Nature of Time Series

Although K-S entropy as estimated from a plot of incremental redundancy versus lag isn't accurate, certain tendencies at high embedding dimensions on that plot can characterize or help distinguish between periodic, chaotic, and random data. Periodic data have a K-S entropy of zero. When H'_KS is zero, (ΔR=c_a-H'_KSm) reduces to just ΔR=c_a at all lags. In other words, the relation between incremental redundancy and lag is a horizontal straight line at a positive and constant ordinate value of c_a. Usually, the overall trend with lag is only generally horizontal and can include periodic spikes (not shown here). An asymptotic accumulation line for such data also is roughly horizontal.
For chaotic data, K-S entropy H'_KS is positive. Then is ΔR=c_a-(+H'_KS)m=c_a-H'_KSm. That equation says that, on the plot of ΔR versus lag, we get a straight line sloping downward. Finally, as discussed earlier, incremental redundancies for random data are virtually zero, regardless of lag.