Exploration of Seismic Data From the Kobe Earthquake

To render the data stationary we have logarithmically rescaled it, standardized it to zero mean and unit variance (SLR = "Standardized Logarathmically Rescaled"), and then computed the difference series using Difference[data,n] for n = 1, 2, yielding the original data plus the three associated time series, SLRdata, DifferenceSLRdata, and Difference2SLRdata.

\(\(DisplayTogetherArray[

      	{{ListPlot[Standardize[data], options1], 
          ListPlot[SLRdata, options2]}, {
          ListPlot[DifferenceSLRdata, options3], 
          ListPlot[Difference2SLRdata, options4]}}]; \)\)

The power spectrum of the data shows a pronounced peak at around 140 and a less pronounced one near 550, corresponding respectively to approximate periods of 22 and 5.5 respectively. These periodicities can also be seen in the correlation functions, which are now plotted.

DisplayTogetherArray[{{ListPlot[CorrelationFunction[SLRdata, 100]], 
          ListPlot[PartialCorrelationFunction[SLRdata, 100]]}, {
          ListPlot[CorrelationFunction[Difference2SLRdata, 100]], 
          ListPlot[PartialCorrelationFunction[Difference2SLRdata, 100]]}}];

Because the partial correlation function decays so rapidly, the Burg algorithm is used initially to estimate the parameters of AR(p) models for p = 1, ..., 25. We show all 25  correlation functions (out to a lag of 100) for the resulting models using ListPlot3D.

\(burgModels = BurgEstimate[Difference2SLRdata, 25]; 

    ListPlot3D[\((CorrelationFunction[#1, 100]&)\)/@burgModels, 
      AxesLabel -> {"Lag", "Model Number", None}]; \)