Modelling and simulation of seasonal rainfall

Transcription

1 Modelling and simulation of seasonal rainfall Phil Howlett a, Julia Piantadosi a,1,, Jonathan Borwein b a Scheduling and Control Group, Centre for Industrial and Applied Mathematics (CIAM), University of South Australia, Mawson Lakes, SA 595, Australia b Centre for Computer Assisted Research Mathematics and its Applications (CARMA), University of Newcastle, Callaghan, NSW 238, Australia Abstract We use multi-dimensional checkerboard copulas to construct joint probability density functions for seasonal rainfall at specific locations on the east coast of Australia. The joint distribution preserves the known marginal monthly rainfall distributions and matches the observed monthly rank correlation coefficients. The method can be used with any desired marginal distributions and provides a much improved model for the variance of the total seasonal rainfall. We apply the new models to two particular case studies and present a collection of Matlab programs that enable easy implementation of the proposed method. Keywords: Maximum entropy methods, multiply stochastic hyper-matrices, multi-dimensional copulas, seasonal rainfall, rank correlation coefficients 2 MSC: 15B51, 6E5, 65C5, 52A Introduction We describe a new method for modelling and simulation of seasonal rainfall totals. The proposed model uses a multi-dimensional checkerboard copula of maximum entropy or a multi-dimensional normal checkerboard copula to construct a joint probability distribution that preserves the known marginal monthly rainfall distributions and matches the observed rank correlation coefficients. The procedure was initially presented by Piantadosi et al. (212 Corresponding author 1 Tel.: ; fax: julia.piantadosi@unisa.edu.au Preprint submitted to Journal of Hydrology August 15, 212

2 a, 211, 212 b) in a mathematical context as the solution to a challenging optimization problem. Our aim in this paper is to explain the new models to a broad range of climate scientists. We will discuss two particular case studies in detail and present a collection of Matlab algorithms that enable easy implementation of the model. The method can be applied to any number of months but numerical complexity of the distribution increases as the number of months increases. Indeed the joint probability density is defined by a multi-dimensional hyper-matrix with l = m n elements where m is the number of months and n is the number of subintervals used to partition the data. In our case studies l = 4 3 = 64. The mathematical foundations will be outlined in an appendix where we explain the notation, key equations and crucial formulæ. The models were tested at two specific locations in New South Wales Sydney and Kempsey where we have comprehensive rainfall data over periods of 15 years and 122 years respectively. In each case we consider a three month period during the wettest part of the year the months March April May at Sydney and the months February March April at Kempsey where the observed data shows consistently positive correlation between the monthly totals. Repeated simulations provide two key insights. Checkerboard copulas can be used to build models for seasonal rainfall distributions that incorporate known marginal monthly distributions and provide much improved estimates of seasonal variation. Rainfall statistics at near-coastal locations in eastern Australia are inherently unstable when viewed over time periods of the same duration as the observed data. A significant problem in modelling seasonal rainfall arises in the following way. If one wishes to simulate rainfall totals for a fixed period, say for one month, then conventional wisdom suggests that a random variable with probability density defined by a gamma distribution provides an excellent model. In order to extend the model to describe rainfall accumulations over several months it is necessary to construct a multi-dimensional probability distribution that will describe the joint behaviour of the monthly rainfall totals. In general this is not an easy task. In the special case where the monthly rainfall totals are mutually independent the joint probability density is simply a product of individual densities. For correlated monthly rainfall totals this simple procedure is no longer 2

3 valid. In general there may be macroscopic climatic factors that cause correlation between individual monthly rainfall totals. For instance, in northern and eastern Australia including New South Wales it is thought that the quasi-periodic occurrence of El Niño or La Niña events will cause, respectively, significantly drier or wetter summer and autumn seasons than one would normally expect. Thus it is no surprise that we found predominantly positive correlation between individual monthly totals during this time of the year at both locations. With positive monthly correlations one would expect to see a significantly higher variance in the seasonal totals than would otherwise be the case. This expectation is confirmed by examining the observed data. Our original challenge was to construct a mathematical model that respected the marginal gamma distributions for the individual monthly totals and allowed us to incorporate the observed correlation. Our task now is to apply the new methods to specific case studies and explain the rationale to a wider audience. The mathematical difficulty is that there is no classical multi-dimensional joint probability density with correlated marginal gamma distributions. The solution proposed recently in Piantadosi et al. (212 a,b) is to construct a multi-dimensional checkerboard copula either a copula of maximum entropy or a normal copula that will join together the desired marginal gamma distributions. In this paper we show that the new models enable simulations that respect the known monthly marginal distributions and provide much improved estimates for the seasonal variance. We calculate both copulas for two specific case studies but for the detailed simulations we restrict our attention to the copula of maximum entropy. Simulations with the normal copula give similar results. We will outline the structure of the copula of maximum entropy in Appendix A and present Matlab algorithms in Appendix C that will enable interested readers to construct these copulas for other case studies should they so wish. The mathematical basis for the copula of maximum entropy is described in detail in Piantadosi et al. (212 a,b) while details of the mathematical basis for the normal copula can be found in Piantadosi et al. (212 b). See Nelsen (1999) for general information about copulas and Wang (26) for details of the multi-variate normal distribution. 2. A brief review of rainfall modelling It has been usual to model both short-term and long-term rainfall accumulations at a specific location by a gamma or related distributions. We cite, 3

