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Jan 11, 2010

Can the future be modelled?

[This post is based on an earlier one posted, in French, about a month ago]

My principal research area is complex systems modelling. Without getting too technical, the main purpose is to develop computer-based representation of some interesting phenomena.
There is no definite consensus in the scientific community as to when a system becomes complex, but you will probably agree that genetics, the climate, the immune system, or even financial interactions between stock markets, are "complex".

In future posts, I will probably try to detail some of the techniques that can be used, (as many are quite interesting!), but today's focus is on the concept of future.
The main goal of our work is to better understand these systems. In this process, proposing predictive tools, (and validating them), is often essential.

The main limitation is that, by nature, any model corresponds to a choice: it does not contain all the components of the real system. Otherwise, this would imply a complete understanding of that system, which would contradict the original motivation for this work.
Any model is, therefore, an imperfect representation of the corresponding system, and is generally valid only under certain conditions.

As a consequence, a model does not predict *the* future, but provides a set of possible scenarios, (these being more or less probable, depending on model realism).
On one of my projects, I am working on models of epidemic spreads. These models, (mine as well as those developed by colleagues elsewhere), will never "predict" the exact evolution of an ongoing infectious outbreak. Even a perfect model would be unable to provide this.

Let us assume, for the sake of the argument, that such a model exists, and that a simulation is run, based on the infection of an avatar for Mr Smith, who was just diagnosed with an emerging disease. The "perfect model" returns terrible news: according to the latest simulation, Mrs Smith probably is already infected, and the disease will rapidly spread through the entire population.
Yet, a week later, no further case was diagnosed. After extensive investigations, it becomes apparent that, at the very moment Mr Smith was put in quarantine, his wife, (who was very worried about these developments), suffered from a heart attack and passed away. She never had a chance to infect others, and the outbreak never occurred.

This is a made-up example, but the conclusion is nonetheless very clear: the future does not exist. There is a set of possible evolutions, and the present is the result of one of these taking shape. A predictive model "only" reduces the set and identifies the most likely outcomes, (which is already very useful!).

Another crucial aspect is the time scale used for the model. For instance, in the context of climate change, a long-term temperature increase does not mean that that every day will be warmer than the previous one, or even that every year will be warmer than the previous one.

Let us consider an artificial system, in which we introduced a forced increase, and some random variations. We could for instance generate a series of values where the i-th element is given by the equation below, (based on a normal distribution). In short, if Mathematics are not really your thing, the first value is obtained from small variations around 995, the second from similar variations around 996, the third around 998, the fourth 1001, and so on.

This results in a typical evolution, shown in the figure below. On the one hand, from one value to the next, the difference can be quite large, and can be positive or negative, (dotted line). On the other hand, the average value over the last ten values, (red line), is more stable, and largely on the increase.
In particular, it is interesting to look at values 40 to 60: variations are significant, and include severe dips, but the average value keeps increasing.
With a little imagination, it is easy to understand why denying global warming on the basis of a poor summer or a very cold winter does not make any sense, (nor, conversely, does it make any sense to point to a single warm summer or mild winter to "prove" global warming).

In some systems, there might be a positive feedback: the higher the current value, the more likely its increase becomes, (and the lower it is, the more likely it is to decrease further). This can for instance be taken into account using the following equation. Instead of using 995+i for the i-th value, we take the previous value as a reference point.

The overall behaviour is more stable, but periods of "negative growth" also tend to last longer, (see values 25-30 below). This model is very crude of course, but does not seem too far from some phenomena observed with stock values, (Finance is not my strong point, so correct me if I am wrong).

The time scale is, therefore, crucial. A trader will monitor the stock values by the minute, while a pensioner will be worried about the long-term evolution, over several years. Similarly, climate science will investigate changes over decades and centuries, while meteorology focuses on the hours and days immediately ahead.

This implies the development of specific models, which are developed to answer precise questions about the system. As a result, you should always be careful when someone uses the fact that weather forecasts are not accurate for more than a few days ahead as an argument to doubt the validity of climate model.