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New Arcsine Climate Forecast: Hot and Cold!

If you weren’t worried before, then take a look at this shocking new climate forecast!
Arcsine climate forecast

No, only kidding. This is the real forecast:
Arcsine climate forecast

Sorry. Can’t help myself. Here are four more “forecasts”.

Arcsine climate forecast

Each of the “forecasts” were generated by what is called a “random walk.” Here is what that is. Grab a coin and go out and stand on a corner of some sidewalk that stretches for a long way in both directions. Call one direction “positive” and the other “negative”. The corner you start at will be called “zero”. Flip your coin: If it is heads, then take one step forward toward positive; if tails, then take one backward toward negative. Keep doing this for a long time and soon you will find…that your neighbors think you are crazy.

But that’s a random walk. If you do the coin flips and steps for a long enough time, you’ll find that you spend a heck of lot more time than you might have guessed on either the positive or the negative side. You will probably find that, when you quit, you are way up along the positive side, or way down along the negative. This is true even though the average of those coin flips, the +1s and -1s that make up your steps, is pretty near 0; and even though the average goes to 0 the longer you flip the coin.

The “climate model forecasts” generated above were done so by reference to a paper by A.H. Gordon, available here, called “Global Warming as a Manifestation of a Random Walk”. It is a very readable paper that bears attention.

Gordon proposed that a climate could be made by generating random “shocks” to a climate system. What’s that? Well, imagine the climate is going along peacefully, maintaining its temperature and minding it’s own business, when suddenly—bame!—some external force causes it to change its temperature up or down. An external force might be a change in the Earth’s orbit, or a shifting in cloud cover, or a flock of birds flying this way or that, or anything. This shock persists in the system for some time; little shocks build up and over the course of a year the climate—the mean temperature—changes. It is just as likely for this random-walk climate’s temperature to go up as it is to go down,.

Random walks have some surprising properties which, by virtue of being surprising, are not intuitive. The first is that we’d expect adding random ups and downs (1s and -1s) together would get us a bunch of no changes (or 0s). We don’t get 0s, but numbers which travel far from 0 as time goes on. In fact, it can be shown—via something called the arcsine law—that it’s more probable that this climate will be at an extreme value whenever the series stops, and will not be near 0. The pictures show this.

What about the real climate, the one we actually live in? It’s certainly true that the real climate experiences external shocks of every kind. Gordon found (over the period he looked and with one particular, often used data set) that temperatures went up about just as many times as it went down, just like what would be expected in a random walk climate. He found that the value of the temperature at the end of the series he had was an extreme one, just like we would be expect in a random walk climate. He made a lot of pictures, like we have, and noticed that a lot of them look just like our real climate.

The pictures that make up our and Gordon’s “arcsine forecast”, for technical reasons, aren’t 1s and -1s, but numbers simulated from a normal distribution with a central parameter of 0, which means the numbers are equally likely to be above 0 as below 0 just like in the -1/+1 random walk, but here they can be any number greater or less than 0 (the standard deviation parameter, for those who know of such things, is set at 0.12, which is the same as the estimated standard deviation parameter for actual global mean temperature; see Gordon’s paper for a fuller description).

What does all this say about the real climate? That it happens to look just like a bunch of random numbers. Gordon cautions, “That is not to say that the temperature record is a random walk, but that it does possess similar features.” The surface temperature records “also exhibits properties of the arc-sine law. It is concluded that the global series could have arisen from random fluctuations and could therefore be analogous to arc-sine law governed by random walks.”

This means the climate we have might be less controllable than we thought it was (controllable negatively or positively through man’s activities).

He ends with some sage advice:

It is important to examine all ways and means by which the observed data series develop trends before facing hard and fast conclusions that any particular activity is the one and only responsible agent.

Below is the code where you can generate your own arcsine climate model forecast in R: