Gaussian Processes on Graphs#

The module implements methods to approximate Gaussian processes using Sparse Approximations. The functions of immediate use to practitioners are gp_graph_exp_quad_cov_lpdf(vector, vector, array [] vector, real, real, array [,] int) to evaluate the likelihood of a one-dimensional signal and gp_inv_graph_exp_quad_cov(vector, vector, array [] vector, real, real, array[,] int) to construct a non-centered parameterization using white noise. Analogues are implemented for Matérn kernels with \(\nu=3/2\) and \(\nu=5/2\).

Function Reference#

array [] int out_degrees(int n, array [,] int edge_index)#

Evaluate the out-degree of nodes in the directed acyclic graph induced by edge_index. The node labels of successors must be ordered and predecessors must have index less than successors.

Parameters:

n – Number of nodes.
edge_index – Directed edges as a tuple of predecessor and successor indices, i.e., the node labelled edge_index[1, i] is the predecessor of the node labelled edge_index[2, i].

Returns:

Out-degree of each node.

vector gp_graph_conditional_loc_scale(vector y, array [] vector x, int kernel, real sigma, array [] real length_scale, int node, array [] int predecessors, real epsilon)#

Evaluate the location and scale for a node given its predecessors, assuming zero mean.

Parameters:

y – State of each node.
x – Array of k + 1 locations in p dimensions.
kernel – Kernel to use (0 for squared exponential, 1 for Matern 3/2, 2 for Matern 5/2).
sigma – Marginal scale of the kernel.
length_scale – Correlation length of the kernel.
predecessors – Labels of k predecessors of the target node.
epsilon – Nugget variance for numerical stability.

Returns:

Location and scale parameters for the distribution of the node given its predecessors.

real gp_graph_lpdf(vector y, vector loc, array [] vector x, int kernel, real sigma, array [] real length_scale, array [,] int edges, array [] int degrees, real epsilon)#

Evaluate the log probability of a graph Gaussian process.

Parameters:

y – State of each node.
loc – Mean of each node.
x – Position of each node.
kernel – Kernel to use (0 for squared exponential, 1 for Matern 3/2, 2 for Matern 5/2).
sigma – Marginal scale of the kernel.
length_scale – Correlation length of the kernel.
edges – Directed edges between nodes constituting a directed acyclic graph. Edges are stored as a matrix with shape (2, m), where m is the number of edges. The first row comprises parents of children in the second row. The first row can have arbitrary order, but the second row must be sorted.
degrees – Out-degree of each node.
epsilon – Nugget variance for numerical stability.

Returns:

Log probability of the graph Gaussian process.

real gp_graph_lpdf(vector y, vector loc, array [] vector x, int kernel, real sigma, real length_scale, array [,] int edges, array [] int degrees, real epsilon)#

Evaluate the log probability of a graph Gaussian process.

Parameters:

y – State of each node.
loc – Mean of each node.
x – Position of each node.
kernel – Kernel to use (0 for squared exponential, 1 for Matern 3/2, 2 for Matern 5/2).
sigma – Marginal scale of the kernel.
length_scale – Correlation length of the kernel for each dimension.
edges – Directed edges between nodes constituting a directed acyclic graph. Edges are stored as a matrix with shape (2, m), where m is the number of edges. The first row comprises parents of children in the second row. The first row can have arbitrary order, but the second row must be sorted.
degrees – Out-degree of each node.
epsilon – Nugget variance for numerical stability.

Returns:

Log probability of the graph Gaussian process.

real gp_graph_lpdf(vector y, vector loc, array [] vector x, int kernel, real sigma, real length_scale, array [,] int edges)#

Evaluate the log probability of a graph Gaussian process.

Parameters:

y – State of each node.
loc – Mean of each node.
x – Position of each node.
kernel – Kernel to use (0 for squared exponential, 1 for Matern 3/2, 2 for Matern 5/2).
sigma – Marginal scale of the kernel.
length_scale – Correlation length of the kernel.
edges – Directed edges between nodes constituting a directed acyclic graph. Edges are stored as a matrix with shape (2, m), where m is the number of edges. The first row comprises parents of children in the second row. The first row can have arbitrary order, but the second row must be sorted.

