Fitting the Bartlett-Lewis rainfall model using Approximate Bayesian Computation

Abstract

The Barlett-Lewis (BL) rainfall model is a stochastic model for the rainfall at a single point in space, constructed using a cluster point process. The cluster process is constructed by taking a primary/parent process, called the storm arrival process in our context, and then attaching to each storm point a finite secondary/daughter point process, called a cell arrival process. To each cell arrival point we then attach a rain cell, with an associated rainfall duration and intensity. The total rainfall at time t is then the sum of the intensities from all active cells at that time. Following Rodriguez-Iturbe et al. (1987), we suppose that the storm arrival process is a Poisson process, and that the cell arrival processes are independent Poisson processes, truncated after an exponentially distributed time (the storm duration). Rain cells are all i.i.d., with independent exponentially distributed duration and intensity. Because it has an intractible likelihood function, in the past the BL model has been fitted using the Generalized Method of Moments (GMM). The puprose of this paper is to show that Approximate Bayesian Computation (ABC) can also be used to fit this model, and moreover that it gives a better fit than GMM. GMM fitting matches theoretical and observed moments of the process, and thus is restricted to moments for which you have an analytic expression. ABC fitting compares the observed process to simulations, and thus places no restrictions on the statistics used to compare them. The penalty we pay for this increased flexibility is an increase in computational time. The ABC methodology supposes that we have an observation D from some model f(·|θ), depending on parameters θ, and that we are able to simulate from f. Let π be the prior distribution for θ and S = S(D) a vector of summary statistics for D, then ABC generates samples from f(θ|ρ(S(D∗), S(D)) < ∊), where D∗ ∼ f(·|θ), θ ∼ π, and ρ is some distance function. If S is a sufficient statistic, then as ∊ → 0 this will converge to the posterior f(θ|D). The choice of good summary statistics is important to the success of ABC fitting. To fit the BL model we used rainfall aggregated over six-minute and hourly intervals, and then compared the mean, standard deviation, auto-correlation at lags 1 and 2, probability of no rain, mean length of wet and dry periods, standard deviation of wet and dry periods, and the total number of wet and dry periods We note that for GMM fitting we can only use the first five of these statistics, because we do not have analytic expressions for the others. Using a simulation study we demonstrate that ABC fitting can give less biased and less variable estimates than GMM. We also give an application to rainfall data from Bass River, Victoria, July 2010. Again we see that the ABC fit is better than the GMM fit. An important advantage of ABC fitting over GMM fitting is that we can use summaries of the data that capture useful information, whether or not we have an expression for their expectation. Moreover, this means that ABC can be used for models for which GMM fitting is not available. For example, if we used a gamma distribution for the duration of a rain cell, rather than an exponential distribution, then we would not be able to calculate the second order statistics of the model, making GMM fitting impossible. However ABC fitting would proceed as before, with the addition of a single parameter. This opens up the possibility of fitting much more realistic stochastic rainfall models.