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

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

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

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

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

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

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

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

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

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

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

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

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

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