Improving forecasts of imperfect models using piecewise stochastic processes

Forecasting complex systems is important for understanding and predicting phenomena. Due to the complexity and error sensitivity inherent in these predictive models, forecasting proves challenging. This paper presents a novel approach to assimilate system observations into predictive models. The approach makes use of a recursive partitioning algorithm to facilitate the computation of local sets of model corrections as well as provide a data structure to traverse the model space. These local sets of corrections act as a sample from a piecewise stochastic process. Appending these corrections to the predictive model incorporates hidden residual dynamics, resulting in improved forecasting performance. Numerical experiments demonstrate that this approach results in improved forecasting for the Lorenz 1963 model. In addition, comparisons are made between two types of corrections: Vector Difference and Gaussian. Vector Difference corrections provide the best computational efficiency and forecasting performance. To further justify the effectiveness of this approach it is successfully applied to more complex systems such as Lorenz for various chaotic parameterizations, coupled Lorenz, and cubic Lorenz.