# Problem 3 of Climate Science

Problem 3. Why is the concept of â€˜climateâ€™ distinguished from the concept of â€˜weatherâ€™ by an arbitrary free parameter, usually involved in averaging or smoothing or â€™scaleâ€™ transformations of 10 to 30 years?

The recent article on Question #9 by Meiers and response by Stephen Goddard used a coin toss analogy to answer this question. Meiers states that while the uncertainty of the probability of heads in the short term is high, over the long term we expect the answer to become more certain as we average over more examples. This, he says, is an analogy to weather and climate, and why climate models can be poor at predicting the immediate future, yet can be reliable in the more distant future.

A series containing a unit root contradicts this intuition.

To illustrate I have developed an excel spreadsheet DStrends with a deterministic and stochastic trend (call them I(0) deterministic and I(1) stochastic). It generates a new set of random numbers and trends every time it is reset.

Given a coin toss at time t of Y(t) with equal probability (values -1, 0 and 1) the two series are given by:

D(t) = 0.01t + Y(t)
S(t) = 0.01 + Y(t) + S(t-1)

There is an underlying trend of 0.01t in each series. The two series are quite different as can be seen in the figure below. While the deterministic series shows a slight upward trend, the trend is drowned out by the variability in the series with the unit root.

The uncertainty in each series can be seen by plotting the standard error of the mean of each series for increasing observations. The standard error is given by the standard deviation divided by the square root of the number of observations, and represents the uncertainty in the estimation of the mean value.

The standard error of the deterministic series starts high and decreases. With a low trend (0.01 per year) the uncertainty will keep decreasing. With higher trends (0.1 per year) there will be a minimum around thirty observations (years?) before increasing again as the trend becomes more important.

The standard error of the stochastic series is constant over the range of observations (run the spreadsheet a number of times for yourself). Constant uncertainty is theoretically predicted for a series with a unit root, no-matter what scale we look at the observations.

This illustrates clearly that the notion of uncertain â€˜weatherâ€™ and more certain â€˜climateâ€™ only makes sense for a deterministic series (as shown by the local minimum of uncertainty around 30 observations). There is no sense in a weather/climate distinction when a unit root is present as there is no scale over which estimates of the mean become less uncertain.

So why is there a distinction between weather and climate? The answer is because of the assumption that global temperature is a deterministic trend. However, all the empirical evidence points to the overwhelming effects of a unit (or near unit) root in the global temperature series, which means the deterministic trend is a false assumption.

Dr Meiers proposed a bet which illustrates his erroneous assumption:

How can a more complex situation be modeled more easily and accurately than a simpler situation? Let me answer that with a couple more questions:1. You are given the opportunity to bet on a coin flip. Heads you win a million dollars. Tails you die. You are assured that it is a completely fair and unbiased coin. Would you take the bet? I certainly wouldnâ€™t, as much as itâ€™d be nice to have a million dollars.2. You are given the opportunity to bet on 10000 coin flips. If heads comes up between 4000 and 6000 times, you win a million dollars. If heads comes up less than 4000 or more than 6000 times, you die. Again, you are assured that the coin is completely fair and unbiased. Would you take this bet? I think I would.

If the question was, “Is the coin biased?” and the observable (like temperature) was the final value of an integrating variable I(1), then Dr Meiers’ bet after 10000 would actually be very unwise, as the value of the observable (positive or negative) after 10000 flips is no more certain than after 1 or 100.

## 0 thoughts on “Problem 3 of Climate Science”

1. guest says:

Doesn’t that 0.01t term in the S(t) equation mean that it has a quadratic trend, rather than a linear trend? It seems to me that the equation should read S(t) = 0.01 + Y(t) +S(t-1).

davids99us: Fixed, thanks.

2. guest says:

Doesn't that 0.01t term in the S(t) equation mean that it has a quadratic trend, rather than a linear trend? It seems to me that the equation should read S(t) = 0.01 + Y(t) +S(t-1).

3. Anonymous says:

David, tonight I was on Google Earth at the big Chinese earthquake at 33 16 34, 96 37 38. If you go to any of several places about 40 km N-W of there, you will see a number of patterns of long mountain ridges with numerous, roughly parallel incised creeks. Looks a bit like a coarse comb, but there are several similar systems. I have often wondered why the creeks have such regularity of spacing. One set I’m looking at has an 800m separation. Indeed much of the aerial pattern of this part of the world shows this type of repetition. There’s a wonderful example at 26 46 50, 98 56 00 and points further north, the famed three rivers parallel system, view from eye height of 800 km or closer. I have never seen a power spectrum type exercise done on the frequency of the separation distances, but I’d guess that there were some prominent spikes. To me, such spikes divide one type of mechanism from another. In that way, they are roughtly analagous to your problem of separation of climate from weather. Personally, I feel that weather grades into climate and there is no useful cyclicity period that sets them apart. But that would be my approach, a naive power spectrum on numerous stations over 100 years+ to see if there is another repetition besides night/day, Milankovitch, annual seasons and all those other terms already known. If there is no data discontinuity, I do not see the need for a nomenclature discontinuity.

That Meiers coin flip example above was naive, as I posted the other day on WUWT, stipulating a perfect coin. (Looking at the results of flips is not forecasting, it is looking at past data. Besides, a 10,000 flip will not produce a range from 4,000 to 6,000 except with a hugely small probability, so the time it would take to get the experiment right is time enough for weather to change to climate).

I’m really enjoying your examples of the use of unit roots as I have not been there before.

Have you already solved problem #2?

• Anonymous says:

Geoff, a cyclic would create a small blip down in the uncertainty, as for example with an 11 year cycle the value would be a bit more certain 11 years out, but overall the pattern would be the same and not produce thresholds or steps.

