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Disaggregation of rainfall time series via Gibbs sampling
Abstract
We propose a MCMC methodology to estimate all the components of the RodriguezIturbe model. This parametric model is associated with a likelihood function, and we use the Gibbs sampler to draw posterior deviates of the parameters in a Bayesian framework, conditionally on the data. The Gibbs sampler incorporates a Metropolis-Hastings step to sample the internal features (cell durations, cell lengths, etc.) of the model. The methodology is associated with reversible jumps and birth and death steps. Noninformative priors are used. We perform the simulations on a real data set of hourly rainfall measurements, and we exhibit a lack of fit with the theoretical model. To improve some of the inefficiencies of the model, we generalize it, considering a joint bivariate negatively correlated exponential distribution for the cell durations and cell lengths. The MCMC methodology is then extended to handle this new model. Model fitting is still further improved by applying post-processing rescaling and smoothing-sharpening techniques on the unconditional simulated rainfall amounts. Finally, disaggregation of daily amounts into hourly amounts is considered at the end of the paper. Some of the issues considered here are missing data, assessing the convergence of the algorithm, asymptotic relationships and prediction.
... Unfortunately, there exists no multivariate extension of the exponential distribution that allows for a straightforward specification of a covariance structure. Some suggestions how to model correlation in this framework are given in [2], in [38], or in the technical report by Granville and Smith [21]. The latter is of particular interest to us, since they also use MCMC methods for estimation. ...
... Reversible-jump MCMC was introduced by Green [22]. Some applications are discussed e.g. in [7] or in the book by Frühwirth-Schnatter [12], while Granville and Smith apply it to rainfall disaggregation [21] using an extended version of the model found in [38]. ...
A stochastic model for rainfall at a single site is studied in which storms arrive in a Poisson process, each storm consisting of a cluster of a random number of rain cells, each cell having random duration and depth. A model studied in an earlier paper is extended to provide a better fit to empirical experience, the extension being based on the attachment of a single random variable to each storm to achieve in particular some correlation between the durations of the cells within a single storm. The properties of the new model are developed, its fitting to two sets of empirical data is described and the examination of adequacy of fit is studied in some detail via properties not used in the fitting procedure. Finally a theoretical study is made of short-term prediction from the model.
Markov chain Monte Carlo (MCMC) methods have been used extensively in statistical physics over the last 40 years, in spatial statistics for the past 20 and in Bayesian image analysis over the last decade. In the last five years, MCMC has been introduced into significance testing, general Bayesian inference and maximum likelihood estimation. This paper presents basic methodology of MCMC, emphasizing the Bayesian paradigm, conditional probability and the intimate relationship with Markov random fields in spatial statistics. Hastings algorithms are discussed, including Gibbs, Metropolis and some other variations. Pairwise difference priors are described and are used subsequently in three Bayesian applications, in each of which there is a pronounced spatial or temporal aspect to the modeling. The examples involve logistic regression in the presence of unobserved covariates and ordinal factors; the analysis of agricultural field experiments, with adjustment for fertility gradients; and processing of low-resolution medical images obtained by a gamma camera. Additional methodological issues arise in each of these applications and in the Appendices. The paper lays particular emphasis on the calculation of posterior probabilities and concurs with others in its view that MCMC facilitates a fundamental breakthrough in applied Bayesian modeling. Comments: Arnoldo Frigessi (41–43), Alan E. Gelfand, Bradley P. Carlin (43–46), Charles J. Geyer (46–48), G. O. Roberts, S. K. Sahu, W. R. Gilks (49–51), Wing Hung Wong (52–53), Bin Yu (54–58), Julian Besag, Peter Green, David Higdon, Kerrie Mengersen (58–66).