Returns:

Log probability of the graph Gaussian process.

real gp_graph_lpdf(vector y, vector loc, array [] vector x, int kernel, real sigma, array [] real length_scale, array [,] int edges)#

Evaluate the log probability of a graph Gaussian process.

Parameters:

y – State of each node.
loc – Mean of each node.
x – Position of each node.
kernel – Kernel to use (0 for squared exponential, 1 for Matern 3/2, 2 for Matern 5/2).
sigma – Marginal scale of the kernel.
length_scale – Correlation length of the kernel for each dimension.
edges – Directed edges between nodes constituting a directed acyclic graph. Edges are stored as a matrix with shape (2, m), where m is the number of edges. The first row comprises parents of children in the second row. The first row can have arbitrary order, but the second row must be sorted.

Returns:

Log probability of the graph Gaussian process.

vector gp_inv_graph(vector z, vector loc, array [] vector x, int kernel, real sigma, array [] real length_scale, array [,] int edges, array [] int degrees, real epsilon)#

Transform white noise to a sample from a graph Gaussian process

Parameters:

z – White noise for each node.
loc – Mean of each node.
x – Position of each node.
kernel – Kernel to use (0 for squared exponential, 1 for Matern 3/2, 2 for Matern 5/2).
sigma – Marginal scale of the kernel.
length_scale – Correlation length of the kernel for each dimension.
edges – Directed edges between nodes constituting a directed acyclic graph. Edges are stored as a matrix with shape (2, m), where m is the number of edges. The first row comprises parents of children in the second row. The first row can have arbitrary order, but the second row must be sorted.
degrees – Out-degree of each node.
epsilon – Nugget variance for numerical stability.

Returns:

Sample from the Graph gaussian process.

vector gp_inv_graph(vector z, vector loc, array [] vector x, int kernel, real sigma, real length_scale, array [,] int edges, array [] int degrees, real epsilon)#

Transform white noise to a sample from a graph Gaussian process

Parameters:

z – White noise for each node.
loc – Mean of each node.
x – Position of each node.
kernel – Kernel to use (0 for squared exponential, 1 for Matern 3/2, 2 for Matern 5/2).
sigma – Marginal scale of the kernel.
length_scale – Correlation length of the kernel.
edges – Directed edges between nodes constituting a directed acyclic graph. Edges are stored as a matrix with shape (2, m), where m is the number of edges. The first row comprises parents of children in the second row. The first row can have arbitrary order, but the second row must be sorted.
degrees – Out-degree of each node.
epsilon – Nugget variance for numerical stability.

Returns:

Sample from the Graph gaussian process.

vector gp_inv_graph(vector z, vector loc, array [] vector x, int kernel, real sigma, array [] real length_scale, array [,] int edges)#

Transform white noise to a sample from a graph Gaussian process

Parameters:

z – White noise for each node.
loc – Mean of each node.
x – Position of each node.
kernel – Kernel to use (0 for squared exponential, 1 for Matern 3/2, 2 for Matern 5/2).
sigma – Marginal scale of the kernel.
length_scale – Correlation length of the kernel for each dimension.
edges – Directed edges between nodes constituting a directed acyclic graph. Edges are stored as a matrix with shape (2, m), where m is the number of edges. The first row comprises parents of children in the second row. The first row can have arbitrary order, but the second row must be sorted.

Returns:

Sample from the Graph gaussian process.

vector gp_inv_graph(vector z, vector loc, array [] vector x, int kernel, real sigma, real length_scale, array [,] int edges)#

Transform white noise to a sample from a graph Gaussian process

Parameters:

z – White noise for each node.
loc – Mean of each node.
x – Position of each node.
kernel – Kernel to use (0 for squared exponential, 1 for Matern 3/2, 2 for Matern 5/2).
sigma – Marginal scale of the kernel.
length_scale – Correlation length of the kernel.
edges – Directed edges between nodes constituting a directed acyclic graph. Edges are stored as a matrix with shape (2, m), where m is the number of edges. The first row comprises parents of children in the second row. The first row can have arbitrary order, but the second row must be sorted.

Returns:

Sample from the Graph gaussian process.

real gp_graph_exp_quad_cov_lpdf(vector y, vector loc, array [] vector x, real sigma, real length_scale, array [,] int edges, array [] int degrees, real epsilon)#

Evaluate the log probability of a graph Gaussian with squared exponential kernel.

