local_covariance_matrix {SpatialBSS}

Computation of Local Covariance Matrices

Description

local_covariance_matrix computes local covariance matrices for a random field based on a given set of spatial kernel matrices.

Usage

local_covariance_matrix(x, kernel_list, lcov = c('lcov', 'ldiff', 'lcov_norm'), 
                        center = TRUE)

Arguments

x

a numeric matrix of dimension c(n, p) where the p columns correspond to the entries of the random field and the n rows are the observations.

kernel_list

a list with spatial kernel matrices of dimension c(n, n). This list is usually computed with the function spatial_kernel_matrix.

lcov

a string indicating which type of local covariance matrix to use. Either 'lcov' (default) or 'ldiff'.

center

logical. If TRUE the data x is centered prior computing the local covariance matrices. Default is TRUE.

Details

Two versions of local covariance matrices are implemented, the argument lcov determines which version is used:

• 'lcov':

  LCov(f) = 1/n \sum_{i,j} f(d_{i,j}) (x(s_i)-\bar{x}) (x(s_j)-\bar{x})' ,

• 'ldiff':

  LDiff(f) = 1/n \sum_{i,j} f(d_{i,j}) (x(s_i)-x(s_j)) (x(s_i)-x(s_j))',

• 'lcov_norm':

  LCov^*(f) = 1/(n F^{1/2}_{f,n}) \sum_{i,j} f(d_{i,j}) (x(s_i)-\bar{x}) (x(s_j)-\bar{x})',

  with

  F_{f,n} = 1 / n \sum_{i,j} f^2(d_{i,j}).

Where d_{i,j} \ge 0 correspond to the pairwise distances between coordinates, x(s_i) are the p random field values at location s_i, \bar{x} is the sample mean vector, and the kernel function f(d) determines the locality. The choice 'lcov_norm' is useful when testing for the actual signal dimension of the latent field, see sbss_asymp and sbss_boot. The function local_covariance_matrix computes local covariance matrices for a given random field and given spatial kernel matrices, the type of computed local covariance matrices is determined by the argument 'lcov'. If the argument center equals FALSE then the centering in the above formula for LCov(f) is not carried out. See also spatial_kernel_matrix for details.

Value

local_covariance_matrix returns a list of equal length as the argument kernel_list. Each list entry is a numeric matrix of dimension c(p, p) corresponding to a local covariance matrix. The list has the attribute 'lcov' which equals the function argument lcov.

References

Muehlmann, C., Filzmoser, P. and Nordhausen, K. (2021), Spatial Blind Source Separation in the Presence of a Drift, Submitted for publication. Preprint available at https://arxiv.org/abs/2108.13813.

Bachoc, F., Genton, M. G, Nordhausen, K., Ruiz-Gazen, A. and Virta, J. (2020), Spatial Blind Source Separation, Biometrika, 107, 627-646, doi:10.1093/biomet/asz079.

See Also

spatial_kernel_matrix, sbss

Examples

# simulate coordinates
coords <- runif(1000 * 2) * 20
dim(coords) <- c(1000, 2)
coords_df <- as.data.frame(coords)
names(coords_df) <- c("x", "y")
# simulate random field
if (!requireNamespace('gstat', quietly = TRUE)) {
  message('Please install the package gstat to run the example code.')
} else {
  library(gstat)
  model_1 <- gstat(formula = z ~ 1, locations = ~ x + y, dummy = TRUE, beta = 0, 
                   model = vgm(psill = 0.025, range = 1, model = 'Exp'), nmax = 20)
  model_2 <- gstat(formula = z ~ 1, locations = ~ x + y, dummy = TRUE, beta = 0, 
                   model = vgm(psill = 0.025, range = 1, kappa = 2, model = 'Mat'), 
                   nmax = 20)
  model_3 <- gstat(formula = z ~ 1, locations = ~ x + y, dummy = TRUE, beta = 0, 
                   model = vgm(psill = 0.025, range = 1, model = 'Gau'), nmax = 20)
  field_1 <- predict(model_1, newdata = coords_df, nsim = 1)$sim1
  field_2 <- predict(model_2, newdata = coords_df, nsim = 1)$sim1
  field_3 <- predict(model_3, newdata = coords_df, nsim = 1)$sim1
  field <- as.matrix(cbind(field_1, field_2, field_3))

  # computing two ring kernel matrices and corresponding local covariance matrices
  kernel_params_ring <- c(0, 0.5, 0.5, 2)
  ring_kernel_list <- 
    spatial_kernel_matrix(coords, 'ring', kernel_params_ring)
  loc_cov_ring <- 
    local_covariance_matrix(x = field, kernel_list = ring_kernel_list)
    
  # computing two ring kernel matrices and corresponding local difference matrices
  kernel_params_ring <- c(0, 0.5, 0.5, 2)
  ring_kernel_list <- 
    spatial_kernel_matrix(coords, 'ring', kernel_params_ring)
  loc_cov_ring <- 
    local_covariance_matrix(x = field, kernel_list = ring_kernel_list, lcov = 'ldiff')
  
  # computing three ball kernel matrices and corresponding local covariance matrices
  kernel_params_ball <- c(0.5, 1, 2)
  ball_kernel_list <- 
    spatial_kernel_matrix(coords, 'ball', kernel_params_ball)
  loc_cov_ball <- 
    local_covariance_matrix(x = field, kernel_list = ball_kernel_list)
  
  # computing three gauss kernel matrices and corresponding local covariance matrices
  kernel_params_gauss <- c(0.5, 1, 2)
  gauss_kernel_list <- 
    spatial_kernel_matrix(coords, 'gauss', kernel_params_gauss)
  loc_cov_gauss <- 
    local_covariance_matrix(x = field, kernel_list = gauss_kernel_list)
}