Gaussian Graphical Models

gaussian_graphical_model(G::Graph{Directed}; s_varname::VarName="s", l_varname::VarName="l", w_varname::VarName=:w)
gaussian_graphical_model(G::MixedGraph; s_varname::VarName="s", l_varname::VarName="l", w_varname::VarName="w")
gaussian_graphical_model(G::Graph{Undirected}; s_varname::VarName="s", k_varname::VarName="k")

Given a graph G construct a gaussian graphical model for G. Optionally one can set the letter used for the variable of the covariance matrix by setting s_varname.

Examples

julia> G = graph_from_edges(Directed, [[1, 2], [2, 3]])
Directed graph with 3 nodes and the following edges:
(1, 2)(2, 3)

julia> GM = gaussian_graphical_model(G)
Gaussian Graphical Model on a Directed graph with 3 nodes and 2 edges

julia> typeof(GM)
GaussianGraphicalModel{Graph{Directed}, Nothing}

julia> CG = graph_from_labeled_edges(Dict((1, 2) => "red", (2, 3) => "green"); name=:color)
Undirected graph with 3 nodes and the following labeling(s):
label: color
(2, 1) -> red
(3, 2) -> green

julia> GM = gaussian_graphical_model(CG)
Gaussian Graphical Model on a Undirected graph with 3 nodes and 2 edges with labeling(s) [:color]

directed_edges_matrix(M::GaussianGraphicalModel{<:AbstractGraph{T}}) where {T <: Union{Mixed, Directed}}

Create the weighted adjacency matrix $\Lambda$ of a directed graph G whose entries are the parameter ring of the graphical model M.

Examples

julia> GM = gaussian_graphical_model(graph_from_edges(Directed, [[1,2], [2,3]]))
Gaussian Graphical Model on a Directed graph with 3 nodes and 2 edges

julia> directed_edges_matrix(GM)
[0   l[1, 2]         0]
[0         0   l[2, 3]]
[0         0         0]

error_covariance_matrix(M::GaussianGraphicalModel{Graph{Directed}, L}) where L

Create the covariance matrix $ \Omega $ of the independent error terms in a directed Gaussian graphical model M

Examples

julia> M = gaussian_graphical_model(graph_from_edges(Directed, [[1,2], [2,3]]))
Gaussian Graphical Model on a Directed graph with 3 nodes and 2 edges

julia> error_covariance_matrix(M)
[w[1]      0      0]
[   0   w[2]      0]
[   0      0   w[3]]

error_covariance_matrix(M::GaussianGraphicalModel{Graphh{Directed}, L}) where L

Create the covariance matrix $ \Omega $ of the independent error terms in a directed Gaussian graphical model M

Examples

julia> M = gaussian_graphical_model(graph_from_edges(Directed, [[1,2], [2,3]]))
Gaussian Graphical Model on a Directed graph with 3 nodes and 2 edges

julia> error_covariance_matrix(M)
[w[1]      0      0]
[   0   w[2]      0]
[   0      0   w[3]]

concentration_matrix(M::GaussianGraphicalModel{Graph{Undirected}, T}) where T

Create the concentration matrix K of an undirected Gaussian graphical model which is a symmetric positive definite matrix whose nonzero entries correspond to the edges of the associated graph.

Examples

julia> M = gaussian_graphical_model(graph_from_edges([[1,2], [2,3]]))
Gaussian Graphical Model on a Undirected graph with 3 nodes and 2 edges

julia> concentration_matrix(M)
[k[1, 1]   k[2, 1]         0]
[k[2, 1]   k[2, 2]   k[3, 2]]
[      0   k[3, 2]   k[3, 3]]

gaussian_ring(F::Field, n::Int; s_var_name::VarName=:s, cached=false)
gaussian_ring(n::Int; s_var_name::VarName=:s, cached=false)
gaussian_ring(GM::GaussianGraphicalmodel)

A polynomial ring whose variables correspond to the entries of a covariance matrix of n Gaussian random variables. It is a multivariate polynomial ring whose variables are named s[i,j]and whose coefficient field is QQ by default or one can specify the field F by passing it as the first argument.

If cached is true, the internally generated polynomial ring will be cached.

Examples

julia> R = gaussian_ring(3)
GaussianRing over Rational field in 6 variables
s[1, 1], s[1, 2], s[1, 3], s[2, 2], s[2, 3], s[3, 3]

julia> M = gaussian_graphical_model(graph_from_edges([[1,2], [2,3]]))
Gaussian Graphical Model on a Undirected graph with 3 nodes and 2 edges

julia> gaussian_ring(M)
GaussianRing over Rational field in 6 variables
s[1, 1], s[1, 2], s[1, 3], s[2, 2], s[2, 3], s[3, 3]

covariance_matrix(GM::GaussianGraphicalModel)

Return the covariance matrix associated to the graphical model GM as a matrix over the underlying model ring of GM, see model_ring

Examples

julia> GM = gaussian_graphical_model(graph_from_edges(Directed, [[1,2], [2,3]]))
Gaussian Graphical Model on a Directed graph with 3 nodes and 2 edges

julia> covariance_matrix(GM)
[s[1, 1]   s[1, 2]   s[1, 3]]
[s[1, 2]   s[2, 2]   s[2, 3]]
[s[1, 3]   s[2, 3]   s[3, 3]]