Rather than arbitrary risk heuristics, Earth Force computes rigorous, closed-form parametric distributions from first mathematical principles:
Extreme Heat & Photochemical Smog
Generalized Extreme Value (GEV) Distribution via L-Moments
Hosking (1990) Formulation
Atmospheric temperature and ground ozone spikes exhibit heavy, asymmetric tails. Earth Force computes unbiased sample L-moments (λ₁, λ₂, λ₃) to fit GEV location (μ), scale (σ), and shape (ξ) parameters without iterative convergence failures:
F(x) = exp( - [ 1 + ξ · (x - μ) / σ ]^(-1 / ξ) )
// Inverse exceedance on the sample-day basis of the ~30-day daily fit:
T(x) = 1 / (1 - F(x))
time basis = one calendar day in the 30-day window, not a year
Heat and ozone use T = 1 / (1 − F(x)) as a rarity score inside that short daily sample. A gate of T ≥ 20 means p ≤ 0.05 in the 30-day daily GEV — not a 20-year climatological return period. We do not publish year-based return periods until a longer historical model justifies that time basis. Magnitude gaps still apply: ozone +50% and ≥ 2.5σ, or temperature at least +8°C above the 30-day mean.
Precipitation Deluges & Flash Flooding
Standardized Precipitation Index (SPI) via 2-Parameter Gamma
Thom (1958) Maximum Likelihood
Precipitation is zero-bounded and positively skewed. Earth Force fits a 2-parameter Gamma distribution with shape α and scale β using Thom’s maximum likelihood estimate:
A = ln(x̄) - (1/n) ∑ ln(x_i), α = [1 + √(1 + 4A/3)] / (4A), β = x̄ / α
Cumulative Probability: H(x) = q + (1 - q) · Γ(α, x / β)
SPI = Φ⁻¹(H(x)) (Abramowitz & Stegun Rational Approximation)
Precipitation anomalies are flagged when SPI ≥ +2.5 with at least 40 mm in 24h, or when 24h accumulation exceeds 60 mm.
Multiple Hypothesis Elimination
Benjamini-Hochberg q-values (diagnostic, not a 5% FDR)
Benjamini & Hochberg (1995)
Testing 500 cities at once needs a multiple-testing diagnostic. Earth Force computes BH q-values across the full city family, then displays them. Magnitude gates decide the radar flag. Because those gates are functions of the same series as the p-value, selecting first and then promising FDR would not be valid. We therefore do not claim a 5% FDR on the flagged set.
1. Sort all city p-values: p_(1) ≤ p_(2) ≤ ... ≤ p_(m)
2. q_(i) = min_(j ≥ i) [ min(1.0, (m / j) · p_(j)) ]
3. q is shown on the city page. Flags do not require q ≤ 0.05.
Fire FRP ≥ 500 MW and rain ≥ 60 mm remain physical detections. Isolation Forest is a separate unsupervised layer and is labeled as such — not an FDR discovery.
Silent Multivariate Compound Traps
Isolation Forest (Unsupervised ML)
Liu, Ting & Zhou (2008) · 100 trees · contamination 0.03 · hard cap 10–15
Single-metric gates miss weather traps where every reading looks ordinary but the joint combination is rare — for example stagnant wind plus high humidity holding ground emissions in place. Earth Force standardizes a 6-feature vector (temperature, humidity, wind, PM2.5, ozone, soil moisture), fits an Isolation Forest across the 500-city matrix, and ranks cities by isolation score.
s(x, n) = 2 ^ ( - E[h(x)] / c(n) )
c(n) = 2 H(n - 1) - 2(n - 1)/n
Flag only the top 10–15 eligible cities (never more than 20)
Cities already flagged by the univariate magnitude gates are excluded so Isolation Forest only surfaces silent compound traps. The two largest |z| features become the public driver line.