Parameters:

y – State of each node.
loc – Mean of each node.
x – Position of each node.
sigma – Marginal scale of the kernel.
length_scale – Correlation length of the kernel.
edges – Directed edges between nodes constituting a directed acyclic graph. Edges are stored as a matrix with shape (2, m), where m is the number of edges. The first row comprises parents of children in the second row. The first row can have arbitrary order, but the second row must be sorted.
degrees – Out-degree of each node.
epsilon – Nugget variance for numerical stability.

Returns:

Log probability of the graph Gaussian process.

real gp_graph_exp_quad_cov_lpdf(vector y, vector loc, array [] vector x, real sigma, array [] real length_scale, array [,] int edges, array [] int degrees, real epsilon)#

Evaluate the log probability of a graph Gaussian with squared exponential kernel.

Parameters:

y – State of each node.
loc – Mean of each node.
x – Position of each node.
sigma – Marginal scale of the kernel.
length_scale – Correlation length of the kernel for each dimension.
edges – Directed edges between nodes constituting a directed acyclic graph. Edges are stored as a matrix with shape (2, m), where m is the number of edges. The first row comprises parents of children in the second row. The first row can have arbitrary order, but the second row must be sorted.
degrees – Out-degree of each node.
epsilon – Nugget variance for numerical stability.

Returns:

Log probability of the graph Gaussian process.

real gp_graph_exp_quad_cov_lpdf(vector y, vector loc, array [] vector x, real sigma, real length_scale, array [,] int edges)#

Evaluate the log probability of a graph Gaussian with squared exponential kernel.

Parameters:

y – State of each node.
loc – Mean of each node.
x – Position of each node.
sigma – Marginal scale of the kernel.
length_scale – Correlation length of the kernel.
edges – Directed edges between nodes constituting a directed acyclic graph. Edges are stored as a matrix with shape (2, m), where m is the number of edges. The first row comprises parents of children in the second row. The first row can have arbitrary order, but the second row must be sorted.

Returns:

Log probability of the graph Gaussian process.

vector gp_inv_graph_exp_quad_cov(vector z, vector loc, array [] vector x, real sigma, real length_scale, array [,] int edges, array [] int degrees, real epsilon)#

Transform white noise to a sample from a graph Gaussian process with squared exponential kernel.

Parameters:

z – White noise for each node.
loc – Mean of each node.
x – Position of each node.
sigma – Marginal scale of the kernel.
length_scale – Correlation length of the kernel.
edges – Directed edges between nodes constituting a directed acyclic graph. Edges are stored as a matrix with shape (2, m), where m is the number of edges. The first row comprises parents of children in the second row. The first row can have arbitrary order, but the second row must be sorted.
degrees – Out-degree of each node.
epsilon – Nugget variance for numerical stability.

Returns:

Sample from the Graph gaussian process.

vector gp_inv_graph_exp_quad_cov(vector z, vector loc, array [] vector x, real sigma, array [] real length_scale, array [,] int edges, array [] int degrees, real epsilon)#

Transform white noise to a sample from a graph Gaussian process with squared exponential kernel and different length scales along each feature dimension.

Parameters:

z – White noise for each node.
loc – Mean of each node.
x – Position of each node.
sigma – Marginal scale of the kernel.
length_scale – Correlation lengths of the kernel for each dimension.
edges – Directed edges between nodes constituting a directed acyclic graph. Edges are stored as a matrix with shape (2, m), where m is the number of edges. The first row comprises parents of children in the second row. The first row can have arbitrary order, but the second row must be sorted.
degrees – Out-degree of each node.
epsilon – Nugget variance for numerical stability.

Returns:

Sample from the Graph gaussian process.

vector gp_inv_graph_exp_quad_cov(vector z, vector loc, array [] vector x, real sigma, real length_scale, array [,] int edges)#

Transform white noise to a sample from a graph Gaussian process with squared exponential kernel.

Parameters:

z – White noise for each node.
loc – Mean of each node.
x – Position of each node.
sigma – Marginal scale of the kernel.
length_scale – Correlation length of the kernel.
edges – Directed edges between nodes constituting a directed acyclic graph. Edges are stored as a matrix with shape (2, m), where m is the number of edges. The first row comprises parents of children in the second row. The first row can have arbitrary order, but the second row must be sorted.

