# Methodology — Earthquake Occurrence Probabilities This document specifies, at a level intended to survive academic scrutiny, exactly how every number shown by the Disaster Probability Tracker is computed: the data, the estimators, the model-selection rules, the validation, and the limitations. Nothing in the app is produced by any method not described here. **One sentence version:** the app reports *long-run statistical occurrence frequencies* of earthquakes, computed from complete portions of merged instrumental catalogs with exact uncertainty, under a homogeneous Poisson model (default) and a Gamma renewal model (time-dependent view) — it does not, and cannot, predict individual earthquakes. --- ## 1. Data ### 1.1 Sources | Era | Source | Magnitudes | License | |---|---|---|---| | 2000-01-01 → 2024-01-08 | USGS ANSS Comprehensive Catalog (ComCat), M4.5+, all depths, `type=earthquake` only | ComCat preferred magnitudes (mww/mwc/mwb/mb…) | Public domain (US Gov) | | 1900 → 1999 | ISC-GEM Global Instrumental Earthquake Catalogue (uniformly re-processed Mw), via the Swiss Re Historical Earthquake Statistics Dataset | Mw | CC BY-SA 3.0 | | 2024-01-09 → 2025-06-05 ("bridge") | ComCat M5.5+, depth ≤ 70 km, via the Swiss Re dataset | ComCat preferred | Public domain | | runtime → today | Live USGS FDSN event query (M5+), fetched by the browser and cached locally; **replaces** the bridge segment entirely when online | ComCat preferred | Public domain | | 10 AD → 1899 | NGDC/WDS Significant Earthquake Database, via the Swiss Re dataset | mixed/unknown | Public domain | Merge rules are hard cutovers by date (no fuzzy dedup needed): ISC-GEM before 2000, ComCat CSV from 2000, Swiss Re/ComCat bridge after the CSV ends, live top-up after that. Rows without a usable magnitude are excluded from all statistics. The pre-1900 NGDC record is **display context only** — it is a *damage-selected* catalog (deaths, damage, tsunamis), not a complete sample of seismicity, and it never enters any rate computation. ### 1.2 Completeness windows A catalog is "complete at magnitude m" from the date after which essentially every event ≥ m on Earth was detected and cataloged. Using data before that date biases rates low. Following the ISC-GEM documentation (Storchak et al. 2013) and standard practice: | Threshold | Complete from | |---|---| | M ≥ 7.5 | 1900 | | M ≥ 6.5 | 1918 | | M ≥ 5.5 | 1964 | | M ≥ 4.5 | 2000 (ComCat era) | Every rate for threshold m is computed **only inside the longest window complete at m**; this is why different magnitudes legitimately use different observation spans. Additionally, the [4.5, 5.0) band is capped at the bundled ComCat end date (2024-01-08) because the live top-up fetches M5+ only; and when the app is fully offline, all statistics are evaluated *as of the bundled data edge* rather than today, so quiet time that is merely un-cataloged is never counted as real quiet time. These windows are conservative globally. For a specific well-instrumented region (California, Japan) local completeness is better than this; for mid-ocean ridges before 1964 it can be worse. ## 2. Estimators ### 2.1 Empirical exceedance rates (the primary object) For threshold m with n events observed over T complete years, the annual rate is λ̂ = n/T, with the **exact (Garwood 1936) Poisson confidence interval**: ``` λ_lo = χ²(α/2, 2n) / 2T (0 when n = 0) λ_hi = χ²(1 − α/2, 2n + 2) / 2T ``` computed from a full implementation of the regularized incomplete gamma function (verified in the test suite against standard χ² tables). This interval is honest at any n, including n = 0 and n = 4 — which matters, because the interesting thresholds are exactly the rare ones. ### 2.2 Gutenberg–Richter b-value The magnitude–frequency relation log₁₀ N(≥M) = a − bM (Gutenberg & Richter 1944) is fitted by the Aki–Utsu maximum-likelihood estimator with the Tinti & Mulargia (1987) correction for 0.1-magnitude binning: ``` b̂ = log₁₀(e) / ( M̄ − (Mc − ΔM/2) ), ΔM = 0.1 ``` with the Shi & Bolt (1982) standard error. The fit uses **only the homogeneous modern era** (2000 → catalog end, magnitudes from a single