Biology · Systems & Quantitative Biology

Fitting models, identifiability, sensitivity and uncertainty

Three assigned parameter pairs all predict an equilibrium state of two. An exact equilibrium observation cannot distinguish them. Yet their predictions for a change from zero differ. A model can fit a record without every parameter being separately known.

Research Design, Data & Scientific Practice introduces observations, comparisons and uncertainty. Gene Regulation & Cell Signalling develops biological feedback, while Population Ecology & Life-History Patterns develops rates and population boundaries. Their English chapters are available in this working manuscript; publication remains pending. Start with arithmetic, averages and ordinary graph reading. A derivative is introduced as an instantaneous rate of change. Exponentials and the circular sine–cosine example are explained where used; a first reading can follow their values and arrows, then return to their formulas with further mathematical preparation.

Back to the Biology map

By the end, you can

Separate fitting, identifiability, sensitivity and uncertainty, and identify when another observable can distinguish equally fitting parameter choices.

Compare a prediction with the quantity actually measured

Fitting chooses parameter values to make the model's predicted observables agree with records under a specified comparison rule. The observable matters: a concentration, a signal and a count are not interchangeable. Units, timing and an observation rule belong in the comparison. A residual is the observed value minus the predicted value under the convention used here.

An objective such as a sum of squared residuals defines what better agreement means for that fit. It also makes assumptions about which discrepancies receive greater weight. Measurement variation, unequal precision or dependence between records may require additional modelling. Minimising a chosen numerical objective does not by itself verify the measurement rule or establish a biological mechanism.

A record constrains what it can actually observe. Internal states may be unmeasured, or several internal quantities may enter the same observable. Repeatedly measuring the same ratio very precisely can still leave its numerator and denominator unresolved. It is useful to ask which combinations of parameters the available record distinguishes before interpreting a fitted parameter table.

Different parameter pairs can predict the same record

In the original rule dx/dt = a - kx, both a and k are positive constants. Its equilibrium solves 0 = a - kx, so x* = a/k. Observing only x* = 2 is consistent with (a, k) = (2, 1), (4, 2) or (6, 3), and many other pairs with the same ratio. Perfect precision in that one equilibrium observable does not separate a from k.

This is structural non-identifiability relative to the equilibrium-only observation setup: distinct parameter pairs give the same observable. Practical non-identifiability concerns the limitations of finite, noisy or incomplete data even when an ideal observation setup could in principle distinguish parameters. Both concepts depend on what is observed, when, and under which model assumptions.

For a separate hypothetical start x (0) = 0, the same rule predicts x (t) = (a/k)(1 - e⁻ᵏᵗ). At time one, the three chosen pairs predict about 1.264, 1.729 and 1.900. That transient observable would distinguish these three predictions in an ideal noise-free comparison with known starting conditions. It does not guarantee that a real measurement has sufficient precision or that this rule describes a particular organism.

One equilibrium can fit several parameter pairs

Equilibrium-only observations and the known-start transient are separate setups. No fitted uncertainty interval or empirical measurement is supplied.

Read the diagram in words

The pairs (2, 1), (4, 2) and (6, 3) all predict equilibrium two because their ratios coincide. Starting separately from zero, their time-one predictions are approximately 1.264, 1.729 and 1.900. The additional observable separates these ideal predictions under the specified rule.

Assigned parameter pair
Assigned parameter pair	Input coefficient in state units per time unit	Proportional coefficient in inverse time units	Equilibrium prediction	Time-one prediction under the separate zero start
First parameter pair	2	1	2	1.2642411176571153
Second parameter pair	4	2	2	1.7293294335267746
Third parameter pair	6	3	2	1.900425863264272

Time-one prediction under the separate zero start
x(0)	t
0	1

dx/dt=a-kx; x*=a/k; x(t)=(a/k)(1-exp(-kt))

Original scalar rule dx/dt=a-kx with positive constants, arbitrary compatible units and equilibrium-only observations in the first setup. A separate ideal transient setup specifies x (0)= 0. No actual observation, source product-ten example, parameter confidence interval or successful biological measurement is asserted.

