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Closed-form marginal variance of the directional OU model

Source: R/covariance_directional_ou.R
directional_ou_variance.Rd

r(x,x) at each evaluation point – the diagonal of directional_ou_covariance(), computed directly via the marginal variance recursion along each edge, without any last-common-ancestor or transfer-factor work.

Usage

directional_ou_variance(
  graph,
  kappa,
  tau,
  PtE = NULL,
  sigma_source = NULL,
  normalized = TRUE,
  cpp = TRUE
)

Arguments

graph

A metric_graph object with directional edge weights set (set_edge_weights(directional_weights = ...)) and the vertex weight functions set (setDirectionalWeightFunction()).

kappa

Rate parameter of the OU process (kappa > 0).

tau

White-noise scale. The stationary marginal variance is 1/(2*kappa*tau^2) – the same convention as MetricGraph:::r_1(). Note this is NOT reciprocal_tau (used elsewhere in the package as tau = 1/reciprocal_tau); pass tau directly here.

PtE

Evaluation points, a matrix with columns (edge_number, distance_on_edge). Defaults to graph$get_PtE() (the observation locations).

sigma_source

Anchoring variance at the source vertices (indegree 0). NULL (default) anchors every source at the stationary variance 1/(2*kappa*tau^2), matching Qalpha1_edges_custom(..., stationary_points = "all"). Otherwise a numeric vector giving Var(u) at each source vertex, either named by vertex index (as character) or positional in the order which(graph$get_degrees("indegree") == 0).

normalized

If TRUE (default), distance_on_edge in PtE/PtE2 is in [0, 1]; if FALSE, distances are absolute.

cpp

If TRUE (default), use the C++ numeric setup and the shared C++ covariance fill for oriented in-trees and out-trees. Irregular trees use the generic R fallback. Set to FALSE to use the pure-R implementation throughout.

Value

A numeric vector of length nrow(PtE).

See also

directional_ou_covariance() for the full closed-form covariance (including the \deqn{r(s,t) = r(a,a) A(a,s) A(a,t)} formula and the tree/acyclicity requirements).

Examples

edge1 <- rbind(c(0, 0), c(1, 0))
edge2 <- rbind(c(1, 0), c(2, 0))
edge3 <- rbind(c(1, 0), c(1, 1))
graph <- metric_graph$new(edges = list(edge1, edge2, edge3))
#> Starting graph creation...
#> LongLat is set to FALSE
#> Creating edges...
#> Setting edge weights...
#> Computing bounding box...
#> Setting up edges
#> Merging close vertices
#> Total construction time: 0.24 secs
#> Creating and updating vertices...
#> Storing the initial graph...
#> Computing the relative positions of the edges...
graph$set_edge_weights(weights = data.frame(w = c(1, 1, 1)),
                       directional_weights = "w")
graph$setDirectionalWeightFunction(f_in = function(x) sqrt(x / sum(x)))
directional_ou_variance(graph, kappa = 1, tau = 1,
                        PtE = rbind(c(1, 0.5), c(2, 0.5)))
#> [1] 0.5 0.5