cataloging regime) above the completeness magnitude Mc. Globally Mc = 4.5 (catalog floor); regionally Mc is estimated by maximum curvature (Wiemer & Wyss 2000) + 0.2. The global fit on the shipped data gives **b = 1.14 ± 0.003** — slightly above the canonical ~1.0 because ComCat preferred magnitudes mix mb (which saturates) with Mw; this is a documented property of mixed-type catalogs, and the b-value is used only for interpolation and regional scaling, never to override empirical counts (§ 2.4). ### 2.3 Tapered Gutenberg–Richter (large-magnitude tail) A pure G-R line overpredicts M ≥ 9 rates because the Earth's plate system imposes a finite corner moment. For extrapolation beyond robust empirical counts the app fits the **tapered Pareto/Gutenberg–Richter distribution** (Kagan 2002) on seismic moments (Hanks & Kanamori 1979, log₁₀M₀ = 1.5Mw + 9.05) of all M5.5+ events in the complete record: β fixed at (2/3)·b̂ (profile likelihood), corner moment maximized by golden-section search. On the shipped data the corner magnitude comes out **Mc ≈ 9.2**, consistent with published global estimates (Kagan's global corner Mw ≈ 8.5–9.6 depending on zonation and era). ### 2.4 Rate-selection policy For the headline number at threshold m: 1. **n ≥ 3 events in the complete window → the empirical rate**, with its exact CI. Counted reality beats any curve. 2. **n < 3, global scope → tapered G-R extrapolation**, flagged in the UI, with a widened uncertainty band (never narrower than the empirical exact CI). 3. **n < 3, local scope → fixed-b scaling**: the regional empirical rate at the largest locally robust threshold, scaled down by 10^(−b·Δm) with the global b (regional b is used instead when the region itself has ≥ 50 events above its Mc). Flagged in the UI. The magnitude–frequency chart always shows the empirical points *and* the fitted curves together with confidence whiskers, so the extrapolation is visually auditable. ## 3. Probability models ### 3.1 Poisson (default) P(≥1 event in the next T years) = 1 − e^(−λT). The homogeneous Poisson model is the standard null model for large-earthquake occurrence at global and regional scale (Kagan & Jackson 1991; Michael 2011). The 95% band shown is the rate CI propagated through this formula. Scrubbing the time slider widens T — the probability rise you see is *cumulative window probability*, which is the statistically meaningful quantity (under Poisson, the hazard per unit time never rises no matter how long it has been quiet). ### 3.2 Renewal ("overdue" view) — and why it usually refuses to say "overdue" The colloquial "overdue" intuition is a renewal-process claim: inter-event times follow some distribution, and the conditional probability of an event given elapsed quiet time τ is ``` P(event in (τ, τ+T] | quiet until τ) = [F(τ+T) − F(τ)] / [1 − F(τ)] ``` The app fits a **Gamma distribution to the observed inter-event times** by full MLE (Newton iteration on the digamma equation, Minka initialization), inside the same completeness window, and evaluates that conditional probability at the actual current open interval. It requires ≥ 8 observed intervals; below that it reports "n/a" rather than fitting noise. The shape parameter is the verdict. CV = 1/√k: - **CV ≈ 1** → exponential → memoryless → "overdue" is meaningless. - **CV > 1** → *clustered* (aftershock sequences, triggering): a long quiet spell makes the near-term probability slightly **lower**, the opposite of overdue. - **CV < 1** → quasi-periodic: quiet time genuinely raises the hazard. On real data, global inter-event CVs at every threshold are **> 1** (1.2–1.9 on the shipped catalog): world seismicity clusters. Genuine quasi-periodic "overdue" behavior is an individual-fault phenomenon (e.g., the Brownian Passage Time models used for specific fault segments in UCERF3, Matthews et al. 2002; Field et al. 2015) and requires paleoseismic recurrence data that no instrumental catalog contains. The app says exactly this in the UI instead of manufacturing a rising "overdue meter" that the data do not support. This is the difference between a tool that can be scrutinized and one