lacov | R Documentation |
Computation of local autocovariance matrices for a multivariate space-time dataset based on a given set of spatio-temporal kernel functions.
lacov(x, coords, time, kernel_type, kernel_parameters,
lags, kernel_list = NULL, center = TRUE)
x |
either a numeric matrix of dimension |
coords |
a numeric matrix of dimension |
time |
a numeric vector of length |
kernel_type |
either a string or a string vector of length |
kernel_parameters |
a numeric vector of length |
lags |
an integer vector of length |
kernel_list |
a list of spatio-temporal kernel matrices with dimension |
center |
logical. If |
Local autocovariance matrices 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. See examples below.
Alternatively, a list of kernel matrices can be given directly to the function lacov
through the kernel_list
argument. A list with kernel matrices can be computed with the function stkmat
.
lacov
returns a list of length K
where each entry is a numeric matrix of dimension c(p, p)
corresponding to a local autocovariance matrix.
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>.
stkmat
, stbss
# space and time coordinates
n_t <- 50
n_sp <- 10
st_coords <- as.matrix(expand.grid(1:n_sp, 1:n_sp, 1:n_t))
# simulate three latent white noise fields
field_1 <- rnorm(nrow(st_coords))
field_2 <- rnorm(nrow(st_coords))
field_3 <- rnorm(nrow(st_coords))
# compute the observed field
latent_field <- cbind(field_1, field_2, field_3)
mixing_matrix <- matrix(rnorm(9), 3, 3)
observed_field <- latent_field
# lacov with different ring kernels and same lags
lacov_r <- lacov(observed_field, coords = st_coords[, 1:2], time = st_coords[, 3],
kernel_type = 'ring',
kernel_parameters = list(c(0, 1), c(1, 2)), lags = 1)
# lacov with same ball kernels and different lags
lacov_b <- lacov(observed_field, coords = st_coords[, 1:2], time = st_coords[, 3],
kernel_type = 'ball', kernel_parameters = 1, lags = c(1, 2, 3))
# lacov with different gauss kernels and different lags
lacov_g <- lacov(observed_field, coords = st_coords[, 1:2], time = st_coords[, 3],
kernel_type = 'gauss', kernel_parameters = 1, lags = 1:3)
# lacov mixed kernels
lacov_m <- lacov(observed_field, 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)
# lacov with a kernel list
kernel_list <- stkmat(coords = st_coords[, 1:2], time = st_coords[, 3],
kernel_type = 'ring',
kernel_parameters = list(c(0, 1)), lags = 1)
lacov_k <- lacov(observed_field, kernel_list = kernel_list)
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