] [,3] [,4]
#> [1,] 0 2 1 0
#> [2,] 2 0 0 1
#> [3,] 2 0 0 1
#> [4,] 0 2 1 0
#>
#> , , 3
#>
#> [,1] [,2] [,3] [,4]
#> [1,] 1 0 0 1
#> [2,] 0 1 1 0
#> [3,] 0 2 0 0
#> [4,] 2 0 0 0
# Example 3:
qk_coeff(2, 1/2) # For quaternion Wishart distribution with k = 2
#> , , 1
#>
#> [,1] [,2]
#> [1,] 1 0
#> [2,] 0 1
#>
#> , , 2
#>
#> [,1] [,2]
#> [1,] -0.5 1
#> [2,] 0.5 0
wish_ps()
The function wish_ps() computes the coefficient matrix
$\mathcal{H}k$ that allows us to compute $\mathbb{E}[p{\kappa}(W)]$, which is obtained based
on Proposition 5 of Hillier and Kan (2024), after a modification for the general $\beta$-Wishart case. $\mathcal{H}_k$ is represented as a 3-dimensional array where each slice along the third dimension represents a coefficient matrix of the polynomial in descending powers of $n$.
Arguments
k: The order of the $\mathcal{H}_k$ matrixalpha: The type of Wishart distribution ($\alpha=2/\beta$):1/2: Quaternion Wishart1: Complex Wishart2: Real Wishart (default)
Output
A 3-dimensional array representing $\mathcal{H}k$, a matrix of constants that allows us to obtain $\mathbb{E}[p{\kappa}(W)]$, where $|\kappa|=k$ and $W \sim W_m^{\beta}(n,\Sigma)$.
Examples
# Example 1:
wish_ps(3) # For real Wishart distribution with k = 3
#> , , 1
#>
#> [,1] [,2] [,3]
#> [1,] 1 0 0
#> [2,] 0 1 0
#> [3,] 0 0 1
#>
#> , , 2
#>
#> [,1] [,2] [,3]
#> [1,] 3 3 0
#> [2,] 4 1 1
#> [3,] 0 6 0
#>
#> , , 3
#>
#> [,1] [,2] [,3]
#> [1,] 4 3 1
#> [2,] 4 4 0
#> [3,] 8 0 0
# Example 2:
wish_ps(4, 1) # For complex Wishart distribution with k = 4
#> , , 1
#>
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 1 0 0 0 0
#> [2,] 0 1 0 0 0
#> [3,] 0 0 1 0 0
#> [4,] 0 0 0 1 0
#> [5,] 0 0 0 0 1
#>
#> , , 2
#>
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 0 4 2 0 0
#> [2,] 3 0 0 3 0
#> [3,] 4 0 0 2 0
#> [4,] 0 4 1 0 1
#> [5,] 0 0 0 6 0
#>
#> , , 3
#>
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 5 0 0 6 0
#> [2,] 0 7 3 0 1
#> [3,] 0 8 2 0 1
#> [4,] 6 0 0 5 0
#> [5,] 0 8 3 0 0
#>
#> , , 4
#>
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 0 4 1 0 1
#> [2,] 3 0 0 3 0
#> [3,] 2 0 0 4 0
#> [4,] 0 4 2 0 0
#> [5,] 6 0 0 0 0
# Example 3:
wish_ps(2, 1/2) # For quaternion Wishart distribution with k = 2
#> , , 1
#>
#> [,1] [,2]
#> [1,] 1 0
#> [2,] 0 1
#>
#> , , 2
#>
#> [,1] [,2]
#> [1,] -0.5 1
#> [2,] 0.5 0
qkn_coeff()
The function qkn_coeff() computes the inverse of the coefficient matrix $\tilde{\mathcal{C}}_k$, which is obtained based on Corollary 2 of Hillier and Kan (2024), after a modification for the general $\beta$-Wishart case. $\tilde{\mathcal{C}}_k^{-1}$ is represented as a 3-dimensional array where each slice along the third dimension represents a coefficient matrix of the polynomial in descending powers of $\tilde{n}$.
Arguments
k: The order of the $\tilde{\mathcal{C}}_k$ matrixalpha: The type of Wishart distribution ($\alpha=2/\beta$):1/2: Quaternion Wishart1: Complex Wishart2: Real Wishart (default)
Output
A 3-dimensional array representing $\tilde{\mathcal{C}}k^{-1}$, a matrix of constants that allow us to obtain $\mathbb{E}[p{\lambda}(W^{-1})W^{-r}]$, where
$r+|\lambda|=k$ and $W \sim W_m^{\beta}(n,\Sigma)$.
Examples
# Example 1:
qkn_coeff(2) # For real Wishart distribution with k = 2
#> , , 1
#>
#> [,1] [,2]
#> [1,] 1 0
#> [2,] 0 1
#>
#> , , 2
#>
#> [,1] [,2]
#> [1,] -1 -1
#> [2,] -2 0
# Example 2:
qkn_coeff(3, 1) # For complex Wishart distribution with k = 3
#> , , 1
#>
#> [,1] [,2] [,3] [,4]
#> [1,] 1 0 0 0
#> [2,] 0 1 0 0
#> [3,] 0 0 1 0
#> [4,] 0 0 0 1
#>
#> , , 2
#>
#> [,1] [,2] [,3] [,4]
#> [1,] 0 -2 -1 0
#> [2,] -2 0 0 -1
#> [3,] -2 0 0 -1
#> [4,] 0 -2 -1 0
#>
#> , , 3
#>
#> [,1] [,2] [,3] [,4]
#> [1,] 1 0 0 1
#> [2,] 0 1 1 0
#> [3,] 0 2 0 0
#> [4,] 2 0 0 0
# Example 3:
qkn_coeff(2, 1/2) # For quaternion Wishart distribution with k = 2
#> , , 1
#>
#> [,1] [,2]
#> [1,] 1 0
#> [2,] 0 1
#>
#> , , 2
#>
#> [,1] [,2]
#> [1,] 0.5 -1
#> [2,] -0.5 0
iwish_ps()
The function iwish_ps() computes the inverse of the coefficient matrix $\tilde{\mathcal{H}}k$ that allows us to compute $\mathbb{E}[p{\kappa}(W^{-1})]$, which is obtained based
on Eq.(82) of Hillier and Kan (2024),
after a modification for the general $\beta$-Wishart case. $\tilde{\mathcal{H}}_k^{-1}$ is represented as a 3-dimensional array where each slice along the third dimension represents a
coefficient matrix of the polynomial in descending powers of $\tilde{n}$.
Arguments
k: The order of the $\tilde{\mathcal{H}}_k$ matrixalpha: The type of Wishart distribution ($\alpha=2/\beta$):1/2: Quaternion Wishart1: Complex Wishart2: Real Wishart (default)
Output
A 3-dimensional array representing $\tilde{\mathcal{H}}k^{-1}$, a matrix of constants that allows us to obtain $\mathbb{E}[p{\kappa}(W^{-1})]$, where $|\kappa|=k$ and $W \sim W_m^{\beta}(n,\Sigma)$.
