Tuesday, August 12, 2008

Climate Change Detection

You'll want some dice or a random number generator for our first efforts to think about climate change detection. Start with one six-sided die. We'll assume that it's a fair die -- that each face is just as likely as any other to turn up. If we toss it many times, the average of the numbers that shows up will be 3.5.

Weather and Climate
This is one of our distinctions between weather and climate. Weather is what we saw on any given throw and climate is that average. But 3.5 is not a number that you can get on any single throw. What's up with this? It turns out that this, too, is a reasonable thing for thinking about climate. If you look at some very small area, and only one parameter (say temperature), it's possible that you'll see the 'climatic norm' occur. But only for that small area and limited look. As soon as you look at a large scale, you find that although the weather (instantaneous state) can be generally close to climate, it is seldom close everywhere. See, for example, this sea surface temperature anomaly map(difference from climatology).

The 1 die model of climate doesn't work very well. On the map, we see that most of the area is near climatology, and that the farther away we are (hot or cold), the less area is present. A better model then is to use 5 dice (again 6 sided). In this case our 'climate' is a total of 17.5, which still doesn't happen as weather, although you can get close. The maximum is 30, and the minimum is 5. The maximum difference from climatology is 12.5. But most of the time we'll be close to climatology. Take 5 dice, throw them, add them up and record the result for a couple hundred throws. (If you're moderately quick with your addition, this takes only a few minutes; I've done it.)

Within your 200 totals, you'll find some runs of constantly increasing values, and about as many runs of constantly decreasing values. Some of the runs will be long, and some short. But the number of each is about the same whether it's increasing or decreasing. If you compute the average of the first 5 sums, then 10 sums, ... out to the whole 200, you'll see that the average wobbles around. It tends to be closer to 17.5 as time goes on (as you average more tosses) but only in a very jerky fashion.

Climate Change
Suppose I take one of your dice and tape a 6 over the side that should read 1. The average throw now totals 18.3333 (repeating), instead of 17.5. The minimum is now 6 instead of 5, but the maximum is unchanged. What will happen, though, is we'll see the high totals more often. Each of these is more or less a fair description of what we've seen in the climate of the last 120 years -- warmer minimum temperatures, higher averages, and not much change in maximum temperatures. (Not a perfect analog, but pretty good for only 5 dice.)

Repeat the exercise of tossing the dice and adding them up. How long do you have to do it before the average is obviously different from the first run? Remember that the first time the average jumped around for a while. You'll have to go on for longer than that this time.

How many throws do you need to make before you can tell that there are fewer low numbers? More high numbers?

In doing climate change detection professionally, these sorts of analyses are applied. Are there more extreme highs? Fewer extreme lows? Higher average? Just how many data points do we need to detect that in the statistics?

Final question: when did the climate change? When you have a long enough series of throws to detect it in the statistics reliably, or when you saw me tape over the 1 face with a 6?

7 comments:

silence said...

Of course, these are independent events, so you can sit down and use Student's T without any complications...

tamino said...

This is a nice illustration of the fact that there's both signal and noise in climate data, and it can take quite a while for the signal to emerge from the noise. I posted with the same theme here.

But I'm skeptical of your claim that there's been not much change in maximum temperatures. My examination of European daily station data indicates a pronounced increase in very hot daily high temperatures. Do you have contradictory evidence?

Penguindreams said...

Silence: You can, once you decide that you can apply a test aimed at a continuum in a discrete case.

Tamino: We don't have a mutual contradiction. Your analysis nicely shows an example of the heatwave problem -- that there are more high temperatures than previously. The loaded dice will do this as well. What you don't have, on large scale, is something showing that the highest highs are different than before.

The paper I've got in mind is Tom Karl and company's which showed that the minimum temperatures had been rising much faster than the maxima (0.84 vs. 0.28 C between 1951 and 1990). There are several other papers on the topic as a quick look in Google scholar will show. A search on: Karl minimum temperature, will get you started. (Full text of this paper is available.)

So yes, maximum temperatures are up. But not nearly as much as minimum. For my 5 dice example, it seems sufficient to get the effect on minimum values.

Michael Hauber said...

A lazier alternative of the dice throwing is to generate some random numbers in MS Excel, or whatever other tool you may have.

Generate 5 random numbers from 0 to 1 for each month, add them all up. Chart the result, and be surprised at how often you see something that looks like it might just be a statistically significant trend in data you know should be random and trendless.

Or is the Excel random number generator flawed?

Penguindreams said...

I don't know of any flaws in the Excel random number generator, but I also haven't checked. I'd probably use a spread sheet or short program myself.

In envisioning a class activity, I think there are some pluses to having actual dice being thrown and added by the students.

Ian O said...

If you substitute one 1 with a 6 doesn't that make the minimum 10?
Sorta like what is happening in the Arctic.....

Penguindreams said...

IanO: No; I guess I didn't show it clearly enough. We have 5 dice. 4 of them we leave unchanged, showing 1,2,3,4,5,6.

The 5th, we change to show 6,2,3,4,5,6. The minimum total will be 1 (4 times) plus 2 (the minimum on our loaded die), so 6. If we loaded the 5th die so that all faces were 6, you're right.

It's possible that that version would be more accurate than the one I meant. But I think my original is better ... for now.