Spatial Blind Source Separation Model

In the Spatial Blind Source Separation (SBSS) model it is assumed that a \(p\)-variate random field \(X(s) = \{ X_1(s),\dots,X_p(s) \}\) is a linear mixture of an unobserved random field \(Z(s) = \{ Z_1(s),\dots,Z_p(s) \}\). Here \(s\) denotes spatial locations in a domain \(s \in S \subseteq \mathbb{R}^d\), 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

\[ X(s) = \Omega Z(s) + \mu, \] where \(\Omega\) is the so called mixing matrix and \(\mu\) 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 \(\text{E}\{ Z(s) \} = 0\), \(\text{Cov} \{ Z(s) \} = \text{E} \{ Z(s) Z(s)^\top \} = I_p\) for all \(s \in \mathcal{S}\) and \(\text{Cov} \{ Z(s_1), Z(s_2) \} = \text{E} \{ Z(s_1) Z(s_2)^\top \} = D(\lVert s_1 - s_2 \rVert)\) for all \(s_1, s_2 \in \mathcal{S}\) where \(D\) is some diagonal matrix. Hence, \(Z(s)\) is also a weakly-stationary random field.

The SBSS approach uses the spatially weighted local scatter matrices to identify a transformation that separates the original multivariate process into uncorrelated or independent latent fields. These fields are then assumed to exhibit different spatial dependence structures, which can be analyzed independently. SBSS can be applied under the assumption of isotropy or anisotropy, hence the latent fields can be estimated assuming isotropic or anisotropic covariance functions.

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 \(W = \Omega ^ {-1}\) 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 \(W = \Omega ^ {-1}\) 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 \(\bar{X}\). Note that if \(f(x) = f_0(x) = I(x = 0)\) where \(I(x)\) is the indicator function, then \(LCov(f_0)\) reduces to an ordinary covariance estimator. Three options for the kernel function are suggested:

• Ball: \(f(d;r) = I(\lVert d \rVert \le r)\), where \(I(\cdot)\) denotes the indicator function and \(r \ge 0\),

• Ring: \(f(d;r_{in}, r_{out}) = I(r_{in} < \lVert d \rVert \le r_{out})\), where \(I(\cdot)\) denotes the indicator function and \(r_{out} > r_{in} \ge 0\),

• Gauss: \(f(d;r) = \exp(-0.5 (\Phi^{-1}(0.95) \lVert d \rVert/r)^2)\), where \(\Phi^{-1}(0.95)\) is the 95% quantile of the standard normal distribution and \(r \ge 0\).

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 \(s_i\).

It is worth pointing out that the above mentioned local covariance matrices implicitly assume isotropy, hence a proper generalization has been proposed to identify latent fields that exhibit anisotropic spatial dependencies: \[ LCov(\alpha, f) = \frac{1}{n} \sum_{i=1}^n \sum_{j=1}^n f(\mathbf{h}_{ij})I(\mathbf{h}_{ij} \angle \mathbf{e}_{\alpha_1} \in [0, \alpha_2 ]) (X(s_i)-\bar{X}) (X(s_j)-\bar{X})^\top , \] where \(\mathbf{h}_{ij} = \mathbf{s}_i - \mathbf{s}_j\), \(I(\cdot)\) is the indicator function \(\boldsymbol{\alpha} =(\alpha_1, \alpha_2)^\top\) and \(\angle\) denotes the angle between two vectors, \(\mathbf{e}_{\alpha_1} = (\cos{\alpha_1}, \sin{\alpha_1})^\top\) with \(\alpha_1 \in [0, 2 \pi]\), \(\alpha_2 \in [0, \pi / 2]\) and \(f\) is a spatial kernel function, as previously detailed. Note that the indicator function \(I(\cdot)\) excludes coordinate pairs where the angle of the separation vector \(\mathbf{h}_{ij}\) relative to the x-axis does not lie between \(\alpha_1\) and \(\alpha_2\),

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 \(X^{st}(s) = \Sigma^{-1/2} ( X(s) - \text{E}(X(s)))\) then \(X^{st}(s) = U Z(s)\), where \(U\) is an orthogonal matrix and \(\Sigma^{-1/2}\) 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 \(LDiff(f)\), and \(M(\boldsymbol{\alpha}, f)\) refers to \(LCov(\alpha, f)\) 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 \(f_1,\dots,f_k\) the corresponding local scatter matrices \(M(f_1),\dots,M(f_k)\) are computed with the whitened data \(X^{st}(s)\) from the first step. Then an orthogonal matrix \(U\) is computed by:

   • the eigen-decomposition of \(M(f_1) = U \Lambda U^\top\), for the case of \(k=1\).