Returns:

Sample from the Graph gaussian process.

vector gp_inv_graph_exp_quad_cov(vector z, vector loc, array [] vector x, real sigma, array [] real length_scale, array [,] int edges)#

Transform white noise to a sample from a graph Gaussian process with squared exponential kernel.

Parameters:

z – White noise for each node.
loc – Mean of each node.
x – Position of each node.
sigma – Marginal scale of the kernel.
length_scale – Correlation length of the kernel for each dimension.
edges – Directed edges between nodes constituting a directed acyclic graph. Edges are stored as a matrix with shape (2, m), where m is the number of edges. The first row comprises parents of children in the second row. The first row can have arbitrary order, but the second row must be sorted.

Returns:

Sample from the Graph gaussian process.

real gp_graph_matern32_cov_lpdf(vector y, vector loc, array [] vector x, real sigma, real length_scale, array [,] int edges, array [] int degrees, real epsilon)#

Evaluate the log probability of a graph Gaussian with Matern 3 / 2 kernel.

Parameters:

y – State of each node.
loc – Mean of each node.
x – Position of each node.
sigma – Marginal scale of the kernel.
length_scale – Correlation length of the kernel.
edges – Directed edges between nodes constituting a directed acyclic graph. Edges are stored as a matrix with shape (2, m), where m is the number of edges. The first row comprises parents of children in the second row. The first row can have arbitrary order, but the second row must be sorted.
degrees – Out-degree of each node.
epsilon – Nugget variance for numerical stability.

Returns:

Log probability of the graph Gaussian process.

real gp_graph_matern32_cov_lpdf(vector y, vector loc, array [] vector x, real sigma, real length_scale, array [,] int edges)#

Evaluate the log probability of a graph Gaussian with Matern 3 / 2 kernel.

Parameters:

y – State of each node.
loc – Mean of each node.
x – Position of each node.
sigma – Marginal scale of the kernel.
length_scale – Correlation length of the kernel.
edges – Directed edges between nodes constituting a directed acyclic graph. Edges are stored as a matrix with shape (2, m), where m is the number of edges. The first row comprises parents of children in the second row. The first row can have arbitrary order, but the second row must be sorted.

Returns:

Log probability of the graph Gaussian process.

real gp_graph_matern32_cov_lpdf(vector y, vector loc, array [] vector x, real sigma, array [] real length_scale, array [,] int edges, array [] int degrees, real epsilon)#

Evaluate the log probability of a graph Gaussian with Matern 3 / 2 kernel.

Parameters:

y – State of each node.
loc – Mean of each node.
x – Position of each node.
sigma – Marginal scale of the kernel.
length_scale – Correlation length of the kernel for each dimension.
edges – Directed edges between nodes constituting a directed acyclic graph. Edges are stored as a matrix with shape (2, m), where m is the number of edges. The first row comprises parents of children in the second row. The first row can have arbitrary order, but the second row must be sorted.
degrees – Out-degree of each node.
epsilon – Nugget variance for numerical stability.

Returns:

Log probability of the graph Gaussian process.

real gp_graph_matern32_cov_lpdf(vector y, vector loc, array [] vector x, real sigma, array [] real length_scale, array [,] int edges)#

Evaluate the log probability of a graph Gaussian with Matern 3 / 2 kernel.

Parameters:

y – State of each node.
loc – Mean of each node.
x – Position of each node.
sigma – Marginal scale of the kernel.
length_scale – Correlation length of the kernel for each dimension.
edges – Directed edges between nodes constituting a directed acyclic graph. Edges are stored as a matrix with shape (2, m), where m is the number of edges. The first row comprises parents of children in the second row. The first row can have arbitrary order, but the second row must be sorted.

Returns:

Log probability of the graph Gaussian process.

vector gp_inv_graph_matern32_cov(vector z, vector loc, array [] vector x, real sigma, real length_scale, array [,] int edges, array [] int degrees, real epsilon)#

Transform white noise to a sample from a graph Gaussian process with Matern 3 / 2 kernel.

Parameters:

z – White noise for each node.
loc – Mean of each node.
x – Position of each node.
sigma – Marginal scale of the kernel.
length_scale – Correlation length of the kernel.
edges – Directed edges between nodes constituting a directed acyclic graph. Edges are stored as a matrix with shape (2, m), where m is the number of edges. The first row comprises parents of children in the second row. The first row can have arbitrary order, but the second row must be sorted.
degrees – Out-degree of each node.
epsilon – Nugget variance for numerical stability.

