Distinguishing between low-dimensional dynamics and randomness in measured time series

The success of current attempts to distinguish between low-dimensional chaos and random behavior in a time series of observations is considered. First we discuss stationary stochastic processes which produce finite numerical estimates of the correlation dimension and K2 entropy under naive application of correlation integral methods. We then consider several straightforward tests to evaluate whether correlation integral methods reflect the global geometry or the local fractal structure of the trajectory. This determines whether the methods are applicable to a given series; if they are we evaluate the significance of a particular result, for example, by considering the results of the analysis of stochastic signals with statistical properties similar to those of observed series. From the examples considered, it is clear that the correlation integral should not be used in isolation, but as one of a collection of tools to distinguish chaos from stochasticity.