Why Allow Space-Time Dependence to Change?
For a latent spatial effect observed repeatedly over time, a separable covariance assumes that spatial dependence has the same pattern at every temporal separation. That can be restrictive. For example, nearby locations might remain relatively similar across a long time gap even as the overall association between observations declines. A nonseparable model allows the spatial covariance profile to change with temporal lag.
In a GLMM, this covariance describes dependence in a latent random effect, not necessarily the correlation of the observed responses. For example, a linear predictor may contain a latent field \(W(s,t)\) at location \(s\) and time \(t\). The focus here is the covariance structure of that field.
Nonseparability is not just a label for a complicated formula. Its practical implication is that the spatial covariance profile may depend on the temporal lag. For a stationary model with \(C(0,u)>0\), define the spatial profile normalized at lag \(u\) as \(R_u(h)=C(h,u)/C(0,u)\). Under separability, \(R_u(h)=C_S(h)/C_S(0)\), which does not depend on \(u\). A change in \(R_u(h)\) across lags is therefore evidence against a separable representation.
Validity Comes Before Interpretation
A proposed function is a covariance function only if it generates valid covariance matrices for every finite set of observation points. Specifically, for any points \((s_i,t_i)\) and any real numbers \(a_1,\ldots,a_n\), it must satisfy
Equivalently, every finite matrix with entries \(C(s_i-s_j,t_i-t_j)\) must be positive semidefinite. This requirement is essential: a formula that gives plausible-looking correlations at selected lags can still fail to define a valid joint random field.
For a stationary Gaussian latent effect, this condition ensures that the joint Gaussian distributions implied at any finite collection of space-time points have valid covariance matrices. Stationarity and isotropy are modeling assumptions, not consequences of nonseparability. In an isotropic spatial model, \(C\) depends on the distance \(\lVert h\rVert\) rather than the direction of \(h\). The model should be used only over the spatial and temporal domain for which these assumptions are reasonable.
A Valid Class With Lag-Dependent Spatial Profiles
One widely used valid family is the Gneiting class. For \(d\)-dimensional Euclidean space, one member of the class is
where \(\sigma_W^2>0\), \(a>0\), \(c>0\), \(0<\alpha\leq 1\), \(0<\gamma\leq 1\), and \(0<\beta\leq 1\). Under these restrictions this is a valid stationary space-time covariance. The parameter \(a\) has units of time raised to \(-2\alpha\), and \(c\) has units of distance raised to \(-2\gamma\), so both terms in the formula are dimensionless. Its normalized spatial profile is
Because the denominator changes with \(|u|\), this profile generally changes with temporal lag. The prefactor also changes with lag, so the overall covariance can decline while the normalized spatial profile becomes broader.
Worked Example: Spatial Dependence Under a Gneiting Covariance
Consider a one-dimensional spatial domain and choose \(\sigma_W^2=a=c=1\), \(\alpha=\beta=\gamma=1\). These values meet the stated parameter restrictions. With distance and time measured in the corresponding unit scales, the covariance is
At temporal lag \(u=0\), the covariance between locations separated by one distance unit is
At lag \(u=1\), the covariance at zero spatial separation is \(C(0,1)=1/\sqrt{2}\approx 0.7071\). At distance \(h=1\),
Thus the covariance at distance one is larger at lag one than at lag zero in this example, even though the zero-distance covariance has declined from 1 to about 0.7071. The normalized spatial profile also changes, from about 0.3679 to about 0.6065. Those calculations demonstrate nonseparability: a separable covariance would have the same normalized spatial profile at both temporal lags.
Building Valid Models From Separable Components
A useful construction is a nonnegative weighted sum of valid separable covariance functions. For example, if \(S_k(h)\) and \(T_k(u)\) are valid spatial and temporal covariance functions, then
is valid. To see why, take any finite set of space-time points. For component \(k\), the spatial and temporal covariance matrices on those points are positive semidefinite. Their entrywise product is positive semidefinite: if their entries are inner products of vectors \(a_i\) and \(b_i\), the product entries are inner products of \(a_i\otimes b_i\) and \(a_j\otimes b_j\). Multiplying this matrix by \(w_k\geq 0\) preserves positive semidefiniteness, as does summing the component matrices. The result is therefore a valid covariance matrix for every finite set of points. The sum need not itself factor into one spatial term and one temporal term.
Worked Example: A Valid Mixture and Its Lag-Dependent Spatial Profile
Consider the stationary covariance
Each term is a product of valid exponential spatial and temporal covariance functions, and both weights are nonnegative. The construction above therefore guarantees validity. Also, \(C(0,0)=0.6+0.4=1\), so the latent field has marginal variance one.