   • the joint diagonalization of the local scatter matrices \(M(f_1),\dots,M(f_k)\), for the case of \(k > 1\). Formally, this is done by maximizing \[ \sum_{l=1}^k \left \| \text{diag} \left( {{U}^\top M(f_l) { U }} \right) \right \|^2 \] for isotropic random fields, or \[ \sum_{l=1}^k \left \| \text{diag} \left( {{U^\top} M^S_{(\boldsymbol{\alpha}, f)_l} {U}} \right) \right \|^2 \] if the anisotropic assumption hold, where \(M^S\) are the symmetric local covariance matrices, obtained as

\[ \mathbf{M}_{\boldsymbol{\alpha},f}^S(\mathbf{x}) \rightarrow 2^{-1} (\mathbf{M}_{\boldsymbol{\alpha},f}(\mathbf{x}) + \mathbf{M}_{\boldsymbol{\alpha},f}(\mathbf{x})^\top). \]

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 \(\hat{W} = U^\top LCov(f_0)^{-1/2}\). 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.

Computing spatial kernel matrices

Internally, sbss computes for each chosen kernel function a matrix of dimension c(n,n) where each entry corresponds to the kernel function evaluated at the distance between two points. Hence, the entry i,j corresponds to \(f(s_i - s_j)\) in the definition of local covariance matrices above. spatial_kernel_matrices is called inside sbss and therefore needs coords, kernel_type and kernel_parameters as described above. For the former example:

ring_kernel_matrices <- spatial_kernel_matrix(coords, kernel_type, kernel_parameters)

ring_kernel_matrices is a list with the four corresponding ring kernel matrices, which can also be given to sbss via the kernel_list argument. This procedure avoids unnecessary computation of kernel matrices when sbss is called numerous times with the same coordinates and kernel function setting. For the above example:

sbss_k <- sbss(x = field, kernel_list = ring_kernel_matrices)

Note that the coords argument is not needed as the spatial kernel matrices are already computed.

Computing local covariance matrices

With the function local_covariance_matrix local covariance matrices can be computed based on spatial kernel matrices. This function gets the values of the random field and kernel matrices (usually the output of spatial_kernel_matrix) as input arguments. Furthermore, with the argument center it can be set if the random field is centered prior computing local covariance matrices. For the former example:

local_cov <- local_covariance_matrix(field, kernel_list = ring_kernel_matrices, 
                                     center = TRUE)

The output of this function is a list of same length as kernel_list, where each entry is a local covariance matrix for the corresponding spatial kernel matrix. Again local_covariance_matrix has the argument lcov that determines which version of the local scatter matrices is computed. 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 is the same call of local_covariance_matrix but \(LDiff(f)\) matrices are computed (note that for \(LDiff(f)\) matrices centering has no impact):

local_diff <- local_covariance_matrix(field, kernel_list =  ring_kernel_matrices, 
                                      lcov = 'ldiff', center = TRUE)

Anisotropic Local Covariance Matrices

The sbss function of the SpatialBSS package allows also the identification of latent fields that exhibit anisotropic spatial dependencies. To show this functionality the veneto_weather dataset (object of class sf), which contains the weekly aggregated data for six meteorological variables collected from 75 monitoring stations in the Veneto region, Italy, during the year 2021, is used.

Similarly to the previous example, spatial kernel functions and their corresponding parameters have to be set, in order to estimate the unmixing matrix. In this example, ring kernels have been selected, where the non-overlapping rings do not extend beyond \(80000\) meters, as the maximum distance between the spatial point pairs is \(181299.9\) meters.

In addition, the angles argument, which is a vector of two values (i.e., the main direction \(\alpha_1\) and the angular tolerance \(\alpha_2\)) has to be set in order to include the anisotropic constraint .

library('SpatialBSS')

data(veneto_weather)

kernel_type <- 'ring'
kernel_parameters <-c(0, 40000, 40000, 65000, 65000, 80000)
                                          
angles_1 <- list(c(0, pi / 8), c(pi / 4, pi / 8), c(pi / 2, pi / 8), c(3 * pi / 4, pi / 8))

sbss_anis <- sbss(x = veneto_weather, kernel_type = kernel_type, 
                  kernel_parameters = kernel_parameters, 
                  angles = angles_1)

The output of sbss is a list of class 'sbss' as previously detailed.

The found latent components (i.e., entries s of the output), which are spatially uncorrelated, can be used to predict the original variables on a grid or at desired spatial points using some interpolation methods.

For example, the predict function have to be recalled if the IDW method is used to interpolate the estimated latent fields. On the other hand, if the kriging method is applied for prediction purposes, the structural analysis for each component has to be conducted separately and a proper model has to be fitted for each latent random field. Based on the fitted models, the individual kriging predictions for the found components have to be aggregated into a vector and then multiplied by w_inv, i.e., the inverse of the loading matrix, to obtain the prediction of the original data at the spatial locations of interest.