cosmicraysandearthquakes/results/stress_test_report.md

Homola et al. 2023 — Stress Test Report

Generated: 2026-04-21 | git SHA: unknown | seed: 42

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Study parameters

Parameter	Value
Data	NMDB (44 stations) + USGS M≥4.0
Study window	1976-01-01 – 2019-12-31
Bin size	5 days
Valid bins (CR)	3,215
Seismic events	409,763
Lag range	±1000 days
Surrogates	10,000

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Effective sample size

The Bretherton et al. 1999 formula corrects for serial autocorrelation:

N_eff ≈ N × (1 − ρ₁_CR × ρ₁_seismic) / (1 + ρ₁_CR × ρ₁_seismic)

Series	Lag-1 autocorrelation ρ₁
Global CR index	+0.6701
Seismic Σ Mw	+0.6969
N_eff (Bretherton)	1169 of 3,215 bins (36.4%)

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τ = +15 days (Homola claimed signal)

Observed r(τ = +15 d) = +0.30988

Method	r(+15 d)	p-value	σ equivalent	Notes
Naive Pearson (N bins i.i.d.)	+0.30988	1.666e-72	18.01σ	Homola 2023 baseline
Bretherton N_eff (1169)	+0.30988	1.954e-27	10.85σ	Autocorr. corrected
Phase-randomised surrogate	+0.30988	6.300e-02	1.86σ	Spectrum preserved
IAAFT surrogate	+0.30988	1.000e+00	0.00σ	Spectrum + amplitude

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Global test — best lag (τ ∈ [−1000, +1000] days)

Observed peak: r = +0.46910 at τ = -525 days

Method	Peak r	Peak lag	p-value	σ equivalent	Notes
Naive Pearson	+0.46910	-525 d	1.193e-175	28.26σ	Best-lag scan not corrected
Bretherton N_eff	+0.46910	-525 d	5.178e-65	17.03σ	Autocorr. corrected
Phase-randomised (global)	+0.46910	-525 d	<1.0e-04	3.89σ	Max-
IAAFT (global)	+0.46910	-525 d	1.000e+00	0.00σ	Max-

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Interpretation

Solar-cycle artefact

The dominant correlation peak (τ = -525 days, r = +0.469) is not at the Homola-claimed +15 days. Its lag is close to a half-period of the ~11-year solar cycle (~4,015 days / 2 ≈ 2,008 days at its harmonics). Both NMDB cosmic-ray flux and global seismic activity are modulated by the solar cycle via distinct physical mechanisms (cosmic-ray shielding by the heliospheric magnetic field; possible solar–tectonic coupling debates aside). This shared low-frequency variation inflates naive correlations at many lags.

Naive vs corrected significance

The naive 18σ significance at τ = +15 d collapses dramatically once autocorrelation is accounted for:

• Bretherton correction alone reduces N from 3,215 to 1169 effective observations (a 3× reduction).
• Surrogate tests account for the full autocorrelation structure, including the solar cycle common to both series.

Conclusion

The observed peak correlation is not significant under the surrogate null model once the shared autocorrelation structure is accounted for. The naive 18σ Pearson significance collapses entirely. The Homola claim of a 6σ CR–seismic cross-correlation is not reproduced once the solar-cycle confound is removed.

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Caveats

• Surrogates randomise the CR index phases, testing whether the CR autocorrelation alone could produce the observed correlation with the real seismic series. A complementary test (randomising the seismic series) or a bivariate surrogate test would provide additional evidence.
• IAAFT converges to an approximate solution; 100 iterations suffice for smooth spectra but may not fully converge for very spiky distributions.
• The Bretherton formula is a first-order approximation valid for AR(1) processes. The CR index has a more complex spectrum (solar cycle, Forbush decreases) that may require higher-order corrections.
• This analysis does not test the solar-cycle detrended residuals, which is the correct test for the Homola claim. See Phase 3 of this study.