Examples
# Example 1:
iwish_ps(3) # For real Wishart distribution with k = 3
#> , , 1
#>
#> [,1] [,2] [,3]
#> [1,] 1 0 0
#> [2,] 0 1 0
#> [3,] 0 0 1
#>
#> , , 2
#>
#> [,1] [,2] [,3]
#> [1,] -3 -3 0
#> [2,] -4 -1 -1
#> [3,] 0 -6 0
#>
#> , , 3
#>
#> [,1] [,2] [,3]
#> [1,] 4 3 1
#> [2,] 4 4 0
#> [3,] 8 0 0
# Example 2:
iwish_ps(4, 1) # For complex Wishart distribution with k = 4
#> , , 1
#>
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 1 0 0 0 0
#> [2,] 0 1 0 0 0
#> [3,] 0 0 1 0 0
#> [4,] 0 0 0 1 0
#> [5,] 0 0 0 0 1
#>
#> , , 2
#>
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 0 -4 -2 0 0
#> [2,] -3 0 0 -3 0
#> [3,] -4 0 0 -2 0
#> [4,] 0 -4 -1 0 -1
#> [5,] 0 0 0 -6 0
#>
#> , , 3
#>
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 5 0 0 6 0
#> [2,] 0 7 3 0 1
#> [3,] 0 8 2 0 1
#> [4,] 6 0 0 5 0
#> [5,] 0 8 3 0 0
#>
#> , , 4
#>
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 0 -4 -1 0 -1
#> [2,] -3 0 0 -3 0
#> [3,] -2 0 0 -4 0
#> [4,] 0 -4 -2 0 0
#> [5,] -6 0 0 0 0
# Example 3:
iwish_ps(2, 1/2) # For quaternion Wishart distribution with k = 2
#> , , 1
#>
#> [,1] [,2]
#> [1,] 1 0
#> [2,] 0 1
#>
#> , , 2
#>
#> [,1] [,2]
#> [1,] 0.5 -1
#> [2,] -0.5 0
qkn_coeffr()
The function qkn_coeffr() computes the coefficient matrix $\tilde{\mathcal{C}}_k$ for the general $\beta$-Wishart case.
Elements of $\tilde{\mathcal{C}}_k$ are rational
polynomials of $\tilde{n}$. The output contains two components:
c and den. c is a 3-dimensional array where each slice along the third dimension represents a coefficient matrix of the numerator polynomial in descending powers of $\tilde{n}$,
and den is a vector that represents the coefficients
of the denominator polynomial in descending power of $\tilde{n}$.
Arguments
k: The order of the $\tilde{\mathcal{C}}_k$ matrixalpha: The type of Wishart distribution ($\alpha=2/\beta$):1/2: Quaternion Wishart1: Complex Wishart2: Real Wishart (default)
Output
The output has two components: c and den. c is
a 3-dimensional array representing the numerator polynomial
of $\tilde{\mathcal{C}}k$, and den is a vector representing the denominator polynomial of $\tilde{\mathcal{C}}_k$, where $\tilde{\mathcal{C}}_k$ is a matrix of constants that allow us to obtain $\mathbb{E}[p{\lambda}(W^{-1})W^{-r}]$, where
$r+|\lambda|=k$ and $W \sim W_m^{\beta}(n,\Sigma)$.
Examples
# Example 1:
qkn_coeffr(2) # For real Wishart distribution with k = 2
#> $c
#> , , 1
#>
#> [,1] [,2]
#> [1,] 1 0
#> [2,] 0 1
#>
#> , , 2
#>
#> [,1] [,2]
#> [1,] 0 1
#> [2,] 2 -1
#>
#>
#> $den
#> [1] 1 -1 -2
# Example 2:
qkn_coeffr(3, 1) # For complex Wishart distribution with k = 3
#> $c
#> , , 1
#>
#> [,1] [,2] [,3] [,4]
#> [1,] 1 0 0 0
#> [2,] 0 1 0 0
#> [3,] 0 0 1 0
#> [4,] 0 0 0 1
#>
#> , , 2
#>
#> [,1] [,2] [,3] [,4]
#> [1,] 0 2 1 0
#> [2,] 2 0 0 1
#> [3,] 2 0 0 1
#> [4,] 0 2 1 0
#>
#> , , 3
#>
#> [,1] [,2] [,3] [,4]
#> [1,] 0 0 0 2
#> [2,] 0 0 2 0
#> [3,] 0 4 -2 0
#> [4,] 4 0 0 -2
#>
#>
#> $den
#> [1] 1 0 -5 0 4
# Example 3:
qkn_coeffr(2, 1/2) # For quaternion Wishart distribution with k = 2
#> $c
#> , , 1
#>
#> [,1] [,2]
#> [1,] 1 0
#> [2,] 0 1
#>
#> , , 2
#>
#> [,1] [,2]
#> [1,] 0.0 1.0
#> [2,] 0.5 0.5
#>
#>
#> $den
#> [1] 1.0 0.5 -0.5
iwish_psr()
The function iwish_psr() computes the coefficient matrix $\tilde{\mathcal{H}}_k$ for the general $\beta$-Wishart case.
Elements of $\tilde{\mathcal{H}}_k$ are rational
polynomials of $\tilde{n}$. The output contains two components:
c and den. c is a 3-dimensional array where each slice along the third dimension represents a coefficient matrix of the numerator polynomial in descending powers of $\tilde{n}$,
and den is a vector that represents the coefficients
of the denominator polynomial in descending power of $\tilde{n}$.
Arguments
k: The order of the $\tilde{\mathcal{C}}_k$ matrixalpha: The type of Wishart distribution ($\alpha=2/\beta$):1/2: Quaternion Wishart1: Complex Wishart2: Real Wishart (default)
Output
The output has two components: c and den. c is
a 3-dimensional array representing the numerator polynomial
of $\tilde{\mathcal{H}}k$, and den is a vector representing the denominator polynomial of $\tilde{\mathcal{H}}_k$, where $\tilde{\mathcal{H}}_k$ is a matrix of constants that allow us to obtain $\mathbb{E}[p{\lambda}(W^{-1})]$, where
$|\lambda|=k$ and $W \sim W_m^{\beta}(n,\Sigma)$.
Examples
# Example 1:
iwsih_psr(2) # For real Wishart distribution with k = 2
#> $c
#> , , 1
#>
#> [,1] [,2]
#> [1,] 1 0
#> [2,] 0 1
#>
#> , , 2
#>
#> [,1] [,2]
#> [1,] 0 1
#> [2,] 2 -1
#>
#>
#> $den
#> [1] 1 -1 -2
# Example 2:
iwish_psr(4, 1) # For complex Wishart distribution with k = 4
#> $c
#> , , 1
#>
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 1 0 0 0 0
#> [2,] 0 1 0 0 0
#> [3,] 0 0 1 0 0
#> [4,] 0 0 0 1 0
#> [5,] 0 0 0 0 1
#>
#> , , 2
#>
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 0 4 2 0 0
#> [2,] 3 0 0 3 0
#> [3,] 4 0 0 2 0
#> [4,] 0 4 1 0 1
#> [5,] 0 0 0 6 0
#>
#> , , 3
#>
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 1 0 0 10 0
#> [2,] 0 3 6 0 2
#> [3,] 0 16 -6 0 1
#> [4,] 10 0 0 1 0
#> [5,] 0 16 3 0 -8
#>
#> , , 4
#>
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 0 4 -3 0 5
#> [2,] 3 0 0 3 0
#> [3,] -6 0 0 12 0
#> [4,] 0 4 6 0 -4
#> [5,] 30 0 0 -24 0
#>
#> , , 5
#>
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 0 0 0 0 0
#> [2,] 0 12 -9 0 -3
#> [3,] 0 -24 18 0 6
#> [4,] 0 0 0 0 0
#> [5,] 0 -24 18 0 6
#>
#>
#> $den
#> [1] 1 0 -14 0 49 0 -36 0
# Example 3:
iwish_psr(2, 1/2) # For quaternion Wishart distribution with k = 2
#> $c
#> , , 1
#>
#> [,1] [,2]
#> [1,] 1 0
#> [2,] 0 1
#>
#> , , 2
#>
#> [,1] [,2]
#> [1,] 0.0 1.0
#> [2,] 0.5 0.5
#>
#>
#> $den
#> [1] 1.0 0.5 -0.5
References
Díaz-García, José and Gutiérrez-Jáimez,
Ramón (2011). On Wishart
distribution: som extension.
Linear Algebra and its Applications, 435, 1296-1310.
Drensky, Vesselin, Edelman, Alan, Genoar, Tierney, Kan, Raymond,
and Koev, Plamen (2021). The Densities and Distributions of the Largest Eigenvalue and the Trace of a Beta-Wishart Matrix.
