local_gss_covariance_matrix {SpatialBSS}

Computation of Robust Local Covariance Matrices

Description

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

Usage

local_gss_covariance_matrix(x, kernel_list, lcov = c('norm', 'winsor', 'qwinsor'), 
                        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 robust local covariance matrix to use. Either 'norm' (default), 'winsor' or 'qwinsor'.

center

logical. If TRUE the data x is robustly centered prior computing the local covariance matrices. Default is TRUE. See also white_data.

Details

Generalized local sign matrices are determined by radial functions w(l_i), where l_i = ||x(s_i)-T(x)|| and T(x) is Hettmansperger Randles location estimator (Hettmansperger & Randles, 2002), and kernel functions f(d_{i,j}), where d_{i,j}=||s_i - s_j||. Generalized local sign covariance (gLSCM) matrix is then calculated as

gLSCM(f,w) = 1/(n F^{1/2}_{f,n}) \sum_{i,j} f(d_{i,j}) w(l_i)w(l_j)(x(s_i)-T(x)) (x(s_j)-T(x))'

with

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

Three radial functions w(l_i) (Raymaekers & Rousseeuw, 2019) are implemented, the parameter lcov defines which is used:

• 'norm':

  w(l_i) = 1/l_i

• 'winsor':

  w(l_i) = Q/l_i

• 'qwinsor':

  w(l_i) = Q^2/l_i^2.

The cutoff Q is defined as Q = l_{(h)}, where l_{(h)} is hth order statistic of \{l_1, ..., l_n\} and h = (n + p + 1)/2. If the argument center equals FALSE then the centering in the above formula for gLSCM(f,w) is not carried out. See also spatial_kernel_matrix for details.

Value

local_gss_covariance_matrix returns a list with two entries:

cov_sp_list

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

weights

numeric vector of length(n) giving the weights for each observation for the robust local covariance estimation.

References

Hettmansperger, T. P., & Randles, R. H. (2002). A practical affine equivariant multivariate median. Biometrika, 89 , 851-860. doi:10.1093/biomet/89.4.851.

Raymaekers, J., & Rousseeuw, P. (2019). A generalized spatial sign covariance matrix. Journal of Multivariate Analysis, 171 , 94-111. doi:10.1016/j.jmva.2018.11.010.

Sipila, M., Muehlmann, C. Nordhausen, K. & Taskinen, S. (2022). Robust second order stationary spatial blind source separation using generalized sign matrices. Manuscript.

See Also

spatial_kernel_matrix, robsbss

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 <- cbind(field_1, field_2, field_3)
  
  # computing two ring kernel matrices and corresponding 
  # robust local covariance matrices using 'norm' radial function:
  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_gss_covariance_matrix(x = field, kernel_list = ring_kernel_list, 
                               lcov = 'norm')
    
  # computing three ball kernel matrices and corresponding 
  # robust local covariance matrices using 'winsor' radial function:
  kernel_params_ball <- c(0.5, 1, 2)
  ball_kernel_list <- 
    spatial_kernel_matrix(coords, 'ball', kernel_params_ball)
  loc_cov_ball <- 
    local_gss_covariance_matrix(x = field, kernel_list = ball_kernel_list, 
                               lcov = 'winsor')
    
  # computing three gauss kernel matrices and corresponding 
  # robust local covariance matrices using 'qwinsor' radial function:
  kernel_params_gauss <- c(0.5, 1, 2)
  gauss_kernel_list <- 
    spatial_kernel_matrix(coords, 'gauss', kernel_params_gauss)
  loc_cov_gauss <- 
    local_gss_covariance_matrix(x = field, kernel_list = gauss_kernel_list, 
                               lcov = 'qwinsor')
}