20-EFT.WP.Metrology.TimeBase v1.0 | Appendix E — Error & Uncertainty Propagation (Time-Base Edition)

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Appendix E — Error & Uncertainty Propagation (Time-Base Edition)

One-line objective: Define the error-source decomposition and uncertainty propagation from acquisition through link and servo to publication, and standardize external reporting as U = k * u_c with audit-ready persistence.

I. Purpose & Scope

• Applies to
  Uncertainty evaluation and publication for TE/TIE/MTIE, offset/skew/J, adev/mdev/tdev, arrival-time T_arr and the dual-form delta_form.
• Publication convention
  Compute on tau_mono, publish on ts; report all uncertainties as U = k * u_c, default k = 2 (~95% coverage).

II. Terms & Symbols

• TE(t) = ts(t) - ts_ref(t)；y(t) = d( TE(t) ) / dt。
• J(t) (high-frequency jitter), W(t) (low-frequency wander), TE = W + J。
• TIE(t, tau) = TE(t) - TE(t - tau)；MTIE(tau) = max_t | TIE(t, tau) |。
• Allan family: adev(tau), mdev(tau), tdev(tau) = ( tau / sqrt(3) ) * mdev(tau)。
• Combined standard uncertainty u_c; expanded uncertainty U = k * u_c。
• Effective sample size n_eff = N / c_corr (c_corr ≥ 1 is the correlation correction).
• Dual-form arrival:
  form_const = ( 1 / c_ref ) * ( ∫ n_eff d ell )，
  form_general = ( ∫ ( n_eff / c_ref ) d ell )，
  delta_form = | form_const - form_general |，with path gamma(ell) and measure d ell。

III. Axioms P50E- **

• P50E-1 (Explicit source decomposition):
  u^2(TE) = u^2(ref) + u^2(link) + u^2(node) + 2 * cross_terms; evaluate cross-terms when identifiable, otherwise apply defensible upper bounds.
• P50E-2 (Correlation correction): Replace N by n_eff wherever sample statistics are used; document how c_corr was estimated.
• P50E-3 (Frequency-domain consistency): For servo-shaped variances, use PSD integration: var_out = ( ∫ |H(f)|^2 S_in(f) df ).
• P50E-4 (Arrival mandatory): Any T_arr metric must compute both forms and publish delta_form and U(delta_form).
• P50E-5 (Dimensional integrity): Run check_dim(expr) before any propagation; declare unit(x), dim(x).

IV. Minimal Equations S50E- **

• S50E-1 (First-order Δ-method)
  For z = f(x), with sensitivity g = grad_x f,
  u^2(z) = g^T * Cov(x) * g。
• S50E-2 (Mean & variance of samples)
  u(mean(TE)) = s_TE / sqrt( n_eff )，u(s_TE^2) ≈ s_TE^2 * sqrt( 2 / ( n_eff - 1 ) )。
• S50E-3 (Quantile q_p uncertainty, large-sample)
  u(q_p) ≈ sqrt( p * ( 1 - p ) ) / ( sqrt(n_eff) * f_TE( q_p ) )，with f_TE from spacing or kernel density.
• S50E-4 (RMS jitter)
  rms(J) = sqrt( mean( J^2 ) )，u( rms(J) ) ≈ rms(J) / sqrt( 2 * n_eff ) (inflate by sqrt(c_corr) when correlated)。
• S50E-5 (MTIE intervals)
  MTIE is extreme-value–biased; prefer block bootstrap: U( MTIE ) = k * sd_boot( MTIE ), block length B ≈ tau, with B_rep ≥ 1000.
• S50E-6 (Allan-family effective DOF)
  nu_eff(tau) = ( T_total / tau ) / c_nu；
  u( adev(tau) ) = adev(tau) / sqrt( 2 * nu_eff(tau) )；
  u( mdev(tau) ) = mdev(tau) / sqrt( 2 * nu_eff(tau) )；
  u( tdev(tau) ) = ( tau / sqrt(3) ) * u( mdev(tau) )。
• S50E-7 (Variance transfer through servo/link)
  var_out = ( ∫ |H(f)|^2 S_in(f) df )；u( var_out ) from numerical integration and PSD-estimation uncertainty.
• S50E-8 (Uncertainty of the dual-form difference)
  Let a = form_const, b = form_general, delta_form = | a - b |;
  u^2( delta_form ) ≈ u^2(a) + u^2(b) - 2 * cov(a,b) (obtain u^2(a), u^2(b) via S50E-1; cov from shared variables such as n_eff, c_ref).

