white_data {SpatialBSS}

Different Approaches of Data Whitening

Description

white_data whites the data with respect to the sample covariance matrix, or different spatial scatter matrices.

Usage

white_data(x, whitening = c("standard", "rob", "hr"), 
           lcov = c('lcov', 'ldiff', 'lcov_norm'), 
           kernel_mat = numeric(0))

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.

whitening

a string indicating the whitening method. If 'standard' then the whitening is carried out with respect to sample covariance matrix, if 'rob' then the first spatial scatter matrix is used instead of sample the covariance matrix and if 'hr' then the Hettmansperger-Randles location and scatter estimates are used for whitening. See details for more. Default is 'standard'.

lcov

a string indicating which type of local covariance matrix is used for whitening, when the whitening method 'rob' is used. Either 'lcov' (default) or 'ldiff'.

kernel_mat

a spatial kernel matrix with dimension c(n,n), see details. Usually computed by the function spatial_kernel_matrix.

Details

The inverse square root of a positive definite matrix M(x) with eigenvalue decomposition UDU' is defined as M(x)^{-1/2} = UD^{-1/2}U'. white_data whitens the data by M(x)^{-1/2}(x - T(x)) where T(x) is a location functional of x and the matrix M(x) is a scatter functional. If the argument whitening is 'standard', M(x) is the sample covariance matrix and T(x) is a vector of column means of x. If the argument whitening is 'hr', the Hettmansperger-Randles location and scatter estimates (Hettmansperger & Randles, 2002) are used as location functional T(x) and scatter functional M(x). The Hettmansperger-Randles location and scatter estimates are robust variants of sample mean and covariance matrices, that are used for whitening in robsbss. If the argument whitening is 'rob', the argument lcov determines the scatter functional M(x) to be one of the following local scatter matrices:

• '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. See also sbss for details.

Note that LCov(f) are usually not positive definite, therefore in that case the matrix cannot be inverted and an error is produced. Whitening with LCov(f) matrices might be favorable in the presence of spatially uncorrelated noise, and whitening with LDiff(f) might be favorable when a non-constant smooth drift is present in the data.

The argument kernel_mat is a matrix of dimension c(n,n) where each entry corresponds to the spatial kernel function evaluated at the distance between two sample locations, mathematically the entry ij of each kernel matrix is f(d_{i,j}). This matrix is usually computed with the function spatial_kernel_matrix.

Value

white_data returns a list with the following entries:

mu

a numeric vector of length ncol(x) containing the column means of the data matrix x.

x_0

a numeric matrix of dimension c(n, p) containing the columns centered data of x.

x_w

a numeric matrix of dimension c(n, p) containing the whitened data of x.

s

a numeric matrix of dimension c(p, p) which is the scatter matrix M.

s_inv_sqrt

a numeric matrix of dimension c(p, p) which equals the inverse square root of the scatter matrix M used for whitening.

s_sqrt

a numeric matrix of dimension c(p, p) which equals the square root of the scatter matrix M.

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.

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

See Also

sbss, spatial_kernel_matrix

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)
  X <- as.matrix(field)

  # white the data with the usual sample covariance 
  x_w_1 <- white_data(X)
  
  # white the data with a ldiff matrix and ring kernel
  kernel_params_ring <- c(0, 1)
  ring_kernel_list <- 
    spatial_kernel_matrix(coords, 'ring', kernel_params_ring)
  x_w_2 <- white_data(field, whitening = 'rob',
    lcov = 'ldiff', kernel_mat = ring_kernel_list[[1]])
  
  # Generate 5 % of global outliers to data
  field_cont <- gen_glob_outl(field)[,1:3]
  X <- as.matrix(field_cont)
  # white the data using Hettmansperger-Randles location and scatter estimates
  x_w_3 <- white_data(X, whitening = 'hr')
}