4 for instance, Stern and Coe (1984); Wilks and Wilby (1999); Srikanthan and McMahon (21); Rosenberg et al. (24); Fowler et al. (25); Hasan and Dunn (211). A few authors have, nevertheless, observed that simulations in which monthly rainfall totals are modelled as mutually independent gamma random variables generate accumulated bi-monthly, quarterly and yearly totals with much lower variance than the observed totals. See papers by Katz and Parlange (1998); Rosenberg et al. (24); Withers and Nadarajah (211). It is reasonable to surmise that the variance of the generated totals will be increased if the model incorporates an appropriate level of positive correlation between monthly totals. There are many reasons to study rainfall modelling. Our particular interest was prompted by problems of water storage and supply. Hence our interest in rainfall accumulations and rainfall variability over medium-term time scales. There are other aspects of rainfall modelling that are less directly relevant to this paper. Models that replicate rainfall characteristics on a sub-daily, daily or weekly timescale at a particular location or over a larger catchment region can be used by councils and local communities to better understand flood risks and develop effective flood mitigation strategies. Some authors regard stochastic rainfall modelling on a daily basis as a two stage process. They use a Markov chain to simulate the occurrence of wet or dry days and a gamma distribution to model rainfall totals on wet days. Models can describe both single and multiple sites. We cite, for instance, Fowler et al. (25); Srikanthan and McMahon (21); Wilks and Wilby (1999). A popular approach for rainfall modelling on a sub-hourly timescale is to use Neyman-Scott and Bartlet-Lewis cluster processes in which rainfall intensity is treated as a random variable that remains constant during the lifetime of a rain cell. See papers by Rodriguez-Iturbe et al. (1987, 1998); Wheater et al. (25). According to Cowpertwait (1994); Verhoest et al. (1997); Onof et al. (1997) the resulting rectangular rainfall profile is adequate for aggregated data at time-scales of an hour or longer but for finer time-scales it is necessary to use disaggregation techniques. See Cowpertwait et al. (1996); Onof et al. (25). Recently Cowpertwait et al. (27) proposed an extended Bartlet-Lewis Pulses model using a primary Poisson process to generate storms with a random finite lifetime; a secondary Poisson process within the storms to generate rain cells; and a tertiary Poisson process within the rain cells to generate instantaneous random pulses of rain. They claim that the three-level Poisson process can generate fluctuations 4

5 over a wide range of time-scales and provides more realistic rainfall profiles. Bartlet-Lewis models with depth-duration dependence have been studied by Kakou (1997) and the clustered Neyman-Scott model has been extended by Evin and Favre (28) to allow dependence between cell intensity and duration. Space-time Neyman-Scott models with defined storm extent and spatial cross-correlations have also been considered in Leonard et al. (28). The theoretical advances in rainfall modelling have been accompanied by more comprehensive simulation packages. For instance Burton et al. (28) proposed a spatial-temporal model using a Poisson cluster process for multisite input into runoff models. Generalized Linear Models work well at the daily level but Chandler et al. (27); Cox and Isham (1998) noted that disaggregation is required for sub-daily use. 3. Monthly rainfall models for near-coastal New South Wales In this section we discuss the basic characteristics of monthly rainfall at two near-coastal sites in New South Wales Sydney on the mid-central coast and Kempsey on the north-central coast. Both Sydney and Kempsey have a humid sub-tropical or temperate climate with no pronounced dry season. The Köppen classification in each case is Cfa. Although the annual rainfall is relatively high and is generally regarded as reliable there are nevertheless high monthly standard deviations and there is now a consensus that summer and autumn rainfall at both locations is strongly influenced on a recurring basis by the quasi-periodic seasonal climatic events El Niño and La Niña. During El Niño rainfall is inhibited and during La Niña it is enhanced. It is therefore not especially surprising to find positive correlation for monthly rainfall at Sydney during the period March-April-May and at Kempsey during the period February-March-April. In each case these three-month periods occur during the wettest time of the year. One of the specific aims of our work is to demonstrate through modelling and simulation that very wet or very dry seasons may be expected to occur on a regular basis in near-coastal New South Wales even without the anticipated additional effects of climate change. The variation observed in our simulations underlines the inherent uncertainty in future rainfall predictions. 5

