Why Identifiability Begins with the Ideal Output
A model may fit observed drug-response data well and still leave a basic question unanswered: could a different parameter value produce exactly the same model output? Structural identifiability addresses this question before measurement noise or estimation algorithms enter the picture. It asks what the model and the specified experiment can distinguish in principle, assuming the output trajectory is known ideally.
The answer depends on what counts as the same output. For this tutorial, compare trajectories over the same time interval, with the same known inputs, output definitions, initial conditions, and other fixed conditions. Parameter values are considered equivalent if they generate identical outputs throughout that interval—not merely similar predictions at a few observed times.
For a scalar parameter, write the ideal output trajectory as \(y(t;\theta)\). With one output, equivalence means \(y(t;\theta')=y(t;\theta)\) for every \(t\) in the observation interval. With multiple outputs, this equality must hold for each output. The equivalence class therefore depends not only on the equations, but also on which outputs and conditions are included in the comparison.
Global, Local, and Nonidentifiability
The distinctions can be made pointwise: fix a value \(\theta\) and examine its equivalence class. Global identifiability asks whether that class contains only \(\theta\). Local identifiability asks whether \(\theta\) is at least isolated from other equivalent values nearby. This pointwise wording is useful when a model has exceptional parameter values with different properties from generic values.
- Globally identifiable: every admissible \(\theta'\) that produces the same ideal output as \(\theta\) satisfies \(\theta'=\theta\).
- Locally identifiable: there is a neighborhood of \(\theta\) containing no other admissible value that produces the same ideal output. A locally identifiable value need not be globally identifiable; equivalent alternatives may exist farther away.
- Structurally nonidentifiable at \(\theta\): every neighborhood of \(\theta\) contains a distinct admissible value that produces the same ideal output. Thus \(\theta\) is not locally identifiable.
In common usage, “globally nonidentifiable” can mean that uniqueness fails, even when a parameter is locally identifiable. To avoid conflating that with a continuum of arbitrarily close alternatives, this tutorial distinguishes not globally identifiable from not locally identifiable. A parameter can have a finite ambiguity—two possible values, for example—yet be locally identifiable at each value.
These definitions are conditional on the chosen admissible domain. If a model’s equations allow both positive and negative parameter values, both must be considered; if scientific knowledge restricts the parameter to positive values, that restriction changes the equivalence class. The restriction must be stated rather than silently imposed after seeing an ambiguity.
A Nonlinear Model and Its Output Equivalence
Consider a one-parameter model with a known input \(u(t)\), state \(y(t)\), and measured output \(y(t)\):
The parameter enters nonlinearly through its square. For a constant input \(u(t)=1\), the solution is \(y(t;\theta)=e^{-\theta^2t}\). Two parameter values are equivalent over any interval containing positive times precisely when their squares agree. Indeed, equality of the trajectories implies equality at any \(t>0\), so \(e^{-\theta'^2t}=e^{-\theta^2t}\), and therefore \(\theta'^2=\theta^2\). Conversely, equal squares give identical trajectories.
This calculation gives a complete equivalence class, not just a local sensitivity statement. For \(\theta=1\), the equivalent values are \(1\) and \(-1\). The value \(1\) is locally identifiable because a sufficiently small neighborhood around \(1\) excludes \(-1\), but it is not globally identifiable because \(-1\) remains an admissible, equivalent value. At \(\theta=0\), the only equivalent value is \(0\), so that point is globally identifiable. The model as a whole is not globally identifiable over \(\mathbb{R}\), since most values have a distinct equivalent partner.
Worked Example: A Parameter That Is Globally Identifiable
Suppose a different one-parameter model predicts \(y(t;\theta)=e^{-\theta t}\), with \(\theta>0\), and ideal output is available over an interval containing a positive time. Determine whether \(\theta=0.40\ \text{h}^{-1}\) is globally identifiable.
If \(\theta'\) produces the same trajectory, then at any fixed \(t>0\), \(e^{-\theta't}=e^{-0.40t}\). Taking natural logarithms gives \(-\theta't=-0.40t\). Since \(t>0\), division by \(-t\) yields \(\theta'=0.40\ \text{h}^{-1}\). No distinct admissible value can produce the same ideal trajectory, so this parameter value is globally identifiable.
For a numerical check at \(t=2\ \text{h}\), the predicted output is \(e^{-(0.40\ \text{h}^{-1})(2\ \text{h})}=e^{-0.80}\approx0.4493\). Equality at that positive time alone already forces \(\theta'=0.40\ \text{h}^{-1}\) in this model. In general, however, equality at a single time need not establish equality of trajectories; the conclusion here follows from the specific one-to-one formula.
Worked Example: Local but Not Global Identifiability
Return to \(y(t;\theta)=e^{-\theta^2t}\) with \(\theta\in\mathbb{R}\) and input \(u(t)=1\). Let the true value be \(\theta=1.5\). Find all equivalent values and classify the parameter.
Equality of ideal trajectories requires \(\theta'^2=1.5^2=2.25\). Hence \(\theta'=1.5\) or \(\theta'=-1.5\). For instance, at \(t=1\ \text{h}\), both predict \(e^{-2.25}\approx0.1054\); because the squared parameters are equal, the trajectories agree for every time, not just this one.
The value \(1.5\) is not globally identifiable because \(-1.5\) is a distinct equivalent value. It is locally identifiable: choose a neighborhood around \(1.5\) with radius less than \(3\), such as radius \(1\). That neighborhood does not contain \(-1.5\), so it contains no distinct equivalent value. The ambiguity is a separated sign alias, not a nearby continuum.
