Package 'wishmom'

Coefficients of the Denominator Polynomial for $\tilde{H}_k$ and $\tilde{C}_k$

Description

This function computes the coefficients of the denominator polynomial for the elements of $\tilde{H}_k$ and $\tilde{C}_k$. The function returns a vector containing the coefficients in descending powers of $\tilde{n}$, with the last element being the coefficient of $\tilde{n}$.

Usage

denpoly(k, alpha = 2)

Arguments

k	The order of the polynomial (a positive integer)
alpha	The type of Wishart distribution $(\alpha=2/\beta)$: 1/2: Quaternion Wishart 1: Complex Wishart 2: Real Wishart (default)

Value

A vector containing the coefficients of the denominator polynomial in descending powers of $\tilde{n}$ for the elements of $\tilde{H}_k$ and $\mathcal{C}_k$.

Examples

# Example 1: Compute the denominator polynomial for k = 3, alpha = 2
# Output corresponds to the polynomial n1^5-3n1^4-8n1^3+12n1^2+16n1,
# where n1 is \eqn{\tilde{n}}
denpoly(3)

# Example 2: Compute the denominator polynomial for k = 2, alpha = 1
# Output corresponds to the polynomial n1^3-n1, where n1 is \eqn{\tilde{n}}
denpoly(2, alpha = 1)

---

Mapping Matrix that maps $Q_{k+1}$ to $Q_k$ for a beta-Wishart Distribution, but without $n$ on the diagonal

Description

This function computes the matrix that maps $Q_{k+1}$ to $Q_k$ when $W \sim W_m^{\beta}(n, \Sigma)$.

Usage

dkmap(k, alpha = 2)

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 Wishart 1: Complex Wishart 2: Real Wishart (default)

Value

A matrix that maps $Q_{k+1}$ to $Q_k$, but without $n$ on the diagonal.

Examples

# Example 1: Compute the mapping matrix for k = 2, real Wishart
dkmap(2)
# Example 2: Compute the mapping matrix for k = 1, complex Wishart
dkmap(1, 1)
# Example 3: Compute the mapping matrix for k = 2, quaternion Wishart
dkmap(2, 1/2)

---

Generate Integer Partitions in Reverse Lexicographical Order

Description

This function generates all integer partitions of a given integer k in reverse lexicographical order. The function is adapted from "Algorithm ZS1" described in Zoghbi and Stojmenovic (1998), "Fast Algorithms for Generating Integer Partitions", International Journal of Computer Mathematics, Volume 70, Issue 2, pages 319-332.

Usage

ip_desc(k)

Arguments

k	An integer to be partitioned

Value

A matrix where each row represents an integer partition of k. The partitions are listed in reverse lexicographical order.

References

Zoghbi, A., & Stojmenovic, I. (1998). Fast Algorithms for Generating Integer Partitions. International Journal of Computer Mathematics, 70(2), 319-332. DOI: 10.1080/00207169808804755

Examples

# Example 1:
ip_desc(3)

# Example 2:
ip_desc(5)

---

Inverse of a Coefficient Matrix $\tilde{\mathcal{H}}_k$

Description

This function computes the inverse of a coefficient matrix $\tilde{\mathcal{H}}_k$ that allows us to compute the expected value of a power-sum symmetric function of $W^{-1}$, where $W \sim W_m^{\beta}(n,\Sigma)$.

Usage

iwish_ps(k, alpha = 2)

Arguments

k	The order of the $\tilde{\mathcal{H}}_k$ matrix (a positive integer)
alpha	The type of Wishart distribution ($\alpha = 2/\beta$): 1/2: Quaternion Wishart 1: Complex Wishart 2: Real Wishart (default)

Value

Inverse of a coefficient matrix $\tilde{\mathcal{H}}_k$ that allows us to compute the expected value of a power-sum symmetric function of $W^{-1}$, where $W \sim W_m^{\beta}(n,\Sigma)$. The matrix 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 $\tilde{n}$.

Examples

# Example 1:
iwish_ps(3) # For real Wishart distribution with k = 3

# Example 2:
iwish_ps(4, 1) # For complex Wishart distribution with k = 4

# Example 3:
iwish_ps(2, 1/2) # For quaternion Wishart distribution with k = 2

---

Coefficient Matrix $\tilde{\mathcal{H}}_k$

Description

This function computes the coefficient matrix $\tilde{\mathcal{H}}_k$ for $W \sim W_m^{\beta}(n, \Sigma)$.

