One of the central problems in science is forecasting. If we have some knowledge about the past behaviour of a system, how can we make meaningful predictions about its future? A standard approach adopted in physics is to construct a mathematical model based on theoretical considerations and, using measured data for specifying the initial conditions, try to integrate the equations of motion forward in time to predict the future state. This procedure, however, is not always feasible. In fields such as economics we still lack the first principles necessary to make good models. In other cases, such as fluid flow, initial data are difficult to obtain. Finally, in complex nonlinear systems with many degrees of freedom, like a turbulent fluid, the weather, or the economy, it is not possible to solve the equations of dynamics explicitly and to keep track of motion in the high dimensional state space. In these cases model-based forecasting becomes impossible and calls for a different prediction paradigm.
KeywordsStochastic Perturbation Embedding Theorem Time Series Prediction Dynamic Noise Observational Noise
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