Skip to main content

Scaling up probabilities in space

Suppose you have some location or small area, call it location A, and you have decided for this location the 1-in-100 year event for some magnitude in that area is ‘x’. That is to say, the probability of an event with magnitude exceeding ‘x’ in the next year at location A is 1/100. For clarity, I would rather state the exact definition, rather than say ‘1-in-100 year event’.

Now suppose you have a second location, call it location B, and you are worried about an event exceeding ‘x’ in the next year at either location A or location B. For simplicity suppose that ‘x’ is the 1-in-100 year event at location B as well, and suppose also that the magnitude of events at the two locations are probabilistically independent. In this case “an event exceeding ‘x’ in the next year at either A or B” is the logical complement of “no event exceeding ‘x’ in the next year at A, AND no event exceeding ‘x’ in the next year at B”; in logic this is known as De Morgan’s Law. This gives us the result:

Pr(an event exceeding ‘x’ in the next year at either A or B) = 1 – (1 – 1/100) * (1 – 1/100).

This argument generalises to any number of locations. Suppose our locations are numbered from 1 up to n, and let ‘p_i’ be the probability that the magnitude exceeds some threshold ‘x’ in the next year at location i. I will write ‘somewhere’ for ‘somewhere in the union of the n locations’. Then, assuming probabilistic independence as before,

Pr(an event exceeding ‘x’ in the next year somewhere) = 1 – (1 – p_1) * … * (1 – p_n).

If the sum of all of the p_i’s is less than about 0.1, then there is a good approximation to this value, namely

Pr(an event exceeding ‘x’ in the next year somewhere) = p_1 + … + p_n, approximately.

But don’t use this approximation if the result is more than about 0.1, use the proper formula instead.

One thing to remember is that if ‘x’ is the 1-in-100 year event for a single location, it is NOT the 1-in-100 year event for two or more locations.  Suppose that you have ten locations, and x is the 1-in-100 year event for each location, and assume probabilistic independence as before.  Then the probability of an event exceeding ‘x’ in the next year somewhere is 1/10. In other words, ‘x’ is the 1-in-10 year event over the union of the ten locations. Conversely, if you want the 1-in-100 year event over the union of the ten locations then you need to find the 1-in-1000 year event at an individual location.

These calculations all assumed that the magnitudes were probabilistically independent across locations. This was for simplicity: the probability calculus tells us exactly how to compute the probability of an event exceeding ‘x’ in the next year somewhere, for any joint distribution of the magnitudes at the locations. This is more complicated: ask your friendly statistician (who will tell you about the awesome inclusion/exclusion formula). The basic message doesn’t change, though. The probability of exceeding ‘x’ somewhere depends on the number of locations you are considering. Or, in terms of areas, the probability of exceeding ‘x’ somewhere depends on the size of the region you are considering.

Blog post by Prof. Jonathan Rougier, Professor of Statistical Science.

First blog in series here.

Second blog in series here.

Third blog in series here.

Fourth blog in series here.

Popular posts from this blog

Bristol Future’s magical places: Sustainability through the eyes of the community

“What is science? Why do we do it?”. I ask these questions to my students a lot, in fact, I spend a lot of time asking myself the same thing.

And of course, as much as philosophy of science has thankfully graced us with a lot of scholars, academics and researchers who have discussed, and even provided answers to these questions, sometimes, when you are buried under piles of papers, staring at your screen for hours and hours on end, it doesn’t feel very science-y, does it?

 As a child I always imagined the scientist constantly surrounded by super cool things like the towers around Nicola Tesla, or Cousteau being surrounded by all those underwater wonders. Reality though, as it often does, may significantly differ from your early life expectations. I should have guessed that Ts and Cs would apply… Because there is nothing magnificent about looking for that one bug in your code that made your entire run plot the earth inside out and upside down, at least not for me.

I know for myself, I…

The new carbon economy - transforming waste into a resource

As part of Green Great Britain Week, supported by BEIS, we are posting a series of blogs throughout the week highlighting what work is going on at the University of Bristol's Cabot Institute for the Environment to help provide up to date climate science, technology and solutions for government and industry.  We will also be highlighting some of the big sustainability actions happening across the University and local community in order to do our part to mitigate the negative effects of global warming. Today our blog will look at 'Technologies of the future: clean growth and innovation'.



On Monday 8 October 2018, the IPCC released a special report which calls upon world governments to enact policies which will limit global warming to 1.5°C compared with pre-industrial levels, failure to do so will drastically increase the probability of ecosystem collapses, extreme weather events and complete melting of Arctic sea ice. Success will require “rapid and far-reaching” actions in…

Will July’s heat become the new normal?

For the past month, Europe has experienced a significant heatwave, with both high temperatures and low levels of rainfall, especially in the North. Over this period, we’ve seen a rise in heat-related deaths in major cities, wildfires in Greece, Spain and Portugal, and a distinct ‘browning’ of the European landscape visible from space.

As we sit sweltering in our offices, the question on everyone’s lips seems to be “are we going to keep experiencing heatwaves like this as the climate changes?” or, to put it another way, “Is this heat the new norm?”

Leo Hickman, Ed Hawkins, and others, have spurred a great deal of social media interest with posts highlighting how climate events that are currently considered ‘extreme’, will at some point be called ‘typical’ as the climate evolves.
In January 2007, the BBC aired a special programme presented by Sir David Attenborough called "Climate Change - Britain Under Threat".

It included this imagined weather forecast for a "typical s…