Logbook Entry 004 — Effect-Size Threshold δ_min Calibration

Date: 2026-03-31 Work package: WP1 (IC Coastline) — final item Decision gate: Not a DG-1 criterion. δ_min is a classification parameter needed before WP2.

Objective

Calibrate the effect-size threshold δ_min that separates structured from unstructured anomalies in the three-way classifier. Below δ_min, an IC-flagged anomaly is classified as unstructured (memoryless); above δ_min, it is classified as structured (persistent temporal trend).

Pre-registered choices (defined before running)

Variance slope: δ_min = k × median( slope ) under hardest null, with k = 3

Autocorrelation: δ_min = 95th percentile of

acf

under hardest null (percentile-based, because acf is bounded to [-1, 1] and a multiplier-based threshold would exceed the domain)

Window: W = 20
Realisations: 300 per model, T = 200 time steps each

What was done

1. Temporal-structure statistics

Implemented compute_temporal_structure(values, window) in src/temporal.py. For each time step t ≥ W, computes over the trailing window:

Variance slope: Linear regression of log(running variance) over sub-windows of size W/4 within the trailing window. Positive slope → growing instability.
Lag-1 autocorrelation: Sample autocorrelation at lag 1. High values → persistent temporal structure (critical slowing down indicator).

2. Null distributions across all ten models

Model	median	var_slope
Gaussian	0.0514	0.150
Heteroscedastic	0.0668	0.139
Student-t(3)	0.0702	0.136
Student-t(5)	0.0619	0.139
AR(1) ρ = 0.7	0.0530	0.530
Pareto α = 2.5	0.0478	0.143
Pareto α = 3.0	0.0423	0.150
fGn H = 0.9	0.0518	0.360
AR(1) ρ = 0.9	0.0549	0.666
Random walk	0.0578	0.715

Hardest null (variance slope): Student-t(3) — heavy tails create sporadic variance spikes in sub-windows.

Hardest null (autocorrelation): Random walk — inherently non-stationary, with strong serial dependence.

3. δ_min values

Statistic	Method	Hardest null	δ_min
Variance slope	k = 3 × median	Student-t(3)	0.2105
Autocorrelation	95th percentile	Random walk	0.8703

Why two different methods: The variance slope is unbounded, so the standard k × median approach works. Autocorrelation is bounded to [-1, 1]; using k × median with the random walk (median 0.72) would give δ_min = 2.15, which exceeds the domain. The 95th percentile is the natural choice for a bounded statistic — it says: “this autocorrelation exceeds what the most correlated null produces 95% of the time.”

4. Figure

Panel (a):

variance slope

distributions under all ten nulls. The red line marks δ_min = 0.2105. Panel (b):

autocorrelation

distributions. The red line marks δ_min = 0.8703. Note the large separation between the light-tailed models (Gaussian, Pareto, Student-t) and the autocorrelated models (AR(1), fGn, random walk) in panel (b).

5. Sanity check: detectability of injected structure

Sinusoidal drift (amplitude 2σ, period 50 steps on Gaussian noise): max

autocorrelation

= 0.93, exceeds δ_min(acf) = 0.87. Detectable.

Growing variance (σ(t) = 1 + 0.03t on Gaussian noise): max variance slope = 0.42, exceeds δ_min(var) = 0.21. Detectable.

Both injected signals are clearly above their respective thresholds. The three-way classifier will not collapse to two-way under these signal strengths.

6. Classification rule (complete)

With δ_min calibrated, the full classification rule from the Projektantrag §3.2 is now operational:

STABLE:              IC < threshold_95
STRUCTURED ANOMALY:  IC ≥ threshold_95 AND (|var_slope| > 0.2105 OR |autocorr| > 0.8703)
UNSTRUCTURED ANOMALY: IC ≥ threshold_95 AND |var_slope| ≤ 0.2105 AND |autocorr| ≤ 0.8703

where threshold_95 is the 95th-percentile AIPP threshold from Entry 001 (calibrated under worst-case σ per Entry 002).

7. Test suite

13 tests in tests/test_temporal.py:

7 unit tests for compute_temporal_structure
3 unit tests for calibrate_delta_min
1 null-distribution test (δ_min positive and finite across all ten models)
2 detectability tests (sinusoidal drift, growing variance)

All passing. Full suite: 76 tests, 74 passing (2 known failures from Entry 002).

WP1 status: complete

All WP1 tasks are done:

Task	Status	Entry
IC implementation	✅ Done	001
AIPP convergence verification	✅ Done	001
Threshold stability (5 models)	✅ Done	001
σ-sensitivity analysis	✅ Done (FAIL on systematic −20%; mitigated)	002
Power-law and 1/f nulls	✅ Done	003
Finite-N bias quantification	✅ Done	003
Threshold stability (all 10 models)	✅ Done	003
Effect-size threshold δ_min	✅ Done	004

DG-1 was closed in Entry 003. δ_min was not a gate criterion — it is a classification parameter. With it calibrated, WP1 is complete and WP2 can begin.

Files changed

File	Change
`src/temporal.py`	New: `compute_temporal_structure`, `calibrate_delta_min`
`tests/test_temporal.py`	New: 13 tests
`scripts/fig06_delta_min_calibration.py`	New: figure generation script
`logbook/figures/fig06_delta_min_calibration.png`	New: δ_min calibration figure

Entry by U. Warring. AI tools (Claude, Anthropic) used for code prototyping and derivation checking.