V. Propagation Atlas & Steps M50E-1

• Source identification & modeling — enumerate ref/link/node with noise models (type spectra, tempco, quantization).
• Time-domain preprocessing — align_timebase → TE(t); band-split J/W with recorded f_c_hp/f_c_lp.
• Statistical estimation — compute offset.*, skew.*, jitter.rms, TIE/MTIE, adev/mdev/tdev.
• Uncertainty propagation — apply S50E-1..S50E-7 to mean/quantile/power-type gauges; use block bootstrap for extremes.
• Dual-form arrival — compute a/b in parallel; evaluate U(delta_form) via S50E-8 and assert with tol_Tarr.
• Synthesis & publication — combine to u_c → U = k * u_c; write to manifest.time.uncertainty.* and sign.

VI. Uncertainty Gauges for Typical Metrics

• offset.p99 — estimate q_0.99 and f_TE(q_0.99); U = k * u(q_0.99) using S50E-3; obtain n_eff via autocorrelation rho(1..L): c_corr = 1 + 2 * ∑_{h=1}^{L} rho(h).
• skew.mean_ppm — from mean of y(t) via S50E-2; include unit-conversion factors in the sensitivity vector g.
• jitter.rms — u( rms(J) ) from S50E-4; if band-limited to [f1,f2], include PSD-estimation bias.
• MTIE(tau) — use S50E-5; B ≈ tau, sliding extraction; publish U and B_rep.
• tdev(tau) — derive mdev & u(mdev) then map via S50E-6; publish nu_eff.
• delta_form — combine discretization u_disc, medium model u(n_eff), and constant u(c_ref); compute cov(a,b) from shared terms.

VII. Synthesis & Budgeting

• Variance composition
  u_c^2 = ∑ ( c_i * u_i )^2 + 2 * ∑_{i<j} c_i c_j cov(i,j) where c_i are sensitivities/gains.
• Budget header (publish to manifest)
  u(ref), u(link), u(node), u(model), u_disc, u_spec_est, plus rho(h) and n_eff.
• External reporting
  Publish only U and the contribution percentages w_i = ( c_i^2 u_i^2 ) / u_c^2.

VIII. Sequential & Streaming Updates

• Sliding-window estimates — exponential decay or fixed windows; sequential n_eff update: n_eff(t) = λ * n_eff(t-1) + (1-λ) * n_eff_window.
• Incremental bootstrap — maintain a fixed-capacity block buffer; perform online block bootstrap to refresh U(MTIE) and high-quantile U(offset.p99).
• Servo noise shaping — when H(f; θ) changes, recompute the integral in S50E-7 and publish ΔU.

IX. Dual-Form Arrival & Uncertainty

• Mandatory fields — form_const, form_general, delta_form, U(delta_form), tol_Tarr, gamma(ell), L_gamma.
• Contract gates — assert( delta_form ≤ tol_Tarr ) and assert( U(delta_form) ≤ tol_Tarr_u ); on failure, enter rollback and re-estimation.

X. Manifest Mapping & Audit

1. Manifest keys
   • manifest.time.uncertainty.offset.mean.U
   • manifest.time.uncertainty.offset.p99.U
   • manifest.time.uncertainty.jitter.rms.U
   • manifest.time.uncertainty.mtie.tau_<v>.U
   • manifest.time.uncertainty.tdev.tau_<v>.U
   • manifest.time.uncertainty.delta_form.U
   • plus n_eff, nu_eff, c_corr, B_rep, f_c_hp/f_c_lp.
2. Audit trail
   Persist hash_sha256(blob), signature, TraceID; record model version and servo parameters θ_servo.

XI. Implementation Bindings I50E- (Summary)*

• propagate_uncertainty_timebase(metrics, sources, cov) -> U_bundle
• estimate_neff(series, method) -> n_eff, c_corr (method ∈ {ACF, block, spectral})
• bootstrap_mtie(series, tau, B, B_rep) -> {est, U}
• allan_uncertainty(series, tau_set) -> {adev, mdev, tdev, U, nu_eff}
• delta_form_uncertainty(path, neff_model, c_ref) -> {delta_form, U}
• Invariants: unit/dim consistent; n_eff ≥ 1; nu_eff ≥ 1; delta_form ≤ tol_Tarr.

XII. Cross-References

• Metrics & drift: Appendix D.
• Sync servos & protocols: Chapter 5.
• Arrival paths & links: Chapter 9.
• Statistical propagation baseline: Methods.CrossStats v1.0 Appendix E.
• Manifest templates: Appendix C.

Summary