Diagnosing Identifiability with Spatial Data

Why spatial identifiability needs more than a good fit

A spatial model can reproduce measured profiles while leaving its parameters uncertain. For example, a shallow concentration gradient might arise from rapid transport combined with rapid loss, or from slower transport combined with slower loss. A fitted curve alone does not show whether the data distinguish those explanations. Identifiability analysis asks whether plausible changes in individual parameters produce distinguishable changes in the observations.

Here the focus is on comparing parameter sensitivities across locations and sampling times. The aim is not to establish that a model is correct, but to find out which parameter combinations the planned spatial measurements can inform. This is especially important when transport, reaction, and binding all affect a measured concentration profile.

A model and a local sensitivity diagnostic

Consider a one-dimensional tissue segment with free signal concentration \(c(x,t)\). Suppose binding is rapid, reversible, unsaturated, and spatially uniform, so that bound signal acts as a buffer that slows changes in free concentration. A simple reduced model is

Here \(D\) is an effective diffusion coefficient with units of length squared per time, \(k\) is a first-order loss rate with units of inverse time, and \(R\) is a dimensionless retardation factor representing equilibrium binding capacity. This reduced representation assumes that binding equilibrates faster than the resolved changes in concentration. It does not represent saturation, spatially varying binding sites, or binding kinetics that are slow relative to transport and sampling.

For observations with different scales or measurement errors, it is useful to weight sensitivities. If observation \(i\) has standard deviation \(\sigma_i\) on the scale being fitted, define \(J_{ij}=S_{ij}/\sigma_i\). Under independent, approximately Gaussian measurement errors with known variances, \(J^\mathsf{T}J\) is the local Fisher information matrix. Correlated errors require a covariance-based weighting instead. A small eigenvalue of the information matrix signals a parameter combination that produces little change relative to the observation noise.

These are local diagnostics: derivatives are evaluated at a particular parameter estimate. Full column rank is evidence that small changes in each parameter can be distinguished locally, under regularity conditions. It does not prove that there is only one parameter set giving the observations globally. Conversely, an exact rank deficiency means that at least one local parameter direction cannot be resolved by those observations.

Why steady spatial profiles can confound parameters

For a steady profile with fixed boundary concentration \(c_0\), the reduced model gives \(D c_{xx}-kc=0\). On a semi-infinite domain, the decaying solution is \(c(x)=c_0\exp[-x\sqrt{k/D}]\). Defining \(z=x\sqrt{k/D}\), the log-concentration sensitivities are

Thus every steady spatial measurement has the same sensitivity direction for \(D\) and \(k\): increasing both by the same percentage leaves their ratio, and therefore this profile, unchanged. The steady data also contain no information about \(R\) in this model, because the time-derivative term vanishes. Adding locations can improve precision on the gradient, but cannot by itself separate parameters whose sensitivity columns remain proportional.

Use locations and times that change the sensitivity pattern

Transient observations can carry information that steady observations lack. In the model, \(R\) multiplies the time derivative, so the approach to steady state depends on binding retardation. Measurements at different times may therefore respond differently to \(R\), \(D\), and \(k\). Spatial locations also matter: a measurement near a source may be sensitive to delivery or boundary conditions, while a measurement farther away may be more sensitive to the shape and propagation of the profile. Which effect dominates depends on the model and experimental conditions; it should be checked using the fitted model rather than assumed.

A practical local analysis proceeds by calculating sensitivities at the proposed sampling locations and times, scaling parameters on a relative scale, and accounting for measurement error. Sensitivities can be computed by differentiating the model equations or by finite differences. For example, a centered finite-difference estimate for a log-parameter sensitivity is

with other parameters held fixed. The perturbation \(\delta\) should be small enough to approximate a derivative but large enough to avoid numerical round-off. Check that the result is stable to reasonable changes in \(\delta\), and verify that it is not a mesh or solver artifact. Earlier material on mesh convergence and spatial boundary artifacts is relevant because numerical errors can masquerade as sensitivity.

Rank is not the same as practical identifiability

With finite precision, nearly proportional sensitivity columns can be almost as problematic as exactly proportional ones. Rank reports whether a direction is mathematically absent; conditioning and uncertainty report whether it is useful in practice. Examine singular values of the error-weighted matrix \(J\), parameter correlations from the local covariance approximation when its assumptions are reasonable, and changes in predictions over plausible parameter ranges. Do not rely on rank alone.

Diagnostics, alternatives, and limitations

A sensitivity analysis is only as credible as its observation model, parameterization, and numerical implementation. Check that parameters are scaled sensibly: derivatives with respect to raw parameters can make parameters with different units appear incomparable. Check the finite-difference step, solver tolerances, spatial mesh, boundary conditions, and whether measurement errors vary by location or time. A sensitivity pattern that changes markedly with mesh refinement or small numerical settings should not be interpreted biologically.

Compare candidate designs by stacking their weighted sensitivity rows and inspecting the resulting information matrix or singular values. Adding nearby locations is useful when it changes the sensitivity pattern or reduces measurement uncertainty; it is not automatically useful if the new rows repeat existing patterns. Likewise, early-time samples, later-time samples, and bound-fraction assays are alternatives to consider, not guaranteed solutions. Their value depends on the model, assay precision, and feasible sampling schedule.

Sensitivity matrices are a local, derivative-based approximation. They may miss nonlinear or distant alternative parameter sets, and uncertainty approximations based on \(J^\mathsf{T}J\) can be unreliable when the model is strongly nonlinear, parameters are near bounds, or errors are non-Gaussian or correlated but untreated. Profile likelihoods or simulation-based parameter recovery can complement the local diagnosis when these issues matter. They answer related but not identical questions: sensitivities reveal local directions of change, while broader analyses examine behavior beyond that local neighborhood.

A common mistake is to conclude that a parameter is identified because a model fits many spatial points. The number of observations alone does not determine identifiability. If all those points respond to two parameters in almost the same way, the data mainly constrain a combination of them. The correction is to compare sensitivity columns across the actual locations and times, include measurement uncertainty, and seek measurements or perturbations that produce distinct responses.