Model-Free Forecasting of Weather Time Series Using Self-Similar Power Transforms

Machine learning for time series today typically employs any particular flavor of recurrent neural networks. In contrast, we hereby suggest to use an approach of analysis [2-4] which stems from renormalization group theory combined with methods of dynamical theory and optimal control theory. We aim at fulfilling the empirical creed for which we argued toward weather prediction without models [1]. This also corresponds to what Sugihara illustrated in complex eco-systems, after which Ye et al. coined the term “equation-free modeling” [5]. Our particular approach invokes resummation techniques and can be translated into a machine learning framework, ultimately giving predictions of the future behavior of weather time series using information solely based on past values. Our implementation in Python (and also in R) exemplifies this way of analysis through functional programming. We interpret the optimization of the machine learning parameters, and show where the forecasting fixed-points (possible scenarios) translate into the geometry of the time series as embedded according to Takens’ theorem [6]. We find the existence of predictability bubbles. Furthermore, we hint to a generic link between the paradigmatic cost function minimization and the minimality strategy taken by Hardin & Taylor [7] in their exposition of a non-constructive proof of the existence of statistical ampliative inference from past events.

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

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

[3] 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

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

[5] DeAngelis DL, Yurek S, Equation-free modeling unravels the behavior of complex ecological systems, Proceedings of the National Academy of Sciences, Vol. 112 (2015), pp. 3856–3857

[6] Takens F, Detecting strange attractors in turbulence, Lecture Notes in Mathematics, Vol. 898 (1981), pp. 366—381

[7] 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