Finite-perturbation reference sensitivity

Finite-perturbation reference sensitivity measures the change in a comparison when one shared reference value is changed by a finite amount everywhere it occurs. It checks both local sensitivity coefficients and their usefulness over the actual reference-uncertainty range. The reference-sensitivity cancellation check is incomplete if it tests only a derivative at one point.

Let $F(R)$ be the final comparison, with all other independent inputs held fixed. A central finite difference and a second difference areFor sufficiently smooth $F$, these approach $F^{\prime }(R_0)$ and $F^{\prime \prime }(R_0)$, with errors of order $h^2$ when the required derivatives exist. Their units differ: comparison per reference unit and comparison per squared reference unit. The GUM defines sensitivity coefficients as partial derivatives and also discusses determining them by varying an input experimentally.^[ref]

An original model writes $x=R-R_0$ andIt gives $C_h=ϵ+\kappa h^2$ and $Q_h=2\beta$. Comparing $C_h$ with $C_{h/2}$ can expose cubic contamination of the local slope. But if $ϵ=\kappa =0$, every central slope is exactly zero although $F(R_0+h)-F(R_0)=\beta h^2$ is nonzero. A symmetric slope check alone misses an even-in-perturbation response.

For $\beta =1\mathrm{m}{\mathrm{m}}^{-1}$ and $h=0.001\mathrm{mm}$, the one-sided change is 1 nm, while $C_h=0$ and $Q_h=2\mathrm{m}{\mathrm{m}}^{-1}$. If the shared reference error instead has a zero-mean normal distribution with standard deviation $s=h$, the quadratic contribution has mean 1 nm and standard deviation approximately 1.414 nm. These are original model calculations. They show why exact cancellation, zero local slope and negligible finite-range influence are distinct claims.

Use step-size stability of a reference perturbation to select a sensible test range. Extremely small steps can magnify numerical subtraction errors; large steps probe a different region of a nonlinear response. Test several scales near the physical standard uncertainty and inspect one-sided changes as well as central slopes. If the model is stochastic, separate simulation noise from the response to the changed reference.

A computational perturbation changes the assigned shared reference in every dependent calculation while keeping its identity shared. Independently perturbing two uses of the same reference changes the covariance model. A physical experiment must additionally hold other influences stable, because changing a standard or its temperature may change contact or instrument behaviour. NIST separates master-block uncertainty from the comparison process in its uncertainty budget.^[ref] Retain those physical distinctions when interpreting a numerical sensitivity test.

Step-size stability of a reference perturbation checks whether a numerical reference-response estimate behaves consistently as the applied step changes. For a final comparison $F(R)$, a central slope isFor sufficiently smooth $F$, its leading truncation error is proportional to $h^2$. Comparing several steps tests that numerical behavior, while choosing steps near the physical reference uncertainty also probes whether a local derivative adequately represents the measurement model.^[ref]

For an original model, let $x=R-R_0$ and $F(R)=D_0+0.2x+\kappa x^3$, with $\kappa =100\mathrm{m}{\mathrm{m}}^{-2}$. For $h=0.010,0.005,0.0025\mathrm{mm}$, the central slopes are0.210,0.2025 and0.200625. Their successive departures from0.2 shrink by four as $h$ halves. Richardson extrapolation removes the cubic contribution exactly in this model:For a general smooth response, it removes the leading $h^2$ term rather than proving exact cancellation of every nonlinear contribution.

Small steps also amplify output error. If each evaluation has absolute error at most $q$, the error in the central slope is at most $q/h$. With $q=0.1\mathrm{nm}=10^{-7}\mathrm{mm}$, this bound is $10^{-5}$ at $h=0.01\mathrm{mm}$ but0.1 at $h=10^{-6}\mathrm{mm}$. The cubic truncation term decreases with $h$, whereas this evaluation-error bound increases. Equating their orders, $|\kappa |h^2\sim q/h$, suggests the scale $h\sim (q/|\kappa |{)}^{1/3}$, here0.001mm. This scale follows from the stipulated error model, not a universal finite-difference setting.

An unchanging central value is also insufficient to establish differentiability. For $F(R)=|R-R_0|$, every $C_h$ is zero, while the forward and backward slopes are $+1$ and $-1$. Inspecting both one-sided changes reveals the cusp. Likewise, a stable local slope does not rule out a quadratic finite-range response, as finite-perturbation reference sensitivity shows.

A meaningful test changes one assigned shared reference consistently in every dependent calculation, holds the other inputs and computational settings fixed, and records the steps, units, slopes and one-sided changes. Shared reference uncertainty and comparison-process variability are distinct physical contributions in the NIST uncertainty budget.^[ref] If evaluation includes simulation noise, separate its repeatability from the response to the changed reference.