that flatters intuition. Aftershock note: rates are **total seismicity, not declustered**. For the question the app answers — "will at least one event ≥ m occur?" — total rates are the right choice; they slightly inflate short-window probabilities immediately after large events (when clustered hazard really is elevated). A declustered/ETAS view is a documented possible extension, not a correction. ### 3.3 Regional method A local area is a great-circle disc (haversine distance ≤ radius). Rates use the same completeness windows, on events inside the disc. b is fitted regionally when the disc has ≥ 50 events above its estimated Mc, otherwise global b with regional anchoring (§ 2.4.3). When the requested magnitude exceeds anything observed in the disc, the number shown is an extrapolation and the UI flags that whether the local tectonics can produce such an event is a *geological* question the catalog cannot answer. ## 4. Validation The test suite (`test/engine.test.mjs`, 24 tests, run in CI-able Node) pins: - special functions against χ² tables, Γ identities, digamma values; - estimator recovery on synthetic data (G-R b on binned exponential samples; Gamma MLE on simulated Gamma variates; memorylessness of the exponential renewal case); - **the shipped catalog against published benchmarks**: | Quantity | This app | Published | |---|---|---| | Global b (modern era) | 1.14 ± 0.003 | ~0.9–1.2 (mixed-magnitude catalogs) | | λ(M ≥ 7) | 13.3 /yr | USGS long-term average ~15 /yr | | λ(M ≥ 8) | 0.88 /yr | ~1 /yr | | M ≥ 9 events since 1900 | 4 (1960, 1964, 2004, 2011; 1952 Kamchatka is Mw 8.8 in this ISC-GEM edition) | 4–5 depending on catalog | | P(M ≥ 9 anywhere, 1 yr) | 3.1% [0.9%, 7.8%] | ~3–5% (Poisson on the same record) | | TGR corner magnitude | 9.2 | 8.5–9.6 (Kagan 2002 and successors) | | P(M ≥ 5 within 200 km of Draper UT, 50 yr) | ~89% | UWG-2016 Wasatch Front M5+ 50-yr: 93% | The Draper M5 agreement is genuine cross-validation: the Utah Working Group number folds in paleoseismology, ours is instrumental-only, and at M5 the instrumental record is dense enough that they converge. At M6.75+ they diverge (UWG 43%/50 yr vs. a few percent here) — exactly the short-record limitation § 5.2 documents, surfaced as a UI caveat within 300 km of the Wasatch. ## 5. Limitations (read before quoting any number) 1. **These are frequencies, not predictions.** Deterministic earthquake prediction (time, place, magnitude of a specific future event) does not exist at any scientific standard. 2. **Short instrumental record vs. slow faults.** 60–125 years of catalog cannot constrain faults with 300–2,000-year recurrence (Wasatch, Cascadia, many intraplate zones). Instrumental rates *underestimate* large-magnitude hazard there; paleoseismic studies are the corrective, and regional hazard authorities (USGS NSHM, UWG, GEM) should be preferred for building-code-grade decisions. 3. **Magnitude heterogeneity.** ISC-GEM Mw before 2000, ComCat preferred magnitudes after. The b-value is fitted on the homogeneous era only, but cross-era rate comparisons at a fixed threshold inherit ~0.1-unit magnitude-scale noise. 4. **Depth truncation in the bridge segment** (M5.5+, ≤ 70 km, 2024→mid-2025): deep-focus events in that 17-month window are missing unless the live top-up is active (it usually is). 5. **Disc geometry** is a blunt instrument: probabilities scale with the area drawn, and a disc answers "near this point," not "on this fault" or "at this site" (no ground-motion modeling — this is occurrence probability, not shaking hazard, i.e., not a PSHA). 6. **No declustering / ETAS**: short-window probabilities right after a large regional event are conservative (elevated), by design. ## 6. References - Aki, K. (1965). Maximum likelihood estimate of b in the formula log N = a − bM. *Bull. Earthq. Res. Inst.