6 Summary statistics for monthly rainfall at Sydney and Kempsey We used official records from the Australian Bureau of Meteorology at stations: 6662 Sydney (Observatory Hill) NSW ( ); and 5917 Kempsey (Wide Street) NSW ( ). The observed monthly means and standard deviations are shown in Tables 1 and 2. Rainfall is measured in millimetres. For all months the standard deviations are relatively high. Table 1: Oberved monthly means (m) and standard deviations (s) at Sydney. Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec m s Table 2: Observed monthly means (m) and standard deviations (s) at Kempsey. Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec m s Monthly rank correlation coefficients at Sydney and Kempsey Tables 3 and 4 show the rank correlation coefficients for all monthly pairs at Sydney and Kempsey. At Sydney the correlation for (Oct,Nov) is significant at the.1 level (2-tailed) and the correlations for (Jan,Feb), (Jan,Apr), (Jan,Oct), (Mar,Jun), (Apr,May) and (Jun,Sep) are significant at the.5 level (2-tailed). At Kempsey the correlations for (Mar,Jul) and (Jul,Sep) are significant at the.1 level (2 tailed) and the correlations for (Feb,Mar), (Jul,Aug) and (Sep,Nov) are significant at the.5 level (2 tailed). The significant correlations are shown in bold print The proposed modelling task Although the average rainfall is high during late summer and autumn in near-coastal New South Wales the observed data suggests that it is also 6

7 Table 3: Monthly rank correlation coefficients at Sydney. Ja Fe Mr Ap Ma Jn Jl Au Se Oc No De Ja Fe Mr Ap Ma Jn Jl Au Se Oc No De Table 4: Monthly rank correlation coefficients at Kempsey. Ja Fe Mr Ap Ma Jn Jl Au Se Oc No De Ja Fe Mr Ap Ma Jn Jl Au Se Oc No De highly variable. The high variability could be partly caused by the quasiperiodic climatic events El Ninõ and La Ninã events that are believed to be associated respectively with drier than normal and wetter than normal summer and autumn seasons in northern and eastern Australia. If so then we might expect to see positive correlation between monthly rainfall totals during the summer or autumn seasons. In any case, no matter the cause, it is true that positive correlation between monthly totals should mean that seasonal totals show higher than expected variation. To analyse the rainfall patterns we decided to model rainfall for March- April-May at Sydney and February-March-April at Kempsey. The observed correlations are (.112,.43,.183) at Sydney and (.22,.112,.152) at Kempsey. In each case all coefficients are positive. Our intuition that the variation in total autumn rainfall is higher than would otherwise be expected is confirmed by the observed data and by our new models in theory and 7

8 in simulation. Our models are based on an objective representation of the historical data and they show that very wet or very dry autumn seasons are likely to be a regular occurrence in near-coastal New South Wales despite consistently high average rainfalls. When this already high variability is reinforced by credible predictions of increasingly extreme weather patterns one can see that rainfall variability should be an important concern for us all governments, scientists and the broader community. Our interest in practical models of Australian rainfall is inspired in part by our belief that objective simulations can be used to demonstrate the variable nature of the Australian climate. Models that accurately reflect observed historical data allow us to generate realistic simulations of equally likely alternative rainfall histories. The simulations enable us to better understand the relationship between what has been and what might have been and thereby allow us to develop a broader view of history and to see more clearly the limitations of a literal interpretation of the existing observations The gamma distribution The gamma distribution is commonly used to model the probability density for monthly rainfall. The cumulative distribution function is defined on (, ) by the formula F [α, β](x) = x ξ α 1 exp( ξ/β) dξ β α Γ(α) where α > and β > are parameters. The mean value is given by µ = αβ and the variance by σ 2 = αβ 2. The parameters α and β are usually determined by the method of maximum likelihood (ML) but they can also be determined by moment matching (MM) Monthly rainfall models We will use the gamma distribution to model monthly rainfall at both Sydney and Kempsey. For each month at Sydney we use ML to find values α S = (1.7413, , ), β S = ( , , ) or MM to determine alternative values α S = (1.578, 1.273, ), β S = ( , , ). 8

