Predictions and errors

By Javier Amezcua

Predicting is one of the most ambitious goals of science. It goes beyond describing and explaining, and it attempts to “tell the future”. The prediction process has the following basic steps:

  1. We have an estimate of the present conditions of a system, for instance, the atmosphere.
  2. We have a model –i.e. a set of mathematical rules coming from physical principles- which we evolve forward (or integrate) in time.
  3. We get an estimate of the future state of our system at any given time.

When computing a prediction, it is very important to provide a measure of the quality of this prediction. Intuition tells us that we are more certain, for example, in predicting the temperature in our neighborhood for tomorrow, than in predicting the temperature in the same place a year from now. Where does this certainty/uncertainty come from? Let us explore this next.

For the sake of this discussion, consider that the model mentioned in step (2) is perfect. That is, let us assert that have completely captured in our equation all the processes we are interested in, and that we can solve these equations perfectly with a computer code (this is not true in reality, but we will leave that for another blog entry). In this case the quality of a prediction is determined by the error of our estimate mentioned in step (1) –i.e. the error in our initial conditions– and the error growth in time.

As it turns out, errors grow differently in different dynamical systems. In some systems, making a tiny mistake is irrelevant for a future prediction, while in other systems a tiny initial error can ruin a forecast after a certain lead-time. Let’s take a quick view at different families of dynamical systems with the help of Figure 1. The figure has four panels; for each panel the x-axis corresponds to time, while the y-axis corresponds to the value of a physical variable (it can be wind speed, temperature, etc). Let us run a trajectory starte from a given initial condition; we label this reference trajectory (shown in black in the figure). Also, let us evolve trajectories initialised from ‘nearby’ initial conditions – i.e. initial conditions with errors; we label these trajectories as perturbed trajectories. In the figure, red lines indicate that initial perturbed values are larger than the initial true value, while blue lines indicate that the initial perturbed values are smaller than the initial true value. The behaviour in error growth is different in each case:

  1. a) In this example, the perturbed trajectories tend towards the reference trajectory. This is a typical dissipative system. Regardless of the initial conditions, the system evolves towards a fixed point, and any initial error disappears. Think of a pendulum with friction: it does not matter at what height you drop it, it will use its gravitational potential energy to swing for a while, but it will eventually stop.
  1. b) In this example, the errors of the perturbed trajectories grow as time increases, and they do not stop growing, instead, the perturbed trajectories tend towards plus and minus infinity. This system –in which errors grow without limit– is not feasible in reality, since it would require infinite energy. However, if we want to make predictions in a finite-time frame, the accuracy in the initial conditions is crucial, and we will see the quality of the forecast decrease with time.
  1. c) In this example, the initial error of the perturbed trajectories is preserved as time evolves; it neither grows nor decreases with time. This is typical for periodic systems, such as those found in celestial mechanics, or physical processes related from them, like the tides. If we are wrong in our position of the moon tonight, and do a forecast for the next days, the error will stay constant as time progresses. There is another type of systems, called quasi-periodic, which have similar characteristics, but I will not discuss them further.
  1. d) The last kind of systems is perhaps the most interesting to us; we are talking about chaotic systems. The atmosphere is a typical forced-dissipative system that presents chaotic behaviour. In this case, errors initially grow slowly, then the error growth turns faster, and eventually the perturbed trajectories do not resemble the reference trajectory at all, and in fact they do not resemble each other. The accuracy of the initial conditions is crucial for a good forecast, and the quality of a forecast decreases with time. In fact, even the tiniest initial errors will ruin a forecast after a given lead-time. What is different with respect to panel (b)? Errors do not grow forever and without limit, instead they saturate. After a long time, the trajectories – both the reference and the perturbed ones – evolve and live within a permissible range of values (without going to plus or minus infinity). This set of values is know as attractor (or climatology).

2016 05 25 Javier Amezcua Fig 1Figure 1. Error growth for different families of dynamical systems.

Let us discuss chaotic systems a little further using our example in panel (d). A forecast for time t=0.5 is more reliable than that for time t=1, and after approximately t=1.5 we have lost our capacity to predict. Something similar happens in the atmosphere. For large scale features, this limit of predictability is about 2 weeks. Operational centres release forecasts for up to 5 or 7 days in advance, and they equip these forecasts with some probabilistic measure (representing, in simple terms, how different are trajectories initiated from similar initial conditions). Unfortunately, some commercial forecast providers give no information on the accuracy of their forecasts at all. Furthermore, they are known for (irresponsibly) releasing ‘valid’ determinisitic forecasts for up to 45 days in advance (do not confuse this with the proper seasonal outlooks generated by meteorological agencies). As expected, these forecasts change considerably when updated every day, and these changes continue until the lead-time is within the predictability window. Such 45 day ‘forecasts’ are not prediction, they can be considered quasi-random draws from the climatology of different regions. At the end these forecasts have no value and they end up stating the obvious: July will be relatively warm and December will be relatively cold.








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