spatiotemporal_gp: Non-separable spatiotemporal GP

In tulpa: Templated Unified Library for Posterior Approximation in Bayesian Hierarchical Models

Non-separable spatiotemporal GP

Description

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.

No tulpa backend fits a joint space-time covariance, so this constructor errors. A spatial GP alongside a temporal field is fitted directly by fit_st_nested() (spatial_type = "hsgp" or "nngp"); tulpa()'s formula front door does not yet route a continuous spatial field together with a temporal field (it does for an areal one – icar/bym2/car_proper – via tulpa(spatial = , temporal = ); gcol33/tulpa#812).

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 = NULL
)

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.

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.

Value

Nothing: the call always signals an error.

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

# No backend fits a joint space-time covariance, so the constructor refuses:
try(spatiotemporal_gp(~ lon + lat, "year", nonsep_type = "gneiting"))

tulpa documentation built on Oct. 19, 2026, 5:09 p.m.