9 For the monthly distributions at Kempsey we use ML to find α K = (1.552, 2.134, ), β K = (1.4753, , ) or MM to determine the alternative values α K = (1.2841, , 1.625), β K = ( , 81.83, ) A simple calculation will show that in each month and at each location ML under-estimates the monthly variance by comparison with the observed data. This is not especially surprising because ML is designed to provide the best possible approximation to the observed probability distribution across the entire event space. However the observation did prompt us to consider the alternative MM estimation. Figures 1 and 2 show respectively histograms of the observed frequencies versus the fitted probability densities at Sydney in the months March, April and May and at Kempsey in the months February, March and April. 4. Seasonal rainfall models for near-coastal New South Wales The next step in the modelling process is to construct a joint probability distribution for each of the three-month time periods. In past studies of rainfall accumulations over several months Katz and Parlange (1998); Rosenberg et al. (24); Withers and Nadarajah (211) have observed that for models with independent monthly marginal distributions the variance of the accumulated totals is often too low. It has been suggested that this happens because there is an overall positive correlation between the individual monthly totals. Since the observed data shows positive correlation for March-April-May at Sydney and for February-March-April at Kempsey our aim will be to construct joint distributions in such a way that the desired marginal gamma distributions are preserved and so that the grade correlation coefficients match the observed rank correlation coefficients. We can do this using either a checkerboard copula of maximum entropy or a checkerboard normal copula. Thus we can construct four different joint distributions two using the ML marginals and two using the MM marginals at each location. We will compare each of these models with a simple independent model using a joint density obtained as the product of the marginal densities. Our aim is to show that the models with correlation provide an improved estimate for the variance of the overall total. 9

10 We will now give a brief outline of the fundamental ideas. In three dimensions a copula is a joint probability distribution on the unit cube [, 1] 3 with uniform marginal distributions. A checkerboard copula is a probability distribution defined by a uniform subdivision of the unit cube into n 3 congruent smaller cubes in such a way that the probability density is constant on each small cube. If the density on cube I ijk is defined by n 2 h ijk then the marginal distributions will be uniform if h ijk = 1 for all i, h ijk = 1 for all j, h ijk = 1 for all k. j,k i,k In such cases we say that the three dimensional matrix h = [h ijk ] is triplystochastic. We wish to construct a density in this form so that the uniform marginal distributions have the desired correlations. For sufficiently large n it turns out that there are many ways that this can be done. Our particular aim is to find the most disordered or least prescriptive solution the triply-stochastic hyper-matrix which has the most equal subdivision of probabilities but still allows the required correlations. In mathematical terminology this is the hyper-matrix that satisfies the constraints and has the highest possible entropy. A condensed description of the construction process for the checkerboard copula of maximum entropy is given in Appendix A. A full description can be found in Piantadosi et al. (212 a,b). A complete description for construction of the checkerboard normal copula is given in Piantadosi et al. (212 b). Matlab algorithms that will enable the reader to construct the checkerboard copula of maximum entropy 2 are given in Appendix C. Once we have constructed a suitable copula with probability density c(u 1, u 2, u 3 ) on the unit cube we can define the desired joint probability density g(x 1, x 2, x 3 ) = c(u 1, u 2, u 3 )f 1 (x 1 )f 2 (x 2 )f 3 (x 3 ) where u i = F i (x i ) is the cumulative distribution function for month i and f i (x i ) = F i (x i ) is the associated marginal density. More details can be found in Appendix A where we outline the basic probability theory and define the relevant terminology. i,j 2 Matlab algorithms to construct the normal checkerboard copula can be obtained from Dr. Julia Piantadosi ( julia.piantadosi@unisa.edu.au). 1

11 Checkerboard copulas for rainfall in March-April-May at Sydney We set m = 3 and n = 4. The multi-stochastic hyper-matrices h, k R describing respectively the tri-variate checkerboard copula of maximum entropy and the tri-variate normal checkerboard copula for March- April-May rainfall at Sydney are shown below to four decimal place accuracy. The copula of maximum entropy was constrained by the observed rank correlation coefficients ρ 12 =.112, ρ 13 =.43 and ρ 23 =.183. We calculate h 1 h , h 2, h ,, where h i = [h ijk ]. The entropy is given by J(h) The normal copula was constrained by setting θ 12 = , θ 13 = and θ 23 = to match the observed rank correlation coefficients ρ , ρ and ρ Our calculations give k 1 k , k 2, k ,, where k i = [k ijk ]. The entropy is given by J(k) Checkerboard copulas for rainfall in February-March-April at Kempsey We set m = 3 and n = 4. The multi-stochastic hyper-matrices h, k R describing respectively the tri-variate checkerboard copula of maximum entropy and the tri-variate normal checkerboard copula for February- March-April rainfall at Kempsey are shown below to four decimal place accuracy. The copula of maximum entropy was constrained by the observed 11

12 rank correlation coefficients ρ 12 =.22, ρ 13 =.112 and ρ 23 =.152. We calculate h 1 h , h 2, h ,, where h i = [h ijk ]. The entropy is given by J(h) The normal copula was constrained by setting θ 12 = 1.318, θ 13 = and θ 23 = to match the observed rank correlation coefficients ρ 12.22, ρ and ρ Our calculations give k 1 k , k 2, k ,, where k i = [k ijk ]. The entropy is given by J(k) Construction of the joint probability densities using either ML or MM fitted marginals is described in Appendix A Subintervals and partial moments The marginal distributions define a subdivision K ijk of the rainfall space (, ) 3 corresponding to the uniform subdivision of the unit hypercube using the correspondence (x 1, x 2, x 3 ) K ijk (F 1 (x 1 ), F 2 (x 2 ), F 3 (x 3 )) = (u 1, u 2, u 3 ) I ijk We calculate moments about the mean on each of these subintervals. The results are shown below. The subdivision points c i (k) are defined by F i (c i (k)) = k/4 for each k =, 1,..., 4. On the table of moments the row 12

