stkmat {SpaceTimeBSS}

Spatio-Temporal Kernel Matrices

Description

Computation of spatio-temporal kernel matrices for given kernel functions.

Usage

stkmat(coords, time, kernel_type, kernel_parameters, lags)

Arguments

coords

a numeric matrix of dimension c(n,2) where each row represents the spatial coordinates of the corresponding observation over a 2D spatial domain.

time

an integer vector of length n where each entry represents the temporal coordinate of the corresponding observation.

kernel_type

either a string or a string vector of length K (or 1) indicating which spatio-temporal kernel function to use. Implemented choices are 'ring', 'ball' or 'gauss'.

kernel_parameters

a numeric vector of length K (or 1) for the 'ball' and 'gauss' kernel function or a list of length K (or 1) for the 'ring' kernel, see details.

lags

an integer vector of length K (or 1) that provides the temporal lags for the spatio-temporal kernel functions, see details.

Details

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. See examples below.

The output of this function can be used with the function stbss to avoid unnecessary computation of kernel matrices when stbss is called multiple times with the same coordinate/kernel function setting. Additionally, the output can be used with the function lacov.

Value

stkmat returns a list of length K containing numeric matrices of dimension c(n,n) corresponding to the spatio-temporal kernel matrices.

References

Muehlmann, C., De Iaco, S. and Nordhausen, K. (2023), Blind Recovery of Sources for Multivariate Space-Time Environmental Data. Stochastic and Environmental Research and Risk Assessment, 37, 1593–1613, <doi:10.1007/s00477-022-02348-2>.

See Also

stbss, lacov

Examples

# space and time coordinates
n_t <- 50
n_sp <- 10
coords <- runif(n_sp ^ 2 * 2) * n_sp
dim(coords) <- c(n_sp ^ 2, 2)
time <- 1:n_t

st_coords <- as.matrix(expand.grid(1:nrow(coords), 1:length(time)))
st_coords <- cbind(coords[st_coords[, 1], ], time[st_coords[, 2]])

# different ring kernels and same lags
stkmat_r <- stkmat(coords = st_coords[, 1:2], time = st_coords[, 3], 
                   kernel_type = 'ring', 
                   kernel_parameters = list(c(0, 1), c(1, 2)), lags = c(1, 1))

# same ball kernels and different lags
stkmat_b <- stkmat(coords = st_coords[, 1:2], time = st_coords[, 3], 
                   kernel_type = 'ball', kernel_parameters = 1:3, lags = c(1, 2, 3))

# different gauss kernels and different lags
stkmat_g <- stkmat(coords = st_coords[, 1:2], time = st_coords[, 3], 
                   kernel_type = 'gauss', kernel_parameters = 1:3, lags = 1:3)

# mixed kernels
stkmat_m <- stkmat(coords = st_coords[, 1:2], time = st_coords[, 3], 
                   kernel_type = c('ball', 'ring', 'gauss'), 
                   kernel_parameters = list(1, c(1:2), 3), lags = 1:3)