* 43. - Field, E. H., et al. (2015). UCERF3: A synthesis. *BSSA* 105(2A). - Garwood, F. (1936). Fiducial limits for the Poisson distribution. *Biometrika* 28. - Gutenberg, B., & Richter, C. F. (1944). Frequency of earthquakes in California. *BSSA* 34. - Hanks, T. C., & Kanamori, H. (1979). A moment magnitude scale. *JGR* 84. - Kagan, Y. Y. (2002). Seismic moment distribution revisited: I. *GJI* 148. - Kagan, Y. Y., & Jackson, D. D. (1991). Long-term earthquake clustering. *GJI* 104. - Matthews, M. V., Ellsworth, W. L., & Reasenberg, P. A. (2002). A Brownian model for recurrent earthquakes. *BSSA* 92. - Michael, A. J. (2011). Random variability explains apparent global clustering of large earthquakes. *GRL* 38. - Shi, Y., & Bolt, B. A. (1982). The standard error of the magnitude-frequency b value. *BSSA* 72. - Storchak, D. A., et al. (2013). Public release of the ISC-GEM Global Instrumental Earthquake Catalogue (1900–2009). *Seism. Res. Lett.* 84. - Tinti, S., & Mulargia, F. (1987). Confidence intervals of b values for grouped magnitudes. *BSSA* 77. - Wiemer, S., & Wyss, M. (2000). Minimum magnitude of completeness in earthquake catalogs. *BSSA* 90. - Working Group on Utah Earthquake Probabilities (2016). *Earthquake probabilities for the Wasatch Front region.* Utah Geological Survey Misc. Pub. 16-3. --- # Part II — Multi-hazard impact model (master dashboard) The dashboard answers a different question than Part I: not "how often does an earthquake of magnitude m occur here," but **"how often does *any* natural disaster exceed a given human or economic impact, anywhere on Earth (or in the US)?"** The statistical object is a marked multi-hazard point process filtered on impact marks. ## 8. Impact data **EM-DAT**, the international disaster database (CRED / UCLouvain, Brussels — www.emdat.be; Delforge et al. 2025), public table including historical events, snapshot **2024-03-26** (obtained via the public copy committed in `com-480-data-visualization/project-2024-DisasterClass`; EM-DAT terms: free for non-commercial use with attribution — to refresh or re-license, register at public.emdat.be and re-run `etl/build-impacts.mjs`). Transformations (see `etl/build-impacts.mjs`): - **Natural physical hazards only.** The Biological subgroup (epidemics, infestations) is excluded; so are technological disasters. Hazard classes: Earthquake (incl. tsunamis), Storm, Flood, Drought, Extreme temperature, Landslide, Wildfire, Volcanic activity, Other. - **Country rows → physical events.** EM-DAT records one row per affected country; rows sharing a `DisNo.` prefix are aggregated (impacts summed) so "an event killing ≥ X" means the physical event (the 2004 Indian Ocean tsunami is one event with ~226k deaths across 12+ country rows, not 12 smaller events). 26,443 rows → 13,571 physical events, 1900 → 2024-02. - **US scope** keeps each event's US-portion impacts (ISO USA + PRI, VIR, GUM, ASM, MNP — Hurricane Maria's Puerto Rico toll belongs to US mode). - **Damages** use EM-DAT's CPI-adjusted series (≈ 2024 US$). ## 9. Completeness windows (impact thresholds) EM-DAT's coverage of small events improves enormously over time (79 recorded events in the 1900s vs ~4,500 in the 2000s); large events are reliably recorded much further back. Windows were chosen from a per-decade rate stabilization analysis (the decade table is reproduced in the ETL comments): | Threshold | Complete from | Empirical basis | |---|---|---| | ≥ 10 deaths | 2000 | rate plateaus ~1990s–2000s (~115–140/yr) | | ≥ 100 deaths | 1990 | plateau ~20–28/yr from the 1980s | | ≥ 1,000 deaths | 1930 | remarkably stable ~2–4/yr across the century | | ≥ 10,000 deaths | 1900 | large events historically well recorded | | ≥ $100M adj. damage | 2000 | damage reporting matures late | | ≥ $1B adj. damage | 1990 | | A threshold uses the window of the largest table entry ≤ it (conservative: shorter windows widen CIs but avoid undercount bias). ## 10. The probability model Per hazard h, the rate λ_h(X) is the count of qualifying events in the window divided by its length, with exact Garwood CIs (§ 2.1). Hazards superpose: ``` P(≥1 event ≥ X within T) = 1 − exp( −Σ_h λ_h(X) · s_h(T) · T ) ``` s_h(T) is a **seasonality factor**: a smoothed monthly climatology per hazard (2000+, floor threshold for stable counts, ±1-month kernel, normalized to mean 1), averaged over the actual forecast window. It matters only for sub-annual windows (hurricane season is real: the Storm September factor roughly doubles May) and converges to 1 for multi-year windows. Per-hazard "share of risk" is λ_h·s_h / Σ λ·s. No tail extrapolation is performed beyond observed impacts: thresholds with zero qualifying events report "too rare to rate empirically" rather than a fitted number. (A tapered severity-tail fit is a possible extension; the heavy right tail of disaster deaths makes naive power-law extrapolation irresponsible at the "quotable number" bar this project sets.) ## 11. Validation (Part II) `test/impact.test.mjs` (14 tests) pins: | Quantity | This app | Check | |---|---|---| | λ(deaths ≥ 100, global) | ~23/yr | decade-table plateau 20–28/yr | | λ(deaths ≥ 10,000, global) | ~0.6/yr | century record 0.3–1.2/yr | | 2004 Indian Ocean tsunami | 1 event, ~226k deaths, 12+ countries | aggregation correctness | | Hurricane Katrina (US scope) | ~1,800 US deaths | EM-DAT US value | | Storm seasonality | Sep ≫ May | NH hurricane season | | Decomposition | Σ per-hazard λ = total λ exactly | superposition consistency | | λ(damage ≥ $10B adj.) | ~2–3/yr | reinsurance-industry reporting range | ## 12. Limitations (Part II) 1. **Fixed snapshot.** The impact catalog ends 2024-02; there is no live feed (EM-DAT requires authenticated download). Long-run rates remain valid; the "most recent event" list does not extend past the snapshot. 2. **Non-stationarity, both directions.** Disaster *mortality* has declined for decades (early warning, response capacity) — death-based rates at extreme thresholds are dominated by early-century famines/floods and lean high as forecasts. *Damages* rise with exposure — damage-based rates lean low. The completeness windows bound but cannot remove this. 3. **Reporting selection.** EM-DAT entry requires ≥10 deaths, ≥100 affected, or a declaration/appeal; damage figures exist for only ~40% of events and skew toward insured, wealthy regions. 4. **Event aggregation** by DisNo prefix is exact for EM-DAT's own event identity; multi-week compound episodes (e.g., a monsoon season) may appear as several events. 5. **US cross-source redundancy** was evaluated against the tidyDisasters compilation (FEMA + EM-DAT merged) and rejected: that dataset duplicates national death tolls across state-level rows (Katrina sums to ~38,000), so it would corrupt rather than corroborate. NOAA Storm Events is the right future US enrichment source. ## 13. Additional references - Delforge, D., et al. (2025). EM-DAT: the Emergency Events Database. *Int. J. Disaster Risk Reduction*. - UNDRR & CRED (2020). *Human cost of disasters 2000–2019.* --- # Part III — Deep history: what centuries can and cannot tell us Part II's impact rates stop at 1900 because EM-DAT does. The deep-history layer extends the record to ~2150 BC (earthquakes) and the early Holocene (eruptions), with a strict separation between two uses: ## 14. Sources - **NCEI/WDS Global Significant Earthquake Database** (NOAA/NCEI, public domain): destructive earthquakes 2150 BC → present with documentary death tolls, magnitudes, tsunami flags. Snapshot 2021-03 via the official TSV in `rsizem2/noaa-earthquakes`. - **Smithsonian Global Volcanism Program, Volcanoes of the World**: confirmed Holocene eruptions with VEI, via the TidyTuesday 2020-05-12 extract. Citation: Global Volcanism Program, Smithsonian Institution (comp. Venzke). ## 15. Occurrence rates for extremes — where the deep record is trustworthy Some events are too large to escape documentation, so their record is complete over centuries even without instruments: | Class | Window | Basis | Result | |---|---|---|---| | VEI ≥ 6 eruption | 1500 → | Brown et al. (2014): the VEI≥6 record is essentially complete post-1500 | 9 events → **~1.7/century** (λ ≈ 0.017/yr) | | VEI ≥ 7 eruption | 900 → | ice cores + global tephra make these unmissable for ~a millennium | 3 events (Paektu 946, Samalas 1257, Tambora 1815) → **~1 per ~370 yrs**, with the very wide CI n=3 deserves | | M ≥ 8.5 earthquake | 1900 → | instrumental only — see § 16 | ~16 events → λ ≈ 0.13/yr | Probabilities are homogeneous Poisson over these rates. These are **occurrence** probabilities: a VEI-7 anywhere on Earth, not a death toll — impact depends entirely on where it happens (Tambora's toll came with a starving year worldwide; a repeat today would be a global agricultural event). ## 16. Where the deep record must NOT feed rates — and why that is a finding, not an assumption Documented M ≥ 8.5 earthquakes per century climb 3 → 6 → 7 → 9 → 11 from the 1500s to the 1900s. Two hypotheses fit that curve: better record-keeping, or genuinely increasing seismicity. We do not assume the answer — we test it against the data (both analyses run in `test/deep.test.mjs` on every build): **Stationarity inside the constant-detection era.** Since ~1918, detection of M ≥ 7 events is complete and unchanging, so any real secular increase must show there. Raw decade counts wiggle upward slightly, but earthquake counts are clustered (aftershocks; the measured inter-event CV > 1 of § 3.2), which inflates variance and mimics trends. After Gardner–Knopoff-style declustering (drop events within 1.5 yr and 300 km after a larger shock), the 1920s–2010s decade counts are 108, 113, 120, 86, 113, 127, 93, 120, 118, 128: homogeneity χ² = 15.1 (df = 9, below the 16.9 five-percent cutoff) and trend z = 1.3 — statistically flat. This reproduces Shearer & Stark (2012), who posed exactly this question after the 2004–2011 cluster of great quakes. **The size of the documentary undercount.** Over 1500–1899 the documentary record captures ~22 M ≥ 7.5 events per century; instruments record 124–184 per century — roughly one event in six was written down, with remote and oceanic earthquakes the largest missing class. For rising seismicity to explain the ramp instead, rates would have had to jump ~6× precisely when seismometers were deployed and then hold flat for the 120 years since — which the stationarity test above rules out. The honest limit of the claim: "no trend across a century of constant measurement" is not proof of stationarity over millennia; it is the strongest statement the data permit, and the model needs nothing stronger — every rate is computed inside a constant-detection window, so no assumption about the documentary era enters the numbers at all. Death tolls face the additional, untestable-by-us confound that a toll from 1556 reflects 1556's buildings and population (~0.4 ≥100k-death quakes/century documented pre-1900 vs ~5/century since — documentation and population growth compounded). Consequently: - occurrence windows for earthquakes stay instrumental; - the dashboard's death/damage probability windows stay at Part II's 1900+ policy; - documentary catastrophes appear only in the era-labeled display record (`centuryRamp()` in `deep.js` reproduces the evidence; the test suite asserts the ramp exists). Pre-1900 death tolls shown (Antioch 115: ~260,000; Shaanxi 1556: ~830,000) are chronicle-derived estimates with order-of-magnitude uncertainty, labeled by source. ## 17. Additional references (Part III) - Brown, S. K., et al. (2014). Characterisation of the Quaternary eruption record: the LaMEVE database. *J. Applied Volcanology* 3:5. - Newhall, C. G., & Self, S. (1982). The Volcanic Explosivity Index (VEI). *JGR* 87. - Shearer, P. M., & Stark, P. B. (2012). Global risk of big earthquakes has not recently increased. *PNAS* 109(3). - Rougier, J., et al. (2018). The global magnitude–frequency relationship for large explosive volcanic eruptions. *EPSL* 482. - National Geophysical Data Center / World Data Service: NCEI/WDS Global Significant Earthquake Database. NOAA. doi:10.7289/V5TD9V7K --- # Part IV — Model evaluation: the accuracy report card Accuracy here is a measured quantity, not a claim. `backtest/backtest.mjs` walks forward through history month by month (2010-01 → 2024-01, 169 months): for each month and target it fits every candidate model using ONLY earlier events (no lookahead), forecasts P(≥1 qualifying event that month), and scores against what happened — Brier score, log score, and Brier Skill Score (BSS) relative to the flat constant-rate baseline M0. Calibration is checked by bucketing forecasts against observed