Basin of Prediction for Seasonal Weather Forecasting 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 [1—3] which stems from renormalization group theory combined with methods of dynamical systems theory and optimal control theory. We aim at fulfilling the empirical creed for which we argued toward numerical weather prediction without models [4]. This also corresponds to what Sugihara illustrated in complex eco-systems, after which Ye et al. coined the term equation-free modeling [5]. Our approach invokes resummation techniques and can be translated into a machine learning framework, ultimately giving predictions of the future behavior of weather time series (air temperature and pressure near different locations) using information which is solely extracted from the respective past values. Iterating a hindcast process, we learn an ordering between distances to the ideal fixed-point condition which underlies convergence in functional space. The iterations vary on the number of data points, the equidistant time interval between points and the offset of the oldest data point. Windows (training and testing) can thus slide and/or expand. In applying the hypothesized forecasting models we reveal the existence of volatile basins of predictive information, predictability bubbles which can grow and burst, around the present time. When the bubbles are sufficiently and consistently wide, we find application of this machine learning scheme to weather forecasting at the subseasonal to seasonal time scale for temperate regions.

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

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

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

[4] 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

[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), p. 3856—3857