Dynamic Spatial Random Effects

Why Evolve Spatial Effects Through Time?

A spatial random effect can represent latent differences among locations that remain after accounting for observed covariates. In repeated observations, however, assuming the spatial pattern is either identical at every time or independently redrawn at each time may be unsuitable. A dynamic spatial effect gives the pattern persistence: the effect at one time partly carries forward, while new spatially structured innovations allow it to change.

The key distinction is between the covariance of the innovations entering at each time and the covariance of the resulting spatial effects. The transition accumulates innovations over time, so its marginal covariance depends on both the spatial structure and the autoregressive parameter.

Definition: Let $u_t=(u_{1t},\ldots,u_{mt})^\mathsf{T}$ be the vector of latent spatial effects at $m$ locations at discrete time $t$. A first-order dynamic spatial model is $$ u_t=\rho u_{t-1}+\varepsilon_t, $$ where $\rho$ is a scalar temporal persistence parameter and $\varepsilon_t$ is a vector of innovations with mean zero and spatial covariance matrix $\Sigma_\varepsilon$. The innovations are independent across time and independent of the initial state $u_0$.

In a GLMM, these effects can enter a linear predictor, for example $\eta_{it}=x_{it}^{\mathsf T}\beta+u_{it}$, where $x_{it}$ is the covariate vector for location $i$ and time $t$. The response distribution and link function belong to the GLMM; the dynamic model specifies the joint distribution of the latent effects.

Assumptions and Covariance Structure

A common specification takes $\varepsilon_t\sim N_m(0,\tau^2 R)$, where $R$ is a positive-definite spatial correlation matrix and $\tau^2>0$ is the innovation variance scale. Alternatively, a proper spatial precision matrix $Q$ can be used, with $\Sigma_\varepsilon=\tau^2Q^{-1}$. The spatial structure may be built from locations or adjacency, as in the earlier material on spatial covariance structures and proper CAR models. The innovations must have the same covariance at each time for the stationary formulas below to apply.

For a stationary process with a scalar transition, require $|\rho|<1$. Also assume that $u_0$ has mean zero, is independent of future innovations, and has the stationary covariance. Gaussian innovations and a Gaussian initial state make the full latent process jointly Gaussian; the covariance derivations themselves require only finite second moments and the stated independence. If the initial covariance is not stationary, the transition can still be used, but the covariance depends on time since initialization.

A useful feature of the scalar transition is that it multiplies every spatial component by the same $\rho$. Consequently, it preserves the spatial covariance shape over time: locations with larger innovation covariance remain more strongly associated in the propagated effects, scaled by temporal persistence.

Deriving the Stationary Covariance

Let $V=\operatorname{Var}(u_t)$ under stationarity. Since $u_t=\rho u_{t-1}+\varepsilon_t$, and the current innovation is independent of the preceding state, the variance recursion is $V=\rho^2V+\Sigma_\varepsilon$. Solving this matrix equation gives the stationary covariance. For the cross-time covariance, expand the transition over a lag $h$ and use the independence of the intervening innovations from $u_{t-h}$.

V=\rho^2V+\Sigma_\varepsilon \quad\Longrightarrow\quad \operatorname{Var}(u_t)=V=\frac{\Sigma_\varepsilon}{1-\rho^2}, \qquad |\rho|<1.

u_t=\rho^h u_{t-h}+\sum_{k=0}^{h-1}\rho^k\varepsilon_{t-k} \quad\Longrightarrow\quad \operatorname{Cov}(u_t,u_{t-h}) =\rho^h V =\rho^h\frac{\Sigma_\varepsilon}{1-\rho^2}.

The sum in the expansion contains innovations occurring after time $t-h$; each is independent of $u_{t-h}$, so it contributes zero to the cross-covariance. In particular, for locations $i,j$ and times $t,s$, stationarity gives

\operatorname{Cov}(u_{it},u_{js}) =\rho^{|t-s|}\frac{(\Sigma_\varepsilon)_{ij}}{1-\rho^2}.