At temporal lag \(u=4\), the zero-distance covariance is
At spatial distance \(h=1\) and the same temporal lag,
The normalized spatial profile at lag four is therefore
For comparison, at lag zero, \(C(0,0)=1\) and
The normalized spatial profile at distance one is thus about 0.6757 at lag four and 0.5322 at lag zero. This ratio describes spatial attenuation relative to the covariance at zero spatial separation for the same temporal lag; it is not, by itself, the correlation coefficient between two observations separated in both space and time.
Checking a Finite Covariance Matrix
Worked Example: Checking Three Space-Time Points
Use the mixture from the previous example at the points \((0,0)\), \((1,0)\), and \((0,1)\). The diagonal entries are all one. The three off-diagonal entries, rounded to four decimal places, are
The resulting covariance matrix is approximately
Each two-by-two principal minor is positive because each off-diagonal entry has absolute value less than one. The determinant is approximately
So this particular matrix is positive definite. Checking one matrix is a useful numerical check, but it does not prove validity for all point sets. Here, validity for every set follows instead from the nonnegative-mixture construction.
Comparison, Diagnostics, and Limitations
| Approach | What it assumes or offers | Main implication |
|---|---|---|
| Separable covariance | A single spatial covariance multiplied by a single temporal covariance | Spatial profile is unchanged across temporal lags; on a complete grid the covariance has Kronecker structure. |
| Nonseparable covariance | A valid joint space-time covariance that does not factor into one spatial and one temporal term | Spatial dependence can vary with temporal separation, at the cost of more modeling and computation. |
| Weighted mixture of separable terms | A nonnegative sum of valid product components | Validity follows from the construction; the mixture can be nonseparable without being a single product. |
In practice, first check that the chosen formula and parameter restrictions guarantee positive semidefiniteness. For a custom implementation, calculate eigenvalues of covariance matrices at representative observed space-time points; materially negative eigenvalues indicate an invalid matrix or a numerical or coding problem. Very small eigenvalues can also signal near-singularity, which may cause unstable fitting.
Compare the fitted covariance profiles across temporal lags that are represented in the study design. If residual spatial patterns differ systematically by lag, a single separable profile may be too restrictive. Conversely, sparse observations at some combinations of distance and lag provide little information about how the profile changes; a flexible nonseparable model can then be weakly identified or poorly estimated. These checks are evidence about model adequacy, not proof that one covariance family is correct.
Nonseparable models can require more covariance parameters and substantially more computation than a separable model. Their interpretation is on the latent-effect scale in a GLMM. A covariance change in the latent field should not be described directly as the same change in response correlation. The model also cannot compensate for unmodeled trends, unsuitable stationarity assumptions, or a poor choice of spatial or temporal domain.
Common Mistakes
- Assuming every plausible formula is valid. A covariance must produce positive semidefinite matrices for all finite point sets. Use a known valid family or establish validity, rather than relying only on a few computed correlations.
- Calling a sum invalid because it is not a product. A nonnegative sum of valid covariance functions is valid, even when it is nonseparable.
- Confusing covariance magnitude with spatial profile. \(C(h,u)\) can decline with \(u\) while \(C(h,u)/C(0,u)\) increases. State which quantity is being interpreted.
- Interpreting latent covariance as response correlation. The covariance here belongs to the latent random effect; the observed-response relationship also depends on the GLMM likelihood and link.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- What condition distinguishes a valid space-time covariance from a formula that merely gives plausible values at selected lags?
- Under a separable covariance, can the normalized spatial profile \(C(h,u)/C(0,u)\) depend on \(u\)? Explain.
- Why is a nonnegative weighted sum of valid separable covariance functions valid?
- In the mixture example, what does \(R_4(1)\approx 0.6757\) represent, and how does it compare with \(R_0(1)\approx 0.5322\)?
Solutions. (1) Every finite covariance matrix formed from the function must be positive semidefinite. (2) No. If \(C(h,u)=C_S(h)C_T(u)\), then \(C(h,u)/C(0,u)=C_S(h)/C_S(0)\), which does not depend on \(u\). (3) Each component matrix is positive semidefinite; multiplication by a nonnegative weight and addition preserve that property. (4) It is the covariance at spatial distance one divided by the covariance at zero spatial distance, both at temporal lag four. The profile is about 0.6757 at lag four versus 0.5322 at lag zero, indicating less spatial attenuation at distance one relative to the zero-distance covariance at lag four.