# American Institute of Mathematical Sciences

2012, 2(2): 233-256. doi: 10.3934/naco.2012.2.233

## Maximum entropy methods for generating simulated rainfall

 1 Centre for Industrial and Applied Mathematics, Mawson Lakes Campus, University of South Australia, Mawson Lakes Boulevard, Mawson Lakes, 5095, Australia 2 Centre for Computer Assisted Research Mathematics and its Applications (CARMA), University of Newcastle, Callaghan, NSW 2308, Australia 3 Data Analysis Australia Pty Ltd, School of Mathematics and Statistics, University of Western Australia, Crawley WA, 6009, Australia

Received  September 2011 Revised  March 2012 Published  May 2012

We desire to generate monthly rainfall totals for a particular location in such a way that the statistics for the simulated data match the statistics for the observed data. We are especially interested in the accumulated rainfall totals over several months. We propose two different ways to construct a joint rainfall probability distribution that matches the observed grade correlation coefficients and preserves the prescribed marginal distributions. Both methods use multi-dimensional checkerboard copulas. In the first case we use the theory of Fenchel duality to construct a copula of maximum entropy and in the second case we use a copula derived from a multi-variate normal distribution. Finally we simulate monthly rainfall totals at a particular location using each method and analyse the statistical behaviour of the corresponding quarterly accumulations.
Citation: Julia Piantadosi, Phil Howlett, Jonathan Borwein, John Henstridge. Maximum entropy methods for generating simulated rainfall. Numerical Algebra, Control and Optimization, 2012, 2 (2) : 233-256. doi: 10.3934/naco.2012.2.233
##### References:
 [1] Jonathan M. Borwein and Adrian S. Lewis, "Convex Analysis and Nonlinear Optimization, Theory and Examples," Second edition. CMS Books in Mathematics/Ouvrages de Mathmatiques de la SMC, Springer, New York, 3, 2006. [2] Jonathan M. Borwein, Maximum entropy and feasibility methods for convex and nonconvex inverse problems, Optimization, Invited survey paper, to appear, pre-print, http://carma.newcastle.edu.au/jon/inverse-paper.pdf. [3] H. J. Fowler, C. G. Kilsby, P. E. O'Connell and A. Burton, A weather-type conditioned multi-site stochastic rainfall model for the generation of scenarios of climatic variability and change, J. Hydrol., 308 (2005), 50-60. doi: 10.1016/j.jhydrol.2004.10.021. [4] Md Masud Hasan and Peter K. Dunn, Two Tweedie distributions that are near optimal for modelling monthly rainfall in Australia, International J Climatology, 2010. doi: 10.1002/joc.2162. [5] R. W. Katz and M. B. Parlange, Overdispersion phenomenon in stochastic modelling of precipitation, J. Climate, 11 (1998), 591-601. doi: 10.1175/1520-0442(1998)011<0591:OPISMO>2.0.CO;2. [6] Roger B. Nelsen, "An Introduction to Copulas," Lecture Notes in Statistics, Springer-Verlag, New York, (139), 1999. [7] Julia Piantadosi, Phil Howlett and John Boland, Matching the grade correlation coefficient using a copula with maximum disorder, J. Ind. Manag. Optim., 3 (2007), 305-312. [8] J. Piantadosi, J. W. Boland and P. G. Howlett, Simulation of rainfall totals on various time scales-daily, monthly and yearly, Environmental Modeling and Assessment, 14 (2009), 431-438. doi: 10.1007/s10666-008-9157-3. [9] Julia Piantadosi, Phil Howlett and Jonathan Borwein, Copulas with maximum entropy, Optimization Letters, 6 (2012), 99-125. doi: 10.1007/s11590-010-0254-2. [10] J. Piantadosi, P. G. Howlett, J. M. Borwein and J. Henstridge, Generation of simulated rainfall data at different time-scales, in "19th International Congress on Modelling and Simulation" (eds. F. Chan, D. Marinova and R. S. Anderssen), MODSIM2011, (2011), 1652-1658. http://www.mssanz.org.au/modsim2011/D10/wongsosaputro.pdf [11] K. Rosenberg, J. Boland and P. G. Howlett, Simulation of monthly rainfall totals, ANZIAM J., 46 (2004), (E), E85-E104. [12] R. Srikanthan and T. A. McMahon, Stochastic generation of annual, monthly and daily climate data: A review, Hydr. and Earth Sys. Sci., 5 (2001), 633-670. doi: 10.5194/hess-5-653-2001. [13] R. D. Stern and R. Coe, A model fitting analysis of daily rainfall, J. Roy. Statist. Soc. A, 147 (1984), 1-34. [14] Ruye Wang , Conditional and marginal of multivariate Gaussian, (2006), http://fourier.eng.hmc.edu/e161/lectures/gaussianprocess/node5.html. [15] Christopher S. Withers and Saralees Nadarajah, On the compound Poisson-gamma distribution, Kybernetika (Prague), 47 (2011), 15-37. [16] , Differential Entropy, http://en.wikipedia.org/wiki/Differential\_entropy. [17] D. S. Wilks and R. L. Wilby, The weather generation game: a review of stochastic weather models, Prog. Phys. Geog., 23 (1999), 329-357.