If $\Sigma_\varepsilon=\tau^2R$, the marginal variance at location $i$ is $\tau^2R_{ii}/(1-\rho^2)$, while the temporal covariance at lag $h$ is multiplied by $\rho^h$. A negative $\rho$ is mathematically allowed under stationarity and produces alternating signs across successive lags; it should be substantively plausible for the application.

Key takeaway: The innovation covariance is not the stationary covariance of the evolving effects. The latter is $\Sigma_\varepsilon/(1-\rho^2)$. Under a scalar transition, stationary space-time covariance factors into a temporal term and a spatial term, even though the model is specified as a dynamic transition.

Worked Covariance Calculations

Worked Example: Specifying a Dynamic Model From a Spatial Correlation

Suppose three locations have the hypothetical spatial correlation matrix $$ R=\begin{pmatrix} 1&0.4&0.2\\ 0.4&1&0.3\\ 0.2&0.3&1 \end{pmatrix}. $$ Use $\rho=0.6$ and $\tau^2=0.64$, and assume the process is initialized in stationarity. This is a valid positive-definite correlation matrix: it is symmetric, has diagonal entries one, and is strictly diagonally dominant with positive diagonal entries. Thus the innovation covariance $\Sigma_\varepsilon=0.64R$ is positive definite.

The transition is $u_t=0.6u_{t-1}+\varepsilon_t$, with $\varepsilon_t\sim N_3(0,0.64R)$ independently over time. Its stationary covariance is $$ \operatorname{Var}(u_t)=\frac{0.64R}{1-0.6^2} =\frac{0.64R}{0.64}=R. $$

Thus the stationary marginal variance at each location is 1, and the stationary covariance between locations 1 and 2 at the same time is 0.4. The innovation variance scale is only 0.64; repeated innovations increase the stationary variance to 1. This calculation also verifies that the chosen $\rho$ is stationary because $|0.6|<1$.

Worked Example: Covariance Across Different Locations and Times

Continue with the hypothetical model above. Find the covariance between location 1 at time 2 and location 3 at time 5. The time gap is $5-2=3$, and $R_{13}=0.2$. The stationary cross-time formula gives $$ \operatorname{Cov}(u_{1,2},u_{3,5}) =0.6^3\left(\frac{0.64(0.2)}{1-0.6^2}\right) =0.216(0.2)=0.0432. $$

The result is positive and in squared units of the latent effect. The spatial covariance at a common time is 0.2; three transitions reduce this cross-time covariance by the factor $0.6^3$. If the time gap were one instead, the covariance would be $0.6(0.2)=0.12$. It would be incorrect to use $0.64(0.2)$ as the stationary spatial covariance: that is the innovation covariance, not the marginal covariance.

Worked Example: A Fixed Initial State Rather Than Stationarity

Suppose instead that $u_0=0$ is fixed, $\rho=0.5$, $\tau^2=1$, and the innovation spatial correlation between locations 1 and 2 is $R_{12}=0.3$. There is no stationary random initial effect in this specification. At time 1, $u_1=\varepsilon_1$, so $\operatorname{Var}(u_1)=R$. At time 3, the variance is $$ \operatorname{Var}(u_3) =(1+0.5^2+0.5^4)R =1.3125R. $$

For $t\geq s$, the transition implies $\operatorname{Cov}(u_t,u_s)=\rho^{t-s}\operatorname{Var}(u_s)$, because the innovations after time $s$ are independent of $u_s$. Therefore, $$ \operatorname{Cov}(u_{1,3},u_{2,1}) =0.5^{3-1}(1)(0.3)=0.075. $$ The value differs from the stationary covariance at lag 2, which would be $0.5^2(1/(1-0.5^2))(0.3)=0.1$. Early-time covariance depends on initialization; using the stationary formula here would be unjustified.