Random Matrices: Theory and Applications, 10(1).
Letac, Gérard, and Massam, Héelène (2004). All invariant moments of the Wishart distribution.
Scandinavian Journal of Statistics, 31, 295-318.
Hillier, Grant, and Kan, Raymond (2024). On the expectations of equivariant matrix-valued functions of Wishart and inverse Wishart Matrices.
Scandinavian Journal of Statistics, 51, 697-723.
Try the wishmom package in your browser
Any scripts or data that you put into this service are public.
wishmom documentation built on Sept. 11, 2024, 8:29 p.m.
R Package Documentation
Browse R Packages
We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
knitr::opts_chunk$set( collapse = TRUE, comment = "#>" )
Introduction
The wishmom package provides functions to compute the expectation of matrix-valued functions of $\beta$-Wishart and inverse $\beta$-Wishart distributions ($\beta=1$: Real Wishart, $\beta=2$: Complex Wishart, $\beta=4$: Quaternion Wishart, $\beta=8$: Octonion Wishart). The main functions in this package are wishmom() and iwishmom(), which handle the numerical computation of moments of $\beta$-Wishart and inverse $\beta$-Wishart distributions, respectively. The corresponding functions for generating analytical expressions of moments
of $\beta$-Wishart and inverse $\beta$-Wishart distributions are wishmom_sym() and iwishmom_sym(). These programs are developed based on the results in Letac and Massam (2004)
and Hillier and Kan (2024).
You can install the package and load it using:
install.packages("wishmom") library("wishmom")
Mathematical Background
$\beta$-Wishart Distribution
The $\beta$-Wishart distribution is a fundamental distribution in multivariate statistics. When $\beta=1,\;2,\;4,\;8$, it is the real Wishart, complex Wishart, quaternion Wishart, and octonion Wishart, respectively. The density function of $W \sim W_m^\beta(n,\Sigma)$, i,e., a $\beta$-Wishart distribution with $n$ degrees of freedom and an $m \times m$ covariance matrix $\Sigma$, is given by (when $n > m-1$) (see Díaz-García and Gutiérrez-Jáimez (2011, Corollary 1))
[ f(W) = \frac{\left(\frac{\beta}{2}\right)^\frac{mn\beta}{2}}{\Gamma_m^{(\beta)}\left(\frac{n \beta}{2}\right) |\Sigma|^\frac{n \beta}{2}}|W|^{\frac{(n-m+1)\beta}{2}-1} \mbox{etr}\left(-\frac{\beta \Sigma^{-1}W}{2}\right), ]
where
[ \Gamma_m^{(\beta)}(a) = \pi^\frac{m(m-1)\beta}{4}\prod_{i=1}^m \Gamma\left(a-\frac{(i-1)\beta}{2}\right). ]
Note that we do not require $n$ to be an integer but the definition of the density of $W$ requires $\beta = 1,\;2,\;4$ or $8$. However, if our interest is only on the functions of eigenvalues of $W$, we can generalize this to any real $\beta>0$. Therefore, for any symmetric functions (say power-sum) of the eigenvalues of $W$, they can be well defined even when $\beta$ is not equal to $1,\; 2,\;4$ or $8$. For $W \sim W_m^\beta(n,\Sigma)$, the joint density of its eigenvalues $\lambda_1 \geq \cdots \geq \lambda_m$ is given by (see Dresnky, Edelman, Genoar, Kan, and Koev (2021))
[ f(\lambda_1,\ldots,\lambda_m) = \frac{| \Sigma|^{-\frac{n\beta}{2}}}{\mathcal{K}m^ {(\beta)}\left(\frac{n\beta}{2}\right)} |\Lambda|^{\frac{(n-m+1)\beta}{2}-1} {}_0^{}F_0^{(\beta)}\left(-\frac{\beta}{2}\Lambda,\Sigma^{-1}\right)\prod{1 \leq i < j \leq m}(\lambda_i-\lambda_j)^{\beta}, ] where $\Lambda = \mbox{Diag}(\lambda_1,\ldots,\lambda_m)$,
[ \mathcal{K}_m^{(\beta)}(a) = \frac{\left(\frac{2}{\beta}\right)^{ma}} {\pi^{\frac{m(m-1)\beta}{2}}} \frac{\Gamma_m^{(\beta)}\left(\frac{m\beta}{2}\right)\Gamma_m^{(\beta)}(a)}{\left[\Gamma\left(\frac{\beta}{2}\right)\right]^m}, ]
[ {}0^{}F_0^{(\beta)}(A,B) = \sum{k=0}^\infty \sum_{\kappa \vdash k} \frac{C_\kappa^{(\beta)}(A)C_\kappa^{(\beta)}(B)} {k!C_\kappa^{(\beta)}(I_m)}, ]
and $C_\kappa^{(\beta)}(X)$ is the Jack function of the eigenvalues of $X$.
Instead of using $\beta$, we use $\alpha = 2/\beta$ in our programs to describe the type of Wishart distribution. Therefore, $\alpha=2$ is for real Wishart, $\alpha=1$ is for complex Wishart, and $\alpha =1/2$ is for quaternion Wishart.
Moments of Matrix-valued Functions of $\beta$-Wishart and Inverse $\beta$-Wishart Distributions
Let $\lambda = (\lambda_1,\ldots, \lambda_k)$ be an integer partition of a positive integer $k$, where $|\lambda|= \lambda_1+\ldots+\lambda_k=k$, with $\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_k \geq 0$. The power-sum symmetric function $p_\lambda(W)$ of $W$ corresponding to a partition $\lambda$ is defined as
[ p_{\lambda}(W) = \prod_{i=1}^{\ell(\lambda)}p_{\lambda_i}(W), ]
where $\ell(\lambda)$ is the number of non-zero parts of $\lambda$, and $p_i(W) = \mbox{tr}(W^i)$. We are interested in computing
[ \mathbb{E}[W^rp_{\lambda}(W)]\;\;\;\;\mbox{and} \;\;\;\; \mathbb{E}[W^{-r}p_{\lambda}(W^{-1})], ]
where $W \sim W^{\beta}_m(n,\Sigma)$.
The method that we use is based on a generalization of the recurrence relations given in Hillier and Kan (2024) for which the cases of $\beta = 1$ and $2$ were developed. Specifically, we have the following recurrence relations:
\begin{align} \mathbb{E}[W^{r+1}p_{\lambda}(W)] & = \left[n+\left(\frac{2}{\beta}-1\right)r\right]\Sigma \mathbb{E}[W^{r} p_{\lambda}(W)]+ \sum_{j=1}^{r} \Sigma \mathbb{E}[W^{r-j}p_j(W)p_{\lambda}(W)] \nonumber \ & \;\;\;\;{}+\frac{2}{\beta}\sum_{i=1}^{\ell(\lambda)}\lambda_i \Sigma \mathbb{E}\left[W^{r+\lambda_i}p_{\lambda_{(i)}}(W)\right], \ \Sigma^{-1}\mathbb{E}[W^{-r}p_{\lambda}(W^{-1})] & = \left[\tilde{n}-\left(\frac{2}{\beta}-1\right)r\right]\mathbb{E}[W^{-(r+1)}p_{\lambda}(W^{-1})] -\sum_{j=1}^{r} \mathbb{E}[W^{-r-1+j}p_j(W^{-1})p_{\lambda}(W^{-1})] \nonumber \ & \;\;\;\;{}-\frac{2}{\beta} \sum_{i=1}^{\ell(\lambda)}\lambda_i \mathbb{E}[W^{-r-1-\lambda_i}p_{\lambda_{(i)}}(W^{-1})], \end{align} where $\tilde{n}= n-m+1-(2/\beta)$ and $\lambda_{(i)}$ is $\lambda$ with its $i$-th element removed. Together with the boundary conditions $\mathbb{E}[W] = n\Sigma$ and $\mathbb{E}[W^{-1}] = \Sigma^{-1}/\tilde{n}$, we can obtain $\mathbb{E}[E^rp_{\lambda}(W)]$ and $\mathbb{E}[W^{-r}p_{\lambda}(W^{-1})]$. Note that $\mathbb{E}[W^{-r}p_{\lambda}(W^{-1})]$ exists if and only if $\tilde{n}>2(r+|\lambda|)$.
