Package {GWnorm}

This function returns the log of the G-Wishart normalising constant I_G(beta, D) = int_[S^p_++(G)] det(K)^beta * exp(-tr(KD)) dK from the transformed version log(C_G(delta, D)) = int_[S^p_++(G)] det(K)^((delta-2)/2) * exp(-tr(KD)/2) dK

Description

This function returns the log of the G-Wishart normalising constant I_G(beta, D) = int_[S^p_++(G)] det(K)^beta * exp(-tr(KD)) dK from the transformed version log(C_G(delta, D)) = int_[S^p_++(G)] det(K)^((delta-2)/2) * exp(-tr(KD)/2) dK

Usage

C_GtoI_G(graph, CGval, delta)

Arguments

Value

A number log(I_G(beta, D))

Clique-completion

Description

This function returns the completion of a matrix with respect to a graph. The entries of the given matrix are such that in the clique decomposition of the chordal completion there is no linear term in the determinant expansion.

Usage

Clique_complete(
  D,
  missing_edges,
  cliques,
  cliques_ann,
  prec_count = 0,
  tol = 1e-12
)

Arguments

Value

A real symmetric positive definite p by p matrix

This function is a wrapper for BDgraph to compare and to avoid the NOTE in the package checks since we only had BDgraph in the examples

Description

This function is a wrapper for BDgraph to compare and to avoid the NOTE in the package checks since we only had BDgraph in the examples

Usage

I_G_BD(graph, beta, D, n_samp = 1000, C_G_flag = FALSE)

Arguments

Value

A number log(I_G(beta, D)) [or log(C_G(delta, D))]

G Wishart normalising constant through MC integration

Description

This function returns the log of transformed G-Wishart normalising constant I_G(beta, D) = int_[S^p_++(G)] det(K)^beta * exp(-tr(KD)) dK for connected graphs G. If there is no explicit formula, or the integral cannot be reduced into lower dimensions, MC integration is used.

Usage

I_G_MC(
  graph,
  beta,
  D,
  n_samp = 1000,
  nu = 10,
  err_flag = FALSE,
  int1d = TRUE,
  robust = FALSE,
  quench_k = 0,
  chordal = NULL
)

Arguments

Value

A number log(I_G(beta, D))

Examples

set.seed(42)
p <- 5
data <- matrix(runif(10*p), ncol = p)
D <- t(data) %*% data
beta <- 1
adj <- matrix(0, nrow = p, ncol = p)
adj[1,5] <- adj[1,2] <- adj[2,3] <- adj[3,4] <- adj[4,5] <- 1 # 5-cycle
graph <- igraph::graph_from_adjacency_matrix(adj, mode = c("max"))
I_G_MC(graph, beta, D) 
# compare with BDgraph gnorm()
I_G_BD(graph, beta, D, n_samp = 1e5) 

G Wishart normalising constant for chordal graphs

Description

This function returns the log of transformed G-Wishart normalising constant I_G(beta, D) = int_[S^p_++(G)] det(K)^beta * exp(-tr(KD)) dK for chordal graphs G, using explicit formula (cliques / separators).

Usage

I_G_chordal(
  graph,
  beta,
  D = NULL,
  cliques = NULL,
  separators = NULL,
  ratio = FALSE
)

Arguments

Value

A number log(I_G(beta, D))

Examples

set.seed(42)
p <- 5
data <- matrix(runif(10*p), ncol = p)
D <- t(data) %*% data
beta <- 1
adj <- matrix(0, nrow = p, ncol = p)
adj[1,c(2:5)] <- adj[2,3] <- adj[3,4] <- adj[4,5] <- 1 # chordal completion of 5-cycle
graph <- igraph::graph_from_adjacency_matrix(adj, mode = c("max"))
I_G_chordal(graph, beta, D) 
# compare with BDgraph gnorm()
I_G_BD(graph, beta, D, n_samp = 1e4) 

G Wishart normalising constant for complete graphs

Description

This function returns the log of the transformed G-Wishart normalising constant I_G(beta, D) = int_[S^p_++(G)] det(K)^beta * exp(-tr(KD)) dK for complete graphs G, using explicit formula det(D)^(-beta-(p+1)/2) Gamma_n(beta+(p+1)/2)

Usage

I_G_complete(p, beta, D = NULL, ratio = FALSE)

Arguments

Value

A number log(I_G(beta, D))

Examples

set.seed(42)
p <- 6
data <- matrix(runif(10*p), ncol = p)
D <- t(data) %*% data
beta <- 1
I_G_complete(6, beta, D)
# compare with BDgraph gnorm()
adj <- matrix(1, nrow = 6, ncol = 6)
graph <- igraph::graph_from_adjacency_matrix(adj, mode = c("max"))
I_G_BD(graph, beta, D)

