Inferring long memory using extreme events

Abstract

Many natural and physical processes display long memory and extreme events. In these systems, the measured time series is invariably contaminated by noise and/or missing data. As the extreme events display a large deviation from the mean behavior, noise and/or missing data do not affect the extreme events as much as it affects the typical values. Since the extreme events also carry the information about correlations in the full-time series, we can use them to infer the correlation properties of the latter. In this work, we construct three modified time series using only the extreme events from a given time series. We show that the correlations in the original time series and in the modified time series are related, as measured by the exponent obtained from the detrended fluctuation analysis technique. Hence, the correlation exponents for a long memory time series can be inferred from its extreme events alone. We demonstrate this approach for several empirical time series.Extreme events display pronounced deviation from their typical behavior, e.g., earthquakes and market crashes. Such events occur in nature and many technological systems, often leading to a significant impact on both nature and society. Most of these systems are long-range correlated (long memory), implying that the correlations decay as power law, which is considerably slower than uncorrelated signals. The presence of long memory is inferred from measured time series representing all the events, both extreme and non-extremes. But can we infer long memory only by examining only the extreme events of a time series by disregarding the non-extremes? This work shows that we can estimate long-range correlations of a time series from extreme events alone. As extreme events are far less affected by noise and/or missing data than non-extreme events in general, this approach can be useful since the measured time series of these systems is invariably contaminated by noise.