23-EFT.WP.Metrology.PathCorrection v1.0 | Appendix D — Metrics and Drift Measures (Path-Specific)

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Appendix D — Metrics and Drift Measures (Path-Specific)

One-Sentence Goal
Define the observability metrics and drift measures for the entire path-correction chain—together with units/dimensions, window lengths, suggested thresholds, and online detection–fallback coupling—so results are monitorable, triggerable, and auditable end to end.

I. Scope and Objects

• Coverage spans geometry → media → dual forms → composition → uncertainty → runtime observability and health measures.
• Outputs: metric definitions, computation windows W, suggested thresholds/actions, and mappings into manifest.path(.rt).
• Constraints: every metric declares unit(x) and dim(x) and passes check_dim; strict adherence to parallel dual-form recording.

II. Terms and Variables (common)

• delta_form(t): dual-form difference (seconds), dim="[T]".
• map_residual(t): mapping/model residual (medium or multipath components, rendered in seconds).
• TEC_bias(t): TEC bias (unit="TECU"), or its delay equivalent
  TEC_bias_T(t) = ( 40.3 / c_ref ) * ( TEC_bias / f^2 ) (seconds).
• temp_coeff(t): fiber thermal coefficient ∂T_env/∂T (seconds/K), and its drift temp_coeff_drift.
• L_rt(t): end-to-end runtime latency (seconds); hit_total: cache hit rate.
• u_c(t), U(t): combined standard uncertainty and coverage interval (seconds).
• age(src): source freshness (seconds); cov(gamma): path coverage (0–1).

III. Axioms P80d-* (non-negotiable)

• P80d-1 Dimensional consistency: for any quantity under comparison or thresholding, check_dim( y - f(x) ) = 0.
• P80d-2 Online dual-form metering: record delta_form(t) and aggregated statistics every publication cycle.
• P80d-3 Drift anchored to RefCond: evaluate all drifts normalized to RefCond (see Chapter 11).
• P80d-4 Replayable metrics: persist windows, parameters, and hashes needed for computation so runs can be reproduced by TraceID.

IV. Minimal Equations S80d-* (metric definitions)

• Dual-form consistency
  delta_form_p99(W) = quantile( { delta_form(t) | t ∈ W }, 0.99 )，unit="s"，dim="[T]"。
  Constraint: delta_form_p99(W) ≤ tol_Tarr (see Appendix B C80-1011).
• Mapping/model residuals (troposphere / ionosphere / multipath)
  map_residual(t) = T_obs(t) - T_model(t; RefCond)（seconds）。
  Aggregation: map_residual_p95(W) = quantile( { |map_residual(t)| }, 0.95 )。
• Ionospheric bias
  TEC_bias(T) = TEC_est - TEC_ref（unit="TECU"）；
  TEC_bias_T = ( 40.3 / c_ref ) * ( TEC_bias / f^2 )（seconds），with f the operating frequency.
• Fiber thermal coefficient and drift
  temp_coeff = ( d T_env / d T )（seconds/K），via sliding regression:
  temp_coeff(t) = argmin_k ( ∑_{τ∈W} w_τ * ( T_env(τ) - ( k * ΔT(τ) + b ) )^2 )。
  Drift: temp_coeff_drift = temp_coeff(W_long) - temp_coeff(W_short)。
• Uncertainty & coverage monitoring
  U_overshoot_rate(W) = |{ t ∈ W : |residual(t)| > U(t) }| / |W|。
  u_c_p95(W) = quantile( { u_c(t) }, 0.95 )。
• Runtime & freshness
  L_rt_p99(W) = quantile( { L_rt(t) }, 0.99 )；
  age_max(W) = max_t age(src_t)；
  hit_total(W) = hits / requests。

V. Measurement Workflow M80-Drift (shared online/offline route)

1. Ready: fix RefCond; choose window parameters { W_short, W_long } (typ. W_short = 5–15 min, W_long = 24 h); select robust stats (Huber / quantiles).
2. Model/Estimate per epoch:
   • Compute delta_form(t), map_residual(t), u_c(t), U(t), L_rt(t), age(src).
   • Maintain aggregates with online quantiles (P² or t-digest) and EWMA smoothing.
3. Checks: run drift detectors (next section), emit drift.flags/tags and suggested actions.
4. Persist:
   • Batch → manifest.path.metrics.* (offline).
   • Stream → manifest.path.rt as rt.metrics.* and dashboards.
   • Record windows, algo.ver, digest.hash.

