Spatial Blind Source Separation Model

In the Spatial Blind Source Separation (SBSS) model it is assumed that a p-variate random field X(s) = {X₁(s), …, X_p(s)} is a linear mixture of an unobserved random field Z(s) = {Z₁(s), …, Z_p(s)}. Here s denotes spatial locations in a domain s ∈ S ⊆ ℝ^d, where d is the dimension of the domain. Usually d equals two, which would for example correspond to x and y coordinates or longitude and latitude values. The following presented method is mainly designed to be used with irregular spatial data. Formally, the SBSS model is given by

X(s) = ΩZ(s) + μ, where Ω is the so called mixing matrix and μ is some constant shift. The latent field Z(s) shows a clearer structure than the original field X(s). In particular, Z(s) is assumed to have zero mean and unit covariance and the components of Z(s) show no spatial cross-dependence. Meaning that E{Z(s)} = 0, Cov{Z(s)} = E{Z(s)Z(s)^⊤} = I_p for all s ∈ 𝒮 and Cov{Z(s₁), Z(s₂)} = E{Z(s₁)Z(s₂)^⊤} = D(∥s₁ − s₂∥) for all s₁, s₂ ∈ 𝒮 where D is some diagonal matrix. Hence, Z(s) is also a weakly-stationary random field.

Recovering the source random field Z(s) is particularly beneficial in spatial prediction tasks such as various kriging methods. As the latent field shows no cross-dependence, univariate kriging can be applied on each component of Z(s) individually. This approach avoids the complex modeling of cross-covariances and the use of CoKriging.

Moreover, an estimate of the unmixing matrix W = Ω⁻¹ can be used for interpretation similar to PCA. The loadings are given by W but in contrast to PCA the present method explicitly uses the spatial dependence of the data.

In order to recover an estimate of the unmixing matrix W = Ω⁻¹ local covariance matrices are used.

Local Covariance Matrices and Spatial Kernel Functions

The aim of local covariance matrices is to quantify spatial dependence. Local covariance matrices are defined as

$$ LCov(f) = \frac{1}{n} \sum_{i=1}^n \sum_{j=1}^n f(s_i - s_j) (X(s_i)-\bar{X}) (X(s_j)-\bar{X})^\top , $$ and local difference matrices write as

$$ LDiff(f) = \frac{1}{n} \sum_{i=1}^n \sum_{j=1}^n f(s_i - s_j) (X(s_i) - X(s_j)) (X(s_i) - X(s_j))^\top. $$

Here, the sums range over all points in the domain and the locality is defined by the kernel function f. For each location s_i in the considered domain spatial dependence to each point s_j is computed by the inner sum. The kernel function f(s_i − s_j) determines which points s_j are considered based on the distance between them. Whereas, the outer sum averages the inner sums of all points s_i which leads to an average local spatial dependence measure for the random field X(s). The difference between LCov(f) and LDiff(f) matrices lies in the fact that LDiff(f) matrices are more robust when the random field shows a smooth trend compared to LCov(f) matrices, as the former is computed by the difference of observations and the latter by differences to the sample mean vector X̄. Note that if f(x) = f₀(x) = I(x = 0) where I(x) is the indicator function, then LCov(f₀) reduces to an ordinary covariance estimator. Three options for the kernel function are suggested:

• Ball: f(d; r) = I(d ≤ h)

• Ring: f(d; r_in, r_out) = I(r_in < d ≤ r_out)

• Gauss: f(d; r) = exp (−0.5(Φ⁻¹(0.95)d/r)²)

where d = ∥s₁ − s₂∥, Φ⁻¹(0.95) is the 95% quantile of the standard normal distribution and r are the parameters for the kernel functions. The ball kernel function considers all coordinates inside a radius r around points s_i, the Gauss kernel can be considered as a smooth version of the ball kernel. And as the name implies, ring kernels consider points s_j that are between r_in and r_out separated from s_i.

How the estimate the unmixing matrix?

One key result that is used in almost every BSS method states that when the data is standardized by X^st(s) = Σ^−1/2(X(s) − E(X(s))) then X^st(s) = UZ(s), where U is an orthogonal matrix and Σ^−1/2 is the inverse square root of the covariance matrix. This result can be used to reduce the task of finding a general unmixing matrix to a two step approach of first standardizing the data and then only finding an orthogonal matrix U. In the following, M(f) denote either LCov(f) or LDiff(f) matrices. These considerations lead to the following algorithm for SBSS:

1. Standardizing the random field X(s) by $X^{st}(s) = LCov(f_0)^{-1/2} ( X(s) - \overline{X})$, where $\overline{X} = \frac{1}{n} \sum_{i = 1}^n X(s_i)$.

