22-EFT.WP.Metrology.Instrument v1.0 | Appendix E — Error & Uncertainty Propagation (GUM / Monte Carlo)

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Appendix E — Error & Uncertainty Propagation (GUM / Monte Carlo)

One-Sentence Objective
Using GUM with a Monte Carlo complement as the baseline, define a unified convention for Type A/B decomposition, correlation-aware and nonlinearity-aware propagation, coverage, and conformity decisions for instrument measurement models—and align all outputs with manifest, contract, and dashboard fields.

I. Scope & Targets

• Covered objects: uncertainty evaluation and propagation for noise, linearity, bandwidth, timebase-related quantities, and environmental corrections appearing throughout this volume.
• Inputs: measurement model y = f(x_1, ..., x_n); input estimates hat{x}; standard uncertainties u(x_i); covariances cov(x_i,x_j); distribution assignments D_i; degrees of freedom nu_i.
• Outputs: u_c(y), U = k * u_c(y), coverage interval [y_L, y_U] or posterior quantiles {q_2.5%, q_97.5%}, and the manifest.instrument.uncert.* fields.
• Cross-references: timebase and arrival-time uncertainty in Metrology.TimeBase v1.0; statistical coverage and power in Methods.CrossStats v1.0.

II. Terms & Variables

• Measurement model & sensitivities: y = f(x), c_i = ( ∂f / ∂x_i ) |_{hat{x}}, J = ∂f/∂x (Jacobian).
• Type A/B: u_A(x) (from sample variance), u_B(x) (from certificates / distribution assignments), u(x) = sqrt( u_A^2 + u_B^2 ).
• Covariance & correlation: V_x = [ cov(x_i,x_j) ], R = [ rho_ij ], cov(x_i,x_j) = rho_ij * u(x_i) * u(x_j).
• Synthesis & coverage: u_c^2(y) = J V_x J^T, U = k * u_c, with k set from nu_eff and target coverage probability.
• Monte Carlo: trials M, y^{(m)} = f( x^{(m)} ), quantiles q_p, quantile standard error SE(q_p).

III. Postulates P70E-*

• P70E-1 (Explicit model): all uncertainty evaluations must be based on an explicit measurement model y = f(x); inputs must pass check_dim(expr) for units and dimensions.
• P70E-2 (Explicit correlation): declare V_x or R; hidden “independence” assumptions are forbidden; correlations from certificates must be populated.
• P70E-3 (Unified timebase): evaluate statistical windows on tau_mono, publish on ts; for T_arr, report uncertainties for both formulations and their delta_form in parallel.
• P70E-4 (Unified coverage convention): use GUM-LPU in linear regimes; use Monte Carlo for significant nonlinearity / bounded / discrete inputs; when the two disagree beyond threshold, Monte Carlo prevails.
• P70E-5 (Co-published metadata): any key metric must co-publish {u_c, U, k, nu_eff} or {q_2.5%, q_97.5%} and method ∈ {GUM, MC}.

IV. Minimal Equations S70E-*

1. S70E-1 (Combined standard uncertainty — scalar):
   u_c^2(y) = ∑_{i=1}^n ∑_{j=1}^n c_i * c_j * cov(x_i, x_j)。
2. S70E-2 (Combined standard uncertainty — vector):
   V_y = J * V_x * J^T。
3. S70E-3 (Welch–Satterthwaite):
   nu_eff = ( u_c^4 ) / ( ∑ ( c_i^4 * u^4(x_i) / nu_i ) )（approximate, no-correlation case）。
4. S70E-4 (Coverage factor):
   U = k(nu_eff, CL) * u_c，where CL is the coverage probability (e.g., 95%).
5. S70E-5 (Typical distribution assignments):
   • Rectangular: u = a / sqrt(3) (half-width a)；Triangular: u = a / sqrt(6)；Normal: use certificate-provided u.
   • Quantization resolution: u_res = q / sqrt(12) (step q).
6. S70E-6 (Environmental-correction propagation):
   y_corr = y_raw + corr_env(x; RefCond)，u_c^2(y_corr) = u_c^2(y_raw) + u_c^2( corr_env ) + 2 * cov( y_raw, corr_env )。
7. S70E-7 (Timebase contribution):
   u_T(y) ≈ | ∂f/∂t | * u(t)；u(t) is synthesized from offset/skew/J。
8. S70E-8 (Two-form arrival-time difference):
   delta_form = | ( 1 / c_ref ) * ( ∫ n_eff d ell ) - ( ∫ ( n_eff / c_ref ) d ell ) |，its uncertainty u(delta_form) acts as an additional publication gate term。
9. S70E-9 (Monte Carlo coverage interval):
   y^{(m)} = f( x^{(m)} )，publish [ q_{(1-CL)/2}, q_{(1+CL)/2} ] and
   SE(q_p) ≈ sqrt( p(1-p) / ( M * f_Y(q_p)^2 ) )（using a kernel-density approximation for f_Y）。

