Set-Theoretical Allegory on the Intrinsic Predictability of Weather Time Series

After decades of model-driven weather forecasting, we originally proposed [1, 1bis] a novel scheme of weather prediction, freed of modelization. We reinforced the very state-of-the-art which Richardson found regretful: "the past history of the atmosphere is used, so to speak, as a full-scale working model of its present self". Works by Robert [2, 2bis] and Orrell [3] demonstrated how sensitive dependence on initial conditions, otherwise called chaoticity, is a byproduct of, e.g., the very geometrical solutions to Navier-Stokes equations and may not be equally present in the atmospheric system. Models may be chaotic but they may not necessarily imbue the system with chaos. The main impediment stopping us from obtaining longer-term accurate weather forecasts would not be the butterfly effect but the errors in the models. Yet, the statistical self-similarity which characterizes weather time-series plays a definite role in reverting the habitually supposed chaos handicap into a major predictive strength. Our scheme thence advocates the prediction of weather time series from the lone knowledge of past raw data values, in the voluntary absence of models. Moreover, this utmost approach found justification [4] in a mathematical result by Hardin & Taylor [5], a non-constructive proof of the existence of statistical ampliative inference from past events. The reasons and heuristics of this reasoning can be explained by way of a prisoners' hats problem. From a conundrum first appraised by Gabay and O’Connor, two graduate students at Cornell in 2004, Hardin & Taylor expounded the following allegory in their 2008 paper. Consider an infinite set of prisoners indexed by ranks. After giving the prisoners an opportunity to formulate a stragegy, the warden places a hat (red or blue, chosen at random) on each prisoner’s head. By definition, a prisoner can only observe the hats on the heads of prisoners with a larger rank. Each prisoner must guess the colour of the hat on his/her head, without hearing any of the other guesses. What strategy would guarantee that only a finite number of prisoners guess incorrectly? We can elegantly prove that the solution relies on a prior agreement between prisoners of how one orders the set of all possible hat assignments. In other words, they agree on how one possible hat assignment is lesser than another assignment. Each prisoner then bases his/her guess on the least such assignment that agrees with what he/she can see. With such a winning strategy, almost all prisoners grasped the color correctly. An analogy can be drawn between this problem and the predictability of time-series from the sole knowledge of past values. A working example of weather forecasting without models will be given and put in perspective with the aforementioned proof.

[1] Lafitte (Levitas) MJ. et al., Of weather prediction without models, 17B.1, 23rd Conference on Weather Analysis and Forecasting/19th Conference on Numerical Weather Prediction, 2009

[1bis] Lafitte (Levitas) MJ., Of weather prediction without models II, 24th Conference on Weather Analysis and Forecasting/20th Conference on Numerical Weather Prediction, 2010

[2] Robert R., L'effet papillion n'existe plus, Gazette des Mathématiciens, Société Mathématique de France, No. 90 (Oct. 2001), p. 11—25

[2bis] Robert R. et al., Long range predictability of atmospheric flows, Non Linear Processes in Geophysics, No. 8 (2001), p. 55—67

[3] Orrell D. et al., Model error in weather forecasting, Nonlinear Processes in Geophysics, Vol. 8 (2001), p. 357—371

[4] George A., A proof of induction?, Philosopher's Imprint, Vol. 7 No. 2 (March 2007), p. 1—5

[5] Hardin CS. et al., A peculiar connection between the axiom of choice and predicting the future, American Mathematical Monthly, Vol. 115 (February 2008), p. 91—96