2. For a certain choice of kernel functions f₁, …, f_k the corresponding local scatter matrices M(f₁), …, M(f_k) are computed with the whitened data X^st(s) from the first step. Then an orthogonal matrix U is computed by:

   • For the case of k = 1: The eigen-decomposition of M(f₁) = UΛU^⊤ is computed to find U.

   • For the case of k > 1: The local scatter matrices M(f₁), …, M(f_k) are approximately jointly diagonalized by maximizing their diagonal elements. Formally, this is done by maximizing $$ \sum_{l=1}^k \left \| \text{diag} \left( {{U}^\top M(f_l) { U }} \right) \right \|_F^2 . $$ Hence, this approach finds an orthogonal matrix U that makes all considered local covariance matrices as diagonal as possible.

The estimated unmixing matrix is given by Ŵ = U^⊤LCov(f₀)^−1/2. In summary, this approach leads to a latent field Z(s) that shows minimal spatial cross-dependence. The package SpatialBSS implements the above discussed approaches and is presented below.

Functions for class 'sbss'

To examine the estimated latent random field, the plot function can be used. Internally, it casts the latent field to an object of class 'SpatialPointsDataFrame' and uses spplot from the package sp. ... can be used to provide further arguments to spplot:

plot(sbss_res, colorkey = TRUE, as.table = TRUE, cex = 1)

SpatialBSS also provides a predict function which uses Inverse Distance Weighting (IDW) to interpolate the estimated latent field on a grid. The power parameter for IDW can be set with the argument p (default is 2). n_grid determines the side lengths of the rectangular shaped grid by the differences of the rounded maximum and minimum values divided by the n_grid argument for each column of the coords argument that was used for the sbss call. Similar as in the plot function, these predictions are plotted with spplot and ... can be used to provide further arguments for plotting.

predict(sbss_res, p = 2, n_grid = 50, colorkey = TRUE, as.table = TRUE, cex = 1)

Spatial data can be handled in R with numerous packages, it seems that the package sp as well as sf are commonly used. sbss can also be used in conjunction with the class 'SpatialPointsDataFrame' from sp and the class 'sf' from sf. For example the same data and kernel setting as above with the package sp.

field_sp <- sp::SpatialPointsDataFrame(coords = coords, data = data.frame(field))
res_sbss_sp <- sbss(x = field_sp, kernel_type = kernel_type, 
                    kernel_parameters = kernel_parameters)

And the same setting from above used with the package sf. Note that sf provides a plot function which is used in the plot and predict methods for 'sbss' objects.

if (requireNamespace('sf', quietly = TRUE)) {
  field_sf <- sf::st_as_sf(data.frame(coords = coords, field), 
                           coords = c(1,2))
  res_sbss_sf <- sbss(x = field_sf, kernel_type = kernel_type,
                      kernel_parameters = kernel_parameters)
} else {
  message('Please install the package sf to run the example code.')
}

For the two examples above coords is not needed as the spatial objects carry the coordinates of field already. Furthermore, the entry s of an object of class 'sbss' is now of the same class as the argument x of the sbss call.

Alternative local scatter matrix

For all the above calls of sbss LCov(f) matrices were used. The argument lcov of sbss defines which type of local scatter matrix is used, the default is 'lcov' which leads to the use of LCov(f) matrices and 'ldiff' leads to the use of LDiff(f) matrices. In the following, the same sbss call is made but with the use of LDiff(f) matrices.

sbss_res_lcov <- sbss(x = field, coords = coords, 
                 kernel_type = kernel_type, lcov = 'ldiff',
                 kernel_parameters = kernel_parameters)

LDiff(f) matrices might be favorable when a smooth drift is present in the data. The idea of LDiff(f) matrices is that the difference of observations in close vicinity cancels out the drift, therefore the change of the drift function in space must be somewhat mild. Still, in the whitening step the estimation of the covariance matrix can be corrupted by the drift. Therefore, whitening can be carried out with respect to some LDiff(f) matrix. This is controlled by the rob_whitening argument.

sbss_res_lcov <- sbss(x = field, coords = coords, rob_whitening = TRUE,
                 kernel_type = kernel_type, lcov = 'ldiff',
                 kernel_parameters = kernel_parameters)

In the call above whitening is done with respect to a LDiff(f) matrix where the kernel function and parameters are the first occurring in the argument kernel_parameters. In principle this can be also done with LCov(f) matrices but as LCov(f) matrices are not necessarily positive definite there is a high chance that the inversion produces an error.