"Unprecedented" warming (or, how certain are you of that trend?)

Global temperatures have been rising for the last 50 years right? That is not to controversial. Look at most data that attempts to produce some average temperature for the earth over time (whatever that actually means) and there is a self evident increase. It has been warmer in recent decades that it was in earlier decades.

Ignoring potential sources of error over time (including unsatisfactory adjustments for urban heat affects, areas that haven't had sufficient measurement coverage in the past or even now, such as oceans and the poles) and just taking data as given, the increase over the last 50 years appears to be around 0.8 degrees. People get hold of this data, plot a line through it, proclaim that there is a positive trend. It is from that observational position that the hypothesis of the enhanced greenhouse effect is employed to explain the trend and then extrapolate into the future.

Think carefully about this. There are two distinct points here.

• Historical observations.
• Hypothesised causes.

The first is just acting as witness to what we can measure. The latter is an application of an hypothesis.

I make this point because, in this post, I want to deal solely with the former. What exactly does the historical data tell us with any certainty in terms of behaviour? Simply saying "it has gone up" hides a multitude of sins. What I am trying to investigate is whether the data we have witnessed is indeed extraordinary. Is it so unusual that we need to search for some, or even any, explanation or cause?

I am going to do this, like all statisticians by employing a model. I am going to start with a basic naive one the explains any monthly temperature anomaly as being zero plus or minus some random element. This random element will be described by a Gaussian (or normal) distribution, be centred around zero and have a variance of about 0.01 (which means a standard deviation of 0.1).

This would be written as Tt = N(0,0.01) ; Tt= temperature anomaly at time t.

Such a model would lead to a history of temperature anomalies over say 30 years that looks something like this:

No need for fancy statistics. There is no trend here. Just stable temperature about some constant average over time, affected by some random noise from month to month. This doesn't look anything like the actual record for global temperature anomalies, which everyone agrees seems to have risen over time. And therein lies a weakness. When people look at the actual temperature series and think "this is definitely going up", they are implicitly comparing it with the model above. Because it is clearly different, an intuitive reaction from most people is that:

"there must be something going on to push up temperatures"

And from that point a leap is made to try and find some hypothesis, possibly any hypothesis, that will "explain" the cause of the obvious positive trend. But what if this model is wrong. What if the nature of temperatures is better described by an alternative model that has different characteristics. Well, you might not be surprised to find out I think that is the case.

Let's think a little more deeply about (monthly) temperature anomalies. Our naive model above assumes that the temperature in one month is unrelated in any way to the temperature in any other month. But it is reasonable to postulate that maybe temperatures are related between months. The global temperature should reflect the transfer of energy around the world. It should operate with lags and so we might expect that the temperature anomaly witnessed in any month is in part affected by the anomaly witnessed in previous months. If it was unseasonally hot in January, then it should be more likely to be unseasonally hot in February. What this describes is an auto regressive process. We can investigate the data to see if there is any reasonable sign of such a phenomenon and to try and estimate the size of it.

So that is what I did. And a simple piece of statistics indicates that there seems to about two months months of inertia. In fact it tends to indicate our model should be more like this (the 0.65 and 0.25 drop out of the analysis:

Tt = 0.65*Tt-1 + 0.25* Tt-2 + N(0,0.01) ; Tt= temperature anomaly at time t.


So, let's now see what sort of series of temperature anomalies this type of model might be expected to produce. Here is simply one example, but note that this model will produce a much greater range of apparently different results:






















Compare this with the first chart. There appears to be a clear trend (I have drawn the linear trend). In fact there is a trend that equates to 1.4 degrees per century - or about 0.7 degrees over 50 years. This is pretty much exactly what we have witnessed from actual temperature measurements over the same sorts of periods. Compare it with an actual temperature series over a similar period (this is from the University of Alabama Huntsville):




















So what is going on? I have created this series of data using a very basic and naive model that reflects the actual measured characteristics and contains in it absolutely nothing that might cause warming. The trick of course is that this is but one potential outcome. I could generate another series and it might have a cooling trend, or no trend. But the salient point remains. Simple random data over time can generate what appears to be a material trend caused by something, when it is in fact simply an random outcome well within the range of what one might expect.

If you conducted some proper statistical tests on that series and the computed trend, you would find that the trend is not statistically significant. It would confirm what I have just said, that such an apparently significant trend - as high as that which is estimated from actual temperatures - is within the range of what might be randomly generated without any physical cause.

So, when I am asked to consider whether we are at risk of producing dangerous climate change, I have trouble getting past even the first obstacle. What does the data show us. Up until now, nothing that doesn't appear within the expected range of outcomes given the characteristics of the data involved. A trend that isn't particularly special.

Now this doesn't rule out human influence on the climate, in the form of CO2 forcing, or other localised influences (such as land use change). However, it does nothing to support the argument that we are definitely experiencing dangerous rates of warming that can only be explained by physical causes over time.