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Of weather prediction without models II

Lorenz had already expressed [2] the existence of what he termed the empirical approach, as opposed to the dynamical. Also called "prediction without laws" [3], such a forecasting would be based on an analogue method, where two states of the atmosphere would be deemed alike. Naturally this was long before we knew [4] how to reconstruct [5] the state space of a dynamical system from a set of observations. Yet this modern embedding approach, void of model usage, could very well be an example of what Lorenz considered but had to relinquish.

Though recognized as the father of the butterfly effect, Lorenz called into question the real limits of predictability. In one of his last papers [6], he propounded that, in order to tackle the veridical forecasting limits, the error growth should result solely from the amplification of already present errors. We formerly mentioned the work of Robert [7] on the dubiety of this sensitive dependency in the prediction of atmospheric flows, we now echo the work of Orrell [8] in stating that the main thing stopping us from getting longer-term accurate weather forecasts is not the butterfly effect but the errors in the models.

Our scheme thence advocates the prediction of weather from the lone knowledge of past raw data values, in the voluntary absence of models. But where in the data can predictability information be found? What evidence do we have for holographic boundaries inside the time-series of atmospheric variability? This is in reality at the heart of Hurst's work [9] on the Nile river records, that of statistical persistence or long-range dependency. The so-called [10] "second law of self-organization" is at play, with the inherent ability of a time-series to store information [11] about bubbles of its own future. How does one draw [e.g., 12, 13] on this information reserve? One example is the aforementioned reconstruction as instanced by Farmer & Sidorowich [14]. Alike methods [e.g., 15, 15bis, 15ter] are left to the discretion of the author who devised and used self-similarity analysis techniques for the past fifteen years. As phrased by Lorenz: "existence of predictability may be established by demonstrating a forecasting procedure which exhibits skill" [16].

Moreover, this utmost approach (a systems approach [17] in its philosophy of counter-reductionism) finds justification [18] in a recent mathematical result [19], a non-constructive proof of the existence of ampliative inference from past events. Hence with no pretension on the how-to, this proof set up by others indicates that inductive extrapolations could exist and thus provides the grounds for the statistical inference we ourselves suggest. The heuristics of this reasoning will be explained by way of a prisoners' game and analogy; a working example will also be given and put in perspective with the proof.

A mathematical seer conceivably interprets dreams of the past, foretelling the weather.

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

[2] Lorenz EN., Three approaches to atmospheric predictability, Bulletin of the American Meteorological Society, Vol. 50 (1969), pp. 345--351

[3] Lorenz EN., How much better can weather prediction become?, Technology Review, July/August (1969), pp. 39--49

[4] Takens F., Detecting strange attractors in turbulence, in Rand DA. & Yound L-S, Dynamical Systems and Turbulence, Lecture Notes in Mathematics, Vol. 898 (1981), pp. 230--242

[5] Packard N. et al., Geometry from a time series, Physical Review Letters, Vol. 45 (1980), pp. 712--716

[6] Lorenz EN., Predictability --- a problem partly solved, in Palmer T. & Hagedorn R., Predictability of Weather and Climate, Cambridge University Press (2006), pp. 40--58

[7] Robert R. et al., Long range predictability of atmospheric flows, Non Linear Processes in Geophysics, Vol. 8 (2001), pp. 55--67

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

[9] Hurst H., Long term storage capacity of reservoirs. Transactions of the American Society of Civil Engineers Vol. 116 (1951), pp. 770--799

[10] Farmer JD., The second law of organization, Chap. 22 in Brockman J., The Third Culture: Beyond the Scientific Revolution, Simon&Schuster (1995)

[11] Farmer JD., Cool is not enough, Nature, Vol. 436 (2005), pp. 627--628

[12] Smith LA, Local optimal prediction: exploiting strangeness and the variation of sensitivity to initial condition, Philosophical Transactions of the Royal Society A, Vol. 348 (1994), pp. 371-381

[13] Sugihara G. et al., Residual delay maps unveil global patterns of atmospheric nonlinearity and produce improved local forecasts, Proceedings of the National Academy of Sciences, Vol. 96 (1999), pp. 210--215

[14] Farmer JD., Sidorowich JJ., Predicting chaotic time series, Physical Review Letters, Vol. 59 (1987), pp. 845--848; Los Alamos National Laboratory Report No. LA-UR-88-901

[15] Gluzman S, Yukalov VI, Resummation methods for analysing time series, Modern Physics Letters B, Vol. 12 (1998), pp. 61-74

[15bis] Yukalov VI, Gluzman S, Weighted fixed points in self-similar analysis of time series, International Journal of Modern Physics B, Vol. 13 (1999), pp. 1463--1476

[15ter] Yukalov VI, Self-similar extrapolation of asymptotic series and forecasting for time series, Modern Physics Letters B, Vol. 14 (2000), pp. 791--800

[16] Lorenz EN., On the existence of extended range predictability, Journal of Applied Meteorology, Vol. 12 (1973), pp. 543--546

[17] Orrell D. et al., A systems approach to forecasting, Foresight, Issue 14 (Summer 2009), pp. 25--30

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

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