snss_sd {SpatialBSS}

Spatial Non-Stationary Source Separation Simultaneous Diagonalization

Description

snss_sd estimates the unmixing matrix assuming a spatial non-stationary source separation model implying non-constant covariance by simultaneously diagonalizing two covariance matrices computed for two corresponding different sub-domains.

Usage

snss_sd(x, ...)

## Default S3 method:
snss_sd(x, coords, direction = c('x', 'y'), 
     ordered = TRUE, ...)
## S3 method for class 'list'
snss_sd(x, coords, ordered = TRUE, ...)
## S3 method for class 'SpatialPointsDataFrame'
snss_sd(x, ...)
## S3 method for class 'sf'
snss_sd(x, ...)

Arguments

x

either 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, a list of length two defining the subdivision of the domain, an object of class sf or an object of class SpatialPointsDataFrame.

coords

a numeric matrix of dimension c(n,2) when x is a matrix where each row represents the sample location of a point in the spatial domain or a list of length two if x is a list which defines the subdivision of the domain. Not needed otherwise.

direction

a string indicating on which coordinate axis the domain is halved. Either 'x' (default) or 'y'.

ordered

logical. If TRUE the entries of the latent field are ordered according to the decreasingly ordered eigenvalues. Default is TRUE.

...

further arguments to be passed to or from methods.

Details

This function assumes that the random field x is formed by

x(t) = A s(t) + b,

where A is the deterministic p \times p mixing matrix, b is the p-dimensional location vector, x is the observable p-variate random field given by the argument x, t are the spatial locations given by the argument coords and s is the latent p-variate random field assumed to consist of uncorrelated entries that have zero mean but non-constant variances. This function aims to recover s by

W(x(t) - \bar{x}),

where W is the p \times p unmixing matrix and \bar{x} is the sample mean. The function does this by splitting the given spatial domain in half according to the first coordinate (argument direction equals 'x') or the second coodinate (argument direction equals 'y') and simultaneously diagonalizing the sample covariance matrices for each of the two sub-domains.

Alternatively the domain subdivison can be defined by providing lists of length two for the arguments x and coords where the first list entries correspond to the values and coordinates of the first sub-domain and the second entries to the values and coordinates of the second sub-domain.

Value

Similarly as sbss the function snss_sd returns a list of class 'snss' and 'sbss' with the following entries:

s

object of class(x) containing the estimated source random field.

coords

coordinates of the observations. Only given if x is a matrix or list.

w

estimated unmixing matrix.

w_inv

inverse of the estimated unmixing matrix.

d

diagonal matrix containing the eigenvalues of the eigendecomposition.

x_mu

columnmeans of x.

cov_inv_sqrt

square root of the inverse sample covariance matrix for the first sub-domain.

References

Muehlmann, C., Bachoc, F. and Nordhausen, K. (2022), Blind Source Separation for Non-Stationary Random Fields, Spatial Statistics, 47, 100574, doi:10.1016/j.spasta.2021.100574.

See Also

sbss, sp, sf

Examples

# simulate coordinates
n <- 1000
coords <- runif(n * 2) * 20
dim(coords) <- c(n, 2)

# simulate random field
field_1 <- rnorm(n)
field_2 <- 2 * sin(pi / 20 * coords[, 1]) * rnorm(n)
field_3 <- rnorm(n) * (coords[, 1] < 10) + rnorm(n, 0, 3) * (coords[, 1] >= 10)

latent_field <- cbind(field_1, field_2, field_3)
mixing_matrix <- matrix(rnorm(9), 3, 3)
observed_field <- latent_field %*% t(mixing_matrix)

observed_field_sp <- sp::SpatialPointsDataFrame(coords = coords, 
                                              data = data.frame(observed_field))
sp::spplot(observed_field_sp, colorkey = TRUE, as.table = TRUE, cex = 1)

# apply snss_sd with split in x 
res_x <- snss_sd(observed_field, coords, direction = 'x')
JADE::MD(W.hat = coef(res_x), A = mixing_matrix)

# apply snss_sd with split in y
# should be much worse as field shows only variation in x
res_y <- snss_sd(observed_field, coords, direction = 'y')
JADE::MD(W.hat = coef(res_y), A = mixing_matrix)

# print object
print(res_x)

# plot latent field
plot(res_x, colorkey = TRUE, as.table = TRUE, cex = 1)

# predict latent fields on grid
predict(res_x, colorkey = TRUE, as.table = TRUE, cex = 1)

# unmixing matrix
w_unmix <- coef(res_x)

# apply snss_sd with SpatialPointsDataFrame object 
res_x_sp <- snss_sd(observed_field_sp, direction = 'x')

# apply with list arguments
# first axis split by 5
flag_coords <- coords[, 1] < 5
coords_list <- list(coords[flag_coords, ],
                    coords[!flag_coords, ])
field_list <- list(observed_field[flag_coords, ],
                   observed_field[!flag_coords, ])
plot(coords, col = flag_coords + 1)

res_list <- snss_sd(x = field_list,
                    coords = coords_list)
plot(res_list, colorkey = TRUE, as.table = TRUE, cex = 1)
JADE::MD(W.hat = coef(res_list), A = mixing_matrix)