G Wishart normalising constant

Description

This function returns the approximation of the ratio of log transformed G-Wishart normalising constants I_G(beta, D) / I_G(beta, I) for graphs G. Graphs are decomposed into connected prime components. For each component if there is no explicit formula, the approximation is used. Note that this is the same as the ratio C_G(delta, D) / C_G(delta, I) with delta = 2*beta + 2

Usage

I_G_ratio_approx(
  graph,
  beta,
  D,
  connected = FALSE,
  prime = FALSE,
  chordal = NULL,
  use_comp = NULL
)

Arguments

Value

A number approximating log(I_G(beta, D)) - log(I_G(beta, I))

Examples

set.seed(42)
p <- 11
data <- matrix(runif(10*p), ncol = p)
D <- t(data) %*% data
beta <- 1
graph <- igraph::make_graph(edges = c(1,2, 2,4, 2,5, 1,3, 3,5, # disconnected 
                                     5,6, 6,7, 5,7, 4,7, # with prime decomposition too
                                     8,9, 8,11, 9,10, 10,11), n = 11, directed = FALSE)
I_G_ratio_approx(graph, beta, D) 
# compare with MC estimates
I_G_MC(graph, beta, D, n_samp = 1e5) - I_G_MC(graph, beta, diag(p), n_samp = 1e5)

This function returns the approximation of the ratio of log transformed G-Wishart normalising constants I_G(beta, D) / I_G(beta, I) for connected prime graphs G. If there is no explicit formula, the approximation is used. Note that this is the same as the ratio C_G(delta, D) / C_G(delta, I) with delta = 2*beta + 2

Description

This function returns the approximation of the ratio of log transformed G-Wishart normalising constants I_G(beta, D) / I_G(beta, I) for connected prime graphs G. If there is no explicit formula, the approximation is used. Note that this is the same as the ratio C_G(delta, D) / C_G(delta, I) with delta = 2*beta + 2

Usage

I_G_ratio_approx_prime(graph, beta, D, chordal = NULL, use_comp = NULL)

Arguments

Value

A number approximating log(I_G(beta, D)) - log(I_G(beta, I))

Examples

set.seed(42)
p <- 5
data <- matrix(runif(10*p), ncol = p)
D <- t(data) %*% data
beta <- 1
adj <- matrix(0, nrow = p, ncol = p)
adj[1,5] <- adj[1,2] <- adj[2,3] <- adj[3,4] <- adj[4,5] <- 1 # 5-cycle
graph <- igraph::graph_from_adjacency_matrix(adj, mode = c("max"))
I_G_ratio_approx_prime(graph, beta, D, use_comp = TRUE) 
# compare with MC estimates
I_G_MC(graph, beta, D, n_samp = 1e5) - I_G_MC(graph, beta, diag(p), n_samp = 1e5)

G Wishart normalising constant for special cases

Description

This function returns the log of transformed G-Wishart normalising constant I_G(beta, I) = int_[S^p_++(G)] det(K)^beta * exp(-tr(K)) dK for connected graphs G for special cases where there is an explicit formula, including involving a low-dimensional integral.

Usage

I_G_special(
  graph,
  beta,
  connected = FALSE,
  prime = FALSE,
  test_6_cycle = TRUE,
  test_k_partite = TRUE,
  chordal = NULL
)

Arguments

Value

A number log(I_G(beta, I))

Examples

beta <- 1
p <- 5
adj <- matrix(0, nrow = p, ncol = p)
adj[1,5] <- adj[1,2] <- adj[2,3] <- adj[3,4] <- adj[4,5] <- 1 # 5-cycle
graph <- igraph::graph_from_adjacency_matrix(adj, mode = c("max"))
I_G_special(graph, beta) 
# compare with BDgraph gnorm()
I_G_BD(graph, beta, diag(p), n_samp = 1e5)
p <- 6
adj <- matrix(0, nrow = p, ncol = p)
adj[1,6] <- adj[1,2] <- adj[2,3] <- adj[3,4] <- adj[4,5] <- adj[5,6] <- 1 # 6-cycle
graph <- igraph::graph_from_adjacency_matrix(adj, mode = c("max"))
I_G_special(graph, beta) 
# compare with BDgraph gnorm()
I_G_BD(graph, beta, diag(p), n_samp = 1e5)
graph <- igraph::complementer(graph) # complement of 6-cycle
I_G_special(graph, beta) 
# compare with BDgraph gnorm()
I_G_BD(graph, beta, diag(p), n_samp = 1e5)
adj[upper.tri(adj)] <- 1
adj[2,3] <- adj[1,4] <- adj[5,6] <- 0 # complete graph with some edges missing
graph <- igraph::graph_from_adjacency_matrix(adj, mode = c("max"))
I_G_special(graph, beta) 
# compare with BDgraph gnorm()
I_G_BD(graph, beta, diag(p), n_samp = 1e5)