When a Parameter Is Not Locally Identifiable
The same model also shows why identifiability depends on experimental conditions. If the known input is \(u(t)=0\) throughout the observation interval, the differential equation becomes \(dy/dt=0\) and its solution is \(y(t;\theta)=1\) for every real \(\theta\). The parameter then has no effect on the ideal output under this condition.
Worked Example: An Uninformative Input Produces Nonidentifiability
Use \(dy/dt=-\theta^2u(t)y(t)\), \(y(0)=1\), with \(u(t)=0\). Classify \(\theta=0.7\), and verify the conclusion using a nearby value.
Since \(u(t)=0\), the model gives \(dy/dt=0\), so \(y(t;\theta)=1\) at every time for every \(\theta\). In particular, \(\theta=0.7\) and \(\theta'=0.71\) both produce the constant trajectory \(1\). More generally, for any neighborhood around \(0.7\), one can choose a distinct \(\theta'\) inside it, and the outputs remain equal. Thus \(\theta=0.7\) is structurally nonidentifiable under this input.
This conclusion is about the model under the specified input, not a claim that the parameter can never be identified under any experiment. With \(u(t)=1\), the output depends on \(\theta^2\) and distinguishes different magnitudes, though not the sign when the domain is all of \(\mathbb{R}\).
A Short Justification: Why Equality Must Hold Over Time
The equivalence test concerns trajectories, not a particular summary or a handful of measurements. For the constant-input nonlinear model, suppose the trajectories agree for all \(t\in[0,T]\), with \(T>0\). Select any \(t_*>0\) in that interval. Then \(e^{-\theta'^2t_*}=e^{-\theta^2t_*}\). The exponential function is one-to-one, so \(-\theta'^2t_*=-\theta^2t_*\). Since \(t_*\) is positive, division gives \(\theta'^2=\theta^2\), hence \(\theta'=\theta\) or \(\theta'=-\theta\). Conversely, either of those parameter values gives the same exponent at every time, proving that both are equivalent and that there are no other alternatives.
This proof also illustrates a useful discipline: first express the equality condition implied by the outputs, then solve it over the entire admissible parameter domain. A derivative or a numerical optimizer may suggest ambiguity, but it does not by itself enumerate all equivalent values.
Diagnostics, Alternatives, and Limits
A practical identifiability check begins by fixing the model conditions and parameter domain, then asking whether the output map is one-to-one. Useful warning signs include a parameter that disappears from the output equation under the chosen input, a symmetry such as \(\theta^2=(-\theta)^2\), and a computed equality condition that leaves a range of parameter values free. When analytic solutions are unavailable, symbolic or numerical exploration can help reveal candidate equivalences, but a numerical search alone may miss alternatives or fail to establish equality for all times.
Structural identifiability is different from practical identifiability. A structurally identifiable parameter has a unique value given ideal outputs under the specified conditions. Real data are finite and noisy, so the estimate may still be imprecise. Conversely, a structurally ambiguous model can produce a precise-looking numerical estimate if an optimizer selects one of several equivalent solutions; that apparent precision does not remove the ambiguity.
| Question | Structural identifiability | Practical estimation |
|---|---|---|
| What is compared? | Ideal model output trajectories | Observed data, with their noise and sampling |
| Main issue | Can distinct parameter values produce exactly equal outputs? | How uncertain or unstable is an estimate from available data? |
| Typical evidence of difficulty | Equivalent parameter values or an output independent of the parameter | Wide uncertainty, strong sensitivity to data, or unstable estimates |
A common mistake is to call the sign-ambiguous parameter in the second worked example “nonidentifiable” without qualification. The reasoning is that two values give the same output, so the parameter cannot be identified. The correction is to specify the level: it is not globally identifiable, but it is locally identifiable at \(1.5\) because the alternative \(-1.5\) is separated from it. Another mistake is to infer structural nonidentifiability merely because observed data are noisy or sparse. Noise and sparse sampling may make estimation difficult, but structural conclusions require equality of ideal trajectories under the stated conditions.
Check Your Understanding
For each question, use equality of ideal outputs and state the relevant parameter domain or experimental condition.
- For \(y(t;\theta)=e^{-\theta t}\), with \(\theta\in\mathbb{R}\) and an observation interval containing positive times, is \(\theta=0.3\) globally identifiable?
- For \(y(t;\theta)=e^{-\theta^2t}\), with \(\theta\in\mathbb{R}\), list all values equivalent to \(\theta=-2\). Is this value locally identifiable?
- For \(dy/dt=-\theta^2u(t)y\), \(y(0)=1\), what happens to identifiability if \(u(t)=0\) throughout the interval?
- In one sentence, distinguish structural identifiability from practical identifiability.
Solutions. (1) Equality at any \(t>0\) gives \(e^{-\theta't}=e^{-0.3t}\), hence \(\theta'=0.3\); it is globally identifiable. (2) Equality requires \(\theta'^2=4\), so the equivalent values are \(-2\) and \(2\). The value \(-2\) is locally identifiable because \(2\) is separated from it, but it is not globally identifiable. (3) The output is \(y(t)=1\) for every \(\theta\); every neighborhood contains distinct equivalent values, so the parameter is not locally identifiable. (4) Structural identifiability asks whether distinct values can give exactly the same ideal trajectories, whereas practical identifiability concerns the uncertainty of estimating parameters from finite, noisy data.