La proprietà di dipendenza di lungo raggio nei terremoti

The goal of this thesis is detecting long-memory for the daily number of earthquakes and daily amount of released energy time series. These time series were taken by the seismic induced catalogue. The catalogue takes in account the magnitude of induced earthquakes in a geyser field, nevertheless it is considered equivalent to a natural one by seismologists. It collects data from April 2003 to June 2016. The Robinson test for no autocorrelated residuals, Robinson test for shortmemory autocorrelated residuals, the R-S statistic modified by Lo and the Robinson test based on Whittle likelihood are used to estimate the differencing parameter d, or the self-similarity parameter H. The Robinson tests are parametric methods that impose the model of fractional differencing of order d developed by Granger to the residuals of a linear fitting of data, whose coefficients depend on d. The test for no autocorrelated residuals imposes that residuals do not have correlation. It estimates the value of d. Conversely, the test for autocorrelated residuals estimates the value of d, assuming that residuals have also the structure of an autoregressive process of order one. The real short-memory of data could bias the estimates. The R-S analysis modified by Lo is a non-parametric method that estimates the self-similarity parameter H of the time series, modifying the classical R-S analysis to avoid that short memory bias the estimate of H. Anyway, this method is biased in favor of accepting the null hypothesis of no long memory. The Robinson test based on Whittle likelihood is a semiparametric method that selects the d which maximizes an approximate form of frequency domain Gaussian likelihood calculated in an neighborhood of the zero frequency. The results depend on the selection of the parameter that defines the width of the neighborhood, making them not always clear. All these methods were implemented on R studio and showed in the Appendix A. Long-memory is detected for all the time series.