Scaling Symmetries That Hide Parameters

Why a Good Fit May Not Reveal Every Parameter

A model can reproduce measured concentrations exactly while leaving important physiological quantities unresolved. In a production-clearance system, for instance, a high production rate in a large distribution volume can generate the same concentration trajectory as a lower production rate in a smaller volume. This is not necessarily a problem of noisy data or an unsuccessful optimizer. It can be an exact symmetry of the model: a continuous change in parameters and hidden states that leaves every measured output unchanged.

The earlier material on identifiable parameter combinations distinguishes individual parameters from combinations that outputs can determine. Here the focus is a constructive way to prove that certain quantities cannot be distinguished: exhibit a continuous rescaling that preserves the model’s outputs. The proof applies under stated assumptions, not to every production-clearance model or every measurement design.

“Continuous” means that the transformation depends on a continuously varying quantity, such as a positive scale factor \(\lambda\), rather than only exchanging a finite number of parameter values. A symmetry therefore produces a family of indistinguishable parameter sets. Structural nonidentifiability concerns exact equality of ideal model outputs; it is different from practical difficulty estimating a parameter because measurements are sparse or noisy.

A Production-Clearance Model with Concentration Output

Let \(A(t)\) be the amount of a substance in a single well-mixed compartment, measured in mg. Let \(C(t)=A(t)/V\) be its concentration, in mg/L, where \(V\) is a constant volume in L. Assume constant production at rate \(q\), measured in mg/h, and linear clearance with clearance \(CL\), measured in L/h. The amount balance and measured output are

Assume \(q>0\), \(CL>0\), and \(V>0\); production and clearance are constant over the observation period; the compartment is well mixed; and there are no additional inputs, outputs, or nonlinear processes. The parameters are unknown, and concentration is the only measured output. The initial amount \(A(0)=A_0\) is also not known independently. Even if the initial concentration is measured, that supplies \(A_0/V\), not \(A_0\) and \(V\) separately.

Writing the dynamics directly in concentration makes the ratios visible:

Thus the concentration equation contains \(q/V\), \(CL/V\), and \(A_0/V\). This observation suggests which quantities may be recoverable, but the symmetry proof establishes the ambiguity directly.

Why the Rescaling Preserves the Output

The proof must check both the differential equation and the observation function. Let \(A(t)\) be a solution under \((q,CL,V,A_0)\). Define a transformed state and parameters by \(A_\lambda(t)=\lambda A(t)\), \(q_\lambda=\lambda q\), \(CL_\lambda=\lambda CL\), \(V_\lambda=\lambda V\), and \(A_{0,\lambda}=\lambda A_0\). The transformed initial condition holds because \(A_\lambda(0)=\lambda A_0=A_{0,\lambda}\). For the dynamics,

The transformed state therefore satisfies the same model form with the transformed parameters. Its measured concentration is unchanged:

Every positive \(\lambda\) gives an admissible parameter set and exactly the same output trajectory. Values near \(\lambda=1\) already provide a continuous local family, while any positive \(\lambda\) gives a globally indistinguishable set under these assumptions. The ambiguity is not removed by collecting more precise concentration measurements of the same output: those measurements still cannot distinguish members of this family.

What the Concentration Trajectory Can Determine

For positive \(CL/V\), the solution of the concentration equation is

A sufficiently informative ideal trajectory can reveal the steady concentration \(q/CL\), the exponential rate \(CL/V\), and the initial concentration \(A_0/V\). These quantities are unchanged by the scaling symmetry. They are not all independent: \(q/CL=(q/V)/(CL/V)\). In particular, identifying a ratio such as \(q/CL\) does not identify either \(q\) or \(CL\) separately.

This description presumes that the trajectory contains enough information to distinguish its steady level and time course. For example, if the initial concentration already equals the steady concentration, the output is constant; the observed constant alone cannot reveal the exponential rate. The symmetry result remains valid regardless of this additional loss of information: it proves that even an informative concentration trajectory cannot separate the parameters along the scaling family.

How Additional Information Can Break the Symmetry

A symmetry is defined relative to both the model and the observations. Adding a measurement or a known quantity can invalidate the transformation, but the added information must actually change under that transformation. A known volume, for example, cannot remain fixed when the proposed rescaling sends \(V\) to \(\lambda V\), unless \(\lambda=1\). Likewise, an independently known initial amount prevents rescaling \(A_0\).

A different observation may also help. If the amount \(A(t)\) is measured as well as concentration, then \(V=A(t)/C(t)\) wherever \(C(t)\ne0\), so the rescaling no longer preserves both outputs. In practice, independent volume measurements, amount measurements, or other validated information may be more realistic than directly measuring every quantity. Each proposed addition should be checked against the transformation rather than assumed to solve identifiability automatically.

Diagnostics, Alternatives, and Common Pitfalls

To test for a scaling symmetry, write the transformation explicitly and substitute it into both the state equations and every observation equation. Check initial conditions, parameter constraints, units, and any known constants. A candidate rescaling that preserves the differential equation but changes a measured output is not a symmetry of the observation model. Nor is a rescaling valid if it violates a fixed, independently known parameter or state.

This direct symmetry argument is an exact structural check. It complements, rather than replaces, numerical estimation and uncertainty analysis. An optimizer returning one parameter set does not prove that set is unique; it may simply select one point on a family of equivalent solutions. Conversely, a broad confidence interval or unstable estimate can signal practical difficulty without proving an exact symmetry. Approaches based on informative additional outputs or independent parameter measurements can change what the data determine; merely repeating the same concentration observation more precisely cannot remove an exact output-preserving rescaling.

The assumptions matter. If production or clearance depends on another measured quantity, if a known input enters the amount balance, or if volume is independently fixed, the displayed transformation may cease to preserve the model or observations. For instance, a known amount added at a specified time is not automatically rescaled with \(q\), \(CL\), and \(V\); its effect must be checked in the equations. Such modifications require a fresh symmetry analysis.

• Mistake: treating a successful fit as proof of unique parameters. A single fitted tuple shows that the model can match the data; it does not show that no other tuple gives the same output. Substitute a candidate transformation into the full model and output map.
• Mistake: claiming that concentration observations identify production and clearance separately because both affect the curve. In this model, the curve depends on ratios such as \(q/V\) and \(CL/V\). The rescaling proof shows that changing all three quantities proportionally leaves the curve unchanged.
• Mistake: assuming more precise measurements resolve every ambiguity. Precision reduces measurement uncertainty, but an exact symmetry persists even with ideal measurements of the same output. Additional information must break the symmetry.
• Mistake: applying the result to every production-clearance model. The conclusion follows from the stated equations, unknown volume, and concentration-only output. Different dynamics, known quantities, or additional observations can alter it.