Application of the fractional Levy motion to precipitation data

We applied the fractional Lévy motion model to precipitation data, referring to Lavallée (2004) and Lavallée (2008). The data we used were from the Global Preciptiation Climatology Centre (GPCC) monthly precipitation dataset. These data consist of 360 (longitude) × 180 (latitude) × 1336 (monthly, 1901-2012). First, we constructed four datasets: time series of average monthly precipitation of the top (maximum) 1000 precipitation observation stations, top 10, top 100, and top 500. Next, according to Lavallée (2004) and Lavallée (2008), using Fourier transformation, convolution (filtering) and inverse Fourier transformation, we obtained random variables Xt (Lavallée, 2004) from Yt (precipitation). We transformed from Yt to Xt. Finally, we fitted the Lévy law to Xt. As a preliminary result, we present examples of the values of the Lévy law parameters: alpha, beta, gamma, and delta for the "top 100" dataset. Parameters obtained were (1.17, 0.0, 257.6, 0.28; maximum likelihood), (1.10, 0.0, 250.0, -0.99; quantile algorithm), and (1.20, 0.0, 265.1, 0.57; empirical characteristic function algorithm). We used J. P. Nolan's algorithm. The values are quite sensitive to the algorithm that is used. At the Fall meeting, we will present considerations and results obtained using precipitation data other than those of the GPCC. J. P. Nolan, http://academic2.american.edu/~jpnolan/stable/stable.html Lavallée (2004), Stochastic modeling of climatic variability in dendrochronology, GRL, 31, L15202. Lavallée (2008), On the random nature of earthquake sources and ground motions; a unified theory, Advances in Geophysics, 50, chapter 16. Acknowledgement: We thank Dr. D. Lavallee for his comments and suggestions.; An example of results which we obtained. On a log-log plot, PDF of the Lévy law (red line) is more appropriate than the Gaussian law (blue line) in terms of heavy tail or extreme values. This is consistent with Lavallée (2004) and Lavallée (2008) who used slip distribution and climate (dendrochronology) data.