Returns:

Sample from the Graph gaussian process.

vector gp_inv_graph_matern32_cov(vector z, vector loc, array [] vector x, real sigma, real length_scale, array [,] int edges)#

Transform white noise to a sample from a graph Gaussian process with Matern 3 / 2 kernel.

Parameters:

z – White noise for each node.
loc – Mean of each node.
x – Position of each node.
sigma – Marginal scale of the kernel.
length_scale – Correlation length of the kernel.
edges – Directed edges between nodes constituting a directed acyclic graph. Edges are stored as a matrix with shape (2, m), where m is the number of edges. The first row comprises parents of children in the second row. The first row can have arbitrary order, but the second row must be sorted.

Returns:

Sample from the Graph gaussian process.

vector gp_inv_graph_matern32_cov(vector z, vector loc, array [] vector x, real sigma, array [] real length_scale, array [,] int edges, array [] int degrees, real epsilon)#

Transform white noise to a sample from a graph Gaussian process with Matern 3 / 2 kernel.

Parameters:

z – White noise for each node.
loc – Mean of each node.
x – Position of each node.
sigma – Marginal scale of the kernel.
length_scale – Correlation length of the kernel for each dimension.
edges – Directed edges between nodes constituting a directed acyclic graph. Edges are stored as a matrix with shape (2, m), where m is the number of edges. The first row comprises parents of children in the second row. The first row can have arbitrary order, but the second row must be sorted.
degrees – Out-degree of each node.
epsilon – Nugget variance for numerical stability.

Returns:

Sample from the Graph gaussian process.

vector gp_inv_graph_matern32_cov(vector z, vector loc, array [] vector x, real sigma, array [] real length_scale, array [,] int edges)#

Transform white noise to a sample from a graph Gaussian process with Matern 3 / 2 kernel.

Parameters:

z – White noise for each node.
loc – Mean of each node.
x – Position of each node.
sigma – Marginal scale of the kernel.
length_scale – Correlation length of the kernel for each dimension.
edges – Directed edges between nodes constituting a directed acyclic graph. Edges are stored as a matrix with shape (2, m), where m is the number of edges. The first row comprises parents of children in the second row. The first row can have arbitrary order, but the second row must be sorted.

Returns:

Sample from the Graph gaussian process.

real gp_graph_matern52_cov_lpdf(vector y, vector loc, array [] vector x, real sigma, real length_scale, array [,] int edges, array [] int degrees, real epsilon)#

Evaluate the log probability of a graph Gaussian with Matern 5 / 2 kernel.

Parameters:

y – State of each node.
loc – Mean of each node.
x – Position of each node.
sigma – Marginal scale of the kernel.
length_scale – Correlation length of the kernel.
edges – Directed edges between nodes constituting a directed acyclic graph. Edges are stored as a matrix with shape (2, m), where m is the number of edges. The first row comprises parents of children in the second row. The first row can have arbitrary order, but the second row must be sorted.
degrees – Out-degree of each node.
epsilon – Nugget variance for numerical stability.

Returns:

Log probability of the graph Gaussian process.

real gp_graph_matern52_cov_lpdf(vector y, vector loc, array [] vector x, real sigma, real length_scale, array [,] int edges)#

Evaluate the log probability of a graph Gaussian with Matern 5 / 2 kernel.

Parameters:

y – State of each node.
loc – Mean of each node.
x – Position of each node.
sigma – Marginal scale of the kernel.
length_scale – Correlation length of the kernel.
edges – Directed edges between nodes constituting a directed acyclic graph. Edges are stored as a matrix with shape (2, m), where m is the number of edges. The first row comprises parents of children in the second row. The first row can have arbitrary order, but the second row must be sorted.

Returns:

Log probability of the graph Gaussian process.

real gp_graph_matern52_cov_lpdf(vector y, vector loc, array [] vector x, real sigma, array [] real length_scale, array [,] int edges, array [] int degrees, real epsilon)#

Evaluate the log probability of a graph Gaussian with Matern 5 / 2 kernel.

