Correlation Integral as a Tool for Distinguishing between Dynamics and Statistics in Time Series Data

We propose a new test for time series data for proper choice of processing technique. It is based upon the normalized slope of the correlation integral $\phi$($\epsilon$,m)=m^-1<i>d</i>(lnC($\epsilon$))/<i>d</i>ln$\epsilon$, where m is the embedding dimension. It is shown that when j does not tend to 0 on the resolved range of scales as m grows, then there will be serious limitations for dynamical methods even if the data are dynamical by nature. In the latter case it means that the length of time series does not allow to resolve small scales, and on large scales the delay reconstruction for any m mixes true and false neighbours of points and therefore restricts the application of dynamical techniques, such as estimating Lyapunov exponents or predicting time series.