Source: https://jmrplens.github.io/phonometry/guides/gum-uncertainty/ # Measurement uncertainty (GUM and Monte Carlo) Every measurement result is incomplete without a statement of its uncertainty. The *Guide to the Expression of Uncertainty in Measurement* gives two ways to propagate the uncertainties of the inputs of a measurement model $y = f(x_1, \dots, x_N)$ to the output: the **law of propagation of uncertainty** (**ISO/IEC Guide 98-3:2008**, the GUM, clause 5) and the **Monte Carlo method** (**ISO/IEC Guide 98-3-1:2008**, Supplement 1, clause 7). The first combines standard uncertainties analytically through the sensitivity coefficients; the second propagates the whole probability distributions numerically. They agree for linear models and diverge, informatively, when the model is non-linear or the inputs are far from Gaussian. ## 1. The law of propagation of uncertainty (GUM clause 5) For a model $y = f(x_1, \dots, x_N)$ with uncorrelated inputs, the combined standard uncertainty is the quadrature sum of the input contributions $u_i(y) = |c_i|\,u(x_i)$ weighted by the sensitivity coefficients $c_i = \partial f / \partial x_i$: $$ u_c^2(y) = \sum_{i=1}^{N} \left(\frac{\partial f}{\partial x_i}\right)^2 u^2(x_i). $$ `combine_uncertainty` evaluates the sensitivities by central differences, so any callable model works, with no hand-derived partials. Input quantities are described by a `Quantity` (best estimate, standard uncertainty, PDF, degrees of freedom). Type B evaluations from a half-width $a$ are built by `rectangular` ($a/\sqrt{3}$), `triangular` ($a/\sqrt{6}$) and `u_shaped` ($a/\sqrt{2}$) (GUM clause 4.3). ```python from phonometry import metrology # A-weighted level: a reading plus zero-mean calibration, instrument and # positional corrections. The model is their sum. quantities = [ metrology.Quantity(74.0, 0.0, name="Reading"), metrology.rectangular(0.0, 0.20, name="Calibration"), metrology.rectangular(0.0, 0.30, name="Instrument"), metrology.Quantity(0.0, 0.35, dof=9, name="Position (Type A)"), ] result = metrology.combine_uncertainty(lambda a, b, c, d: a + b + c + d, quantities) print(round(result.value, 2)) # 74.0 print(round(result.combined_uncertainty, 3)) # 0.407 dB print(result.contributions.round(3)) # [0. 0.115 0.173 0.35 ] print(round(result.effective_dof, 1)) # 16.5 k, U = result.expanded(0.95) print(round(k, 2), round(U, 2)) # 2.11 0.86 -> Y = 74.0 ± 0.9 dB ``` The **expanded uncertainty** $U = k\,u_c$ scales the combined uncertainty by a coverage factor $k = t_p(\nu_\text{eff})$ from the $t$-distribution at the **effective degrees of freedom** given by the Welch–Satterthwaite formula (Annex G.4). Here the single Type A input (9 degrees of freedom) pulls $\nu_\text{eff}$ down to about 16, so $k = 2.11$ rather than the large-sample 1.96. Correlated inputs are handled by passing a correlation matrix; a fully correlated sum then adds linearly instead of in quadrature. Correlation is the classic silent error in a budget: two corrections traceable to the *same* calibrator, or two channels sharing one instrument, do not average away the way independent terms would, and combining them in quadrature as if they were uncorrelated typically understates $u_c$ (with sensitivities of opposite sign the bias can point the other way). Because the GUM defines Welch–Satterthwaite for *independent* inputs only, a correlated budget with finite input degrees of freedom carries no effective degrees of freedom at all: `effective_dof` is NaN, a warning is issued, and `expanded()` requires an explicit `coverage_factor_override` (e.g. $k = 2$) rather than inventing one. Only when every input is Type B with infinite degrees of freedom does a correlated budget keep $\nu_\text{eff} = \infty$ and use the normal-distribution coverage factor. This chain reproduces the GUM's own worked examples end to end: the Annex H.1 end-gauge budget ($u_c = 31.7$ nm, $U_{99} = 92$ nm against the printed 32/93) and the Annex H.2 correlated resistance measurement ($u_c(R) = 0.071\ \Omega$, $u_c(X) = 0.295\ \Omega$, $u_c(Z) = 0.236\ \Omega$ from Table H.3). ## 2. The Monte Carlo method (Supplement 1) When the model is non-linear or the inputs are markedly non-Gaussian, the GUM Gaussian assumption for the output can be inaccurate. `monte_carlo` instead draws samples of each input