Problem 3. Why is the concept of ‘climate’ distinguished from the concept of ‘weather’ by an arbitrary free parameter, usually involved in averaging or smoothing or ’scale’ 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 ‘weather’ and more certain ‘climate’ 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 wouldn’t, as much as it’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.