Statistical tests for deterministic effects in broad band time series

Abstract

We derive a normalized version of the indicators of Savit and Green, and prove that these normalized statistics have, asymptotically, a normal distribution with a mean of zero and standard deviation of one if the time series is random in the sense of being IID (independent and identically distributed). We verify this result numerically, and study the magnitude of the finite size effects. We also show that these statistics are very sensitive to the existence of deterministic effects in the series, even if the underlying deterministic structure is complex, such as those generated by a chaotic system. We show that with moderate amounts of data, the statistics can easily indicate the presence of an underlying attractor even in the presence of IID noise which is as large as, or greater than the signal. Finally, we discuss the generalization of our approach to include (1) other null hypotheses besides IID which express assumptions of specific dependencies and (2) the study of deterministic effects between more than one time series.