13 sums of the first moments give the total first moment and the row sums of the variances give the total variance for each of the three variables. These moments are used in conjunction with the relevant hyper-matrices to calculate the theoretical variances for each of the two copulas. See formula (A.5) in Appendix A ML subintervals and corresponding partial moments for Sydney Set m = 3 and n = 4 and use the ML fitted marginal gamma distributions for Sydney. The subintervals K ijk are defined by r c r (1) c r (2) c r (3) c r (4) c r (5) with corresponding moments about the mean and variances r m r (1) m r (2) m r (3) m r (4) sum r σ r (1) 2 σ r (2) 2 σ r (3) 2 σ r (4) 2 2 σ r MM subintervals and corresponding partial moments for Sydney Set m = 3 and n = 4 and use the MM fitted marginal gamma distributions for Sydney. The subintervals K ijk are defined by r c r (1) c r (2) c r (3) c r (4) c r (5)

14 3 31 with corresponding moments about the mean and variances r m r (1) m r (2) m r (3) m r (4) sum r σ r (1) 2 σ r (2) 2 σ r (3) 2 σ r (4) 2 2 σ r ML subintervals and corresponding partial moments for Kempsey Set m = 3 and n = 4 and use the ML fitted marginal gamma distributions for Kempsey. The subintervals K ijk are defined by r c r (1) c r (2) c r (3) c r (4) c r (5) with corresponding moments about the mean and variances r m r (1) m r (2) m r (3) m r (4) sum r σ r (1) 2 σ r (2) 2 σ r (3) 2 σ r (4) 2 2 σ r

15 MM subintervals and corresponding partial moments for Kempsey Set m = 3 and n = 4 and use the MM fitted marginal gamma distributions for Kempsey. The subintervals K ijk are defined by r c r (1) c r (2) c r (3) c r (4) c r (5) with corresponding moments about the mean and variances r m r (1) m r (2) m r (3) m r (4) sum r σ r (1) 2 σ r (2) 2 σ r (3) 2 σ r (4) 2 2 σ r Summary statistics In Tables 5 and 6 the summary statistics for the observed total rainfall in March-April-May at Sydney and for February-March-April at Kempsey are compared to the summary population statistics for the independent model and for the correlated models using the two checkerboard copulas with either the ML or MM fitted marginal distributions. All models estimate the overall mean exactly. The independent models clearly under-estimate the overall variance at both locations. At Sydney the copula models both over-estimate the variance whichever marginal distributions are used. This result is slightly unexpected especially in the case of the ML fitted marginals where the individual monthly variances are all smaller than the observed monthly variances. Nevertheless the overall variance depends on the weighted probabilities for each subinterval and on the 15

16 pairwise correlations between the subintervals. If the pairwise correlations are not uniform across all subintervals then one might expect the overall variance to be biassed accordingly. At Kempsey the results are much more in line with what we expect. Both copula models under-estimate the overall variance when the ML fitted marginals are used. This is no surprise because the ML marginals consistently under-estimate the monthly variances. When the MM marginals are used the overall variance for both copula models is very close to the observed overall variance. In every case the copula models do much better than the independent models. Table 5: Model comparison for total rainfall at Sydney. Sydney mean variance observed independent (ML) independent (MM) maximum entropy (ML) maximum entropy (MM) normal (ML) normal (MM) Table 6: Model comparison for total rainfall at Kempsey. Kempsey mean variance observed independent (ML) independent (MM) maximum entropy (ML) maximum entropy (MM) normal (ML) normal (MM) In general terms these results support earlier findings in Piantadosi et al. (212 b) for September-October-November rainfall in Sydney. 16

