Modelling and Simulation of Seasonal Rainfall Using the Principle of Maximum Entropy

We use the principle of maximum entropy to propose a parsimonious model for the generation of simulated rainfall during the wettest three-month season at a typical location on the east coast of Australia. The model uses a checkerboard copula of maximum entropy to model the joint probability distribution for total seasonal rainfall and a set of two-parameter gamma distributions to model each of the marginal monthly rainfall totals. The model allows us to match the grade correlation coefficients for the checkerboard copula to the observed Spearman rank correlation coefficients for the monthly rainfalls and, hence, provides a model that correctly describes the mean and variance for each of the monthly totals and also for the overall seasonal total. Thus, we avoid the need for a posteriori adjustment of simulated monthly totals in order to correctly simulate the observed seasonal statistics. Detailed results are presented for the modelling and simulation of seasonal rainfall in the town of Kempsey on the mid-north coast of New South Wales. Empirical evidence from extensive simulations is used to validate this application of the model. A similar analysis for Sydney is also described.

[1]  Wayne M Getz,et al.  Correlative coherence analysis: variation from intrinsic and extrinsic sources in competing populations. , 2003, Theoretical population biology.

[2]  Bill Ravens,et al.  An Introduction to Copulas , 2000, Technometrics.

[3]  R. Nelsen.  An Introduction to Copulas , 1998 .

[4]  D. Wilks,et al.  The weather generation game: a review of stochastic weather models , 1999 .

[5]  J. Rissanen.  A UNIVERSAL PRIOR FOR INTEGERS AND ESTIMATION BY MINIMUM DESCRIPTION LENGTH , 1983 .

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

[7]  Jonathan M. Borwein,et al.  Maximum entropy methods for generating simulated rainfall , 2012 .

[8]  J. Neumann.  Mathematical Foundations of Quantum Mechanics , 1955 .

[9]  Jonathan M. Borwein,et al.  Copulas with maximum entropy , 2012, Optim. Lett..

[10]  Tom Fleischer,et al.  Applied Functional Analysis , 2016 .

[11]  Jonathan M. Borwein,et al.  PSEUDO MATHEMATICS AND FINANCIAL CHARLATANISM: BACKTEST OVERFITTING AND OUT-OF-SAMPLE PERFORMANCE , 2013 .

[12]  Peter K. Dunn,et al.  Two Tweedie distributions that are near‐optimal for modelling monthly rainfall in Australia , 2011 .

[13]  M. Parlange,et al.  Overdispersion phenomenon in stochastic modeling of precipitation , 1998 .

[14]  T. McMahon,et al.  Stochastic generation of annual, monthly and daily climate data: A review , 2001 .

[15]  Tomasz Kulpa,et al.  On approximation of copulas , 1999 .

[16]  J. Borwein,et al.  Convex Functions: Constructions, Characterizations and Counterexamples , 2010 .

[17]  Phil Howlett,et al.  Simulation of monthly rainfall totals , 2008 .

[18]  Demetris Koutsoyiannis,et al.  Hurst‐Kolmogorov Dynamics and Uncertainty 1 , 2011 .

[19]  Demetris Koutsoyiannis.  “Hurst-Kolomogorov Dynamics and Uncertainty” , 2010 .

[20]  Richard Coe,et al.  A Model Fitting Analysis of Daily Rainfall Data , 1984 .

[21]  J. Rissanen,et al.  Modeling By Shortest Data Description* , 1978, Autom..

[22]  H. Akaike.  A new look at the statistical model identification , 1974 .

[23]  J. Borwein,et al.  Convex Analysis And Nonlinear Optimization , 2000 .

[24]  T. McMahon,et al.  Updated world map of the Köppen-Geiger climate classification , 2007 .

[25]  Eberhard Zeidler,et al.  Applied Functional Analysis: Applications to Mathematical Physics , 1995 .

[26]  R Srikanthan.  Stochastic Generation of Daily Rainfall Data , 2005 .

[27]  Marc Parlange,et al.  HydroZIP: How Hydrological Knowledge can Be Used to Improve Compression of Hydrological Data , 2013, Entropy.

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