Gaussian Graphical Models

gaussian_ring(n::Int; s_var_name::VarName="s", K::Field=QQ, cached=false)

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 K is by default QQ.

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

Examples

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

gens(R::GaussianRing)

Return the generators of the multivariate polynomial ring inside the GaussianRing as a dictionary which can be easily indexed

Examples

julia> R = gaussian_ring(3)
Gaussian ring 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> gens(R)
Dict{Tuple{Int64, Int64}, QQMPolyRingElem} with 6 entries:
  (1, 2) => s[1, 2]
  (1, 1) => s[1, 1]
  (3, 3) => s[3, 3]
  (1, 3) => s[1, 3]
  (2, 2) => s[2, 2]
  (2, 3) => s[2, 3]

covariance_matrix(R::GaussianRing)

Return the covariance matrix associated to R as a matrix over the underlying polynomial ring of R

Examples

julia> R = gaussian_ring(3)
Gaussian ring 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> covariance_matrix(R)
[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]]

graphical_model(G::Graph{Directed}, S::GaussianRing; l_var_name::VarName="l", w_var_name::VarName="w", cached=false)

A parametric statistical model associated to a directed acyclic graph. It contains a directed acylic graph G, a GaussianRing S where the vanishing ideal of the model naturally lives, and a parameter ring whose variables l[i,j] correspond to the directed edges i -> j in G.

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

Examples

julia> M = graphical_model(graph_from_edges(Directed, [[1,2], [2,3]]), gaussian_ring(3))
Gaussian graphical model on a directed graph with edges:
(1, 2), (2, 3)

directed_edges_matrix(M::GraphicalModel{Graph{Directed}, GaussianRing})

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

Examples

julia> M = graphical_model(graph_from_edges(Directed, [[1,2], [2,3]]), gaussian_ring(3))
Gaussian graphical model on a directed graph with edges:
(1, 2), (2, 3)

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

error_covariance_matrix(M::GraphicalModel{Graph{Directed}, GaussianRing})

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

Examples

julia> M = graphical_model(graph_from_edges(Directed, [[1,2], [2,3]]), gaussian_ring(3))
Gaussian graphical model on a directed graph with edges:
(1, 2), (2, 3)

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

parametrization(M::GraphicalModel{Graph{Directed}, GaussianRing})

Creates the polynomial map which parametrizes the vanishing ideal of the directed Gaussian graphical model M. The vanishing ideal of the statistical model is the kernel of this map. This ring map is the pull back of the parametrization $\phi_G$ given by $(Id - \Lambda)^{-T} \Omega (Id - \Lambda)^{T} \mapsto \Sigma$ where $\Lambda =$ directed_edges_matrix(M) and $ \Omega = $ error_covariance_matrix(M).

Examples

julia> M = graphical_model(graph_from_edges(Directed, [[1,2], [2,3]]), gaussian_ring(3))
Gaussian graphical model on a directed graph with edges:
(1, 2), (2, 3)

julia> parametrization(M)
Ring homomorphism
  from multivariate polynomial ring in 6 variables over QQ
  to multivariate polynomial ring in 5 variables over QQ
defined by
  s[1, 1] -> w[1]
  s[1, 2] -> l[1, 2]*w[1]
  s[1, 3] -> l[1, 2]*l[2, 3]*w[1]
  s[2, 2] -> l[1, 2]^2*w[1] + w[2]
  s[2, 3] -> l[1, 2]^2*l[2, 3]*w[1] + l[2, 3]*w[2]
  s[3, 3] -> l[1, 2]^2*l[2, 3]^2*w[1] + l[2, 3]^2*w[2] + w[3]

vanishing_ideal(M::GraphicalModel)

Computes the vanishing ideal for a graphical model M. This is done by computing the kernel of the parametrization.

