It is assumed that the p-variate space-time random field \(x(s,t)\) is formed by $$x(s,t) = A z(s,t) + b,$$ where \(z(s,t)\) is the latent p-variate space-time random field, \(A\) and \(b\) are the mixing matrix and a location vector and \(s\) and \(t\) are the space and time coordinates. Furthermore, it is assumed that \(z(s,t)\) is white and consists of space-time uncorrelated components. The goal is to reverse the linear form by estimating an unmixing matrix and the location vector. This is done by simultaneously/jointly diagonalizing local autocovariance matrices which are defined by $$ LACov(f) = 1/(n F_{f,n}) \sum_{i,j} f(s_i-s_j,t_i-t_j) (x(s_i,t_i)-\bar{x}) (x(s_j,t_j)-\bar{x})',$$ with $$ F^2_{f,n} = 1 / n \sum_{i,j} f^2(s_i-s_j,t_i-t_j).$$ Here, \(x(s_i,t_i)\) are the p random field values at location \(s_i,t_i\), \(\bar{x}\) is the sample mean vector, and the space-time kernel function \(f\) determines the locality. The following kernel functions are implemented and chosen with the argument kernel_type:
'ring': the spatial parameters are inner radius \(r_i\) and outer radius \(r_o\), with \(r_i < r_o\), and \(r_i, r_o \ge 0\), the temporal parameter is the temporal lag \(u\): $$f(d_s,d_t) = I(r_i < d_s \le r_o)I(d_t = u)$$
'ball': the spatial parameter is the radius \(r\), with \(r \ge 0\), the temporal parameter is the temporal lag \(u\): $$f(d_s,d_t) = I(d_s \le r)I(d_t = u)$$
'gauss': Gaussian function where 95% of the mass is inside the spatial parameter \(r\), with \(r \ge 0\), the temporal parameter is the temporal lag \(u\): $$f(d_s,d_t) = exp(-0.5 (\Phi^{-1}(0.95) d_s/r)^2)I(d_t = u)$$
Above, \(I()\) represents the indicator function. The argument kernel_type determines the used kernel function as presented above, the argument lags provides the used temporal lags for the kernel functions (\(u\) in the above formulas) and the argument kernel_parameters gives the spatial parameters for the kernel function. Each of the arguments kernel_type, lags and kernel_parameters can be of length K or 1. Specifically, kernel_type can be either one kernel, then each local autocovariance matrix use the same kernel type, or of length K which leads to different kernel functions for the provided kernel parameters. lags can be either one integer, then for each kernel the same temporal lag is used, or an integer vector of length K which leads to different temporal lags. In the same fashion kernel_parameters is a vector of length K or 1. If kernel_type equals 'ball' or 'gauss' then the corresponding entry of kernel_parameters gives the single spatial radius parameter. In contrast, if (at least one entry of) kernel_type equals 'ring', then kernel_parameters must be a list of length K (or 1) where each entry is a numeric vector of length 2 defining the inner and outer spatial radius.
Internally, stbss calls stkmat to compute a list of c(n,n) kernel matrices based on the parameters given, where each entry of those matrices corresponds to \(f(s_i-s_j,t_i-t_j)\). Alternatively, such a list of kernel matrices can be given directly to the function stbss through the kernel_list argument. This is useful when stbss is called numerous times with the same coordinates/kernel functions as the computation of the kernel matrices is then only done once prior the actual stbss calls. For details see also lacov.
If more than one local autocovariance matrix is used stbss jointly diagonalizes these matrices with the function frjd. ... provides arguments for frjd, useful arguments might be: