extreme-weather-stats

Overview

Extremes (heatwaves, heavy rainfall, droughts, windstorms) cause most weather-related damage, but they're rare by definition — standard statistics understate them. This skill covers extreme value theory (GEV, GPD), return-period estimation, threshold selection, non-stationary methods for a changing climate, and the basics of event attribution.

When to use

• Estimating the 100-year rainfall or wind speed for infrastructure design
• Defining heatwave/cold-wave indices for health early-warning systems
• Testing whether extremes are intensifying in observations or models
• Attributing a specific event (flood, heatwave) to climate change
• Reviewing a "1-in-1000-year event" claim in the media or a report

Core concepts

• Block maxima (GEV): fit the Generalized Extreme Value distribution to annual maxima — the classical route to return levels. Needs long records; wastes data (one value/year).

• Peaks over threshold (GPD): fit the Generalized Pareto Distribution to exceedances of a high threshold — uses data more efficiently; threshold choice is the critical judgment (stability plots guide it).

• Return period: the average waiting time between exceedances (a "100-year event" has a 1% annual chance — and a ~63% chance of occurring at least once in 100 years). Return periods extrapolate beyond the record — uncertainty explodes past ~2–3× the record length.

• Non-stationarity: in a warming climate, distribution parameters drift with time or global temperature; fit μ(t) or σ(t) as functions of a covariate rather than assuming stationarity.

• Attribution framing: probabilistic (how did climate change alter the event's probability/intensity? — the FAR/RR approach) vs storyline (how did known dynamics produce this event?). Both are legitimate; they answer different questions.

• Compound events: damaging extremes often combine (heat + drought, rain + storm surge); univariate return periods understate joint risk.

• Block-maxima vs POT efficiency: POT uses data more efficiently but introduces threshold selection and declustering judgments — for short records the efficiency gain usually wins; for long clean records, GEV on annual maxima is simpler and robust.

• Covariate choice in non-stationary models: global mean temperature (physical, smooth) vs time (a proxy for everything) — temperature covariates generalize better and connect to attribution; time covariates are easier but less interpretable.

• Spatial pooling: regional frequency analysis (index-flood style) borrows strength across sites — essential when single-site records are short; requires homogeneity testing of the pooled region.

Practical workflow

1. Prepare the series

1. Use homogenized, complete records; extremes are sensitive to instrument changes and missing data.
2. Declutter (for POT): ensure exceedances are independent events (e.g., separate rainfall by dry intervals, heatwaves by cool breaks).
3. Check for obvious non-stationarity first (plot annual maxima vs time) — it decides the modeling approach.

2. Fit the distribution

# Pseudocode: GEV fit to annual maxima with a time-varying location
# mu(t) = mu0 + mu1 * t ; maximize likelihood; compare AIC vs stationary fit

1. GEV on annual maxima for a first estimate; GPD on threshold exceedances when the record is short or you need efficiency.
2. Choose thresholds via parameter-stability and mean-residual-life plots — not by round numbers.
3. Fit non-stationary models with covariates (time, global mean temperature); compare against stationary fits with AIC/BIC.
4. Validate with Q–Q plots and return-level plots — a good fit hugs the diagonal; deviations at the tail are warnings.

3. Estimate and communicate return levels

1. Report the estimate with confidence intervals (profile likelihood or bootstrap) — the interval, not the point value, is the honest answer.
2. Don't extrapolate beyond ~3× the record length; say "beyond the observable range" instead of quoting a 10,000-year level from 40 years of data.
3. Translate for non-specialists: "1% annual chance" beats "100-year event" (which people misread as "once per century").

4. Attribute an event (probabilistic)

1. Define the event class (magnitude, duration, region) before looking at model results.
2. Fit the same statistical model to observations and to model ensembles with/without anthropogenic forcing (factual vs counterfactual).
3. Report the probability ratio (RR) and the change in intensity, each with uncertainty — and the model-evaluation step that justifies trusting the models for this event type.

5. Stress-test the return-level estimate

1. Vary the threshold (POT), block definition, and covariate choice — the return level should be stable across reasonable choices; instability is the real uncertainty.
2. Compare stationary vs non-stationary fits with AIC/BIC and likelihood-ratio tests — let the data adjudicate, but prefer the physically motivated covariate.
3. Bootstrap the full pipeline (resampling years/blocks) to get confidence intervals that include the methodological choices, not just the sampling noise.

6. Quick-reference checklist

• Records homogenized and complete; missing data handled explicitly
• Exceedances decluttered into independent events (POT)
• Threshold chosen via stability plots, not round numbers
• Stationarity tested; non-stationary model fit with physical covariate if needed
• Fit validated with Q–Q and return-level plots
• Confidence intervals computed (profile likelihood or bootstrap)
• No extrapolation beyond ~3× the record length
• Results communicated as annual probabilities, not just "N-year events"

Common pitfalls

• Stationarity by default: fitting a stationary GEV to a warming-era record biases return levels low for heat, high for cold.
• Extrapolation hubris: 50 years of data cannot pin down a 1000-year return level — the confidence interval will tell you if you compute it.
• Threshold hacking: tuning the POT threshold until the trend looks significant — pre-register or show stability across thresholds.
• Confusing return period with schedule: clustering happens; three "100-year" floods in a decade is unlikely but not impossible, especially under non-stationarity.
• Attribution without model evaluation: if models can't reproduce the observed statistics of the event type, the attribution numbers are decorative.
• Single-station generalization: an extreme at one gauge is not a regional trend — check spatial coherence before generalizing.
• Declustering by calendar: splitting exceedances by fixed time windows instead of physical event separation — merges distinct storms or splits single ones; use meteorologically meaningful separation criteria.
• Presenting the upper confidence bound as "the" estimate: the precautionary principle doesn't license quoting the 95th percentile as the best estimate — report the central estimate with its interval.