Causality in long-term predictions, past-value problems and a stochastic-deterministic hybrid

Over time scales between 10 days and 10-20 years – the macroweather regime – atmospheric fields, including the temperature, respect statistical scale symmetries, such as power-law correlations, that imply the existence of a huge memory in the system that can be exploited for long-term forecasts. The Stochastic Seasonal to Interannual Prediction System (StocSIPS) is a stochastic model that exploits these symmetries to perform long-term forecasts. It models the temperature as the high-frequency limit of the fractional energy balance equation (fractional Gaussian noise) which governs radiative equilibrium processes when the relevant equilibrium relaxation processes are power law, rather than exponential.

The multivariate version of the model (m-StocSIPS), exploits the space-time statistics of the temperature field to produce realistic global simulations, including realistic teleconnection networks and El Niño events and indices. One of the implications of this model is the lack of Granger-causality: the optimal predictor at gridpoint i is obtained from the past of the timeseries i and cannot be improved using past temperatures from any other location j. This allows to treat predictions for long-memory processes as “past value” problems rather than the conventional initial value approach that uses the current state of the atmosphere to produce ensemble forecasts.

To improve the stochastic predictions, a zero-lag independent (non-stochastic) predictor is needed. Here we use the Canadian Seasonal to Interannual prediction System (CanSIPS), as a deterministic co-predictor. CanSIPS is a long-term multi-model ensemble (MME) system using two climate models developed by the Canadian Centre for Climate Modelling and Analysis (CCCma). The optimal linear combination of CanSIPS and StocSIPS (CanStoc) was based on minimizing the square error of the final predictor in the common hindcast period 1981-2010 using different out-of-sample validations. Global time series and regional maps at 2.5ºx2.5º resolution show that the skill of CanStoc is better than that of each individual model for most of the regions when non-overlapping training and verification periods are used.