Usage

iwish_psr(k, alpha = 2)

Arguments

k	The order of the $\tilde{\mathcal{H}}_k$ matrix (a positive integer)
alpha	The type of Wishart distribution ($\alpha = 2/\beta$): 1/2: Quaternion Wishart 1: Complex Wishart 2: Real Wishart (default)

Value

A list with two elements:

• c: A 3-dimensional array containing the coefficient matrices of the numerator of $\tilde{\mathcal{H}}_k$ in descending powers of $n1$, where $n1 = n - m + 1 - \alpha$

• den: A vector containing the coefficients of the denominator of $\tilde{\mathcal{H}}_k$, in descending powers of $n1$

Examples

# Example 1:
iwish_psr(2) # For real Wishart distribution with k = 2

# Example 2:
iwish_psr(4, 1) # For complex Wishart distribution with k = 4

# Example 3:
iwish_psr(2, 1/2) # For quaterion Wishart distribution with k = 2

---

Expectation of a Matrix-valued Function of an Inverse beta-Wishart Distribution

Description

When iw = 0, the function calculates $E[\prod_{j=1}^r \mbox{tr}(W^{-j})^{f_j}]$, where $W \sim W_m^{\beta}(n, S)$. When iw != 0, the function calculates $E[\prod_{j=1}^r \mbox{tr}(W^{-j})^{f_j}W^{-iw}]$.

Usage

iwishmom(n, S, f, iw = 0, alpha = 2)

Arguments

n	The degrees of freedom of the beta-Wishart matrix $W$
S	The covariance matrix of the beta-Wishart matrix $W$
f	A vector of nonnegative integers $f_j$ that represents the power of $\mbox{tr}(W^{-j})$, where $j=1, \ldots, r$
iw	The power of the inverse beta-Wishart matrix $W^{-1}$ (0 by default)
alpha	The type of Wishart distribution $(\alpha=2/\beta)$: 1/2: Quaternion Wishart 1: Complex Wishart 2: Real Wishart (default)

Value

When iw = 0, it returns $E[\prod_{j=1}^r \mbox{tr}(W^{-j})^{f_j}]$. When iw != 0, it returns $E[\prod_{j=1}^r \mbox{tr}(W^{-j})^{f_j}W^{-iw}]$.

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

# 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

# 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:
# Hermitian S for the complex case
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

# 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

---

Symbolic Expectation of a Matrix-valued Function of an Inverse beta-Wishart Distribution

Description

When iw = 0, the function returns an analytical expression of $E[\prod_{j=1}^r \mbox{tr}(W^{-j})^{f_j}]$, where $W \sim W_m^{\beta}(n, S)$. When iw != 0, the function returns an analytical expression of $E[\prod_{j=1}^r \mbox{tr}(W^{-j})^{f_j}W^{-iw}]$. For a given f, iw, and alpha, this function provides the aforementioned expectations in terms of the variables $\tilde{n}$ and $\Sigma$.

Usage

iwishmom_sym(f, iw = 0, alpha = 2, latex = FALSE)

Arguments

f	A vector of nonnegative integers $f_j$ that represents the power of $\mbox{tr}(W^{-j})$, where $j=1, \ldots, r$
iw	The power of the inverse beta-Wishart matrix $W^{-1}$ (0 by default)
alpha	The type of Wishart distribution $(\alpha=2/\beta)$: 1/2: Quaternion Wishart 1: Complex Wishart 2: Real Wishart (default)
latex	A Boolean indicating whether the output will be a LaTeX string or dataframe (FALSE by default)

Value

When iw = 0, it returns an analytical expression of $E[\prod_{j=1}^r \mbox{tr}(W^{-j})^{f_j}]$. When iw != 0, it returns an analytical expression of $E[\prod_{j=1}^r \mbox{tr}(W^{-j})^{f_j}W^{-iw}]$. If latex = FALSE, the output is a data frame that stores the coefficients for calculating the result. If latex = TRUE, the output is a LaTeX formatted string of the result in terms of $\tilde{n}$ and $\Sigma$.