G Wishart normalising constant

Description

This function returns the log of transformed G-Wishart normalising constant I_G(beta, D) = int_[S^p_++(G)] det(K)^beta * exp(-tr(KD)) dK for graphs G. Graphs are decomposed into connected prime components. For each component if there is no explicit formula, or the integral cannot be reduced into one dimension, MC integration is used.

Usage

I_Gnorm(
  graph,
  beta,
  D,
  connected = FALSE,
  prime = FALSE,
  chordal = NULL,
  n_samp = 1000,
  nu = 10,
  robust = FALSE,
  quench_k = 0,
  err_flag = FALSE,
  useBDgraph = FALSE
)

Arguments

Value

A number log(I_G(beta, D))

Examples

set.seed(42)
p <- 11
data <- matrix(runif(10*p), ncol = p)
D <- t(data) %*% data
beta <- 1
graph <- igraph::make_graph(edges = c(1,2, 2,4, 2,5, 1,3, 3,5, # disconnected 
                                     5,6, 6,7, 5,7, 4,7, # with prime decomposition too
                                     8,9, 8,11, 9,10, 10,11), n = 11, directed = FALSE)
I_Gnorm(graph, beta, D)
# compare with BDgraph 
I_Gnorm(graph, beta, D, n_samp = 1e5, useBDgraph = TRUE)

This function returns the log of the G-Wishart normalising constant log(C_G(delta, D)) = int_[S^p_++(G)] det(K)^((delta-2)/2) * exp(-tr(KD)/2) dK from the transformed version I_G(beta, D) = int_[S^p_++(G)] det(K)^beta * exp(-tr(KD)) dK

Description

This function returns the log of the G-Wishart normalising constant log(C_G(delta, D)) = int_[S^p_++(G)] det(K)^((delta-2)/2) * exp(-tr(KD)/2) dK from the transformed version I_G(beta, D) = int_[S^p_++(G)] det(K)^beta * exp(-tr(KD)) dK

Usage

I_GtoC_G(graph, IGval, beta)

Arguments

Value

A number log(C_G(delta, D))

Isserlis complement matrix

Description

This function returns the Isserlis matrix (rows/columns arranged in some order) with respect to a matrix and a graph for the missing edges (complement of the graph).

Usage

Iss_cmat(M, graph)

Arguments

Value

An m by m matrix, where m is the number of missing edges in the graph

Examples

# example code
adj <- matrix(0, nrow = 5, ncol = 5)
adj[1,5] <- adj[1,2] <- adj[2,3] <- adj[3,4] <- adj[4,5] <- 1 # 5-cycle
graph <- igraph::graph_from_adjacency_matrix(adj, mode = c("max"))
Iss_cmat(matrix(1:25, nrow = 5, byrow = TRUE), graph)

Isserlis matrix

Description

This function returns the Isserlis matrix (rows/columns arranged in some order) with respect to a matrix and a graph

Usage

Iss_mat(M, graph)

Arguments

Value

An m by m matrix, where m is the number of vertices and edges in the graph

Examples

# example code
adj <- matrix(0, nrow = 5, ncol = 5)
adj[1,5] <- adj[1,2] <- adj[2,3] <- adj[3,4] <- adj[4,5] <- 1 # 5-cycle
graph <- igraph::graph_from_adjacency_matrix(adj, mode = c("max"))
Iss_mat(matrix(1:25, nrow = 5, byrow = TRUE), graph)

PD-completion

Description

This function returns the PD-completion of a matrix with respect to a graph. The entries of the given matrix corresponding to the missing edges are modified such that the inverse has zeros in the places of missing edges.