from its PDF, evaluates the model over all trials and reports the mean, the standard deviation and the **probabilistically symmetric coverage interval** (equal probability in each tail, clause 7.7). The inputs are sampled independently; the Supplement's multivariate-Gaussian path for non-independent quantities (6.4.8) is not implemented, so correlated budgets belong to `combine_uncertainty`. The number of trials is fixed (no adaptive 7.9 procedure, at least 2 trials) and the interval is the symmetric one, not the 5.3.4 shortest interval. ```python from phonometry import metrology quantities = [ metrology.Quantity(74.0, 0.0, name="Reading"), metrology.rectangular(0.0, 0.20, name="Calibration"), metrology.rectangular(0.0, 0.30, name="Instrument"), metrology.Quantity(0.0, 0.35, dof=9, name="Position (Type A)"), ] mc = metrology.monte_carlo(lambda a, b, c, d: a + b + c + d, quantities, trials=1_000_000, coverage=0.95, seed=1) print(round(mc.value, 2)) # 74.0 print(round(mc.standard_uncertainty, 3)) # 0.407 dB (matches uc above) print([round(x, 2) for x in mc.interval]) # [73.2, 74.8] ``` For this near-linear model the Monte Carlo standard uncertainty reproduces the GUM $u_c$ to three digits and the 95 % interval matches $Y \pm U$. The two methods are validated against the Guides' own worked examples: the additive model of four unit inputs gives $u_c = 2.0$ (Supplement 1 clause 9.2), and four rectangular inputs give a Monte Carlo interval of $[-3.88,\, 3.88]$ (Supplement 1 clause 9.2.3). When does the Monte Carlo method earn its extra cost? Whenever either of the GUM's two simplifications fails: the model is replaced by its first-order expansion, and the output distribution by a Gaussian (or a $t$). Both hold well for the additive level model above, which is why the two methods agree to three digits. They stop holding when the model is strongly non-linear over the span of the input uncertainties (energy-to-level conversions with wide inputs, products and quotients with large relative uncertainties), when a single non-Gaussian input dominates the budget (one large rectangular term makes the output nearly rectangular, and a Gaussian $\pm\,2u_c$ interval overcovers it), or when the output sits near a physical bound (an absorption coefficient near 0 or 1, a level correction that cannot cross zero), where the true coverage interval is asymmetric and no $Y \pm U$ statement can represent it. In those regimes the Monte Carlo interval is the reference: Supplement 1 (clause 8) treats the GUM framework as validated precisely when it agrees with the Monte Carlo result, and as superseded by it when it does not.

Show the code for this figure

```python import matplotlib.pyplot as plt import numpy as np from phonometry import metrology quantities = [ metrology.Quantity(74.0, 0.0, name="Reading"), metrology.rectangular(0.0, 0.20, name="Calibration"), metrology.rectangular(0.0, 0.30, name="Instrument"), metrology.Quantity(0.0, 0.35, dof=9, name="Position (Type A)"), ] model = lambda a, b, c, d: a + b + c + d result = metrology.combine_uncertainty(model, quantities) mc = metrology.monte_carlo(model, quantities, trials=1_000_000, coverage=0.95, seed=1, keep_samples=True) k, U = result.expanded(0.95) # One line per panel — the budget bars, and the Monte Carlo histogram with # its coverage interval (the committed figure also overlays the GUM Gaussian): result.plot() mc.plot() plt.show() # By hand, both panels — budget bars and the Monte Carlo output distribution: fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12.5, 5.4)) ax1.barh(result.names, result.contributions) ax1.axvline(result.combined_uncertainty, ls="--", label=f"$u_c$ = {result.combined_uncertainty:.3f} dB") ax1.invert_yaxis(); ax1.legend() rng = np.random.default_rng(1) samples = model(np.full(200_000, 74.0), rng.uniform(-0.20, 0.20, 200_000), rng.uniform(-0.30, 0.30, 200_000), rng.normal(0.0, 0.35, 200_000)) ax2.hist(samples, bins=120, density=True, alpha=0.35, label="Monte Carlo") ax2.axvspan(*mc.interval, alpha=0.12, label="95 % coverage interval") ax2.set_title(f"Y = {result.value:.2f} dB, U = {U:.2f} dB (k = {k:.2f})") ax2.legend() plt.show() ```

