Rogue Wave

A 25.6-metre wave that the ocean models said could not happen, and a week-long freeze in Texas that the grid models rated very unlikely. Both come from the same assumption, and that assumption is what this book sets out to test.

Seven weather stations, fifty-one years of daily lows, and one question: how often are several stations cold for days on end? A gentle version of the question barely notices dependence between stations. A severe version depends on it entirely.

Duration is the weapon. One cold day is a nuisance, but seven in a row freeze pipes, drain gas pressure and take down power. A second shuffle test isolates how much of the error comes from cold days following cold days.

When one driver hits every part of a system at the same time, as the polar vortex did to Texas and Omicron did to American hospitals, the slack that was meant to be somewhere else is nowhere. A third test isolates that synchronization, and the three ratios multiply.

The Texas freeze was a weather event that became a power event, a gas event, a water event and a medical event. The models priced each risk separately and added them. The risks do not add. They multiply.

The law The same data, the same model and the same dependence give a pricing error anywhere from 2× to 54,000×, depending only on the question. It grows exponentially with how many things must go wrong at once. That is the law, R ≈ dⁿ, and common corrections such as copulas fix only part of it.

What sets d? The overall growth rate turns out to be a simple combination of the spatial and the persistence growth rates, a fit with R² of 0.998 across eleven domains from weather to finance to epidemics.

Counting the cost. A risk assessed as 50,000 times too low makes any spending on winterization look pointless. The chapter asks how much of the 246 deaths and $195 billion traces back to the pricing error, and shows that, in simulation, an insurer pricing with the independent model is ruined in every domain tested.

Given $10 billion, what should a regulator fix first? The shuffles split the error into persistence, spatial spread and cross-domain coupling, so each part can be matched to a physical mechanism. The Netherlands did something like this for compound flooding.

The whole method as forty lines of Python. It computes the independence, persistence and synchronization ratios for any table of data and any cost function. If the ratio exceeds 10, the book says the independent model is not adequate. If you do not program, skip to the output.

The same code on twenty-five datasets, from weather and rivers to finance, epidemics, earthquakes and drought. Some are dominated by persistence, some by synchronization, some by both. The law says any system with dependent parts and a threshold cost is mispriced.

Not yet computed An AI safety evaluation is also a matrix. Each dangerous capability is tested on its own, but the risk is several at once for several turns in a row. The law predicts the independent check underestimates it by dⁿ. The chapter says plainly that the computation has not been done.

Falsifiable prediction Why the law hid for ninety-one years, from Fisher's p-value in 1935 to Markowitz's choice of variance in 1952, and a prediction that can fail: if the next compound infrastructure failure shows the independent model off by less than 100×, the law is wrong.

Where the events, data and figures in each chapter come from.

The full reference list.

Plain definitions of the book's terms, such as column shuffle, row shuffle, lag shift, pricing ratio, persistence and dependence strength, each with the chapter where it appears.