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

# Example 2: For E[tr(W^{-1})*tr(W^{-2})W^{-1}] with W ~ W_m^1(n,S), represented as a dataframe:
iwishmom_sym(c(1, 1), 1) # iw = 1, for real Wishart distribution

# Example 3: For E[tr(W^{-1})^4] with W ~ W_m^2(n,S), represented as a LaTeX string:
# Using writeLines() to format
writeLines(iwishmom_sym(4, 0, 1, latex=TRUE)) # iw = 0, for complex Wishart distribution

# Example 4: For E[tr(W^{-1})*tr(W^{-2})W^{-1}] with W ~ W_m^2(n,S), represented as a LaTeX string:
# Using writeLines() to format
writeLines(iwishmom_sym(c(1, 1), 1, 1, latex=TRUE)) # iw = 1, for real Wishart distribution

---

Coefficient Matrix $\mathcal{C}_k$

Description

This function computes the coefficient matrix $\mathcal{C}_k$, which is a matrix of constants that allows us to obtain $E[p_{\lambda}(W)W^r]$, where $r+|\lambda|=k$ and $W \sim W_m^{\beta}(n, \Sigma)$.

Usage

qk_coeff(k, alpha = 2)

Arguments

k	The order of the $\mathcal{C}_k$ matrix
alpha	The type of Wishart distribution ($\alpha=2/\beta$): 1/2: Quaternion Wishart 1: Complex Wishart 2: Real Wishart (default)

Value

$\mathcal{C}_k$, a matrix that allows us to obtain $E[p_{\lambda}(W)W^r]$, where $r+|\lambda|=k$ and $W \sim W_m^{\beta}(n, \Sigma)$. The matrix 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 $n$.

Examples

# Example 1:
qk_coeff(2) # For real Wishart distribution with k = 2

# Example 2:
qk_coeff(3, 1) # For complex Wishart distribution with k = 3

# Example 3:
qk_coeff(2, 1/2) # For quaternion Wishart distribution with k = 2

---

Inverse of a Coefficient Matrix $\tilde{\mathcal{C}}_k$

Description

This function computes the inverse of the coefficient matrix $\tilde{\mathcal{C}}_k$

Usage

qkn_coeff(k, alpha = 2)

Arguments

k	The order of the $\tilde{\mathcal{C}}_k$ matrix
alpha	The type of beta-Wishart distribution ($\alpha=2/\beta$): 1/2: Quaternion Wishart 1: Complex Wishart 2: Real Wishart (default)

Value

Inverse of a coefficient matrix $\tilde{\mathcal{C}}_k$ that allows us to obtain $E[p_{\lambda}(W^{-1})W^{-r}]$, where $r+|\lambda|=k$ and $W ~ W_m^{\beta}(n,\Sigma)$. The matrix 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 $\tilde{n}$.

Examples

# Example 1:
qkn_coeff(2) # For real Wishart distribution with k = 2
# Example 2:
qkn_coeff(3, 1) # For complex Wishart distribution with k = 3

# Example 3:
qkn_coeff(2, 1/2) # For quaternion Wishart distribution with k = 2

---

Coefficient Matrix $\tilde{C}_k$

Description

This function computes the coefficient matrix for $\tilde{\mathcal{C}}_k$ for $W \sim W_m^{\beta}(n, \Sigma)$.

Usage

qkn_coeffr(k, alpha = 2)

Arguments

k	The order of the $\tilde{\mathcal{C}}_k$ matrix (a positive integer)
alpha	The type of Wishart distribution ($\alpha = 2/\beta$): 1/2: Quaternion Wishart 1: Complex Wishart 2: Real Wishart (default)

Value

A list with two elements:

• c: A 3-dimensional array containing the coefficient matrices of the numerator of $\tilde{\mathcal{C}}_k$ in descending powers of $n1$, where $n1 = n - m + 1 - \alpha$.

• den: A vector containing the coefficients of the denominator of $\tilde{\mathcal{C}}_k$, in descending powers of $n1$.

Examples

# Example 1:
qkn_coeffr(2) # For real Wishart distribution with k = 2

# Example 2:
qkn_coeffr(3, 1) # For complex Wishart distribution with k = 3

# Example 3:
qkn_coeffr(2, 1/2) # For quaternion Wishart distribution with k = 2

---

Coefficient Matrix $\mathcal{H}_k$

Description

This function computes the coefficient matrix $\mathcal{H}_k$ that allows us to compute the expected value of a power-sum symmetric function of $W$, where $W \sim W_m^{\beta}(n,\Sigma)$.

