Most methods of analysing time series carry out the numerical analysis on the time series itself. This is because the investigator is basically interested in the variable x = f(t) that has actually been measured. But an alternative approach is to suppose that the variable that was measured is merely one of several that might be measured as output from a dynamical system, which might indeed be better characterized using some other variable. For example, fluid converting in a box might be characterized by a probe measuring temperature or velocity at one point in the box, or by some other measurement, as a function of time. If the governing equations for the system are not known, it is also not possible to identify the fundamental variables which should be measured, or even how many of them there are. In this case, one might think that it is impossible to use measurements made on a single, arbitrarily chosen variable to reconstruct any important properties of the full multidimensional system. As we have seen, it is generally not even clear from simply examining the output x, whether or not the system producing x is stochastic or a nonlinear dynamical system of relatively low dimension.
But if, in fact, the signal is a product of a low dimension deterministic system then it turns out that it is possible to reconstruct all the major topological properties of the system, by a technique known as embedding. The basic idea seems to have been discovered independently
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Nonlinear Dynamics and Fractals: New Numerical Techniques for Sedimentary Data
The intention of these notes is to provide sedimentary geologists with an introduction to the new techniques for analyzing experimental and observational data provided by the rapid development of those disciplines generally known as Fractals and Nonlinear Dynamics (chaos theory). A general introduction to a minimum of theory is given, but most of the space is devoted to show how these ideas are useful for interpreting sedimentary data. The main applications are likely to be time series or spatial profiles or two-dimensional maps or images. Sedimentary geologists deal every day with actual time series, such as measurements of current velocity or suspended concentration at a station, or with virtual time series, such as stratigraphic sections, well logs, or topographic profiles yet few geologists know much about the new numerical techniques available to analyze such data.