17 Simulations One of the important applications of our new models is the creation of simulated rainfall data. The observed rainfall data from the Bureau of Meteorology at Sydney covers a time span of 15 years and so it is natural to run simulations over the same period of time. The simulations confirm that positive correlation between individual months increases the expected variation in seasonal rainfall but the simulations also show that rainfall patterns for a time span of this duration are inherently unstable. Indeed our simulations show that sample statistics for March-April-May rainfall at Sydney over the next 15 years are quite likely to vary significantly from the observed statistics over the previous 15 years. At Kempsey the observed rainfall data covers a period of 122 years and the simulation statistics lead to a similar conclusion that monthly rainfall patterns over a time period of this duration are inherently unstable. We confirm the statistical instability by a technical argument in which we show that the total probability error for a simulation over N years is approximately 1/ N. If one wishes to decrease the variation in simulation results over a period of N years by a factor of ten then one would need to run the simulations over a period of 1N years. We will demonstrate the dramatically increased stability by also presenting simulation results for a period of 15 years at Sydney and 122 years at Kempsey. These results should sound a note of caution about any model our models included that is based on monthly rainfall statistics taken over a period of 15 years or less statistics which should be regarded as somewhat less than prescriptive Simulating rainfall using a checkerboard copula Suppose we have a checkerboard copula C h defined by a matrix h = [h i ] R l where l = n 3 and i = (i, j, k) {1,..., n} 3 on a uniform partition {I i } of the unit cube (, 1) 3. Simulated data for monthly rainfall triples may be generated as follows. Define an order for the indices i = (i, j, k) by saying that (i, j, k) (i, j, k ) if i < i or if i = i and j < j or if i = i and j = j and k < k. For each pseudo-random number r (, 1) select the interval I i j k = (a(i ), a(i + 1)) (a(j ), a(j + 1)) (a(k ), a(k + 1)) if h ijk < nr < + h i j k. (i,j,k) (i,j,k ) (i,j,k) (i,j,k ) h ijk 17

18 Once the interval I i j k has been selected the precise position of the pseudorandom point (u r, v r, w r ) I i j k is fixed by generating three (independent) pseudo-random numbers (q r, s r, t r ) (, 1) 3 and setting ( (i 1) + q r (u r, v r, w r ) =, (j 1) + s r, (k ) 1) + t r n n n and the corresponding rainfall triple is defined by (x r, y r, z r ) = ( F 1 1 (u r ), F 1 2 (v r ), F 1 3 (w r ) ) where F 1, F 2 and F 3 are the given marginal distributions Simulations for total rainfall in March-April-May at Sydney The simulations over 15 years at Sydney show considerable variation. In ten consecutive simulations the sample mean of the total rainfall in February- March-April varied from a minimum of 361 mm in S15.7 to a maximum of 392 mm in S15.2. The sample variance ranged from a minimum of mm 2 in S15.7 to a maximum of 5668 mm 2 in K The histograms in Figure 3 for the simulations over 15 years show that samples of this size are statistically unstable. By comparison the histograms in Figure 4 for the simulations over 15 years demonstrate that these much larger samples are statistically stable. Indeed the sample means and variances are both close to the theoretical values (377, 399) for these much larger sample sizes Simulations for total rainfall in February-March-April at Kempsey The simulations over 122 years at Kempsey also show considerable variation. In ten consecutive simulations the sample mean varied from a minimum of 41 mm in K122.9 to a maximum of 464 mm in K The sample variance ranged from a minimum of mm 2 in K122.5 to a maximum of mm 2 in K The histograms in Figures 5 and 6 show that samples over 122 years are statistically unstable but much larger samples over 122 years are statistically stable. For the larger samples the means and variances are both close to the theoretical values (427, 54443). There are two key points to make about interpretation of these simulations. The first point is this. Although the theoretical population statistics for the models are an excellent approximation to the observed statistics there is considerable 18

19 variation in the simulation statistics over a time period equal to the time period of the observations. Thus, even if there is no change in climatic conditions, we should not expect the observed statistics in the next comparable time period to be necessarily the same. The second point turns the question around in the following way. We should not assume that the observed sample statistics are necessarily an accurate representation of the true population statistics. They are our best estimate of the population statistics but the simulations show there is a significant degree of uncertainty associated with samples of this size. Thus we should remember that our model is based on uncertain statistics Error estimation for the simulations The simulations show us there is significant variation in the basic overall statistics for simulations covering a period of 15 years the maximum length of existing reliable rainfall records in Australia. We explain this in the following way. The error vector e is defined as the difference between the theoretical probabilities defined for the intervals I ijk by the relevant triplystochastic hyper-matrix and the corresponding relative frequencies generated by the pseudo-random simulation. The probability error e = e is the Euclidean length of this error vector. We can analyse the error e more precisely in the following way. If there are N realizations and if we renumber the intervals I ijk where (i, j, k) {1, 2, 3, 4} 3 in the form I 1, I 2,..., I l where l = 4 3 and if p r = h r /4 is the probability of selecting I r and N r is the actual number of times I r is selected for each r = 1,..., l then we have [ (Nr ) ] 2 E[e 2 r] = E N p r ( ) 2 ( ) Nr = N p N r p N 1 1 p N l N 1 N l l N 1 + +N l =N = p r(1 p r ) N since p p l = 1 and hence the expected square error is E[e 2 ] = l E[e 2 r] = 1 N r=1 l p r (1 p r ) 1 N. r=1 19

