From time to time, there is discussion about whether the recent warming trend is due just to chance. We have heard arguments that so-called ‘random walk‘ can produce similar hikes in temperature (any reason why the global mean temperature should behave like the displacement of a molecule in Brownian motion?). The latest in this category of discussions was provided by Cohn and Lins (2005), who in essence pitch statistics against physics. They observe that tests for trends are sensitive to the expectations, or the choice of the null-hypothesis .
Cohn and Lins argue that long-term persistence (LTP) makes standard hypothesis testing difficult. While it is true that statistical tests do depend on the underlying assumptions, it is not given that chosen statistical models such as AR (autoregressive), ARMA, ARIMA, or FARIMA do provide an adequate representation of the null-distribution. All these statistical models do represent a type of structure in time, be it as simple as a serial correlation, persistence, or more complex recurring patterns. Thus, the choice of model determines what kind of temporal pattern one expects to be present in the process analysed. Although these models tend to be referred to as ‘stochastic models’ (a random number generator is usually used to provide underlying input for their behaviour), I think this is a misnomer, and think that the labels ‘pseudo-stochastic’ or ‘semi-stochastic’ are more appropriate. It is important to keep in mind that these models are not necessarily representative of nature – just convenient models which to some degree mimic the empirical data. In fact, I would argue that all these models are far inferior compared to the general circulation models (GCMs) for the study of our climate, and that the most appropriate null-distributions are derived from long control simulations performed with such GCMs. The GCMs embody much more physically-based information, and do provide a physically consistent representation of the radiative balance, energy distribution and dynamical processes in our climate system. No GCM does suggest a global mean temperature hike as observed, unless an enhanced greenhouse effect is taken into account. The question whether the recent global warming is natural or not is an issue that belongs in ‘detection and attribution’ topic in climate research.
One difficulty with the notion that the global mean temperature behaves like a random walk is that it then would imply a more unstable system with similar hikes as we now observe throughout our history. However, the indications are that the historical climate has been fairly stable. An even more serious problem with Cohn and Lins’ paper as well as the random walk notion is that a hike in the global surface temperature would have physical implications – be it energetic (Stefan-Boltzmann, heat budget) or dynamic (vertical stability, circulation). In fact, one may wonder if an underlying assumption of stochastic behaviour is representative, since after all, the laws of physics seem to rule our universe. On the very microscopical scales, processes obey quantum physics and events are stochastic. Nevertheless, the probability for their position or occurrence is determined by a set of rules (e.g. the Schrödinger’s equation). Still, on a macroscopic scale, nature follows a set of physical laws, as a consequence of the way the probabilities are detemined. After all, changes in the global mean temperature of a planet must be consistent with the energy budget.
Is the question of LTP then relevant for testing a planet global temperature for trend? To some degree, all processes involving a trend also exhibit some LTP, and it is also important to ask whether the test by Cohn and Lins involves circular logic: For our system, forcings increase LTP and so an LTP derived from the data, already contains the forcings and is not a measure of the intrinsic LTP of the system. The real issue is the true degrees of freedom – number of truely independent observations – and the question of independent and identically distributed (iid) data. Long-term persistence may imply dependency between adjacent measurements, as slow systems may not have had the time to change appreciably between two successive observations (the same state is more or less observed in the successive measurements). Are there reasons to believe that this is the case for our planet? Predictions for subsequent month or season (seasonal forecasting) is tricky at higher latitudes but reasonably skilful regarding El Nino Southern Oscillation (ENSO). However, it is extremely difficult to predict ENSO one or more years ahead. The year-to-year fluctuations thus tend to be difficult to predict, suggesting that LTP is not the ‘problem’ with our climate. On the other hand, there is also the thermal momentum in the oceans which implies that the radiative forcing up to the present time is going has implications for following decades. Thus, in order to be physically consistent, arguing for the presence of LTP also implies an acknowledgement of past radiative forcing in the favour for an enhanced greenhouse effect, since if there were no trend, the oceanic memory would not be very relevant (the short-term effects of ENSO and volcanoes would destroy the LTP).
