Correcting Predictive ModelCorrecting Models of Chaotic Reality

We will assume a chaotic (mixing) reality, can observe a substantially aggregated state vector only and want to predict one or more of its elements using a stochastic model. However, chaotic dynamics can be predicted in a short term only, while in the long term an ergodic distribution is the best predictor. Our stochastic model will thus be considered a local approximation with no predictive ability for the far future. Using an estimate of an ergodic distribution of the predicted scalar (or eventually vector), we get, under additional reasonable assumptions, the uniquely specified resulting model, containing information from both the local model and the ergodic distribution. For a small prediction horizon, if the local model converges in probability to a constant and additional technical assumption is fulfilled, the resulting model converges in L1 norm to the local model. In long term, the resulting model converges in L1 to the ergodic distribution. We propose also a formula for computing the resulting model from the nonparametric specification of the ergodic distribution (using past observations directly). Two examples follow.