V. Workflow M70E-*: Ready → Evaluate → Propagate → Cover → Persist

1. Model & inputs prepared
   • Specify y = f(x) with units/dimensions; record RefCond and timebase fields offset/skew/J, T_arr.form1/2, delta_form.
   • Aggregate Type A: u_A(x) from sample variance; Type B: u_B(x) from certificates/distribution assignments.
2. Correlation & covariance
   • Build V_x; if only rho_ij and u(x) are available, set V_x[i,j] = rho_ij * u(x_i) * u(x_j).
   • For inputs from the same standard, fill correlation coefficients from the certificate.
3. GUM-LPU (linear regime)
   • Compute c_i or J; report u_c, nu_eff, U = k * u_c.
   • Enforce check_dim( y - f(x) ) = "pass".
4. Nonlinear / bounded / discrete (Monte Carlo)
   • Assign D_i (e.g., Normal/Rectangular/Triangular/Lognormal); draw M samples; compute y^{(m)}.
   • Output q_2.5%, q_50%, q_97.5%, u_c = std( y^{(m)} ), and method="MC".
   • Use adaptive stopping: stop when SE(q_97.5%)/u_c ≤ tol_q and |u_c^{(M)} - u_c^{(M/2)}| / u_c^{(M)} ≤ tol_uc.
5. Merge timebase / arrival-time contributions
   • Add u_T(y) and u(delta_form) into u_c(y) or include them via joint MC sampling.
   • Record the coverage interval for the two-form difference and whether it crosses tol_Tarr.
6. Coverage & conformity
   • Choose CL (e.g., 95%); use k(nu_eff, CL) for GUM or quantiles for MC.
   • Apply guardband strategy for conformity (criteria and TUR in Chapter 11).
7. Persistence & signature
   Write manifest.instrument.uncert: {model, inputs, V_x_hash, method, u_c, U, k|CL, nu_eff|M, RefCond, timing, Tarr, delta_form, signature}.

VI. Contracts & Assertions (Mapped to C70 Suite)

• assert C70-UE-1: check_dim(expr) = "pass"; unit(x) consistent.
• assert C70-UE-2: if method="GUM", then nu_eff ≥ nu_min and k matches selected CL.
• assert C70-UE-3: if method="MC", then M ≥ M_min and SE(q_97.5%) ≤ se_q_max.
• assert C70-UE-4: delta_form + z * u(delta_form) ≤ tol_Tarr (with z set by coverage probability).
• assert C70-UE-5: if |U_GUM - U_MC| / U_MC > tol_diff, force adoption of the MC result.

VII. Implementation Bindings I70E-* (Prototypes)

• build_measurement_model(spec) -> f, inputs
• assign_distributions(cert, policy) -> {D_i, u(x_i), nu_i, R}
• propagate_gum(f, x_hat, V_x) -> {u_c, nu_eff, U, k}
• propagate_mc(f, D, R, M, stop_rule) -> {q_low, q_med, q_up, u_c, M_eff}
• merge_timebase_uncert(tb_manifest, f) -> u_T, details
• evaluate_uncert_contracts(results, rules) -> report
• emit_uncert_manifest(results) -> manifest.instrument.uncert
  Invariants: reproducible(seed); V_x positive semi-definite; nu_eff > 0; Delta_t and RefCond persisted; delta_form ≤ tol_Tarr.

VIII. Cross-References

• Timebase uncertainty, two-form T_arr, and delta_form: Metrology.TimeBase v1.0, Appendix E.
• Statistical coverage, power, and drift: Methods.CrossStats v1.0, Appendices E & D.
• Conformity and guardband decisions: Chapter 11 of this volume and Appendix B (C70 Suite).

IX. Quality Metrics & Risk Control

• SLIs: uncert_eval.latency_ms_p99, uncert_eval.error_rate, M_eff, nu_eff_dist.
• SLOs: SE(q_97.5%) ≤ se_q_max, violation rate for tol_diff ≤ p_max.
• Fallbacks: if MC fails to converge or resources are constrained, fall back to GUM with risk="elevated", or extend the window / reduce model dimensionality and re-evaluate.

Summary
This appendix unifies the modeling, correlation handling, coverage, and publication workflow for instrument uncertainty evaluation via P70E-* / S70E-* / M70E-* / I70E-*. It provides MC and coupled-propagation conventions for nonlinearity, correlation, and arrival-time contributions, and closes the loop with manifest, contract, and panel fields for end-to-end auditability.