spatial_kernel_matrix: Computation of Spatial Kernel Matrices

In SpatialBSS: Blind Source Separation for Multivariate Spatial Data

Computation of Spatial Kernel Matrices

Description

spatial_kernel_matrix computes spatial kernel matrices for a given kernel function with its parameters and a set of coordinates.

Usage

spatial_kernel_matrix(coords, kernel_type = c('ring', 'ball', 'gauss'), 
                      kernel_parameters, angles = NULL)

Arguments

coords	a numeric matrix of dimension c(n,2) where each row represents the coordinates of a point in the spatial domain.
kernel_type	a character string indicating which kernel function to use. Either 'ring' (default), 'ball' or 'gauss'.
kernel_parameters	a numeric vector that gives the parameters for the kernel function. At least length of one for 'ball' and 'gauss' or two for 'ring' kernel, see details.
angles	Optional argument specifying the anisotropic constraint. If NULL (default), the function computes isotropic kernels. Otherwise, angles must be a list of numeric vectors, where each vector has length 2: (\alpha_1, \alpha_2). The parameter \alpha_1 must be in the range 0 \leq \alpha_1 \leq 2\pi, and \alpha_2 must satisfy 0 \leq \alpha_2 \leq \pi/2, see details.

Details

Two versions of local covariance matrices can be defined:

• '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 function spatial_kernel_matrix computes a list of c(n,n) matrices where each entry of these matrices correspond to the spatial kernel function evaluated at the distance between two points, mathematically the entry ij of each kernel matrix is f(d_{i,j}). The following kernel functions are implemented and chosen with the argument kernel_type:

• 'ring': parameters are inner radius r_i and outer radius r_o, with r_i < r_o, and r_i, r_o \ge 0:

  f(d;r_i, r_o) = I(r_i < d \le r_o)

• 'ball': parameter is the radius r, with r \ge 0:

  f(d;r) = I(d \le r)

• 'gauss': Gaussian function where 95% of the mass is inside the parameter r, with r \ge 0:

  f(d;r) = exp(-0.5 (\Phi^{-1}(0.95) d/r)^2)

The argument kernel_type determines the used kernel function as presented above, the argument kernel_parameters gives the corresponding parameters for the kernel function. Specifically, if kernel_type equals 'ball' or 'gauss' then kernel_parameters is a numeric vector where each entry corresponds to one parameter. Hence, length(kernel_parameters) spatial kernel matrices of type kernel_type are computed. Whereas, if kernel_type equals 'ring', then kernel_parameters must be a numeric vector of even length where subsequently the inner and outer radii must be given (informally: c(r_i1, r_o1, r_i2, r_o2, ...)). In that case length(kernel_parameters) / 2 spatial kernel matrices of type 'ring' are computed.

If the optional argument angles is provided, the computed spatial kernel matrices incorporate anisotropic constraints. The angles argument must be a list of numeric vectors, where each vector consists of two values: \alpha_1 (the main direction) and \alpha_2 (the angular tolerance). The directional constraint ensures that only spatial relationships within the specified angle range are considered:

• 0 \leq \alpha_1 \leq 2\pi defines the main direction in radians.

• 0 \leq \alpha_2 \leq \pi/2 specifies the angular tolerance.

When angles is not NULL, the function applies the kernel function only to points satisfying d_{ij} \angle e_{\alpha_1} \in [0, \alpha_2]. If angles is NULL, the kernels are computed isotropically.

The output of this function can be used with the function sbss to avoid unnecessary computation of kernel matrices when sbss is called multiple times with the same coordinate/kernel function setting. Additionally, the output can be used with the function local_covariance_matrix to actually compute local covariance matrices as defined above based on a given set of spatial kernel matrices.

Value

spatial_kernel_matrix returns a list with length of length(kernel_parameters) (for 'ball' and 'gauss' kernel functions) or length(kernel_parameters) / 2 (for 'ring' kernel function) containing numeric matrices of dimension c(n,n) corresponding to the spatial kernel matrices.

References

Muehlmann, C., Cappello, C., De Iaco, S. and Nordhausen, K. (2025), Anisotropic Local Covariance Matrices for Spatial Blind Source Separation. Manuscript.

Muehlmann, C., Filzmoser, P. and Nordhausen, K. (2024), Spatial Blind Source Separation in the Presence of a Drift, Austrian Journal of Statistics, 53, 48-68, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.17713/ajs.v53i2.1668")}.

Bachoc, F., Genton, M. G, Nordhausen, K., Ruiz-Gazen, A. and Virta, J. (2020), Spatial Blind Source Separation, Biometrika, 107, 627-646, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1093/biomet/asz079")}.

See Also

sbss, local_covariance_matrix

Examples

# simulate a set of coordinates
coords <- rnorm(100 * 2)
dim(coords) <- c(100, 2)

# computing two ring kernel matrices
kernel_params_ring <- c(0, 0.5, 0.5, 2)
ring_kernel_list <- 
  spatial_kernel_matrix(coords, 'ring', kernel_params_ring)

# computing three ball kernel matrices
kernel_params_ball <- c(0.5, 1, 2)
ball_kernel_list <- 
  spatial_kernel_matrix(coords, 'ball', kernel_params_ball)

# computing three gauss kernel matrices
kernel_params_gauss <- c(0.5, 1, 2)
gauss_kernel_list <- 
  spatial_kernel_matrix(coords, 'gauss', kernel_params_gauss)

# anisotropic examlple
# computing two ring kernel matrices each with two angles
kernel_params_ring <- c(0, 0.5, 0.5, 2)
angles_params <- list(c(pi / 4, pi / 4), c(3 * pi / 4, pi / 4))
ring_aniso_kernel_list <- 
  spatial_kernel_matrix(coords, 'ring', kernel_params_ring, angles_params)

SpatialBSS documentation built on April 4, 2025, 1:07 a.m.