Non-separable spatiotemporal GP

Specify a non-separable Gaussian Process for spatiotemporal effects. Unlike separable models where the covariance factors as \(C_s \otimes C_t\), non-separable models allow for direct space-time interaction in the covariance.

Usage

spatiotemporal_gp(
  coords,
  time_var,
  cov_space = c("exponential", "matern", "gaussian", "spherical"),
  cov_time = c("exponential", "matern", "gaussian"),
  nonsep_type = c("product", "sum", "gneiting", "cressie_huang"),
  nn = 15,
  shared = TRUE
)

Arguments

coords

A one-sided formula specifying coordinate columns (e.g., ~ lon + lat), or a character vector of length 2.

time_var

Name of the time variable in data.

cov_space

Spatial covariance: "exponential" (default), "matern", "gaussian", or "spherical".

cov_time

Temporal covariance: "exponential" (default), "matern", or "gaussian".

nonsep_type

Non-separability type:

• "product": \(C_{st} = C_s \cdot C_t\) (separable, for reference)

• "sum": \(C_{st} = C_s + C_t\)

• "gneiting": Gneiting (2002) non-separable class

• "cressie_huang": Cressie-Huang (1999) non-separable class

nn

Number of nearest neighbors for NNGP approximation. Default 15.

shared

Logical; if TRUE (default), effect enters both processes.

Value

A ratiod_st_gp object

Details

The non-separable covariance functions allow for more flexible space-time dependence:

Gneiting class: $$C(h, u) = \frac{\sigma^2}{(a|u|^{2\alpha} + 1)^{\tau}} \exp\left(-\frac{c\|h\|^{2\gamma}}{(a|u|^{2\alpha} + 1)^{\beta\gamma}}\right)$$

where h is spatial lag, u is temporal lag, and parameters control the space-time interaction.

Cressie-Huang class: Constructed via Fourier transform methods to ensure positive definiteness.

References

Gneiting, T. (2002). Nonseparable, stationary covariance functions for space-time data. Journal of the American Statistical Association, 97(458), 590-600.

Cressie, N., & Huang, H. C. (1999). Classes of nonseparable, spatio-temporal stationary covariance functions. Journal of the American Statistical Association, 94(448), 1330-1340.

Examples

# Non-separable spatiotemporal GP
st_gp <- spatiotemporal_gp(
  ~ lon + lat,
  time_var = "year",
  nonsep_type = "gneiting"
)
print(st_gp)
#> tulpaRatio Non-Separable Spatiotemporal GP
#> ======================================
#> 
#> Coordinates: lon, lat 
#> Time variable: year 
#> Spatial covariance: exponential 
#> Temporal covariance: exponential 
#> Non-separability: Gneiting non-separable 
#> Neighbors (NNGP): 15 
#> Shared: Yes