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##### References:
 [1] Jonathan M. Borwein and Adrian S. Lewis, "Convex Analysis and Nonlinear Optimization, Theory and Examples," Second edition. CMS Books in Mathematics/Ouvrages de Mathmatiques de la SMC, Springer, New York, 3, 2006. [2] Jonathan M. Borwein, Maximum entropy and feasibility methods for convex and nonconvex inverse problems, Optimization, Invited survey paper, to appear, pre-print, http://carma.newcastle.edu.au/jon/inverse-paper.pdf. [3] H. J. Fowler, C. G. Kilsby, P. E. O'Connell and A. Burton, A weather-type conditioned multi-site stochastic rainfall model for the generation of scenarios of climatic variability and change, J. Hydrol., 308 (2005), 50-60. doi: 10.1016/j.jhydrol.2004.10.021. [4] Md Masud Hasan and Peter K. Dunn, Two Tweedie distributions that are near optimal for modelling monthly rainfall in Australia, International J Climatology, 2010. doi: 10.1002/joc.2162. [5] R. W. Katz and M. B. Parlange, Overdispersion phenomenon in stochastic modelling of precipitation, J. Climate, 11 (1998), 591-601. doi: 10.1175/1520-0442(1998)011<0591:OPISMO>2.0.CO;2. [6] Roger B. Nelsen, "An Introduction to Copulas," Lecture Notes in Statistics, Springer-Verlag, New York, (139), 1999. [7] Julia Piantadosi, Phil Howlett and John Boland, Matching the grade correlation coefficient using a copula with maximum disorder, J. Ind. Manag. Optim., 3 (2007), 305-312. [8] J. Piantadosi, J. W. Boland and P. G. Howlett, Simulation of rainfall totals on various time scales-daily, monthly and yearly, Environmental Modeling and Assessment, 14 (2009), 431-438. doi: 10.1007/s10666-008-9157-3. [9] Julia Piantadosi, Phil Howlett and Jonathan Borwein, Copulas with maximum entropy, Optimization Letters, 6 (2012), 99-125. doi: 10.1007/s11590-010-0254-2. [10] J. Piantadosi, P. G. Howlett, J. M. Borwein and J. Henstridge, Generation of simulated rainfall data at different time-scales, in "19th International Congress on Modelling and Simulation" (eds. F. Chan, D. Marinova and R. S. Anderssen), MODSIM2011, (2011), 1652-1658. http://www.mssanz.org.au/modsim2011/D10/wongsosaputro.pdf [11] K. Rosenberg, J. Boland and P. G. Howlett, Simulation of monthly rainfall totals, ANZIAM J., 46 (2004), (E), E85-E104. [12] R. Srikanthan and T. A. McMahon, Stochastic generation of annual, monthly and daily climate data: A review, Hydr. and Earth Sys. Sci., 5 (2001), 633-670. doi: 10.5194/hess-5-653-2001. [13] R. D. Stern and R. Coe, A model fitting analysis of daily rainfall, J. Roy. Statist. Soc. A, 147 (1984), 1-34. [14] Ruye Wang , Conditional and marginal of multivariate Gaussian, (2006), http://fourier.eng.hmc.edu/e161/lectures/gaussianprocess/node5.html. [15] Christopher S. Withers and Saralees Nadarajah, On the compound Poisson-gamma distribution, Kybernetika (Prague), 47 (2011), 15-37. [16] , Differential Entropy, http://en.wikipedia.org/wiki/Differential\_entropy. [17] D. S. Wilks and R. L. Wilby, The weather generation game: a review of stochastic weather models, Prog. Phys. Geog., 23 (1999), 329-357.
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