20 It follows that E[e 2 ] 1/ N. Note that 1/ and 1/ The simulation statistics and probability errors are shown in Tables 7 and 8. We show statistics at Sydney for 1 consecutive runs of S15 over 15 years and S15 over 15 years and at Kempsey for 1 consecutive runs of K122 over 122 years and K122 over 122 years. In addition to our earlier observations about instability of the statistics we note that although the grade correlation coefficients are all positive there are a small number of negative rank correlation coefficients in the S15 and K122 simulations. Table 7: Statistics and probability errors for simulations S15 (above) and S15 (below). run mean var r12 r13 r23 error run mean var r12 r13 r23 error Comparison of simulated and observed data for total rainfall In Tables 9 and 1 we have compared simulated data from K122.4 with the observed data for February-March-April at Kempsey. In general terms we see that although K122.4 was slightly wetter than average and also with a 2

21 Table 8: Statistics and probability errors for simulations K122 (above) and K122 (below). run mean var r12 r13 r23 error run mean var r12 r13 r23 error higher variance the microscopic behaviour of the two data sets is similar. The observed data shows prolonged sequences of below average rainfall ( ) and above average rainfall ( ); and short sequences of very low rainfall (19 192) and very high rainfall ( ). The lowest total was 82 mm (1944) and the highest was 1217 mm (1929). For the simulated data there are also prolonged sequences of below average rainfall (years ) and above average rainfall (years 57 65); and short sequences of very low rainfall (years ) and very high rainfall (years 59 61). The lowest total was 53 mm (year 66) and the highest was 1276 mm (year 59). Similar comparisons can be made between simulated and observed data at Sydney. 21

22 Table 9: Selected years: simulated data K Year February March April Total Conclusions We have shown that both the checkerboard copula of maximum entropy and the checkerboard normal copula can be used to construct a joint probability distribution for seasonal rainfall when there is positive correlation between individual monthly rainfall totals. We have shown that the new models can be easily constructed using the mathematical application package Matlab and that the models can be used to simulate seasonal rainfall on the east coast of Australia. In particular we have shown that the variance of seasonal rainfall is well modelled by this method while the models for the marginal monthly totals are preserved. On a broader front our analysis shows that the 22

23 Table 1: Selected years: Observed data. Year February March April Total

24 statistics for simulated rainfall of similar duration to the observed records are somewhat unstable that is the statistics for repeated simulations show considerable variation from the underlying theoretical values. By implication we conclude that the corresponding statistics from observed rainfall data are also likely to be unstable and hence are not necessarily an accurate reflection of the population statistics. Thus we believe caution should be shown in the interpretation and extrapolation of existing rainfall data. Appendix A. An outline of the basic probability theory Appendix A.1. Multi-dimensional copulas An m-dimensional copula, where m 2, is a cumulative probability distribution C(u) [, 1] defined on the m-dimensional unit hyper-cube u = (u 1, u 2,..., u m ) [, 1] m with uniform marginal probability distributions 3. The grade correlation coefficients are the pairwise correlations of the uniform marginal distributions. Appendix A.1.1. Constructing a joint distribution with prescribed marginals If F r (x r ) [, 1] for each x r R are the prescribed cumulative distributions for the real-valued continuous random variables X r for each r = 1,..., m and C(x) is an m-dimensional copula then the function G(x) [, 1] defined for each x R m by G(x) = C(F 1 (x 1 ),..., F m (x m )) is a joint probability distribution for the vector-valued random variable X = (X 1,..., X m ) with the marginal distribution for X r defined by F r for each r = 1, 2,..., m. The joint density g(x) [, ) is defined by the formula g(x) = c(f 1 (x 1 ),..., F m (x m ))f 1 (x 1 ) f m (x m ) where c(u) [, ) is the density for the joint distribution defined by the copula C(u) and where f r (x r ) [, ) for each r = 1, 2,..., m are the densities for the prescribed marginal distributions. If related real-valued random variables U r = F r (X r ) are defined for each r = 1, 2,..., m then each U r is uniformly distributed on [, 1] and the copula 3 A precise analytic description can be found in Wikipedia (212 a). 24