Parameters:

y – State of each node.
loc – Mean of each node.
x – Position of each node.
sigma – Marginal scale of the kernel.
length_scale – Correlation length of the kernel for each dimension.
edges – Directed edges between nodes constituting a directed acyclic graph. Edges are stored as a matrix with shape (2, m), where m is the number of edges. The first row comprises parents of children in the second row. The first row can have arbitrary order, but the second row must be sorted.
degrees – Out-degree of each node.
epsilon – Nugget variance for numerical stability.

Returns:

Log probability of the graph Gaussian process.

real gp_graph_matern52_cov_lpdf(vector y, vector loc, array [] vector x, real sigma, array [] real length_scale, array [,] int edges)#

Evaluate the log probability of a graph Gaussian with Matern 5 / 2 kernel.

Parameters:

y – State of each node.
loc – Mean of each node.
x – Position of each node.
sigma – Marginal scale of the kernel.
length_scale – Correlation length of the kernel for each dimension.
edges – Directed edges between nodes constituting a directed acyclic graph. Edges are stored as a matrix with shape (2, m), where m is the number of edges. The first row comprises parents of children in the second row. The first row can have arbitrary order, but the second row must be sorted.

Returns:

Log probability of the graph Gaussian process.

vector gp_inv_graph_matern52_cov(vector z, vector loc, array [] vector x, real sigma, real length_scale, array [,] int edges, array [] int degrees, real epsilon)#

Transform white noise to a sample from a graph Gaussian process with Matern 5 / 2 kernel.

Parameters:

z – White noise for each node.
loc – Mean of each node.
x – Position of each node.
sigma – Marginal scale of the kernel.
length_scale – Correlation length of the kernel.
edges – Directed edges between nodes constituting a directed acyclic graph. Edges are stored as a matrix with shape (2, m), where m is the number of edges. The first row comprises parents of children in the second row. The first row can have arbitrary order, but the second row must be sorted.
degrees – Out-degree of each node.
epsilon – Nugget variance for numerical stability.

Returns:

Sample from the Graph gaussian process.

vector gp_inv_graph_matern52_cov(vector z, vector loc, array [] vector x, real sigma, real length_scale, array [,] int edges)#

Transform white noise to a sample from a graph Gaussian process with Matern 5 / 2 kernel.

Parameters:

z – White noise for each node.
loc – Mean of each node.
x – Position of each node.
sigma – Marginal scale of the kernel.
length_scale – Correlation length of the kernel.
edges – Directed edges between nodes constituting a directed acyclic graph. Edges are stored as a matrix with shape (2, m), where m is the number of edges. The first row comprises parents of children in the second row. The first row can have arbitrary order, but the second row must be sorted.

Returns:

Sample from the Graph gaussian process.

vector gp_inv_graph_matern52_cov(vector z, vector loc, array [] vector x, real sigma, array [] real length_scale, array [,] int edges, array [] int degrees, real epsilon)#

Transform white noise to a sample from a graph Gaussian process with Matern 5 / 2 kernel.

Parameters:

z – White noise for each node.
loc – Mean of each node.
x – Position of each node.
sigma – Marginal scale of the kernel.
length_scale – Correlation length of the kernel for each dimension.
edges – Directed edges between nodes constituting a directed acyclic graph. Edges are stored as a matrix with shape (2, m), where m is the number of edges. The first row comprises parents of children in the second row. The first row can have arbitrary order, but the second row must be sorted.
degrees – Out-degree of each node.
epsilon – Nugget variance for numerical stability.

Returns:

Sample from the Graph gaussian process.

vector gp_inv_graph_matern52_cov(vector z, vector loc, array [] vector x, real sigma, array [] real length_scale, array [,] int edges)#

Transform white noise to a sample from a graph Gaussian process with Matern 5 / 2 kernel.

Parameters:

z – White noise for each node.
loc – Mean of each node.
x – Position of each node.
sigma – Marginal scale of the kernel.
length_scale – Correlation length of the kernel for each dimension.
edges – Directed edges between nodes constituting a directed acyclic graph. Edges are stored as a matrix with shape (2, m), where m is the number of edges. The first row comprises parents of children in the second row. The first row can have arbitrary order, but the second row must be sorted.

Returns:

Sample from the Graph gaussian process.

Gaussian Processes on Graphs

Contents

Gaussian Processes on Graphs#

Function Reference#