Deterministic Dynamics Since the time series data are discretely sampled over time, a deterministic model is always a map, and in a delay embedding space it reads Thus we need for a forecast is a prediction of sn+1. If the data are successive measurements are strongly correlated, and it might be advantageous to consider explicitly: To determine the proper construction of the model, we need a cost function or mean squared prediction error, which maximize the likelihood in Gaussian Distribution. Now we have a general form for the function F containing enough freedom of parameters so that it is capable of representing the data.

Local Methods in Phase Space In large data base and small noise level, local methods can be very powerful. Local methods are conceptually simpler than global model, but they require a large numerical efforts. Instead of using a single model for the global dynamics, one can use a new local model for every single data item to be predicted, so that globally arbitrarily nonlinear dynamics is generated: Locally, clean attractors can be embedded in fewer dimensions that are required for a global reconstruction. Covariance matrices of local neighborhood are thus sometimes close to singular and the fits are unstable.

Global nonlinear models The idea of global modelling is to choose an appropriate functional form for F which is flexible enough to model the true function on the whole attractor. Popular strategy is to take F to be a linear superposition of basis functions, The k basis function Φi are kept fixed during the fit and only the coefficients αi are varied. There are three functions that can be used for global model. Polynomials Radial basis functions Neural networks