VI. Drift Detection & Actions (online gating)

1. Statistical detectors
   • EWMA: z_t = λ x_t + (1-λ) z_{t-1}; trigger when |z_t - μ0| > L * σ0.
   • CUSUM (two-sided): C_t^+ = max(0, C_{t-1}^+ + x_t - (μ0 + k)); if C_t^+ > h or C_t^- < -h then trigger.
   • Page–Hinkley: sensitive to slow drift; recommended for temp_coeff.
   • Mann–Kendall: distribution-free trend test for long window W_long.
2. Recommended mapping from observation → detector
   • delta_form_p99 → Shewhart/EWMA (fast shifts).
   • map_residual_p95 → CUSUM (bias) + KS test (distribution drift).
   • TEC_bias_T → dual-window differencing + CUSUM.
   • temp_coeff_drift → Page–Hinkley.
   • U_overshoot_rate → binomial rate gate + Wald interval.
3. Action matrix (summary)
   • Mild: densify integration steps / inflate u_num (degrade/S2).
   • Medium: switch to model or climatology (fallback/S3, set alpha_r).
   • Severe: block publication and hold last good value (block/S4).
   • Concrete thresholds & actions live in the policy card (Appendix B).

VII. Thresholds & Windows (suggested, tiered by scenario)

1. Free space / GNSS
   • tol_Tarr = 5e-12 s; delta_form_p99 ≤ 0.5 * tol_Tarr;
   • map_residual_p95 ≤ 2e-9 s (tropo + iono combined);
   • U_overshoot_rate ≤ 2%, age_max ≤ 1800 s.
2. Fiber / White Rabbit
   • tol_Tarr = 1e-12 s; temp_coeff_drift(W_long,W_short) ≤ 20% * |temp_coeff(W_long)|;
   • Δasym_p95 ≤ 5e-10 s; u_c_p95 ≤ 3e-10 s.
3. Metro NLOS microwave
   • map_residual_p95 ≤ 2e-9 s (post-suppression); specular_k ≤ 3;
   • delta_form_p99 ≤ 1e-11 s; U_overshoot_rate ≤ 5%.

Note: tune thresholds via offline replays, then configure in policy_card.thresholds.* with staged rollout.

VIII. Dashboard & Manifest Mapping (publication fields)

• Required dashboard time series
  metrics.delta_form_p99, metrics.map_residual_p95, metrics.TEC_bias_T,
  metrics.temp_coeff, metrics.temp_coeff_drift, metrics.u_c_p95,
  metrics.U_overshoot_rate, metrics.L_rt_p99, metrics.age_max, metrics.hit_total.
• Suggested manifest.path extensions
  metrics:{ win:{short,long}, delta_form_p99, map_residual_p95, TEC_bias_T, temp_coeff, temp_coeff_drift, u_c_p95, U_overshoot_rate }.
• Runtime (manifest.path.rt)
  rt.metrics:{ L_rt_p99, age_max, hit_total, fallback_rate }.

All numeric values must include unit/dim metadata or be defined globally in the schema annotations.

IX. Quality & Fallback Coupling (with C80 / runtime)

• Execute the policy-card action (degrade / fallback / block).
• Inflate coverage U ← γ * U (for model downgrade or freshness mismatch).
• Tag tags:["degraded", <reason>] and chart the event on dashboards.
• Mark all samples within the affected window as not for training (to avoid “drift poisoning”).

Summary
This appendix defines the core health metrics and drift measures for path correction, provides formulas, dimensions, window lengths, and threshold suggestions, and binds online detectors to policy-card actions.
Minimal go-live set: delta_form_p99, map_residual_p95, TEC_bias_T, temp_coeff_drift, u_c_p95, U_overshoot_rate, L_rt_p99, age_max, hit_total.
Combined with Appendix B (contracts) and Chapter 14 (streaming runtime), it enables a closed loop of monitoring → decision → degrade/fallback → audit, assuring publication quality and traceability.