General Initial Conditions and Practical Checks

For a nonstationary initial state with mean zero and covariance $V_0$, repeated substitution gives, for $t\geq1$,

\operatorname{Var}(u_t) =\rho^{2t}V_0+\sum_{k=0}^{t-1}\rho^{2k}\Sigma_\varepsilon.

For $t\geq s$, the corresponding cross-time covariance is $\operatorname{Cov}(u_t,u_s)=\rho^{t-s}\operatorname{Var}(u_s)$. These expressions converge to the stationary covariance when $|\rho|<1$ and the starting covariance is finite. They also make clear why a fixed zero initial state and a stationary random initial state are different modeling choices.

In practice, check that the innovation covariance is positive definite (or positive semidefinite if the model permits a singular structure), that the time index has a meaningful discrete spacing, and that the assumed innovation structure is defensible. For a fitted model, simulated latent trajectories can be compared with observed spatial-temporal patterns or suitable model-based summaries. Persistent trends in innovations, changing spatial patterns, or markedly different temporal persistence across locations suggest that the assumed transition or common spatial covariance may be inadequate. Near $|\rho|=1$, the stationary variance becomes large for fixed $\Sigma_\varepsilon$, and finite data may provide weak information about the distinction between high persistence and large innovation variation.

A dynamic construction is a mechanistic alternative to specifying an arbitrary joint space-time covariance directly. Under the scalar transition and stationary initialization, its covariance is separable: the temporal factor is $\rho^{|t-s|}$, and the spatial factor is proportional to $R$. This offers a compact interpretation and efficient structure, but it does not produce every possible nonseparable covariance. The earlier material on nonseparable space-time covariance models covers broader covariance forms; here, the focus is the transition mechanism and its consequences. A scalar $\rho$ also assumes equal temporal persistence at every location, and a fixed $R$ assumes the same spatial innovation pattern at each time. These restrictions may be too strong when substantive knowledge or data indicate otherwise.

Common Mistakes and Limitations

Confusing innovation and marginal variance. Using $\Sigma_\varepsilon$ as the stationary covariance omits the accumulation of past innovations. Under stationarity, divide it by $1-\rho^2$.
Applying stationary formulas to a transient process. A fixed initial state generally gives a time-dependent variance. Specify the initialization before calculating covariance.
Ignoring independence assumptions. The covariance recursions rely on innovations being independent over time and independent of the preceding state. Correlated innovations require additional covariance terms.
Assuming a scalar transition permits arbitrary dynamics. It imposes a common persistence parameter and, under stationarity, a separable covariance form. Good fit cannot establish that these restrictions are appropriate in every setting.

Limitation: Dynamic spatial random effects describe dependence in a latent process, not causal effects of one location on another. In a GLMM, dependence in the latent effects also does not by itself determine the response-scale covariance; that depends on the response distribution and link.

Check Your Understanding

Answer each question, then compare with the worked solutions below.

If $\rho=0.4$ and $\Sigma_\varepsilon=0.84R$, what is the stationary covariance?
Under stationary initialization, let $R_{12}=0.5$, $\tau^2=0.75$, and $\rho=0.2$. Find $\operatorname{Cov}(u_{1,t},u_{2,t+2})$.
Why is the stationary covariance formula not automatically appropriate when $u_0=0$ is fixed?
State two substantive restrictions imposed by the scalar-transition dynamic spatial model.

Solutions. (1) Since $1-\rho^2=1-0.16=0.84$, the stationary covariance is $0.84R/0.84=R$. (2) The stationary cross-location covariance at one time is $0.75(0.5)/(1-0.2^2)=0.375/0.96=0.390625$. At lag 2, multiply by $0.2^2$, giving $0.015625$. (3) A fixed initial state has zero initial variance, so the early-time variance is built only from innovations received so far; it generally has not reached the stationary value. (4) It assumes a common temporal persistence parameter across locations and a time-invariant spatial covariance structure for innovations.

What You'll Learn