Usage

PD_complete(graph, D, missing_edges = NULL, tol = 1e-12)

Arguments

Value

A real symmetric positive definite p by p matrix

Examples

set.seed(42)
p <- 5
data <- matrix(runif(10*p), ncol = p)
D <- t(data) %*% data
adj <- matrix(0, nrow = p, ncol = p)
adj[1,5] <- adj[1,2] <- adj[2,3] <- adj[3,4] <- adj[4,5] <- 1 # 5-cycle
graph <- igraph::graph_from_adjacency_matrix(adj, mode = c("max"))
(D_tilde <- PD_complete(graph, D))
(round(solve(D_tilde), 6)) # has zeros at the missing edges

this helper function adds information to cliques related to the missing edges

Description

this helper function adds information to cliques related to the missing edges

Usage

annotate_cliques(cliques, clique_flags, missing_edges, beta)

Arguments

Value

annotated list of cliques

This function just checks if a graph is connected and prime

Description

This function just checks if a graph is connected and prime

Usage

check_prime_connected(graph, check_prime = TRUE)

Arguments

Value

A boolean, TRUE if connected and prime

Chordal Factor # do not export

Description

Chordal Factor # do not export

Usage

chordal_factor(graph)

Arguments

Value

A list of maximal cliques and separators

Newton-Raphson update for the clique-completion

Description

Newton-Raphson update for the clique-completion

Usage

clique_update_D(D, tau, cliques, cliques_ann, prec_flag = TRUE)

Arguments

Value

the update step

Find triangle contains two missing edges # do not export

Description

This function checks whether there is a triangle in the graph(graph2) contains at least two edges from the missing edges (graph1).

Usage

form_triangle(graph1, graph2)

Arguments

Value

TRUE if there is such a triangle

Determine whether the graph is the cycle of length 6 or its complement

Description

Input an igraph object, this function tells you whether graph is the cycle of length 6, or its complement.

Usage

is_6_cycle(graph)

Arguments

Value

An integer, 1 means the graph is the cycle of length 6, 2 means the the graph is the complement of the cycle of length 6, 0 means otherwise

Examples

# C6 example 5 1 3 2 4 6
is_6_cycle(igraph::make_graph(edges = c(1,3, 1,5, 2,3, 2,4, 5,6, 6,4), 
  n = 6, directed = FALSE))
# C6 complement example
is_6_cycle(igraph::make_graph(edges = c(1,2, 1,4, 1,6, 2,5, 2,6, 3,4, 
  3,5, 3,6, 4,5), n = 6, directed = FALSE))
# not C6 examples
is_6_cycle(igraph::make_graph(edges = c(1,3, 1,5, 2,3, 2,4, 5,6, 6,4), 
  n = 7, directed = FALSE))
is_6_cycle(igraph::make_graph(edges = c(1,4, 1,5, 2,3, 2,4, 5,6, 6,4), 
  n = 6, directed = FALSE))

Determine whether the graph is complete k partite

Description

Input an igraph object, this function tells you the size of the partition if the graph is a k partite graph.

Usage

is_k_partite(graph)

Arguments

Value

A vector of the size of the partition. The length of the vector is k. Empty vector means that the graph is not a complete k partite graph.

Examples

# not complete k-partite example
is_k_partite(igraph::make_graph(edges = c(1,2, 2,4, 2,5, 1,3, 3,5, 5,6, 6,7, 
  5,7, 4,7), n = 7, directed = FALSE))
# complete 4-partite example, (1,5), (2,4), (3,7), (6)
is_k_partite(igraph::make_graph(edges = c(1,2, 1,3, 1,4, 1,6, 1,7, 2,3, 2,5, 2,6, 
  2,7, 3,4, 3,5, 3,6, 4,5, 4,6, 4,7, 5,6, 5,7, 6,7), n = 7, directed = FALSE))

Compute means and gradients for the Newton-Raphson update for the clique-completion

Description

Compute means and gradients for the Newton-Raphson update for the clique-completion

Usage

local_mean_grad(D, tau, cliques, cliques_ann, prec_flag = TRUE)

Arguments

Value

the mean and gradient for the update step note this does not take into account the covariance/precision for numerical speed

Compute second moment of the log expansion of the determinant (assuming the mean is 0 from Clique_completion)

Description

Compute second moment of the log expansion of the determinant (assuming the mean is 0 from Clique_completion)

Usage

local_precision(D, tau, cliques, cliques_ann)

Arguments

Value

the precision matrix

Predict Row # do not export

Description

Predict Row # do not export

Usage

predict_row(i, j, D)

Arguments

Value

the predicted values for the vector j

Prime Decomposition

Description

This function returns the prime components and the minimal separators of a connected graph.

Usage

prime_decomp(graph)

Arguments

Value

A list of the prime components and the separators

Examples

graph <- igraph::make_graph(edges = c(1,2, 2,4, 2,5, 1,3, 3,5, 
  5,6, 6,7, 5,7, 4,7), n = 7, directed = FALSE)
prime_decomp(graph)