| Abstrakt: |
We review our studies of the statistics of return intervals and extreme events (block maxima) Extreme event Block maxima in long-term correlated data sets, characterized by a power-law decaying autocorrelation function Function autocorrelation with correlation exponent ν between 0 and 1, for different distributions (Gaussian, exponential, power-law, and log-normal). For the return intervals, the long-term memory leads (i) to a stretched exponential distribution (Weibull distribution), Distribution Weibull with an exponent equal to ν, (ii) to long-term correlations among the return intervals themselves, yielding clustering of both small and large return intervals, Interval return and (iii) to an anomalous behavior of the mean residual time to the next event that depends on the history and increases with the elapsed time in a counterintuitive way. We present an analytical scaling approach Approach scaling and demonstrate that all these features can be seen in long climate records. For the extreme events we studied how the long-term correlations in data sets with Gaussian and exponential distribution densities affect the extreme value statistics, Statistics extreme value i.e., the statistics of maxima values within time segments of fixed duration R. We found numerically that (i) the integrated distribution function of the maxima converges to a Gumbel distribution for large R similar to uncorrelated signals, (ii) the deviations for finite R depend on both the initial distribution of the records and on their correlation properties, (iii) the maxima series exhibit long-term correlations similar to those of the original data, and most notably (iv) the maxima distribution as well as the mean maxima significantly depend on the history, in particular on the previous maximum. Finally we evaluate the effect of long-term correlations on the estimation of centennial events, which is an important task in hydrological risk estimation. We show that most of the effects revealed in artificial data can also be found in real hydro- and climatological data series. [ABSTRACT FROM AUTHOR] |
