Entropy based derivation of probability distributions: A case study to daily rainfall

Abstract The principle of maximum entropy, along with empirical considerations, can provide consistent basis for constructing a consistent probability distribution model for highly varying geophysical processes. Here we examine the potential of using this principle with the Boltzmann–Gibbs–Shannon entropy definition in the probabilistic modeling of rainfall in different areas worldwide. We define and theoretically justify specific simple and general entropy maximization constraints which lead to two flexible distributions, i.e., the three-parameter Generalized Gamma (GG) and the four-parameter Generalized Beta of the second kind (GB2), with the former being a particular limiting case of the latter. We test the theoretical results in 11,519 daily rainfall records across the globe. The GB2 distribution seems to be able to describe all empirical records while two of its specific three-parameter cases, the GG and the Burr Type XII distributions perform very well by describing the 97.6% and 87.7% of the empirical records, respectively.

[1]  J. Hosking.  L‐Moments: Analysis and Estimation of Distributions Using Linear Combinations of Order Statistics , 1990 .

[2]  V. Singh,et al.  Entropy theory for derivation of infiltration equations , 2010 .

[3]  E. Jaynes.  Information Theory and Statistical Mechanics , 1957 .

[4]  P. Mielke,et al.  Some generalized beta distributions of the second kind having desirable application features in hydrology and meteorology , 1974 .

[5]  James B. McDonald,et al.  A generalization of the beta distribution with applications , 1995 .

[6]  J. N. Kapur.  Maximum-entropy models in science and engineering , 1992 .

[7]  C. E. SHANNON,et al.  A mathematical theory of communication , 1948, MOCO.

[8]  Jan Havrda,et al.  Quantification method of classification processes. Concept of structural a-entropy , 1967, Kybernetika.

[9]  Sang Joon Kim,et al.  A Mathematical Theory of Communication , 2006 .

[10]  James B. McDonald,et al.  Some Generalized Functions for the Size Distribution of Income , 1984 .

[11]  Pandu R. Tadikamalla,et al.  A Look at the Burr and Related Distributioni , 1980 .

[12]  Demetris Koutsoyiannis,et al.  Uncertainty, entropy, scaling and hydrological stochastics. 1. Marginal distributional properties of hydrological processes and state scaling / Incertitude, entropie, effet d'échelle et propriétés stochastiques hydrologiques. 1. Propriétés distributionnelles marginales des processus hydrologiques et , 2005 .

[13]  E. Jaynes.  Probability theory : the logic of science , 2003 .

[14]  I. W. Burr.  Cumulative Frequency Functions , 1942 .

[15]  E. Stacy.  A Generalization of the Gamma Distribution , 1962 .

[16]  C. Tsallis.  Possible generalization of Boltzmann-Gibbs statistics , 1988 .

[17]  Domingo Morales,et al.  A summary on entropy statistics , 1995, Kybernetika.

[18]  C. Klüppelberg,et al.  Subexponential distributions , 1998 .

[19]  S. Kotz,et al.  Statistical Size Distributions in Economics and Actuarial Sciences , 2003 .