A responsive output is not the same as a known parameter

Sensitivity asks how a model output changes when a parameter changes while other specified quantities are held fixed. For the equilibrium y = a/k at a = 4 and k = 2, the local derivatives are ∂y/∂a = 1/k = 0.5 and ∂y/∂k = -a/k² = -1. Their units differ. Comparing their raw numerical magnitudes as a universal ranking would ignore those units and the parameter scales.

Relative sensitivity, or elasticity here, compares a small fractional output change with a small fractional parameter change. The local elasticities are (a/y)(∂y/∂a) = +1 and (k/y)(∂y/∂k) = -1. For a finite one-percent increase in a, output increases exactly one percent in this rule. A one-percent increase in k gives a decrease of about 0.9901 percent, approaching the derivative-based result only for sufficiently small changes.

Sensitivity is a conditional property of the chosen rule and output. Identifiability asks whether observations distinguish parameter values. Parameter uncertainty asks what ranges or distributions remain compatible under the inference assumptions; predictive uncertainty also depends on the output and conditions. A ratio can be well constrained while its individual parameters remain uncertain. None of these questions is settled by showing one derivative.

Local relative sensitivity differs from a finite change

Raw derivatives, dimensionless elasticities and finite changes describe separate comparisons. The raw derivatives have different units and cannot supply a universal ranking by magnitude.

Read the diagram in words

At the base pair, the two raw derivatives are 0.5 and -1, while their relative elasticities are +1 and -1. A finite one-percent rise in a gives output 2.02. A finite one-percent rise in k gives about 1.980198, a decrease of about 0.9901 percent.

Assigned equilibrium output
a	k	y
4	2	2

Local fractional sensitivity
Changed parameter	Local derivative with its parameter-dependent units	Local fractional sensitivity
Input parameter	0.5	1
Proportional parameter	-1	-1

Finite one-percent comparison
Changed parameter	a	k	Assigned equilibrium output	Changed parameter · Fractional change	Assigned equilibrium output · Fractional change
Raise input parameter by one percent	4.04	2	2.02	0.01	0.01
Raise proportional parameter by one percent	4	2.02	1.9801980198019802	0.01	-0.00990099009900991

y=a/k; dy/da=1/k; dy/dk=-a/k^2

Original output y=a/k at positive base pair a 4, k 2. Each comparison changes one parameter while holding the other fixed. Raw derivatives have different units; elasticities are local dimensionless quantities. Finite changes are not assumed equal to their infinitesimal approximations.

Worked example

Separate the ratio from its components

For the three pairs, a/k is 2/1, 4/2 and 6/3, each two. Thus their equilibrium predictions coincide. Under the hypothetical zero start, the time-one predictions are 2(1 - e⁻¹), 2(1 - e⁻²) and 2(1 - e⁻³), which differ.

At the base pair (4, 2), y is two. Raising a to 4.04 while k stays two gives 2.02. Raising k to 2.02 while a stays four gives 4/2.02 ≈ 1.980198. The latter fractional change is about -0.009901, not exactly -0.01.

The equilibrium ratio is observable in the first setup, whereas the rate of approach contains additional information under the hypothetical second setup. Whether real records constrain that information depends on their timing, precision, initial-state knowledge and the adequacy of the model.

Check your understanding

Would a perfectly precise equilibrium observation at two distinguish the three displayed parameter pairs?
Show reasoning
No. They all have ratio a/k = 2 and therefore the same equilibrium observable. Precision cannot remove a structural ambiguity in that observation setup.
Why might transient observations add information?
Show reasoning
Their predictions depend on the rate coefficient k as well as the equilibrium ratio. Under the stated known-start model, the three time-one predictions differ, although real inference also depends on noise and model adequacy.
Does a larger raw derivative prove that a parameter is more identifiable?
Show reasoning
No. Raw derivatives have units and describe local output response. Identifiability concerns whether observations distinguish parameter choices; sensitivity alone does not answer it.

What to carry forward

Ask what the observable constrains, then distinguish numerical agreement, parameter distinguishability, response sensitivity and uncertainty.

References

Resources supporting this lesson’s explanations and examples.