CrossCovarianceAG10: Multivariate non-separable cross-covariance function on...

In spamtree: Spatial Multivariate Trees

Description

This function implements the cross-covariance function used in Peruzzi and Dunson (2021), which is derived from eq. 7 in Apanasovich and Genton (2010).

Usage

CrossCovarianceAG10(coords1, mv1, coords2, mv2, 
                    ai1, ai2, phi_i, thetamv, Dmat)

Arguments

coords1	matrix with spatial coordinates
mv1	integer vector with variable IDs. The length must match the number of rows in coords1
coords2	matrix with spatial coordinates
mv2	integer vector with variable IDs. The length must match the number of rows in coords2
ai1	q-dimensional vector
ai2	q-dimensional vector
phi_i	q-dimensional vector
thetamv	for bivariate data (q=2), this is a scalar. For q>2, this is a vector with elements α, β, φ.
Dmat	symmetric matrix of dimension (q, q) with zeroes on the diagonal and whose (i,j) element is δ_{i,j}.

Details

Suppose we have q variables. For h>0 and Δ > 0 define:

C(h, Δ) = \frac{ \exp \{ -φ \|h \| / \exp \{β \log(1+α Δ)/2 \} \} }{ \exp \{ β \log(1 + α Δ) \} }

and for j=1, …, q, define C_j(h) = \exp \{ -φ_j \|h \| \}.

Then the cross-covariance between the ith margin of a q-variate process w(\cdot) at spatial location s and the jth margin at location s' is built as follows. For i = j as

Cov(w(s, ξ_i), w(s', ξ_j)) = σ_{i1}^2 C(h, 0) + σ_{i2}^2 C_i(h) ,

whereas if i \neq j it is defined as

Cov(w(s, ξ_i), w(s', ξ_j)) = σ_{i1} σ_{i2} C(h, δ_{ij}),

where ξ_i and ξ_j are the latent locations of margin i and j in the domain of variables and δ_{ij} = \| ξ_i - ξ_j \| is their distance in such domain.

Value

The cross-covariance matrix for all pairwise locations.

Author(s)

Michele Peruzzi [e-mail]

References

Apanasovich, T. V. and Genton, M. G. (2010) Cross-covariance functions for multivariate random fields based on latent dimensions. Biometrika, 97:15-30. doi: 10.1093/biomet/asp078

Peruzzi, M. and Dunson, D. B. (2021) Spatial Multivariate Trees for Big Data Bayesian Regression. https://arxiv.org/abs/2012.00943

Examples

library(magrittr)
library(dplyr)
library(spamtree)

SS <- 10
xlocs <- seq(0.0, 1, length.out=SS)
coords <- expand.grid(xlocs, xlocs)
c1 <- coords %>% mutate(mv_id=1)
c2 <- coords %>% mutate(mv_id=2)

coords <- bind_rows(c1, c2)
coords_q <- coords %>% dplyr::select(-mv_id)
cx <- coords_q %>% as.matrix()
mv_id <- coords$mv_id

ai1 <- c(1, 1.5)
ai2 <- c(.1, .51)
phi_i <- c(1, 2)
thetamv <- 5

q <- 2
Dmat <- matrix(0, q, q)
Dmat[2,1] <- 1
Dmat[upper.tri(Dmat)] <- Dmat[lower.tri(Dmat)]

CC <- CrossCovarianceAG10(cx, mv_id, cx, mv_id, ai1, ai2, phi_i, thetamv, Dmat)

spamtree documentation built on Dec. 11, 2021, 9:06 a.m.