Usage

wish_ps(k, alpha = 2)

Arguments

k	The order of the $\mathcal{H}_k$ matrix
alpha	The type of Wishart distribution ($\alpha = 2/\beta$): 1/2: Quaternion Wishart 1: Complex Wishart 2: Real Wishart (default)

Value

A coefficient matrix $\mathcal{H}_k$ that allows us to compute the expected value of a power-sum symmetric function of $W$, where $W \sim W_m^{\beta}(n,\Sigma)$. The matrix 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 $n$.

Examples

# Example 1:
wish_ps(3) # For real Wishart distribution with k = 3

# Example 2:
wish_ps(4, 1) # For complex Wishart distribution with k = 4

# Example 3:
wish_ps(2, 1/2) # For quaternion Wishart distribution with k = 2

---

Expectation of a Matrix-valued Function of a beta-Wishart Distribution

Description

When iw = 0, the function calculates $E[\prod_{j=1}^r \mbox{tr}(W^j)^{f_j}]$, where $W \sim W_m^{\beta}(n, S)$. When iw != 0, the function calculates $E[\prod_{j=1}^r \mbox{tr}(W^j)^{f_j}W^{iw}]$

Usage

wishmom(n, S, f, iw = 0, alpha = 2)

Arguments

n	The degrees of freedom of the beta-Wishart matrix $W$
S	The covariance matrix of the beta-Wishart matrix $W$
f	A vector of nonnegative integers $f_j$ that represents the power of $\mbox{tr}(W^j)$, where $j=1, \ldots, r$
iw	The power of the inverse beta-Wishart matrix $W$ (0 by default)
alpha	The type of Wishart distribution $(\alpha=2/\beta)$: 1/2: Quaternion Wishart 1: Complex Wishart 2: Real Wishart (default)

Value

When iw = 0, it returns $E[\prod_{j=1}^r \mbox{tr}(W^j)^{f_j}]$. When iw != 0, it returns $E[\prod_{j=1}^r \mbox{tr}(W^j)^{f_j}W^{iw}]$.

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

# 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

# Example 3: For E[tr(W)^2*tr(W^3)] with W ~ W_m^2(n,S),
# where n and S are defined below:
# Hermitian S for the complex case
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

# Example 4: For E[tr(W)*tr(W^2)^2*tr(W^3)^2*W] 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(1, 2, 2), 1, 1) # iw = 1, for complex Wishart distribution

---

Symbolic Expectation of a Matrix-valued Function of a beta-Wishart Distribution

Description

When iw = 0, the function returns an analytical expression of $E[\prod_{j=1}^r \mbox{tr}(W^j)^{f_j}]$, where $W \sim W_m^{\beta}(n, S)$. When iw != 0, the function returns an analytical expression of $E[\prod_{j=1}^r \mbox{tr}(W^j)^{f_j}W^{iw}]$. For a given f, iw, and alpha, this function provides the aforementioned expectations in terms of the variables $n$ and $\Sigma$.

Usage

wishmom_sym(f, iw = 0, alpha = 2, latex = FALSE)

Arguments

f	A vector of nonnegative integers $f_j$ that represents the power of $\mbox{tr}(W^{-j})$, where $j=1, \ldots, r$
iw	The power of the beta-Wishart matrix $W^{-1}$ (0 by default)
alpha	The type of Wishart distribution $(\alpha=2/\beta)$: 1/2: Quaternion Wishart 1: Complex Wishart 2: Real Wishart (default)
latex	A Boolean indicating whether the output will be a LaTeX string or a dataframe (FALSE by default)

Value

When iw = 0, it returns an analytical expression of $E[\prod_{j=1}^r \mbox{tr}(W^j)^{f_j}]$. When iw != 0, it returns an analytical expression of $E[\prod_{j=1}^r \mbox{tr}(W^j)^{f_j}W^{iw}]$. If latex = FALSE, the output is a data frame that stores the coefficients for calculating the result. 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

# Example 2: For E[tr(W)*tr(W^2)W] with W ~ W_m^1(n,S), represented as a dataframe:
wishmom_sym(c(1, 1), 1) # iw = 1, for real Wishart distribution

# Example 3: For E[tr(W)^4] with W ~ W_m^2(n,S), represented as a LaTeX string:
# Using writeLines() to format
writeLines(wishmom_sym(4, 0, 1, latex=TRUE)) # iw = 0, for complex Wishart distribution

# Example 4: For E[tr(W)*tr(W^2)W] with W ~ W_m^2(n,S), represented as a LaTeX string:
# Using writeLines() to format
writeLines(wishmom_sym(c(1, 1), 1, 1, latex=TRUE)) # iw = 1, for real Wishart distribution

---