25 C(u) describes the distribution of the vector-valued random variable U = (U 1,..., U m ). When the joint distribution G(x) is not known there are many standard choices for C(u) but the best choice will depend on what else is known about the observed distribution. Appendix A.1.2. The grade correlation coefficients It is often the case in practice that the grade correlation coefficients 4 defined by ρ r,s = = E[(F r (X r ) 1/2)(F s (X s ) 1/2)] E[(Fr (X r ) 1/2) 2 ] E[(F s (X s ) 1/2) 2 ] E[(U r 1/2)(U s 1/2)] E[(Ur 1/2) 2 ] E[(U s 1/2) 2 ] = 12E[U r U s ] for each 1 r < s m are specified. It follows from the above definition that the grade correlation coefficients for X are simply the correlations for U. Appendix A.1.3. The entropy The entropy for the copula C(u) with density c(u) is defined by J(C) = ( 1) c(u) log e c(u) du [,1] m where u = (u 1,..., u m ) T [, 1] m. The entropy J(C) of the copula measures the inherent disorder of the distribution. The most disordered copula is the one with c(u) = 1 for all u [, 1] m for which J(C) =. See also Wikipedia (212 b). Appendix A.2. The checkerboard copula of maximum entropy An m-dimensional checkerboard copula is a copula with probability density defined by a step function on an m-uniform subdivision of the hyper-cube 4 In this paper we distinguish between the rank correlation coefficients described in Wikipedia (212 c) which are calculated from the observed data and the grade correlation coefficients which are constructed from the joint probability density using the fitted marginal distributions and matched to the observed rank correlation coefficients. 25

26 [, 1] m. Any continuous copula can be uniformly approximated by a checkerboard copula. Let n N be a natural number and define the partition = a(1) < a(2) < < a(n) < a(n + 1) = 1 of the interval [, 1] by setting a(k) = (k 1)/n for each k = 1,..., n + 1. This partition generates a corresponding m-uniform subdivision of the unit hyper-cube [, 1] m into l = n m congruent hyper-cubes. For each multi-stochastic 5 hyper-matrix h = [h i ] R l where i = (i 1,..., i m ) {1,..., n} m we can define a probability density on the unit hyper-cube [, 1] m in the form of a step function c h (u) R by the formula c h (u) = n m 1 h i (A.1) for each i = (i 1,..., i m ) {1, 2,..., n} m and each u I i = m r=1 (a(i r ), a(i r + 1)) The step function c h (u) [, ) defines a corresponding checkerboard copula C h (u) [, 1] for each u [, 1] m. The grade correlation coefficients for C h are given by ρ r,s = 12 1 n 3 h i (i r 1/2)(i s 1/2) 3 (A.2) i {1,...,n} m and the entropy of h is given by J(h) = ( 1) 1 h n i log e h i + (m 1) log e n. i {1,...,n} m (A.3) If n N is sufficiently large then Piantadosi et al. (212 a) showed that h can be chosen in such a way that the observed grade correlations are imposed and the entropy of the hyper-matrix is maximized. Since entropy is a measure of disorder the solution proposed by Piantadosi et al. for c h can be interpreted as the most disordered or least prescriptive choice of step function for the selected value of n that satisfies the required grade correlation constraints. The corresponding checkerboard copula C = C h is the most disordered such copula. See Piantadosi et al. (212 b) for more information. 5 The hyper-matrix h is said to be multi-stochastic if, for each r = 1, 2,..., m and each fixed p {1, 2,..., n}, the sum over all elements h i with i r = p is equal to 1. 26

27 Appendix A.3. Calculating the variance for the sum of random variables Suppose the random variable X = (X 1,..., X m ) T is distributed according to the joint probability density g(x) (, ) defined by g(x) = n m 1 h i f 1 (x 1 ) f m (x m ), (F 1 (x 1 ),..., F m (x m )) I i (A.4) where i = (i 1,..., i m ) and I i is the usual uniform subdivision of the unit hyper-cube [, 1] m and h = [h i ] R l where l = n m is a multiply-stochastic hyper-matrix. Let S = m r=1 X r be the sum of the random variables and let µ = m r=1 µ r where µ r = E[X r ] for each r = 1,..., m be the mean value of the sum. Define the interval K i as the inverse image of I i under the mapping F (x) = (F 1 (x 1 ),..., F m (x m )) (, 1) m. The variance of the sum is given by m E[(S µ) 2 ] = σr 2 + 2n ( ) h i m r (i r )m s (i s ) (A.5) r=1 i {1,...,n} m 1 r<s m where σr 2 = for each r = 1,..., m and (x r µ r ) 2 f r (x r )dx r (A.6) m r (k) = cr(k+1) c r(k) (x r µ r )f r (x r )dx r (A.7) for each r = 1,..., m and each k = 1, 2,..., n and where we have written K i = (c 1 (i 1 ), c 1 (i 1 + 1)) (c m (i m ), c m (i m + 1)) for each i = (i 1,..., i m ). See Piantadosi et al. (212 b) for details of the theoretical calculations. Appendix A.4. Outline of the mathematical analysis for a copula of maximum entropy The checkerboard copula of maximum entropy is defined by the solution to the following problem. Problem Appendix A.1 (The primal problem). Find the hyper-matrix h = [h i ] R l to maximize the entropy J(h) = ( 1) 1 h n i log e h i + (m 1) log e n (A.8) i {1,...,n} m 27