GMRFs

AbstractGMRF

A Gaussian Markov Random Field (GMRF) is a special case of a multivariate normal distribution where the precision matrix is sparse. The zero entries in the precision correspond to conditional independencies.

GMRF(mean, precision, solver_blueprint=DefaultSolverBlueprint())

A Gaussian Markov Random Field with mean mean and precision matrix precision.

Arguments

• mean::AbstractVector: The mean vector of the GMRF.
• precision::LinearMap: The precision matrix (inverse covariance) of the GMRF.
• solver_blueprint::AbstractSolverBlueprint: Blueprint specifying how to construct the solver.

Type Parameters

• T<:Real: The numeric type (e.g., Float64).
• PrecisionMap<:LinearMap{T}: The type of the precision matrix LinearMap.
• Solver<:AbstractSolver: The type of the solver instance.

Fields

• mean::Vector{T}: The mean vector.
• precision::PrecisionMap: The precision matrix as a LinearMap.
• solver::Solver: The solver instance for computing GMRF quantities.

Notes

The solver is constructed automatically using construct_solver(solver_blueprint, mean, precision) and is used to compute means, variances, samples, and other GMRF quantities efficiently.

precision_map(::AbstractGMRF)

Return the precision (inverse covariance) map of the GMRF.

precision_matrix(::AbstractGMRF)

Return the precision (inverse covariance) matrix of the GMRF.

" conditiononobservations( x::AbstractGMRF, A::Union{AbstractMatrix,LinearMap}, Qϵ::Union{AbstractMatrix,LinearMap,Real}, y::AbstractVector=spzeros(size(A)[1]), b::AbstractVector=spzeros(size(A)[1]); solverblueprint::AbstractSolverBlueprint=CholeskySolverBlueprint() )

Condition a GMRF x on observations y = A * x + b + ϵ where ϵ ~ N(0, Q_ϵ⁻¹).

Arguments

• x::AbstractGMRF: The GMRF to condition on.
• A::Union{AbstractMatrix,LinearMap}: The matrix A.
• Q_ϵ::Union{AbstractMatrix,LinearMap, Real}: The precision matrix of the noise term ϵ. In case a real number is provided, it is interpreted as a scalar multiple of the identity matrix.
• y::AbstractVector=spzeros(size(A)[1]): The observations y; optional.
• b::AbstractVector=spzeros(size(A)[1]): Offset vector b; optional.

Keyword arguments

• solver_blueprint::AbstractSolverBlueprint=CholeskySolverBlueprint(): The solver blueprint; optional.

Returns

A LinearConditionalGMRF object representing the conditional GMRF x | (y = A * x + b + ϵ).

LinearConditionalGMRF{G}(
    prior::G,
    A::Union{AbstractMatrix,LinearMap},
    Q_ϵ::Union{AbstractMatrix,LinearMap},
    y::AbstractVector,
    b::AbstractVector=spzeros(size(A, 1)),
    solver_blueprint::AbstractSolverBlueprint=DefaultSolverBlueprint(),
) where {G<:AbstractGMRF}

A GMRF conditioned on observations y = A * x + b + ϵ where ϵ ~ N(0, Q_ϵ).

Arguments

• prior::G: The prior GMRF.
• A::Union{AbstractMatrix,LinearMap}: The matrix A.
• Q_ϵ::Union{AbstractMatrix,LinearMap}: The precision matrix of the noise term ϵ.
• y::AbstractVector=spzeros(size(A, 1)): The observations y.
• b::AbstractVector=spzeros(size(A, 1)): The offset vector b.
• solver_blueprint::AbstractSolverBlueprint=DefaultSolverBlueprint(): The solver blueprint.

" jointgmrf( x1::AbstractGMRF, A::AbstractMatrix, Qϵ::AbstractMatrix, b::AbstractVector=spzeros(size(A)[1]) )

Return the joint GMRF of x1 and x2 = A * x1 + b + ϵ where ϵ ~ N(0, Q_ϵ⁻¹).

Arguments

• x1::AbstractGMRF: The first GMRF.
• A::AbstractMatrix: The matrix A.
• Q_ϵ::AbstractMatrix: The precision matrix of the noise term ϵ.
• b::AbstractVector=spzeros(size(A)[1]): Offset vector b; optional.

Returns

A GMRF object representing the joint GMRF of x1 and x2 = A * x1 + b + ϵ.

AbstractSpatiotemporalGMRF

A spatiotemporal GMRF is a GMRF that explicitly encodes the spatial and temporal structure of the underlying random field. All time points are modelled in one joint GMRF. It provides utilities to get statistics, draw samples and get the spatial discretization at a given time.

ConstantMeshSTGMRF

A spatiotemporal GMRF with constant spatial discretization.

ImplicitEulerConstantMeshSTGMRF

A spatiotemporal GMRF with constant spatial discretization and an implicit Euler discretization of the temporal dynamics.

ConcreteConstantMeshSTGMRF

A concrete implementation of a spatiotemporal GMRF with constant spatial discretization.

N_t(::AbstractSpatiotemporalGMRF)

Return the number of time points in the spatiotemporal GMRF.

time_means(::AbstractSpatiotemporalGMRF)

Return the means of the spatiotemporal GMRF at each time point.

Returns

• A vector of means of length Nₜ, one for each time point.

time_vars(::AbstractSpatiotemporalGMRF)

Return the marginal variances of the spatiotemporal GMRF at each time point.

Returns

• A vector of marginal variances of length Nₜ, one for each time point.

time_stds(::AbstractSpatiotemporalGMRF)

Return the marginal standard deviations of the spatiotemporal GMRF at each time point.

Returns

• A vector of marginal standard deviations of length Nₜ, one for each time point.

time_rands(::AbstractSpatiotemporalGMRF, rng::AbstractRNG)

Draw samples from the spatiotemporal GMRF at each time point.

Returns

• A vector of sample values of length Nₜ, one sample value vector for each time point.

discretization_at_time(::AbstractSpatiotemporalGMRF, t::Int)

Return the spatial discretization at time t.

spatial_to_spatiotemporal(
    spatial_matrix::AbstractMatrix,
    t_idx::Int,
    N_t::Int,
)

Make a spatial matrix applicable to a spatiotemporal system at time index t_idx. Results in a matrix that selects the spatial information exactly at time t_idx.

Arguments

• spatial_matrix::AbstractMatrix: The spatial matrix.
• t_idx::Integer: The time index.
• N_t::Integer: The number of time points.

kronecker_product_spatiotemporal_model(
    Q_t::AbstractMatrix,
    Q_s::AbstractMatrix,
    spatial_disc::FEMDiscretization;
    solver_blueprint = DefaultSolverBlueprint(),
)

Create a spatiotemporal GMRF through a Kronecker product of the temporal and spatial precision matrices.

Arguments

• Q_t::AbstractMatrix: The temporal precision matrix.
• Q_s::AbstractMatrix: The spatial precision matrix.
• spatial_disc::FEMDiscretization: The spatial discretization.

Keyword arguments

• solver_blueprint::AbstractSolverBlueprint=DefaultSolverBlueprint(): The solver blueprint.