49-Error Budget Card Template v1.0 | Chapter 5 — Correlation Structure & Covariance Modeling

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Chapter 5 — Correlation Structure & Covariance Modeling

I. Purpose & Scope

• Define correlation structures and covariance modeling in path/time/frequency domains, covering kernel selection, parameter estimation, discrete implementations, and numerical stability to support Delta/MC/bootstrap propagation.
• When path quantities (arrival time/phase) are involved, explicitly show gamma(ell) and the measure d ell in text, and record delta_form ∈ {general, factored} in data/metadata; publication requires p_dim = 1.0.

II. Prerequisites & Inputs

• Path consistency: len(gamma_ell) = len(d_ell) = len(n_eff) ≥ 2; Δell satisfies sampling constraints.
• Domain choice: choose correlation domain(s) by task: path ℓ, time t, frequency ν/λ; multi-domain coupling allowed.
• (Local) stationarity: prefer (weak) stationarity within local windows; for nonstationarity use segmentation or kernel tapering.
• Minimal parameter set: variance scale σ^2, correlation length L_c, smoothness/order (e.g., Matérn ν), period P, decay φ, etc.

III. Covariance Kernels (canonical library)

• Exponential (Exp): K(Δx) = σ^2 · exp( −|Δx| / L_c ) (path/time; once differentiable).
• Matérn-ν:
  K(Δx) = σ^2 · (2^{1−ν}/Γ(ν)) ( √(2ν)|Δx|/L_c )^{ν} K_ν( √(2ν)|Δx|/L_c ) (smoothness controlled by ν).
• AR(1) discrete: K[k] = σ^2 · φ^{|k|}, |φ|<1 (time series / within-channel).
• Periodic–exponential composite (PerExp):
  K(Δx) = σ^2 · exp( −2 sin^2(π|Δx|/P)/ℓ_p^2 ) · exp( −|Δx|/L_c ) (periodic + slow trend).
• Frequency tapering: apply high-frequency taper/window to reduce leakage-induced pseudo-correlation.
• Kernel selection principle: prefer physically interpretable and identifiable minimal kernels; compose kernels only as needed and control DoF.

IV. Covariance Structure

1. Path domain: Σ_ij = K(|ℓ_i − ℓ_j|) (Toeplitz approximation); use block and masking for occlusions/broken paths.
2. Time domain: Σ_ij = K(|t_i − t_j|); model unlocked/drift segments independently and zero-correlate with locked blocks.
3. Frequency domain: derive covariance from power spectrum S(ν) and inverse-transform to K(Δt); use quasi-stationary approximation within sub-bands.
4. Multi-channel / cross-domain coupling:
   • Channel correlation: block-diagonal across channels with intra-block Toeplitz; Corr(Δτ_ch^a, Δτ_ch^b)=ρ_ab.
   • Cross-covariance: e.g., Cov(n_eff, α_T·ΔT) = κ·σ_n·σ_T; include cross-partials in the Delta method.

V. Estimation & Diagnostics

• Estimators: MLE, REML, variogram fitting, Yule–Walker (AR).
• Robustification: for heavy tails/outliers use Huber/quantile losses or Winsorize residuals; report robust surrogates and equivalent second-order moments.
• Diagnostics: variogram/correlogram, QQ plot, time–frequency residual spectra, local stationarity tests; provide goodness-of-fit and CIs.
• Nonstationarity: segment-wise estimation of L_c(·) and σ(·), or kernel tapering/local windows; set inter-segment covariance to 0 or small weights.

VI. Numerics & Stability

• Matrix structure: Toeplitz/block-Toeplitz, sparse GMRF, low-rank approximations (Nyström/random SVD).
• Positive-definiteness: add jitter Σ ← Σ + εI (ε in 1e−6 ~ 1e−3 of target variance).
• Complexity control: FFT for convolution/Toeplitz MVMs; block Cholesky; Woodbury for low-rank updates.
• Scale & units: normalize inputs before modeling; restore physical units on output and validate with check_dim.

VII. Interfaces to Propagation

• Delta method: u^2(y) ≈ J Σ Jᵀ; build Σ from kernels/parameters herein (path/time/frequency/channel via block or Kronecker structures).
• MC/bootstrap: sample x ~ (0, Σ) or a robust family; B ≥ 10^4; use spectral/Cholesky/state-space samplers as needed.
• Sensitivity: report numerical ∂u(y)/∂θ for kernel parameters θ = {σ^2, L_c, ν, …} to support DoE tuning.

VIII. Gates & Compliance

• G4 | Dimensional closure: covariance I/O units consistent, p_dim = 1.0.
• G6 | Residual band: residual Q_res under the chosen kernel within admissible band.
• G7 | Conservation: ε_flux within tolerance under paraxial conservation.
• S1/S5 Stops: failure of positive-definiteness / citation non-compliance blocks release.

IX. Machine-Readable (ready to commit)

A. cov_config.yaml

version: "1.0.0"

domains: ["path","time","channel"]

kernels:

path:

name: "exp"

params: { sigma2: 3.0e-3, L_c_m: 25.0 }

time:

name: "ar1"

params: { sigma2: 1.0e-4, phi: 0.92 }

channel:

name: "block_toeplitz"

params: { rho_matrix: [[1,0.35],[0.35,1]] }

nonstationary:

segmentation:

path: [{start:0.0, end:120.0, L_c_m:20.0}, {start:120.0, end:300.0, L_c_m:35.0}]

numerics:

jitter: 1.0e-6

method: "chol_block"

see:

- "EFT.WP.Core.Equations v1.1:S20-1"

- "EFT.WP.Core.Metrology v1.0:check_dim"

B. cov_blocks.json (block structure sketch)

{

"path_indices": [0, 300],

"time_indices": [0, 400],

"channel_blocks": 2,

"structures": ["toeplitz", "ar1", "block"],

"jitter": 1e-6

}

C. fit_report.md (minimal items)

# Covariance Fit Report

- Kernel set: path=exp(L_c=25 m), time=AR1(phi=0.92), channel=block(rho=0.35)

- Estimates: sigma2_path=3.0e-3 ± 0.4e-3, L_c=25.0 ± 3.0 m, ...

- Diagnostics: variogram OK, slight QQ tail, Q_res=0.13 in-band

- Notes: segmented path between [0,120] m / [120,300] m

X. Cross-References

• Coupled with Chapter 3 for T_arr/Phi path covariance.
• Supplies Σ and samplers to Chapter 6 (propagation methods).
• Aligned with Chapter 8 quality gates; outputs Q_res/ε_flux diagnostics.
• Visuals (variogram, correlation curves, fit intervals) appear on the Chapter 11 results page.

XI. Checklist

• gamma(ell), d ell, and delta_form are explicit; path/time/channel indices aligned.
• Kernel(s) and estimation method (MLE/REML/Variogram/AR) selected; robust diagnostics completed.
• Covariance matrix is PD (with jitter) and numerically stable; complexity controls in place.
• Σ generated and validated in Delta/MC; Q_res, ε_flux, and kernel parameter intervals reported.
• cov_config.yaml / fit_report.md exported; citations compliant (volume+version+anchor, coverage ≥ 90%, public v1.* only).