The ‘sweet spot’ at 30 years of minimum uncertainty reflects the sum of the relative variation from the (decreasing) contribution of a well behaved, Gaussian three-sided coin, and the (increasing) contribution from the slope of the trend. However, you have to assume that the world is like that (ie Y=mt+e), which by all indications it is not.

I am glad of the positive feedback, because from the lack of comment I felt noone is getting this, which means I have more work to do explaining it.

I have solved all of the problems but they are ‘up here’. You can take a stab if you like.

4. sherro says:

David, tonight I was on Google Earth at the big Chinese earthquake at 33 16 34, 96 37 38. If you go to any of several places about 40 km N-W of there, you will see a number of patterns of long mountain ridges with numerous, roughly parallel incised creeks. Looks a bit like a coarse comb, but there are several similar systems. I have often wondered why the creeks have such regularity of spacing. One set I'm looking at has an 800m separation. Indeed much of the aerial pattern of this part of the world shows this type of repetition. There's a wonderful example at 26 46 50, 98 56 00 and points further north, the famed three rivers parallel system, view from eye height of 800 km or closer. I have never seen a power spectrum type exercise done on the frequency of the separation distances, but I'd guess that there were some prominent spikes. To me, such spikes divide one type of mechanism from another. In that way, they are roughtly analagous to your problem of separation of climate from weather. Personally, I feel that weather grades into climate and there is no useful cyclicity period that sets them apart. But that would be my approach, a naive power spectrum on numerous stations over 100 years+ to see if there is another repetition besides night/day, Milankovitch, annual seasons and all those other terms already known. If there is no data discontinuity, I do not see the need for a nomenclature discontinuity.That Meiers coin flip example above was naive, as I posted the other day on WUWT, stipulating a perfect coin. (Looking at the results of flips is not forecasting, it is looking at past data. Besides, a 10,000 flip will not produce a range from 4,000 to 6,000 except with a hugely small probability, so the time it would take to get the experiment right is time enough for weather to change to climate).I'm really enjoying your examples of the use of unit roots as I have not been there before.Have you already solved problem #2?

5. davids99us says:

Geoff, a cyclic would create a small blip down in the uncertainty, as for example with an 11 year cycle the value would be a bit more certain 11 years out, but overall the pattern would be the same and not produce thresholds or steps. The 'sweet spot' at 30 years of minimum uncertainty reflects the sum of the relative variation from the (decreasing) contribution of a well behaved, Gaussian three-sided coin, and the (increasing) contribution from the slope of the trend. However, you have to assume that the world is like that (ie Y=mt+e), which by all indications it is not.I am glad of the positive feedback, because from the lack of comment I felt noone is getting this, which means I have more work to do explaining it.

6. Anonymous says:

Thanks, David. Point taken re cyclic.

In concept, I find it difficult to make the jump between mechanism and math analysis, in a worse way than ‘correlation need not be causation’. The example I gave of the Chinese creeks is amenable to math analysis as to frequency, variability etc., but what Nature does to create the pattern remains elusive.

I’m falling into line with the group that suggests that there are multiple principal climate mechanisms, some feeding off each other, some independent; so that an overall analysis of a lone consequence like temperature or rainfall would be formidable unless they could be dissected out cleanly. In such an event 30 years is but a passing figure and it is not useful to talk of a weather/climate divide.

The 1960s divergence in some tree ring studies is an example of a math observation with a puzzling mechanism. The divergence is assumed to be climate related, but it need not be.

No way am I capable of answering Q2. I was joking with you.

• Anonymous says:

In most models, the probability distribution is used to model the errors only, and the ‘signal’ is a deterministic function. But it is possible to model the ‘signal’ with a probability distribution. This was the theme of my book, its the basis of DemetrisK’s approach too I think. Eg. in the case of the unit root:

delta T = probability distribution

Which suggests to me we should be comparing distributions with distribution, not average values with average values. That is what I did with the distribution of species and it works very well.

7. sherro says:

Thanks, David. Point taken re cyclic. In concept, I find it difficult to make the jump between mechanism and math analysis, in a worse way than 'correlation need not be causation'. The example I gave of the Chinese creeks is amenable to math analysis as to frequency, variability etc., but what Nature does to create the pattern remains elusive. I'm falling into line with the group that suggests that there are multiple principal climate mechanisms, some feeding off each other, some independent; so that an overall analysis of a lone consequence like temperature or rainfall would be formidable unless they could be dissected out cleanly. In such an event 30 years is but a passing figure and it is not useful to talk of a weather/climate divide.The 1960s divergence in some tree ring studies is an example of a math observation with a puzzling mechanism. The divergence is assumed to be climate related, but it need not be.No way am I capable of answering Q2. I was joking with you.

8. davids99us says:

In most models, the probability distribution is used to model the errors only, and the 'signal' is a deterministic function. But it is possible to model the 'signal' with a probability distribution. This was the theme of my book, its the basis of DemetrisK's approach too I think. Eg. in the case of the unit root:delta T = probability distributionWhich suggests to me we should be comparing distributions with distribution, not average values with average values. That is what I did with the distribution of species and it works very well.

9. Anonymous says:

David, “we should be comparing distributions with distribution”

Times change. IIRC, we used to derive distributions and either not compare them or adjust them to comparable forms before comparing them. It was one of the first steps our stats people did with new data. Do you mean that this is not standard procedure any more?

10. sherro says:

David, “we should be comparing distributions with distribution”Times change. IIRC, we used to derive distributions and either not compare them or adjust them to comparable forms before comparing them. It was one of the first steps our stats people did with new data. Do you mean that this is not standard procedure any more?

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