Another common false statment, which some contrarians may also find support for from the Cohn and Lins paper, is that the climate system is not well understood. I think this statement is somewhat ironic, but the people who make this statement must be allowed to talk for themselves. If this statement were generally true, then how could climate scientists make complex models – GCMs – that replicate the essential features of our climate system? The fact that GCMs exist and that they provide a realistic description of our climate system, is overwhelming evidence demonstrating that such statement must be false – at least concerning the climate scientists. I’d like to iterate this: If we did not understand our atmosphere very well, then how can a meteorologist make atmospheric models for weather forecasts? It is indeed impressing to see how some state-of-the-art atmopsheric-, oceanic models, and coupled atmospheric-oceanic GCMs reproduce features such as ENSO, the North Atlantic Oscillation (or Arctic or Antarctic Oscillation) on the larger scales, as well as smaller scale systems such as mid-latitude cyclones (the German model ECHAM5 really produces impressive results for the North Atlantic!) and Tropical Instability Waves with such realism. The models are not perfect and have some shortcomings (eg clouds and planetary boundary layer), but these are not necassarily due to a lack of understanding, but rather due to limited computational resources. Take an analogy: how the human body works, conscienceness, and our minds. These are aspects the medical profession does not understand in every detail due to their baffling complexity, but medical doctors nevertheless do a very good job curing us for diseases, and shrinks heal our mental illnesses.
In summary, statistics is a powerful tool, but blind statistics is likely to lead one astray. Statistics does not usually incorporate physically-based information, but derives an answer from a set of given assumptions and mathematical logic. It is important to combine physics with statistics in order to obtain true answers. And, to re-iterate on the issues I began with: It’s natural for molecules under Brownian motion to go on a hike through their random walks (this is known as diffusion), however, it’s quite a different matter if such behaviour was found for the global planetary temperature, as this would have profound physical implications. The nature is not trendy in our case, by the way – because of the laws of physics.
Update & Summary
This post has provoked various responses, both here and on other Internet sites. Some of these responses have been very valuable, but I believe that some of these are based on a misunderstanding. For instance, some seem to think that I am claiming that there is no auto correlation in the temperature record! For those who have this impression, I would urge to please read my post more carefully, because it is not my message. The same comments goes for those who think that I’m arguing that the temperature is iid, as this is definitely not what I say. It is extremely important to be able to understand the message before one can make a sensible response.
I will try to make a summary of my arguments and the same time address some of the comments. Planetary temperatures are governed by physics, and it is crucial that any hypotheses regarding their behaviour are both physically as well as statistically consistent. This does not mean that I’m dismissing statistics as a tool. Setting up such statistical tests is often a very delicate exercise, and I do question whether the ones in this case provide a credible answer.
Some of the response to my post on other Internet sites seem to completely dismiss the physics. Temperature increases involve changes in energy (temperature is a measure for the bulk kinetic energy of the moleclues), thus the first law of thermodynamics must come into consideration. ARIMA models are not based on physics, but GCMs are.
When ARIMA-type models are calibrated on empirical data to provide a null-distribution which is used to test the same data, then the design of the test is likely to be seriously flawed. To re-iterate, since the question is whether the observed trend is significant or not, we cannot derive a null-distribution using statistical models trained on the same data that contain the trend we want to assess. Hence, the use of GCMs, which both incorporates the physics, as well as not being prone to circular logic is the appropriate choice.
There seems to be a mix-up between ‘random walk’ and temperatures. Random walk typically concerns the displacement of a molecule, whereas the temperature is a measure of the average kinetic energy of the molecules. The molecules are free to move away, but the mean energy of the molecules is conserved, unless there is a source (first law of thermodynamics). [Of course, if the average temperature is increased, this affects the random walk as the molecules move faster (higher speed).]