David M. Rubin, 1995. "Forecasting Techniques, Underlying Physics, and Applications", Nonlinear Dynamics and Fractals: New Numerical Techniques for Sedimentary Data, Gerard V. Middleton, Roy E. Plotnick, David M. Rubin
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Science is based on the principle of repeatability: each time a system experiences similar conditions—both internal to the system and forces exerted externally on the system—we expect the system to exhibit a similar response. Forecasting exploits this principle by using the observed behavior of a system to predict behavior when similar conditions recur. Even if the equations describing a system are unknown, we can nevertheless use forecasting to learn about the system. For some purposes— such as weather forecasting, financial forecasting, or noise reduction—predicting the future is the primary goal of the forecasting. For the purpose of characterizing system dynamics, in contrast, predictions are made in an exploratory manner to learn what kinds of models perform best.
For a preview of how the forecasting procedure works, we can consider the Lorenz system described in Chapter 2. Three approaches could be used to predict the future of this 85-system. First, we could measure the initial conditions (nonlinearity of vertical temperature gradient, temperature difference between rising and falling fluid, and intensity of convection) and use the three coupled equations (Equations 2.13) to predict the values of the three variables for successive steps in time.
A second approach could be employed if the governing equations were unknown, but sequential observations of the system were available. We could use the sequential observations to plot the 3-dimensional attractor (Figure 2.1), locate each predictee (a point whose three coordinates are given by the three variables that define the state of the system), identify nearby points on