Let $k=r+|\lambda|$. Hillier and Kan (2024) show that \begin{align} \mathbb{E}[W^{r}p_{\lambda}(W)] & = \sum_{i=1}^k\left[\sum_{\rho \vdash k-i} c_{\lambda,\rho}p_{\rho}(\Sigma)\right]\Sigma^i, \ \mathbb{E}[W^{-r}p_{\lambda}(W^{-1})] & = \sum_{i=1}^k\left[\sum_{\rho \vdash k-i}\tilde{c}{\lambda,\rho}p{\rho}(\Sigma^{-1})\right]\Sigma^{-i}, \end{align} where $c_{\lambda,\rho}$ and $\tilde{c}{\lambda,\rho}$ are constants that depend on $n$ and $\tilde{n}$, respectively, but they do not depend on $\Sigma$. In addition, we have \begin{align} \mathbb{E}[p{\lambda}(W)] & = \sum_{\kappa \vdash k} h_{\kappa}p_{\kappa}(\Sigma), \ \mathbb{E}[p_{\lambda}(W^{-1})] & = \sum_{\kappa \vdash k} \tilde{h}{\kappa}p{\kappa}(\Sigma^{-1}), \end{align} where $h_{\kappa}$ and $\tilde{h}_{\kappa}$ are constants that depend on $n$ and $\tilde{n}$, repsectively, but they do not depend on $\Sigma$.
Using the recurrence relations, Hillier and Kan (2024) develop efficient algorithms for computing the constants $c_{\lambda,\rho}$, $\tilde{c}{\lambda,\rho}$, $h{\kappa}$ and $\tilde{h}_{\kappa}$.
Main Functions in the Package
There are four main functions in this package:
wishmom(), iwishmom(), wishmom_sym(), and iwishmom_sym(). The first two are used to numerically
compute $\mathbb{E}[W^rp_{\lambda}(W)]$
and $\mathbb{E}[W^{-r}p_{\lambda}(W^{-1})]$, respectively.
The last two are used to generate an analytical expression
of $\mathbb{E}[W^rp_{\lambda}(W)]$ and
$\mathbb{E}[W^{-r}p_{\lambda}(W^{-1})]$, respectively.
Moments of $\beta$-Wishart:
wishmom()
The function wishmom() computes
$\mathbb{E}\left[\prod_{j=1}^r \mbox{tr}(W^j)^{f_j}W^{iw}\right]$
numerically, where $W \sim W_m^\beta(n, \Sigma)$. When $iw=0$,
it computes $\mathbb{E}[\prod_{j=1}^r \mbox{tr}(W^j)^{f_j}]$.
Arguments
n: degrees of freedom of the $\beta$-Wishart distributionS: covariance matrix of the $\beta$-Wishart distributionf: a vector of nonnegative integers $f_j$ that represents the power for $\mbox{tr}(W^j)$, $j=1,\ldots, r$iw: Power of $W$alpha: The type of Wishart distribution ($\alpha=2/\beta$):1/2: Quaternion Wishart1: Complex Wishart2: Real Wishart (default)
Output
When $iw=0$, it returns $\mathbb{E}[\prod_{j=1}^r \mbox{tr}(W^j)^{f_j}]$. When $iw \neq 0$, it returns $\mathbb{E}[\prod_{j=1}^r \mbox{tr}(W^j)^{f_j}W^{iw}]$.
Examples
# Example 1: For E[tr(W)^4] with W ~ W_m^1(n,S), where n and S are defined below: n <- 20 S <- matrix(c(25, 49, 49, 109), nrow=2, ncol=2) wishmom(n, S, 4) # iw = 0, for real Wishart distribution #> [1] 8.705084e+13 # Example 2: For E[tr(W)^2*tr(W^3)*W^2] with W ~ W_m^1(n,S), where n and S, are defined below: n <- 20 S <- matrix(c(25, 49, 49, 109), nrow=2, ncol=2) wishmom(n, S, c(2, 0, 1), 2, 2) # iw = 2, for real Wishart distribution #> [,1] [,2] #> [1,] 9.039462e+23 1.956948e+24 #> [2,] 1.956948e+24 4.258714e+24 # Example 3: For E[tr(W)^2*tr(W^3)] with W ~ W_m^2(n,S), where n and S are defined below: n <- 20 S <- matrix(c(25, 49 + 2i, 49 - 2i, 109), nrow=2, ncol=2) wishmom(n, S, c(2, 0, 1), 0, 1) # iw = 0, for complex Wishart distribution #> [1] 2.078126e+17 # Example 4: For E[tr(W)*tr(W^2)^2*tr(W^3)^2*W] with W ~ W_m^2(n,S), where n, S, are defined below: n <- 20 S <- matrix(c(25, 49 + 2i, 49 - 2i, 109), nrow=2, ncol=2) wishmom(n, S, c(1, 2, 2), 1, 1) # iw = 1, for complex Wishart distribution #> [,1] [,2] #> [1,] 3.418999e+41+5.014362e+20i 6.943130e+41-2.833930e+40i #> [2,] 6.943130e+41+2.833930e+40i 1.532151e+42-2.882805e+22i
wishmom_sym()
The function wishmom_sym() generates an analytical expression of
$\mathbb{E}\left[\prod_{j=1}^r \mbox{tr}(W^j)^{f_j}W^{iw}\right]$,
where $W \sim W_m^\beta(n, \Sigma)$. When $iw=0$,
it generates an analytical expression of $\mathbb{E}[\prod_{j=1}^r \mbox{tr}(W^j)^{f_j}]$.
Arguments
f: a vector of nonnegative integers $f_j$ that represents the power for $\mbox{tr}(W^j)$, $j=1,\ldots, r$iw: Power of $W$alpha: The type of Wishart distribution ($\alpha=2/\beta$):1/2: Quaternion Wishart1: Complex Wishart2: Real Wishart (default)latex: The type of outputTRUE: LaTeX expressionFALSE: Dataframe (default)
Output
When $iw = 0$, the output is an analytical expression of $\mathbb{E}[\prod_{j=1}^r \mbox{tr}(W^j)^{f_j}]$.
When $iw \neq 0$, the output is an analytical expression of $\mathbb{E}[\prod_{j=1}^r \mbox{tr}(W^j)^{f_j}W^{iw}]$.
If latex = FALSE (default), the output is a data frame that stores the coefficients for the analytical expression.
If latex = TRUE, the output is a $\LaTeX$ formatted string of the result in terms of $n$ and $\Sigma$.
