## On odds of storms, and extreme precipitation

People talk about “thousand year storms”. Rather than being a storm having a recurrence time of once in a thousand years, these are storms which have a 0.001 chance per year of occurring. Storms aren’t the only weather events of significance which have probabilities of occurrence like these. Consider current precipitation risks for the Town of Westwood, Massachusetts, where I live:

I have highlighted events which have a 0.01 chance per year of occurring, including things like a rainfall of almost an inch in 5 minutes, or 8 inches in a day. Again, the recurrence time is not once in a 100 years. And, note, these are not based upon expected climate change, although there already is some change in the climate baked into these. These are current risks.

So what does 0.01 per year mean? Well, as Radley Horton explains in part below, think of it as rolling a dice (*) having 100 faces, and looking for the event “It rolled the number 10”.

Assuming the rolls are independent, calculating the likelihood of at least once of these events in N years is calculating the upper tail probability of a Binomial Distribution (Lesson 8, page 76).

##### (Figure is from Statistics How To.)

What’s that mean?

It means that, for each successive number of years, the chances of the event happening at least once is as in the following table:

number of years chance of event happening 1 or more times
1 0.010
2 0.020
4 0.039
5 0.049
8 0.077
10 0.096
15 0.140
20 0.182
25 0.222

So by the time 10 years roll by, the 0.01 event has an almost ten times greater chance of happening. If a stormwater management system in the Town of Westwood is effectively destroyed by an 8 inch rain, there’s a 1-in-10 chance of that happening in a 10 year stretch.

As climate chances, extreme precipitation events become more likely. If the 8 inch rain has a 0.01 chance per year now, it will soon have a 1-in-50 chance per year, or 0.02 per year. How does the risk table chance for that?

number of years chance of event happening 1 or more times
1 0.020
2 0.040
4 0.078
5 0.096
8 0.149
10 0.183
15 0.261
20 0.332
25 0.396

Unsurprisingly, that 1-in-10 chance takes 5 years to realize, and in 10 years there’s a slightly less than 1-in-5 chance of it happening. If the stormwater management exceedance costs \$10 million to repair, that means, in the first case that there’s an expected cost per year of \$100,000, and, in 10 years a million dollars. When climate changes to the 1-in-50, these expected losses double.
In the case of weather events, they may not be entirely independent. Events might “bunch up” due to ENSO or other influences. Similarly a big volcanic explosion can affect global weather for a year or two, and depress probabilities of weather events.

When estimating risks of events like these directly from data on occurrences, it’s important to note that Gaussian approximations to distributions or even Poissons will underestimate risk. What’s needed to be used is a Generalized Extreme Value distribution. Lee Fawcett in his article, “A severe forecast” in the current issue of Significance Magazine (December 2019) explains in greater detail. A good book explaining use of the GEV distribution is:

 E. Castillo, A. S. Hadi, N. Balakrishnan, J. M. Sarabia, Extreme Value and Related Models with Applications in Engineering and Science, Wiley, 2005.

The R statistical programming language facility offers a number of packages for doing inference with this distribution.

## About ecoquant

See https://wordpress.com/view/667-per-cm.net/ Retired data scientist and statistician. Now working projects in quantitative ecology and, specifically, phenology of Bryophyta and technical methods for their study.

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