Sensitivity Analysis

Introduction

A network model can contain dozens of rate constants and starting concentrations. Not all influence a chosen prediction equally. Sensitivity analysis asks how an output—such as product yield at ten minutes—changes when an input is perturbed. It helps prioritise measurements, identify fragile conclusions and decide which reaction steps matter for a particular question. Sensitivity is always tied to an output, operating point and perturbation convention.

Core explanation

For output y and parameter p, local sensitivity is ∂y/∂p at a specified parameter set. A normalized form S = (p/y)(∂y/∂p) compares fractional changes when p and y are nonzero. S ≈ 1 means a 1% rise in p causes about a 1% rise in y locally; S ≈ −2 means about a 2% fall. An output could be concentration at one time, peak intermediate level, ignition delay or steady product rate. Different outputs can have different important parameters. An ACS kinetic-modelling study uses normalized local sensitivities to evaluate response of combustion predictions to rate changes.

Estimate a derivative by running the model at p and p + δp while holding other inputs fixed: S ≈ [(y(p + δp) − y(p))/y(p)] ÷ (δp/p). Use a perturbation small enough to represent a local response but large enough to exceed numerical noise. For reversible reactions, changing one forward rate while leaving its reverse rate fixed also changes equilibrium; if the question concerns a kinetic barrier alone, perturb both constants consistently with the equilibrium ratio. State what was held fixed.

Local analysis can miss nonlinear interactions or alternative regimes. If a small increase in temperature activates a new pathway, sensitivity can change sharply. Global sensitivity samples a plausible range of several inputs to see how output variance and interactions behave. It is computationally heavier and depends on chosen input distributions. Neither method establishes the true mechanism by itself; it evaluates the assumed model's behaviour.

Sensitivity and uncertainty are related but distinct. A highly sensitive parameter may be accurately known and contribute little to prediction uncertainty. A poorly known parameter with low sensitivity may also contribute little. A rough first assessment multiplies sensitivity magnitude by fractional parameter uncertainty. Experimental design should focus on parameters that are both influential and uncertain, while remembering that parameter effects can be correlated.

Step-by-step reasoning

1. Specify the output, time and baseline conditions. 2. Choose a parameter and physically consistent perturbation. 3. Recompute the model and calculate absolute or normalized sensitivity. 4. Repeat for key parameters and check perturbation-size dependence. 5. Explore plausible joint ranges if nonlinear interactions could change the conclusion.

Visual explanation

Draw product concentration versus time for a baseline model and two curves with k₂ increased and decreased slightly. The curves may overlap early but diverge later. Draw a bar chart of normalized sensitivity for product at one minute and at one hour; the ranking can change with time. Error bars on rate constants show which sensitivity actually drives prediction uncertainty.

Real-world analogy

A recipe's taste may be very sensitive to salt but hardly affected by a small change in water volume. Knowing sensitivity helps decide which ingredient to measure carefully. But if a completely new cooking method is used, the old sensitivities may not apply. Chemical model sensitivity likewise has a local domain and can change with conditions.

Real-world example

An atmospheric reaction model predicts an ozone peak. Sensitivity analysis shows the peak responds strongly to one photolysis rate and weakly to a slow side reaction. The team prioritises better light-intensity and photolysis measurements. At nighttime conditions, however, the photolysis sensitivity vanishes. The result must be reported with the time and radiation conditions that produced it.

Why?

Why can a parameter with large sensitivity be unimportant to uncertainty? If it is already known within 0.1%, its possible contribution may be much smaller than a moderately sensitive parameter uncertain by 50%. Sensitivity indicates leverage; uncertainty indicates how far the parameter might actually vary.

Common misconception

“Most sensitive parameter is necessarily the rate-determining step” confuses model-output response with a universal mechanism label. “One local derivative represents every condition” is false. “A zero sensitivity proves a reaction never occurs” is false; it may have little effect on the selected output. “Perturbing a reversible rate constant independently always preserves equilibrium” is false.

Worked example

A model predicts y = 10.0 mmol product at 30 minutes when k = 0.50 min⁻¹. Increasing k by 2% to 0.51 min⁻¹ gives y = 10.3 mmol. The fractional output change is 0.3/10.0 = 3%, so normalized finite-difference sensitivity is about 3%/2% = 1.5. If k is uncertain by only ±1%, a local linear estimate suggests about ±1.5% output contribution from k. If another parameter has S = 0.5 but is uncertain by ±20%, its possible effect is roughly ±10%, potentially more important. These estimates ignore interaction and should be tested by rerunning the full model across plausible ranges.

Quick check

1. What does normalized sensitivity S = −1 mean locally? Answer: A small 1% increase in the input tends to decrease the chosen output by about 1% near the baseline.

Exam focus

Compute normalized finite-difference sensitivity and state the output and held-fixed inputs. Distinguish parameter influence from parameter uncertainty. Explain why local sensitivities change with time or operating regime and when a global analysis helps.

Advanced insight

Sensitivity equations can be integrated alongside the model ODEs, or automatic differentiation can compute derivatives without repeated finite-difference runs. These approaches improve numerical efficiency but inherit the assumed mechanism and parameterisation. A precise derivative of a wrong model remains scientifically misleading.

Summary

Sensitivity analysis identifies which model inputs most affect a specified prediction under specified conditions. Combining it with parameter uncertainty and out-of-range tests guides experiments and reveals when a conclusion is fragile.

Practice questions

1. Output rises 4% when a rate constant rises 2%. What is approximate normalized sensitivity? Answer: About 4%/2% = 2. 2. Can sensitivity to a photolysis rate differ between day and night? Answer: Yes. Light-dependent flux can be important by day and absent at night. 3. Does a low-sensitivity reaction necessarily not occur? Answer: No. Its flux may have little effect on the particular output chosen. 4. Why vary several inputs jointly in a global analysis? Answer: It can reveal nonlinear interactions and changed regimes missed by one-at-a-time local perturbations.