Examples
# Example 1: For E[tr(W)^4] with W ~ W_m^1(n,Sigma), represented as a dataframe: wishmom_sym(4) # iw = 0, for real Wishart distribution #> kappa h_kappa #> 1 4 48 #> 2 3,1 32n #> 3 2,2 12n #> 4 2,1,1 12n^2 #> 5 1,1,1,1 n^3 # Example 2: For E[tr(W)*tr(W^2)W] with W ~ W_m^1(n,Sigma), represented as a dataframe: wishmom_sym(c(1,1), 1) # iw = 1, for real Wishart distribution #> i rho c #> 1 1 3 4n^2+4n #> 2 1 2,1 n^3+n^2+4n #> 3 1 1,1,1 n^2 #> 4 2 2 2n^2+2n+8 #> 5 2 1,1 6n #> 6 3 1 4n^2+4n+16 #> 7 4 <NA> 24n+24 # Example 3: For E[tr(W)^4] with W ~ W_m^2(n,Sigma), represented as a LaTeX string: writeLines(wishmom_sym(4, 0, 1, latex=TRUE)) # iw = 0, for complex Wishart distribution #> (6)p_{(4)}(\Sigma) #> +(8n)p_{(3,1)}(\Sigma) #> +(3n)p_{(2,2)}(\Sigma) #> +(6n^2)p_{(2,1,1)}(\Sigma) #> +(n^3)p_{(1,1,1,1)}(\Sigma) # Example 4: For E[tr(W)*tr(W^2)W] with W ~ W_m^2(n,Sigma), represented as a LaTeX string: writeLines(wishmom_sym(c(1, 1), 1, 1, latex=TRUE)) # iw = 1, for complex Wishart distribution #> [(2n^2)p_{(3)}(\Sigma)+(n^3+2n)p_{(2, 1)}(\Sigma)+(n^2)p_{(1, 1, 1)}(\Sigma)]\Sigma #> +[(n^2+2)p_{(2)}(\Sigma)+(3n)p_{(1, 1)}(\Sigma)]\Sigma^2 #> +(2n^2+4)p_{(1)}(\Sigma)\Sigma^3 #> +(6n)\Sigma^4
Moments of Inverse $\beta$-Wishart:
iwishmom()
The function iwishmom() computes
$\mathbb{E}\left[\prod_{j=1}^r \mbox{tr}(W^{-j})^{f_j}W^{-iw}\right]$ numerically,
where $W \sim W_m^\beta(n, \Sigma)$. When $iw=0$,
it computes $\mathbb{E}[\prod_{j=1}^r \mbox{tr}(W^{-j})^{f_j}]$.
Arguments
n: degrees of freedom of the $\beta$-Wishart distributionS: covariance matrix of the $\beta$-Wishart distributionf: a vector of nonnegative integers $f_j$ that represents the power for $\mbox{tr}(W^{-j})$, $j=1,\ldots, r$iw: Power of $W^{-1}$alpha: The type of Wishart distribution ($\alpha=2/\beta$):1/2: Quaternion Wishart1: Complex Wishart2: Real Wishart (default)
Output
When $iw=0$, it returns $\mathbb{E}[\prod_{j=1}^r \mbox{tr}(W^{-j})^{f_j}]$. When $iw \neq 0$, it returns $\mathbb{E}[\prod_{j=1}^r \mbox{tr}(W^{-j})^{f_j}W^{-iw}]$.
Examples
# Example 1: For E[tr(W^{-1})^2] with W ~ W_m^1(n,S), where n and S are defined below: n <- 20 S <- matrix(c(25, 49, 49, 109), nrow=2, ncol=2) iwishmom(n, S, 2) # iw = 0, for real Wishart distribution #> [1] 0.0006680892 # Example 2: For E[tr(W^{-1})^2*tr(W^{-3})W^{-2}] with W ~ W_m^1(n,S), where n and S are defined below: n <- 20 S <- matrix(c(25, 49, 49, 109), nrow=2, ncol=2) iwishmom(n, S, c(2, 0, 1), 2, 2) # iw = 2, for real Wishart distribution #> [,1] [,2] #> [1,] 1.328434e-10 -6.101692e-11 #> [2,] -6.101692e-11 2.824292e-11 # Example 3: For E[tr(W^{-1})^2*tr(W^{-3})] with W ~ W_m^2(n,S), where n and S are defined below: n <- 20 S <- matrix(c(25, 49 + 2i, 49 - 2i, 109), nrow=2, ncol=2) iwishmom(n, S, c(2, 0, 1), 0, 1) # iw = 0, for complex Wishart distribution #> [1] 1.17985e-08 # Example 4: For E[tr(W^{-1})*tr(W^{-2})^2*tr(W^{-3})^2*W^{-1}] with W ~ W_m^2(n,S), where n and S are defined below: n <- 30 S <- matrix(c(25, 49 + 2i, 49 -2i, 109), nrow=2, ncol=2) iwishmom(n, S, c(1, 2, 2), 1, 1) # iw = 1, for complex Wishart distribution #> [,1] [,2] #> [1,] 1.348928e-21+0.000000e+00i -6.116211e-22+2.496413e-23i #> [2,] -6.116211e-22-2.496413e-23i 3.004350e-22+0.000000e+00i
iwishmom_sym()
The function iwishmom_sym() generates an analytical expression of $\mathbb{E}\left[\prod_{j=1}^r \mbox{tr}(W^{-j})^{f_j}W^{-iw}\right]$,
where $W \sim W_m^\beta(n, \Sigma)$. When $iw=0$,
it generates an analytical expression of $\mathbb{E}[\prod_{j=1}^r \mbox{tr}(W^{-j})^{f_j}]$.
Arguments
f: a vector of nonnegative integers $f_j$ that represents the power for $\mbox{tr}(W^{-j})$, $j=1,\ldots, r$iw: Power of $W^{-1}$alpha: The type of Wishart distribution ($\alpha=2/\beta$):1/2: Quaternion Wishart1: Complex Wishart2: Real Wishart (default)latex: The type of outputTRUE: LaTeX expressionFALSE: Dataframe (default)
Output
When $iw = 0$, the output is an analytical expression of $\mathbb{E}[\prod_{j=1}^r \mbox{tr}(W^{-j})^{f_j}]$.
When $iw \neq 0$, the output is an analytical expression of $\mathbb{E}[\prod_{j=1}^r \mbox{tr}(W^{-j})^{f_j}W^{-iw}]$.
If latex = FALSE (default), the output is a data frame that stores the coefficients for the analytical expression. If latex = TRUE, the output is a $\LaTeX$ formatted string of the result in terms of $\tilde{n}$ and $\Sigma$, where $\tilde{n} = n-m+1-\alpha$ and $m$ is the dimension of the $\beta$-Wishart
distribution.
Examples
# Example 1: For E[tr(W^{-1})^4] with W ~ W_m^1(n,Sigma), represented as a dataframe: iwishmom_sym(4) # iw = 0, for real Wishart distribution #> $dataframe #> kappa h_kappa_numerator #> 1 4 240n1-288 #> 2 3,1 64n1^2-256n1 #> 3 2,2 12n1^2-60n1+216 #> 4 2,1,1 12n1^3-72n1^2+36n1+72 #> 5 1,1,1,1 n1^4-7n1^3+n1^2+35n1-6 #> #> $denominator #> [1] "n1^8-7n1^7-11n1^6+107n1^5+34n1^4-388n1^3-24n1^2+288n1" # Example 2: For E[tr(W^{-1})*tr(W^{-2})W^{-1}] with W ~ W_m^1(n,Sigma), represented as a dataframe: iwishmom_sym(c(1,1), 1) # iw = 1, for real Wishart distribution #> $dataframe #> i rho c_numerator #> 1 1 3 4n1^3-16n1^2-20n1+24 #> 2 1 2,1 n1^4-6n1^3+3n1^2+6n1 #> 3 1 1,1,1 n1^3-6n1^2+3n1+6 #> 4 2 2 2n1^3-10n1^2+36n1 #> 5 2 1,1 10n1^2-42n1+36 #> 6 3 1 4n1^3-16n1^2+60n1-72 #> 7 4 <NA> 40n1^2-48n1 #> #> $denominator #> [1] "n1^8-7n1^7-11n1^6+107n1^5+34n1^4-388n1^3-24n1^2+288n1" # Example 3: For E[tr(W^{-1})^4] with W ~ W_m^2(n,Sigma), represented as a LaTeX string: writeLines(iwishmom_sym(4, 0, 1, latex=TRUE)) # iw = 0, for complex Wishart distribution #> [(30\tilde{n})p_{(4)}(\Sigma^{-1}) #> +(16\tilde{n}^2-24)p_{(3,1)}(\Sigma^{-1}) #> +(3\tilde{n}^2+18)p_{(2,2)}(\Sigma^{-1}) #> +(6\tilde{n}^3-24\tilde{n})p_{(2,1,1)}(\Sigma^{-1}) #> +(\tilde{n}^4-8\tilde{n}^2+6)p_{(1,1,1,1)}(\Sigma^{-1})] #> /(\tilde{n}^8-14\tilde{n}^6+49\tilde{n}^4-36\tilde{n}^2) # Example 4: For E[tr(W^{-1})*tr(W^{-2})W^{-1}] with W ~ W_m^2(n,Sigma), represented as a LaTeX string: writeLines(iwishmom_sym(c(1, 1), 1, 1, latex=TRUE)) # iw = 1, for complex Wishart distribution #> [[(2\tilde{n}^3-8\tilde{n})p_{(3)}(\Sigma^{-1})+(\tilde{n}^4-4\tilde{n}^2)p_{(2,1)}(\Sigma^{-1})+(\tilde{n}^3-4\tilde{n})p_{(1,1,1)}(\Sigma^{-1})]\Sigma^{-1} #> +[(\tilde{n}^3+6\tilde{n})p_{(2)}(\Sigma^{-1})+(5\tilde{n}^2)p_{(1,1)}(\Sigma^{-1})]\Sigma^{-2} #> +(2\tilde{n}^3+12\tilde{n})p_{(1)}(\Sigma^{-1})\Sigma^{-3} #> +(10\tilde{n}^2)\Sigma^{-4}] #> /(\tilde{n}^8-14\tilde{n}^6+49\tilde{n}^4-36\tilde{n}^2)
Auxiliary Functions
Below is a list of auxiliary functions that are called by
wishmom, iwishmom, wishmom_sym, and iwishmom_sym.
