Choosing Sampling Times With Fisher Information

Why Sampling Times Matter

In an ODE model, a measurement can be precise yet contribute little to estimating a parameter if the model prediction barely changes with that parameter at the chosen time. Conversely, observations at times when sensitivities differ can help distinguish parameters whose effects otherwise compensate for one another. Sampling-time design asks which feasible observation times are expected to give the most useful information for a stated estimation goal.

The answer depends on the target. One schedule may improve precision for a particular parameter while another better estimates a different parameter or combination. Fisher information makes these comparisons quantitative, but its conclusions are local to a model, parameter value, and error specification.

Information for a Candidate Schedule

This expression assumes that the planned observation times and variances are specified, the observations are conditionally independent given the model, and the mean and variance model are appropriate. For a vector-valued observation with covariance matrix \(R_k\), its contribution is \(J_k^{\mathsf T}R_k^{-1}J_k\). Thus a high-noise observation generally contributes less information than an equally sensitive low-noise observation.

For an ODE state \(x(t)\) satisfying \(x'=g(x,\theta,t)\), sensitivities can be obtained by differentiating the ODE. For parameter \(\theta_j\), \(S_j=\partial x/\partial\theta_j\) satisfies \(S_j'=(\partial g/\partial x)S_j+\partial g/\partial\theta_j\), with the corresponding derivative of the initial condition. If the observation mean is \(h(x(t),\theta)\), its sensitivity is \((\partial h/\partial x)S_j+\partial h/\partial\theta_j\). These sensitivities evaluated at candidate times form the rows \(J_k\).

In a regular model, the inverse information matrix approximates the covariance matrix of an efficient parameter estimate near the parameter value used for the calculation. It is therefore a local precision diagnostic, not a guarantee of realized precision or a substitute for checking model fit and practical identifiability. In a linear Gaussian model with known variance, the corresponding least-squares estimator has covariance exactly equal to \(\mathcal{I}^{-1}\), when the information is nonsingular.

How to Compare Schedules for a Stated Goal

If \(\mathcal{I}\) is nonsingular, the diagonal element \((\mathcal{I}^{-1})_{jj}\) approximates the variance of the estimate of \(\theta_j\); its square root is the approximate standard error. For a scalar target \(q^{\mathsf T}\theta\), the approximate variance is \(q^{\mathsf T}\mathcal{I}^{-1}q\). The entries of \(q\) must make the target dimensionally meaningful. For example, subtracting a rate from an acceleration is not a valid parameter contrast; a time factor or a dimensionless rescaling is needed.

A useful schedule comparison begins with the target, not with a single universal score. A schedule can be judged by the approximate variance of one parameter, a chosen valid combination, or a prespecified collection of targets. Broad summaries such as the determinant of \(\mathcal{I}\) can help compare overall information, but they need not favor the schedule that best estimates a particular parameter. When comparing schedules for the same parameterization, the determinant can be a useful relative measure; its numerical value depends on parameter units.

Why Distinct and Well-Separated Times Can Help

For a two-parameter model, observations whose sensitivity rows point in nearly the same direction provide nearly redundant information. In the preceding example, two distinct positive times give an information determinant proportional to the square of the separation between the times. This can be shown directly. With two times \(t_1,t_2\) and unit observation variance, the sensitivity matrix is \(J=\begin{pmatrix}t_1&t_1^2\\t_2&t_2^2\end{pmatrix}\), and \(\mathcal{I}=J^{\mathsf T}J\). Since \(\det(J^{\mathsf T}J)=\det(J)^2\),

If \(t_1=t_2\), the determinant is zero: repeating a measurement at the same time adds information in one sensitivity direction but cannot identify both parameters separately in this example. If the times are close, the determinant is small and the inverse information can be large. Separation alone is not a universal prescription: the response sensitivities, observation errors, feasible time window, and target all matter.

Diagnostics, Alternatives, and Limitations

Before selecting a schedule, compute sensitivities at the proposed times and inspect the resulting information matrix. A zero or very small eigenvalue, a large condition number, or very large diagonal entries of \(\mathcal{I}^{-1}\) can indicate rank deficiency or near-redundancy. Check that the sensitivities are evaluated at a scientifically plausible parameter value and that their numerical calculation is stable. For nonlinear ODE models, comparing schedules at more than one plausible parameter value can reveal whether a design's apparent advantage is fragile.

The main alternative is not a different formula but a different design objective. Choosing the smallest \((\mathcal{I}^{-1})_{jj}\) targets one parameter; choosing the smallest \(q^{\mathsf T}\mathcal{I}^{-1}q\) targets a specified scalar combination. A determinant-based comparison summarizes joint information but may rank schedules differently, as the first worked example demonstrated. Such summaries are not interchangeable, and parameter scaling or units must be handled consistently when interpreting them.

Fisher information is local: in a nonlinear model, changing the parameter value can change the sensitivities and the preferred schedule. The approximation also relies on the assumed observation model and variances. It does not account automatically for model misspecification, measurement failures, informative dropout, or constraints on sampling operations. A proposed design should therefore be checked against feasible time windows, anticipated error levels, and the precision target that matters scientifically. It describes expected local precision under assumptions, not a promise about any one realized data set.

Common Pitfall

• Wrong reasoning: “Schedule B has the larger information determinant, so it must be better for every parameter and every scientific target.”
• Correction: A determinant summarizes joint information, not the variance of every individual parameter or combination. Compare the relevant diagonal of \(\mathcal{I}^{-1}\) or \(q^{\mathsf T}\mathcal{I}^{-1}q\) for the stated, dimensionally valid target.
• Wrong reasoning: “The individual sensitivity is large, so the parameter must be estimated precisely.”
• Correction: Precision depends on the full set of sensitivity directions and their error variances. Large but nearly collinear sensitivity rows can leave parameter combinations weakly determined.