GMRF

class cuqi.distribution.GMRF(mean=None, prec=None, bc_type='zero', order=1, **kwargs)

Gaussian Markov random field (GMRF).

Parameters:

• mean (array_like) – Mean of the GMRF.

• prec (float) – Precision of the GMRF.

• bc_type (str) – The type of boundary conditions to use. Can be ‘zero’, ‘periodic’ or ‘neumann’.

• order (int) – The order of the GMRF. Can be 0, 1 or 2.

Notes

The GMRF defines a distribution over a set of points where each point conditioned on all the others follow a Gaussian distribution.

For 1D (physical_dim=1), the current implementation provides three different cases:

• Order 0: \(x_i \sim \mathcal{N}(\mu_i, \delta^{-1})\),

• Order 1: \(x_i \mid x_{i-1},x_{i+1} \sim \mathcal{N}(\mu_i+(x_{i-1}+x_{i+1})/2, (2\delta)^{-1}))\),

• Order 2: \(x_i \mid x_{i-1},x_{i+1} \sim \mathcal{N}(\mu_i+(-x_{i-1}+2x_i-x_{i+1})/4, (4\delta)^{-1}))\),

where \(\delta\) is the prec parameter and the mean parameter is the mean \(\mu_i\) for each \(i\).

For 2D (physical_dim=2), order 0, 1, and 2 are also supported in which the differences are defined in both horizontal and vertical directions.

It is possible to define boundary conditions for the GMRF using the bc_type parameter.

Illustration as a Gaussian distribution

It may be beneficial to illustrate the GMRF distribution for a specific parameter setting. In 1D with zero boundary conditions, the GMRF distribution can be represented by a Gaussian, \(\mathcal{N}(\mu, \mathbf{P}^{-1})\), with mean \(\mu\) and the following precision matrices depending on the order:

• Order 0:

\[\mathbf{P} = \delta \mathbf{I}.\]

• Order 1:

\[\mathbf{P} = \delta \begin{bmatrix} 2 & -1 & & \newline -1 & 2 & -1 & \newline & \ddots & \ddots & \ddots \newline & & -1 & 2 \end{bmatrix}.\]

• Order 2:

\[\mathbf{P} = \delta \begin{bmatrix} 6 & -4 & 1 & & & \newline -4 & 6 & -4 & 1 & & \newline 1 & -4 & 6 & -4 & 1 & \newline & \ddots & \ddots & \ddots & \ddots & \ddots \newline & & \ddots & \ddots & \ddots & \ddots \newline & & & 1 & -4 & 6 \newline \end{bmatrix}.\]

General representation

In general we can define the GMRF distribution on each point by

\[x_i \mid \mathbf{x}_{\partial_i} \sim \mathcal{N}\left(\sum_{j \in \partial_i} \beta_{ij} x_j, \kappa_i^{-1}\right),\]

where \(\kappa_i\) is the precision of each Gaussian and \(\beta_{ij}\) are coefficients defining the structure of the GMRF.

For more details see: See Bardsley, J. (2018). Computational Uncertainty Quantification for Inverse Problems, Chapter 4.2.

__init__(mean=None, prec=None, bc_type='zero', order=1, **kwargs)

Initialize the core properties of the distribution.

Parameters:

• name (str, default None) – Name of distribution.

• geometry (Geometry, default _DefaultGeometry (or None)) – Geometry of distribution.

• is_symmetric (bool, default None) – Indicator if distribution is symmetric.

Methods

__init__([mean, prec, bc_type, order])	Initialize the core properties of the distribution.
disable_FD()	Disable finite difference approximation for logd gradient.
enable_FD([epsilon])	Enable finite difference approximation for logd gradient.
get_conditioning_variables()	Return the conditioning variables of this distribution (if any).
get_mutable_variables()	Return any public variable that is mutable (attribute or property) except those in the ignore_vars list
get_parameter_names()	Returns the names of the parameters that the density can be evaluated at or conditioned on.
gradient(*args, **kwargs)	Returns the gradient of the log density at x.
logd(*args, **kwargs)	Evaluate the un-normalized log density function of the distribution.
logpdf(x)	Evaluate the log probability density function of the distribution.
pdf(x)	Evaluate the log probability density function of the distribution.
sample([N])	Sample from the distribution.
to_likelihood(data)	Convert conditional distribution to a likelihood function given observed data

Attributes

FD_enabled	Returns True if finite difference approximation of the logd gradient is enabled.
FD_epsilon	Spacing for the finite difference approximation of the logd gradient.
dim	Return the dimension of the distribution based on the geometry.
geometry	Return the geometry of the distribution.
is_cond	Returns True if instance (self) is a conditional distribution.
mean
name	Name of the random variable associated with the density.
prec
sqrtprec
sqrtprecTimesMean