ip_desc()
The function ip_desc() generates all integer partitions of a given integer k in a reverse lexicographical order.
Arguments
k: A positive integer to be partitioned
Output
A matrix where each row represents an integer partition of k, listed in a reverse lexicographical order.
Examples
# Example 1: List of integer partitions of 3 ip_desc(3) #> [,1] [,2] [,3] #> [1,] 3 0 0 #> [2,] 2 1 0 #> [3,] 1 1 1 # Example 2: List of integer partitions of 5 ip_desc(5) #> [,1] [,2] [,3] [,4] [,5] #> [1,] 5 0 0 0 0 #> [2,] 4 1 0 0 0 #> [3,] 3 2 0 0 0 #> [4,] 3 1 1 0 0 #> [5,] 2 2 1 0 0 #> [6,] 2 1 1 1 0 #> [7,] 1 1 1 1 1
dkmap()
The function dkmap() computes the mapping matrix $D_k$
discussed in Appendix B of Hillier and Kan (2024), modified
for the general $\beta$-Wishart case. The returned matrix
is $D_k$ but with $n$ in the diagonal elements removed.
Arguments
k: The order of the mapping matrix $D_k$ (a positive integer)alpha: The type of $\beta$-Wishart distribution ($\alpha =2/\beta$):1/2: Quaternion Wishart1: Complex Wishart2: Real Wishart (default)
Output
The mapping matrix $D_k$ but with $n$ removed from its diagonal.
Examples
# Example 1: Compute the mapping matrix for k = 2, real Wishart dkmap(2) #> [,1] [,2] [,3] [,4] #> [1,] 2 1 1 0 #> [2,] 2 1 0 1 #> [3,] 4 0 0 0 #> [4,] 0 4 0 0 # Example 2: Compute the mapping matrix for k = 1, complex Wishart dkmap(1, 1) #> [,1] [,2] #> [1,] 0 1 #> [2,] 1 0 # Example 3: Compute the mapping matrix for k = 2, quaternion Wishart dkmap(2, 1/2) #> [,1] [,2] [,3] [,4] #> [1,] -1.0 1.0 1 0 #> [2,] 0.5 -0.5 0 1 #> [3,] 1.0 0.0 0 0 #> [4,] 0.0 1.0 0 0
denpoly()
The function denpoly() computes the coefficients of the denominator polynomial for the elements
$\tilde{\mathcal{H}}_k$ and $\tilde{\mathcal{C}}_k$. The function returns a vector containing the coefficients
in descending powers of $\tilde{n}$, with the last element being the coefficient of $\tilde{n}$.
Arguments
k: The order of the polynomial (a positive integer)alpha: The type of $\beta$-Wishart distribution ($\alpha =2/\beta$):1/2: Quaternion Wishart1: Complex Wishart2: Real Wishart (default)
Output
A vector containing the coefficients of the denominator polynomial in descending powers of $\tilde{n}$ for the elements of $\mathcal{\tilde{H}}_k$ and $\tilde{\mathcal{C}}_k$.
Examples
# Example 1: Compute the denominator polynomial for k = 3 and alpha = 2 # Output corresponds to the polynomial n1^5-3n1^4-8n1^3+12n1^2+16n1, # where n1 is \eqn{\tilde{n}} denpoly(3) #> [1] 1 -3 -8 12 16 # Example 2: Compute the denominator polynomial for k = 2 and alpha = 1 # Output corresponds to the polynomial n1^3-n1, where n1 is \eqn{\tilde{n}} denpoly(2, alpha = 1) #> [1] 1 0 -1
qk_coeff()
The function qk_coeff() computes the coefficient matrix
$\mathcal{C}_k$, which is obtained based on Corollary 1
of Hillier and Kan (2024), after a modification for the
general $\beta$-Wishart case. $\mathcal{C}_k$ is represented as a
3-dimensional array where each slice along the third dimension represents a coefficient matrix of the polynomial in descending powers of $n$.
Arguments
k: The order of the $\mathcal{C}_k$ matrixalpha: The type of Wishart distribution ($\alpha=2/\beta$):1/2: Quaternion Wishart1: Complex Wishart2: Real Wishart (default)
Output
A 3-dimensional array representing $\mathcal{C}k$, a matrix of constants that allow us to obtain $\mathbb{E}[p{\lambda}(W)W^r]$, where $r+|\lambda|=k$ and $W \sim W_m^{\beta}(n,\Sigma)$.
Examples
# Example 1: qk_coeff(2) # For real Wishart distribution with k = 2 #> , , 1 #> #> [,1] [,2] #> [1,] 1 0 #> [2,] 0 1 #> #> , , 2 #> #> [,1] [,2] #> [1,] 1 1 #> [2,] 2 0 # Example 2: qk_coeff(3, 1) # For complex Wishart distribution with k = 3 #> , , 1 #> #> [,1] [,2] [,3] [,4] #> [1,] 1 0 0 0 #> [2,] 0 1 0 0 #> [3,] 0 0 1 0 #> [4,] 0 0 0 1 #> #> , , 2 #> #> [,1] [,2] [,3] [,4] #> [1,] 0 2 1 0 #> [2,] 2 0 0 1 #> [3,] 2 0 0 1 #> [4,] 0 2 1 0 #> #> , , 3 #> #> [,1] [,2] [,3] [,4] #> [1,] 1 0 0 1 #> [2,] 0 1 1 0 #> [3,] 0 2 0 0 #> [4,] 2 0 0 0 # Example 3: qk_coeff(2, 1/2) # For quaternion Wishart distribution with k = 2 #> , , 1 #> #> [,1] [,2] #> [1,] 1 0 #> [2,] 0 1 #> #> , , 2 #> #> [,1] [,2] #> [1,] -0.5 1 #> [2,] 0.5 0
wish_ps()
The function wish_ps() computes the coefficient matrix
$\mathcal{H}k$ that allows us to compute $\mathbb{E}[p{\kappa}(W)]$, which is obtained based
on Proposition 5 of Hillier and Kan (2024), after a modification for the general $\beta$-Wishart case. $\mathcal{H}_k$ is represented as a 3-dimensional array where each slice along the third dimension represents a coefficient matrix of the polynomial in descending powers of $n$.