The `UncertaintyResult` carries the `value`, `combined_uncertainty`, the `sensitivities`, the per-input `contributions` and the `effective_dof`; its `.plot()` draws the budget and `.expanded(coverage)` returns the pair $(k, U)$. The `MonteCarloResult` carries the `value`, `standard_uncertainty`, the coverage `interval` and its `coverage`; with `keep_samples=True` it also retains the output `samples`, and its `.plot()` draws the output histogram with the coverage interval marked (the right panel above). The building-acoustics uncertainty of ISO 12999-1, which combines reproducibility terms for a single-number rating, is a separate, domain-specific budget. A budget is only as honest as the record behind it: every averaged input assumes stationarity, and [Data qualification](https://jmrplens.github.io/phonometry/guides/data-qualification/) provides the Bendat & Piersol tests that check it before the propagation starts. ## Where the library consumes this machinery The domain pages carry their own, standard-prescribed uncertainty clauses, and each is an instance of the clause-5 sum this page implements in general form; use this page whenever your measurement model is not one of those standardized cases: - **Calibration.** The calibrator's class tolerance and the pre/post drift bound of [Calibration](https://jmrplens.github.io/phonometry/reference/api/levels/calibration/) are textbook Type B inputs: propagate them (`Quantity(..., "rectangular")`) through the level model rather than quoting the tolerance alone. - **Environmental levels.** The ISO 1996-2 combined uncertainty of the [Levels](https://jmrplens.github.io/phonometry/reference/api/levels/levels/) page is Formula (2) of that standard: the same quadrature sum with the standard's own sensitivity coefficients. - **Building acoustics.** The per-band and single-number uncertainties of [field insulation measurements](https://jmrplens.github.io/phonometry/guides/insulation-field/) apply the tabulated reproducibility terms of ISO 12999-1, a standardized budget for one specific model. - **Absorption.** The [materials page](https://jmrplens.github.io/phonometry/guides/materials/) carries the ISO 12999-2 absorption uncertainty, the same construction for the reverberation-room method. - **Occupational exposure.** The ISO 9612 uncertainty of [occupational measurements](https://jmrplens.github.io/phonometry/reference/api/hearing/occupational-exposure/) budgets sampling, instrument and position contributions in exactly this way. ## References - Joint Committee for Guides in Metrology. (2008). *Evaluation of measurement data — Guide to the expression of uncertainty in measurement* (JCGM 100:2008, the GUM). BIPM. [doi:10.59161/JCGM100-2008E](https://doi.org/10.59161/JCGM100-2008E), [free PDF](https://www.bipm.org/documents/20126/2071204/JCGM_100_2008_E.pdf). The law of propagation, the Type B evaluations and the Annex H worked examples that section 1 implements and reproduces. - Joint Committee for Guides in Metrology. (2008). *Evaluation of measurement data — Supplement 1 to the "Guide to the expression of uncertainty in measurement" — Propagation of distributions using a Monte Carlo method* (JCGM 101:2008). BIPM. [doi:10.59161/JCGM101-2008](https://doi.org/10.59161/JCGM101-2008), [free PDF](https://www.bipm.org/documents/20126/2071204/JCGM_101_2008_E.pdf). The Monte Carlo propagation of section 2, its probabilistically symmetric coverage interval and the clause 8 validation of the GUM framework against the Monte Carlo result. - International Organization for Standardization. (2020). *Acoustics — Determination and application of measurement uncertainties in building acoustics — Part 1: Sound insulation* (ISO 12999-1:2020). [iso.org catalogue](https://www.iso.org/standard/73930.html). The domain-specific reproducibility budget for building-acoustics single-number ratings, mentioned above as a separate companion to the general GUM machinery. ## Standards ISO/IEC Guide 98-3:2008, *Uncertainty of measurement — Part 3: Guide to the expression of uncertainty in measurement (GUM:1995)*: the law of propagation of uncertainty (clause 5), Type B evaluation (clause 4.3), the expanded uncertainty and coverage factor (clause 6, Annex G) and the Welch–Satterthwaite effective degrees of freedom (Annex G.4). ISO/IEC Guide 98-3-1:2008, *Supplement 1 — Propagation of distributions using a Monte Carlo method*: the numerical propagation and the probabilistically symmetric coverage interval (clause 7).