frequencies. The same run executes in CI (`test/backtest.test.mjs`) so the report card cannot silently rot. ## 18. Results (snapshot data, 169 held-out months) | Target | events | M1 seasonal | M2 + trend | M3 + cluster | |---|---|---|---|---| | deaths ≥ 100, global | 128 | **+5.1%** | **+7.7%** | +4.8% | | deaths ≥ 1,000, global | 30 | +0.6% | +0.6% | +0.6% | | deaths ≥ 10,000, global | 7 | −2.0% | −2.0% | −2.0% | | deaths ≥ 100, US | 10 | +0.7% | +0.7% | +0.7% | | damage ≥ $10B, global | 39 | +3.6% | +3.6% | +2.3% | (Cells are BSS vs the flat baseline; positive = more skillful.) ## 19. What was adopted, and what was rejected - **Seasonality (M1) is validated** and stays. - **The trend adjustment (M2) is adopted**: a Poisson log-linear trend fitted to the aggregate annual counts of qualifying events inside the completeness window, evaluated at the forecast date. Guards (all part of the validated configuration, implemented once in `poissonTrendMultiplier` and shared by the engine and the backtest): the trend applies only when its slope is significant (|z| ≥ 2), the multiplier is clamped to [⅓, 3], and the projection never extends more than 3 years past the data. Where the gate closes, M2 ≡ M1 — which is why most rows tie. Where it opens (deaths ≥ 100: event frequency at that severity has declined significantly across 1990→2024), it corrects exactly the over-forecasting the calibration table shows for the flat model (mean forecast 0.89 vs observed 0.80). - **Monthly clustering (M3) is rejected**: conditioning on the previous month's activity scored below plain seasonality (+4.8% < +5.1% at deaths ≥ 100). Whatever month-to-month dependence exists is already captured by the seasonal cycle at this granularity. Sub-monthly aftershock clustering for the earthquake page remains plausible future work — the harness is ready to judge it. - The small negative skill at deaths ≥ 10,000 (7 events in 14 years) reflects seasonal-factor noise at a threshold too sparse for a monthly climatology; the effect is within noise and the seasonal factor at such thresholds is driven by the shared floor (§ 10), not the sparse events themselves. ## 20. Scoring references - Brier, G. W. (1950). Verification of forecasts expressed in terms of probability. *Mon. Wea. Rev.* 78. - Gneiting, T., & Raftery, A. E. (2007). Strictly proper scoring rules, prediction, and estimation. *JASA* 102. ## 19b. The renewal ("overdue") hypothesis — scored, shipped as comparison, adoption pre-registered Prompted by the observation that no ≥100,000-death event has occurred since 2010 (16+ quiet years against a 4.6-year historical mean gap), model **M4** fits a Gamma renewal distribution to the training inter-event gaps and conditions on elapsed time. Walk-forward results: | Target | best prior model | M4 renewal | |---|---|---| | deaths ≥ 100 | +7.7% (M2) | +1.3% | | deaths ≥ 1,000 | +0.6% | **+1.7%** | | deaths ≥ 10,000 | −2.0% | **+0.8%** (only model above baseline) | | damage ≥ $10B | +3.6% | +0.0% | Renewal conditioning helps precisely at sparse, high-severity thresholds — where gap structure carries information — and hurts at dense ones, where seasonality and trend dominate. At the ≥100k threshold specifically, the fitted gaps are mildly quasi-periodic (CV ≈ 0.8 from 24 gaps), so the conditional probability given the current quiet spell runs ~1.5× the memoryless number. **Status: comparison view, not headline.** The dashboard's "Time since last" panel shows the renewal number beside the validated model wherever ≥ 15 gaps exist. Because promoting M4 for "sparse thresholds" chosen from these same scores would be selection on the test set, adoption is **pre-registered** instead: if M4's advantage at deaths ≥ 1,000 and ≥ 10,000 persists when the EM-DAT catalog is refreshed (2024→present is genuinely out-of-sample), M4 becomes the headline at thresholds whose baseline monthly probability is below 50%. The advantage is also sensitive to the evaluation range (present on 2010–2024, absent on 2012–2024) — additional evidence that it is a weak signal requiring out-of-sample confirmation before it may drive the headline.