Arguments
k: The order of the $\mathcal{H}_k$ matrixalpha: The type of Wishart distribution ($\alpha=2/\beta$):1/2: Quaternion Wishart1: Complex Wishart2: Real Wishart (default)
Output
A 3-dimensional array representing $\mathcal{H}k$, a matrix of constants that allows us to obtain $\mathbb{E}[p{\kappa}(W)]$, where $|\kappa|=k$ and $W \sim W_m^{\beta}(n,\Sigma)$.
Examples
# Example 1: wish_ps(3) # For real Wishart distribution with k = 3 #> , , 1 #> #> [,1] [,2] [,3] #> [1,] 1 0 0 #> [2,] 0 1 0 #> [3,] 0 0 1 #> #> , , 2 #> #> [,1] [,2] [,3] #> [1,] 3 3 0 #> [2,] 4 1 1 #> [3,] 0 6 0 #> #> , , 3 #> #> [,1] [,2] [,3] #> [1,] 4 3 1 #> [2,] 4 4 0 #> [3,] 8 0 0 # Example 2: wish_ps(4, 1) # For complex Wishart distribution with k = 4 #> , , 1 #> #> [,1] [,2] [,3] [,4] [,5] #> [1,] 1 0 0 0 0 #> [2,] 0 1 0 0 0 #> [3,] 0 0 1 0 0 #> [4,] 0 0 0 1 0 #> [5,] 0 0 0 0 1 #> #> , , 2 #> #> [,1] [,2] [,3] [,4] [,5] #> [1,] 0 4 2 0 0 #> [2,] 3 0 0 3 0 #> [3,] 4 0 0 2 0 #> [4,] 0 4 1 0 1 #> [5,] 0 0 0 6 0 #> #> , , 3 #> #> [,1] [,2] [,3] [,4] [,5] #> [1,] 5 0 0 6 0 #> [2,] 0 7 3 0 1 #> [3,] 0 8 2 0 1 #> [4,] 6 0 0 5 0 #> [5,] 0 8 3 0 0 #> #> , , 4 #> #> [,1] [,2] [,3] [,4] [,5] #> [1,] 0 4 1 0 1 #> [2,] 3 0 0 3 0 #> [3,] 2 0 0 4 0 #> [4,] 0 4 2 0 0 #> [5,] 6 0 0 0 0 # Example 3: wish_ps(2, 1/2) # For quaternion Wishart distribution with k = 2 #> , , 1 #> #> [,1] [,2] #> [1,] 1 0 #> [2,] 0 1 #> #> , , 2 #> #> [,1] [,2] #> [1,] -0.5 1 #> [2,] 0.5 0
qkn_coeff()
The function qkn_coeff() computes the inverse of the coefficient matrix $\tilde{\mathcal{C}}_k$, which is obtained based on Corollary 2 of Hillier and Kan (2024), after a modification for the general $\beta$-Wishart case. $\tilde{\mathcal{C}}_k^{-1}$ is represented as a 3-dimensional array where each slice along the third dimension represents a coefficient matrix of the polynomial in descending powers of $\tilde{n}$.
Arguments
k: The order of the $\tilde{\mathcal{C}}_k$ matrixalpha: The type of Wishart distribution ($\alpha=2/\beta$):1/2: Quaternion Wishart1: Complex Wishart2: Real Wishart (default)
Output
A 3-dimensional array representing $\tilde{\mathcal{C}}k^{-1}$, a matrix of constants that allow us to obtain $\mathbb{E}[p{\lambda}(W^{-1})W^{-r}]$, where $r+|\lambda|=k$ and $W \sim W_m^{\beta}(n,\Sigma)$.
Examples
# Example 1: qkn_coeff(2) # For real Wishart distribution with k = 2 #> , , 1 #> #> [,1] [,2] #> [1,] 1 0 #> [2,] 0 1 #> #> , , 2 #> #> [,1] [,2] #> [1,] -1 -1 #> [2,] -2 0 # Example 2: qkn_coeff(3, 1) # For complex Wishart distribution with k = 3 #> , , 1 #> #> [,1] [,2] [,3] [,4] #> [1,] 1 0 0 0 #> [2,] 0 1 0 0 #> [3,] 0 0 1 0 #> [4,] 0 0 0 1 #> #> , , 2 #> #> [,1] [,2] [,3] [,4] #> [1,] 0 -2 -1 0 #> [2,] -2 0 0 -1 #> [3,] -2 0 0 -1 #> [4,] 0 -2 -1 0 #> #> , , 3 #> #> [,1] [,2] [,3] [,4] #> [1,] 1 0 0 1 #> [2,] 0 1 1 0 #> [3,] 0 2 0 0 #> [4,] 2 0 0 0 # Example 3: qkn_coeff(2, 1/2) # For quaternion Wishart distribution with k = 2 #> , , 1 #> #> [,1] [,2] #> [1,] 1 0 #> [2,] 0 1 #> #> , , 2 #> #> [,1] [,2] #> [1,] 0.5 -1 #> [2,] -0.5 0
iwish_ps()
The function iwish_ps() computes the inverse of the coefficient matrix $\tilde{\mathcal{H}}k$ that allows us to compute $\mathbb{E}[p{\kappa}(W^{-1})]$, which is obtained based
on Eq.(82) of Hillier and Kan (2024),
after a modification for the general $\beta$-Wishart case. $\tilde{\mathcal{H}}_k^{-1}$ is represented as a 3-dimensional array where each slice along the third dimension represents a
coefficient matrix of the polynomial in descending powers of $\tilde{n}$.
Arguments
k: The order of the $\tilde{\mathcal{H}}_k$ matrixalpha: The type of Wishart distribution ($\alpha=2/\beta$):1/2: Quaternion Wishart1: Complex Wishart2: Real Wishart (default)
Output
A 3-dimensional array representing $\tilde{\mathcal{H}}k^{-1}$, a matrix of constants that allows us to obtain $\mathbb{E}[p{\kappa}(W^{-1})]$, where $|\kappa|=k$ and $W \sim W_m^{\beta}(n,\Sigma)$.
Examples
# Example 1: iwish_ps(3) # For real Wishart distribution with k = 3 #> , , 1 #> #> [,1] [,2] [,3] #> [1,] 1 0 0 #> [2,] 0 1 0 #> [3,] 0 0 1 #> #> , , 2 #> #> [,1] [,2] [,3] #> [1,] -3 -3 0 #> [2,] -4 -1 -1 #> [3,] 0 -6 0 #> #> , , 3 #> #> [,1] [,2] [,3] #> [1,] 4 3 1 #> [2,] 4 4 0 #> [3,] 8 0 0 # Example 2: iwish_ps(4, 1) # For complex Wishart distribution with k = 4 #> , , 1 #> #> [,1] [,2] [,3] [,4] [,5] #> [1,] 1 0 0 0 0 #> [2,] 0 1 0 0 0 #> [3,] 0 0 1 0 0 #> [4,] 0 0 0 1 0 #> [5,] 0 0 0 0 1 #> #> , , 2 #> #> [,1] [,2] [,3] [,4] [,5] #> [1,] 0 -4 -2 0 0 #> [2,] -3 0 0 -3 0 #> [3,] -4 0 0 -2 0 #> [4,] 0 -4 -1 0 -1 #> [5,] 0 0 0 -6 0 #> #> , , 3 #> #> [,1] [,2] [,3] [,4] [,5] #> [1,] 5 0 0 6 0 #> [2,] 0 7 3 0 1 #> [3,] 0 8 2 0 1 #> [4,] 6 0 0 5 0 #> [5,] 0 8 3 0 0 #> #> , , 4 #> #> [,1] [,2] [,3] [,4] [,5] #> [1,] 0 -4 -1 0 -1 #> [2,] -3 0 0 -3 0 #> [3,] -2 0 0 -4 0 #> [4,] 0 -4 -2 0 0 #> [5,] -6 0 0 0 0 # Example 3: iwish_ps(2, 1/2) # For quaternion Wishart distribution with k = 2 #> , , 1 #> #> [,1] [,2] #> [1,] 1 0 #> [2,] 0 1 #> #> , , 2 #> #> [,1] [,2] #> [1,] 0.5 -1 #> [2,] -0.5 0
qkn_coeffr()
The function qkn_coeffr() computes the coefficient matrix $\tilde{\mathcal{C}}_k$ for the general $\beta$-Wishart case.