Examples

julia> M = graphical_model(graph_from_edges(Directed, [[1,2], [2,3]]), gaussian_ring(3))
Gaussian graphical model on a directed graph with edges:
(1, 2), (2, 3)

julia> vanishing_ideal(M)
Ideal generated by
  -s[1, 2]*s[2, 3] + s[1, 3]*s[2, 2]

graphical_model(G::Graph{Undirected}, S::GaussianRing; k_var_name::VarName="k", cached=false)

A parametric statistical model associated to an undirected graph. It contains an undirected graph G, a GaussianRing S where the vanishing ideal of the model naturally lives, and a parameter ring whose variables k[i,j] correspond to the edges i - j in G.

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

Examples

julia> M = graphical_model(graph_from_edges([[1,2], [2,3]]), gaussian_ring(3))
Gaussian graphical model on an undirected graph with edges:
(1, 2), (2, 3)

concentration_matrix(M::GraphicalModel{Graph{Undirected}, GaussianRing})

Creates 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 = graphical_model(graph_from_edges([[1,2], [2,3]]), gaussian_ring(3))
Gaussian graphical model on an undirected graph with edges:
(1, 2), (2, 3)

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

parametrization(M::GraphicalModel{Graph{Undirected}, GaussianRing})

Creates the polynomial map which parametrizes the vanishing ideal of the undirected Gaussian graphical model M. The vanishing ideal of the statistical model is the kernel of this map. This ring map is the pull back of the parametrization $\phi_G$ given by $ K \mapsto K^{-1}$ where $ K = $ concentration_matrix(M) and the entries of $ K^{-1} $ are given by the standard cofactor formula.

Examples

julia> M = graphical_model(graph_from_edges([[1,2], [2,3]]), gaussian_ring(3))
Gaussian graphical model on an undirected graph with edges:
(1, 2), (2, 3)

julia> parametrization(M)
Ring homomorphism
  from multivariate polynomial ring in 6 variables over QQ
  to fraction field of multivariate polynomial ring
defined by
  s[1, 1] -> (k[2, 2]*k[3, 3] - k[2, 3]^2)//(k[1, 1]*k[2, 2]*k[3, 3] - k[1, 1]*k[2, 3]^2 - k[1, 2]^2*k[3, 3])
  s[1, 2] -> (-k[1, 2]*k[3, 3])//(k[1, 1]*k[2, 2]*k[3, 3] - k[1, 1]*k[2, 3]^2 - k[1, 2]^2*k[3, 3])
  s[1, 3] -> (k[1, 2]*k[2, 3])//(k[1, 1]*k[2, 2]*k[3, 3] - k[1, 1]*k[2, 3]^2 - k[1, 2]^2*k[3, 3])
  s[2, 2] -> (k[1, 1]*k[3, 3])//(k[1, 1]*k[2, 2]*k[3, 3] - k[1, 1]*k[2, 3]^2 - k[1, 2]^2*k[3, 3])
  s[2, 3] -> (-k[1, 1]*k[2, 3])//(k[1, 1]*k[2, 2]*k[3, 3] - k[1, 1]*k[2, 3]^2 - k[1, 2]^2*k[3, 3])
  s[3, 3] -> (k[1, 1]*k[2, 2] - k[1, 2]^2)//(k[1, 1]*k[2, 2]*k[3, 3] - k[1, 1]*k[2, 3]^2 - k[1, 2]^2*k[3, 3])

vanishing_ideal(M::GraphicalModel{Graph{Undirected}, GaussianRing})

Computes the vanishing ideal of the undirected Gaussian graphical model M. This is done by saturating the ideal given by $ \Sigma K - Id $ by the determinant of $ K $ and then eliminating all variables k[i,j] where $ K =$ concentration_matrix(M).

Examples

julia> M = graphical_model(graph_from_edges([[1,2], [2,3]]), gaussian_ring(3))
Gaussian graphical model on an undirected graph with edges:
(1, 2), (2, 3)

julia> vanishing_ideal(M)
Ideal generated by
  -s[1, 2]*s[2, 3] + s[1, 3]*s[2, 2]