On record-breaking extremes

It is a good tradition in science to gain insights and build intuition with the help of thought-experiments. Let’s perform a couple of thought-experiments that shed light on some basic properties of the statistics of record-breaking events, like unprecedented heat waves. I promise it won’t be complicated, but I can’t promise you won’t be surprised.

Assume there is a climate change over time that is U-shaped, like the blue temperature curve shown in Fig. 1. Perhaps a solar cycle might have driven that initial cooling and then warming again – or we might just be looking at part of a seasonal cycle around winter. (In fact what is shown is the lower half of a sinusoidal cycle.) For comparison, the red curve shows a stationary climate. The linear trend in both cases is the same: zero.

Fig. 1.Two idealized climate evolutions.

These climates are both very boring and look nothing like real data, because they lack variability. So let’s add some random noise – stuff that is ubiquitous in the climate system and usually called ‘weather’. Our U-shaped climate then looks like the curve below.

Fig. 2. “Climate is what you expect, weather is what you get.” One realisation of the U-shaped climate with added white noise.

So here comes the question: how many heat records (those are simply data points warmer than any previous data point) do we expect on average in this climate at each point in time? As compared to how many do we expect in the stationary climate? Don’t look at the solution below – first try to guess what the answer might look like, shown as the ratio of records in the changing vs. the stationary climate.

When I say “expected on average” this is like asking how many sixes one expects on average when rolling a dice a thousand times. An easy way to answer this is to just try it out, and that is what the simple computer code appended below does: it takes the climate curve, adds random noise, and then counts the number of records. It repeats that a hundred thousand times (which just takes a few seconds on my old laptop) to get a reliable average.

For the stationary climate, you don’t even have to try it out. If your series is n points long, then the probability that the last point is the hottest (and thus a record) is simply 1/n. (Because in a stationary climate each of those n points must have the same chance of being the hottest.) So the expected number of records declines as 1/n along the time series.

Ready to look at the result? See next graph. The expected record ratio starts off at 1, i.e., initially the number of records is the same in both the U-shaped and the stationary climate. Subsequently, the number of heat records in the U-climate drops down to about a third of what it would be in a stationary climate, which is understandable because there is initial cooling. But near the bottom of the U the number of records starts to increase again as climate starts to warm up, and at the end it is more than three times higher than in a stationary climate.

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