Functions to specify spatiotemporal interaction effects for tulpa models. These capture dependencies that arise when spatial patterns vary over time, or when temporal trends differ across space.
Specify a spatiotemporal interaction effect for tulpa models. The interaction captures structured or unstructured deviation from the additive spatial + temporal model.
No tulpa backend fits an interaction term, so this constructor errors. The
additive space-time model is fitted by tulpa(spatial = , temporal = ) and
by fit_st_nested().
Usage
spatiotemporal(
spatial,
temporal,
type = c("I", "II", "III", "IV", "iid", "separable"),
shared = NULL
)Arguments
- spatial
A spatial specification from
spatial_car(),spatial_bym2(), orspatial_gp().- temporal
A temporal specification from
temporal_rw1(),temporal_rw2(),temporal_ar1(), ortemporal_gp().- type
Interaction type:
"I"or"iid": Unstructured interaction (IID)"II": Structured time at each location"III": Structured space at each time point"IV": Fully structured (Kronecker product of spatial and temporal)"separable": Separable covariance (Kronecker product)
Logical; if TRUE (default), spatiotemporal effect enters both all processes. Set to FALSE for process-specific effects (triggers warning about potential confounding).
Details
Spatiotemporal interactions extend the basic additive model:
$$\eta_{st} = X\beta + f_s(space) + f_t(time)$$
to include interactions:
$$\eta_{st} = X\beta + f_s(space) + f_t(time) + \delta_{st}$$
where \(\delta_{st}\) captures space-time interactions.
Interaction Types (following Knorr-Held, 2000):
Type I: Unstructured interaction - IID \(\delta_{st} \sim N(0, \sigma^2)\)
Type II: Structured time, unstructured space - temporal structure at each location
Type III: Structured space, unstructured time - spatial structure at each time
Type IV: Structured space AND time - full Kronecker interaction
Separable Models:
Separable: Covariance is Kronecker product \(C_{st} = C_s \otimes C_t\)
Non-separable: GP with joint space-time metric
Type I (IID)
Independent random effect for each space-time combination: $$\delta_{st} \stackrel{iid}{\sim} N(0, \sigma^2_\delta)$$
This is the simplest form, requiring S*T parameters but capturing no structured interaction.
Type II (Temporal structure per location)
Each location has its own temporal random effect: $$\delta_{\cdot t}^{(s)} \sim RW(\sigma^2)$$
This captures location-specific temporal trends but assumes independence across locations.
Type III (Spatial structure per time point)
Each time point has its own spatial random effect: $$\delta_{s \cdot}^{(t)} \sim ICAR(\tau)$$
This captures time-specific spatial patterns but assumes independence across time points.
Type IV (Full structure)
Kronecker product of spatial and temporal precision matrices: $$Q_\delta = Q_s \otimes Q_t$$
This is the most constrained model, assuming the interaction has the same structure as the marginal effects.
Separable
For GP-based effects, assumes separable covariance: $$C(\mathbf{s}_1, t_1; \mathbf{s}_2, t_2) = C_s(\mathbf{s}_1, \mathbf{s}_2) \cdot C_t(t_1, t_2)$$
References
Knorr-Held, L. (2000). Bayesian modelling of inseparable space-time variation in disease risk. Statistics in Medicine, 19(17-18), 2555-2567.