Elements of $\tilde{\mathcal{C}}_k$ are rational
polynomials of $\tilde{n}$. The output contains two components:
c and den. c is a 3-dimensional array where each slice along the third dimension represents a coefficient matrix of the numerator polynomial in descending powers of $\tilde{n}$,
and den is a vector that represents the coefficients
of the denominator polynomial in descending power of $\tilde{n}$.
Arguments
k: The order of the $\tilde{\mathcal{C}}_k$ matrixalpha: The type of Wishart distribution ($\alpha=2/\beta$):1/2: Quaternion Wishart1: Complex Wishart2: Real Wishart (default)
Output
The output has two components: c and den. c is
a 3-dimensional array representing the numerator polynomial
of $\tilde{\mathcal{C}}k$, and den is a vector representing the denominator polynomial of $\tilde{\mathcal{C}}_k$, where $\tilde{\mathcal{C}}_k$ is a matrix of constants that allow us to obtain $\mathbb{E}[p{\lambda}(W^{-1})W^{-r}]$, where
$r+|\lambda|=k$ and $W \sim W_m^{\beta}(n,\Sigma)$.
Examples
# Example 1: qkn_coeffr(2) # For real Wishart distribution with k = 2 #> $c #> , , 1 #> #> [,1] [,2] #> [1,] 1 0 #> [2,] 0 1 #> #> , , 2 #> #> [,1] [,2] #> [1,] 0 1 #> [2,] 2 -1 #> #> #> $den #> [1] 1 -1 -2 # Example 2: qkn_coeffr(3, 1) # For complex Wishart distribution with k = 3 #> $c #> , , 1 #> #> [,1] [,2] [,3] [,4] #> [1,] 1 0 0 0 #> [2,] 0 1 0 0 #> [3,] 0 0 1 0 #> [4,] 0 0 0 1 #> #> , , 2 #> #> [,1] [,2] [,3] [,4] #> [1,] 0 2 1 0 #> [2,] 2 0 0 1 #> [3,] 2 0 0 1 #> [4,] 0 2 1 0 #> #> , , 3 #> #> [,1] [,2] [,3] [,4] #> [1,] 0 0 0 2 #> [2,] 0 0 2 0 #> [3,] 0 4 -2 0 #> [4,] 4 0 0 -2 #> #> #> $den #> [1] 1 0 -5 0 4 # Example 3: qkn_coeffr(2, 1/2) # For quaternion Wishart distribution with k = 2 #> $c #> , , 1 #> #> [,1] [,2] #> [1,] 1 0 #> [2,] 0 1 #> #> , , 2 #> #> [,1] [,2] #> [1,] 0.0 1.0 #> [2,] 0.5 0.5 #> #> #> $den #> [1] 1.0 0.5 -0.5
iwish_psr()
The function iwish_psr() computes the coefficient matrix $\tilde{\mathcal{H}}_k$ for the general $\beta$-Wishart case.
Elements of $\tilde{\mathcal{H}}_k$ are rational
polynomials of $\tilde{n}$. The output contains two components:
c and den. c is a 3-dimensional array where each slice along the third dimension represents a coefficient matrix of the numerator polynomial in descending powers of $\tilde{n}$,
and den is a vector that represents the coefficients
of the denominator polynomial in descending power of $\tilde{n}$.
Arguments
k: The order of the $\tilde{\mathcal{C}}_k$ matrixalpha: The type of Wishart distribution ($\alpha=2/\beta$):1/2: Quaternion Wishart1: Complex Wishart2: Real Wishart (default)
Output
The output has two components: c and den. c is
a 3-dimensional array representing the numerator polynomial
of $\tilde{\mathcal{H}}k$, and den is a vector representing the denominator polynomial of $\tilde{\mathcal{H}}_k$, where $\tilde{\mathcal{H}}_k$ is a matrix of constants that allow us to obtain $\mathbb{E}[p{\lambda}(W^{-1})]$, where
$|\lambda|=k$ and $W \sim W_m^{\beta}(n,\Sigma)$.
Examples
# Example 1: iwsih_psr(2) # For real Wishart distribution with k = 2 #> $c #> , , 1 #> #> [,1] [,2] #> [1,] 1 0 #> [2,] 0 1 #> #> , , 2 #> #> [,1] [,2] #> [1,] 0 1 #> [2,] 2 -1 #> #> #> $den #> [1] 1 -1 -2 # Example 2: iwish_psr(4, 1) # For complex Wishart distribution with k = 4 #> $c #> , , 1 #> #> [,1] [,2] [,3] [,4] [,5] #> [1,] 1 0 0 0 0 #> [2,] 0 1 0 0 0 #> [3,] 0 0 1 0 0 #> [4,] 0 0 0 1 0 #> [5,] 0 0 0 0 1 #> #> , , 2 #> #> [,1] [,2] [,3] [,4] [,5] #> [1,] 0 4 2 0 0 #> [2,] 3 0 0 3 0 #> [3,] 4 0 0 2 0 #> [4,] 0 4 1 0 1 #> [5,] 0 0 0 6 0 #> #> , , 3 #> #> [,1] [,2] [,3] [,4] [,5] #> [1,] 1 0 0 10 0 #> [2,] 0 3 6 0 2 #> [3,] 0 16 -6 0 1 #> [4,] 10 0 0 1 0 #> [5,] 0 16 3 0 -8 #> #> , , 4 #> #> [,1] [,2] [,3] [,4] [,5] #> [1,] 0 4 -3 0 5 #> [2,] 3 0 0 3 0 #> [3,] -6 0 0 12 0 #> [4,] 0 4 6 0 -4 #> [5,] 30 0 0 -24 0 #> #> , , 5 #> #> [,1] [,2] [,3] [,4] [,5] #> [1,] 0 0 0 0 0 #> [2,] 0 12 -9 0 -3 #> [3,] 0 -24 18 0 6 #> [4,] 0 0 0 0 0 #> [5,] 0 -24 18 0 6 #> #> #> $den #> [1] 1 0 -14 0 49 0 -36 0 # Example 3: iwish_psr(2, 1/2) # For quaternion Wishart distribution with k = 2 #> $c #> , , 1 #> #> [,1] [,2] #> [1,] 1 0 #> [2,] 0 1 #> #> , , 2 #> #> [,1] [,2] #> [1,] 0.0 1.0 #> [2,] 0.5 0.5 #> #> #> $den #> [1] 1.0 0.5 -0.5
References
Díaz-García, José and Gutiérrez-Jáimez, Ramón (2011). On Wishart distribution: som extension. Linear Algebra and its Applications, 435, 1296-1310.
Drensky, Vesselin, Edelman, Alan, Genoar, Tierney, Kan, Raymond, and Koev, Plamen (2021). The Densities and Distributions of the Largest Eigenvalue and the Trace of a Beta-Wishart Matrix. Random Matrices: Theory and Applications, 10(1).
Letac, Gérard, and Massam, Héelène (2004). All invariant moments of the Wishart distribution. Scandinavian Journal of Statistics, 31, 295-318.
Hillier, Grant, and Kan, Raymond (2024). On the expectations of equivariant matrix-valued functions of Wishart and inverse Wishart Matrices. Scandinavian Journal of Statistics, 51, 697-723.
Try the wishmom package in your browser
Any scripts or data that you put into this service are public.
R Package Documentation
Browse R Packages
We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
Embedding an R snippet on your website
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.