IntroductionInferences Concerning a Single Mean from Observations Assuming Common Known VarianceInferences Concerning the Spread of a Normal Distribution from Observations Having Common Known MeanInferences When Both Mean and Standard Deviation are UnknownInferences Concerning the Difference Between Two MeansInferences Concerning a Variance RatioAnalysis of the Linear ModelA General Discussion of Highest Posterior Density RegionsH.P.D. Regions for the Linear Model: A Bayesian Justification of Analysis of VarianceComparison of ParametersComparison of the Means of k Normal PopulationsComparison of the Spread of k DistributionsSummarized Calculations of Various Posterior Distributions
The available empirical descriptions of extratropical cyclonic storms are employed to formulate a physically realistic stochastic representation of the ground level rainfall intensity field in space and time. The stochastic representation is based on three-component stochastic point processes which possess the general features of the embedding of rain cells within small mesoscale areas within large mesoscale areas within synoptic storms. Certain scale idealizations, and assumptions on functional forms which qualitatively reflect the physical features, lead to a closed form expression for the covariance function, i.e., the real space-time spectrum, of the rainfall intensity field. The theoretical spectrum explains the empirical spectral features observed by Zawadzki almost a decade ago. Of particular interest and importance in this connection is an explanation of the empirical observation that the Taylorian propogation of the fine scale structure, via a transformation of time to space through the storm velocity, holds only for a small time lag and not throughout. The results here indicate the extent of this lag in terms of the characteristic scales associated with cell durations, cellular birthrates and velocities, etc.
Neyman-Scott type cluster point processes have been used in several studies to model temporal rainfall at a single location. In this paper we study the applicability of such models with the rainfall thought of as instantaneous bursts at the points of the Neyman-Scott process. We find that this class of models does not provide adequate fit to some observed rainfall series. We also discuss some estimation problems associated with the fitting procedure, and examine the importance and appropriateness of the distributional assumptions made in the modeling.
ABSTRACT A monotonic transformation is applied to hourly rainfall data to achieve marginal normality. This de nes a latent Gaussian variable, with zero rainfall corresponding to censored values below a threshold. Autocor-relations of the latent variable are estimated by maximum likelihood. The goodness of t of the model to Edinburgh rainfall data is comparable with that of existing point process models. Gibbs sampling is used to disaggre-gate daily rainfall data, to generate typical hourly data conditional on daily totals.
Series of hourly rainfall data are simulated which are consistent with recorded daily totals. A two-stage process is used. First, a long sequence of hourly data is simulated using the Rodriguez-lturbe rainfall model (Rodriguez-Iturbe et al., Proc. R. Soc. London, Ser. A, 417, 283–298, 1988). Then, comparisons are made in daily totals between all observed and all simulated 3 day sequences. For each observed sequence, the simulated rainfall which gives the best match is re-expressed on an hourly basis, and used as one possible realisation of hourly rainfall on those days. For Edinburgh, Turnhouse, for which hourly data are available, the agreement between observed and disaggregated series is good, in terms of both histograms of hourly rainfall and summary statistics.
The objective in this paper is to present and fit a relatively simple stochastic spatial-temporal model of rainfall in which the arrival times of rain cells occur in a clustered point process. In the x-y plane, rain cells are represented as discs; each disc having a random radius; the locations of the disc centres being given by a two-dimensional Poisson process. The intensity of each cell is a random variable that remains constant over the area of the disc and throughout the lifetime of the cell, the lifetime being an exponential random variable. The cells are randomly classified from 1 to n with different parameters for the different cell types, so that the random variables of an arbitrary cell, e.g. radius and intensity, are correlated. Multi-site second-order properties are derived and used to fit the model to hourly rainfall data taken from six sites in the Thames basin, UK.
advantages of a Bayesian approach. Whilst of course we recognize the importance and intellectual standing of the long debate about philosophies of inference at a more fundamental level, nevertheless it is surely true that some of the main historical objections to Bayesian inference have included the difficulty of computation, the need to approximate, the necessity to use stylized priors, and the inability to assess the impact of arbitrary assumptions in prior specifications. MCMC answers these objections amazingly well, and indeed also allows one to perturb the likelihood function. For those of us who were closet Bayesians, or at least are open-minded enough to discover what the paradigm can provide, MCMC does remove reasons not to be Bayesian. Geyer's claim that similar progress has been made in likelihood inference is surely grossly overstated. Integration is central to the Bayesian paradigm but runs into problems for almost any moderately complicated
The use of the Gibbs sampler for Bayesian computation is reviewed and illustrated in the context of some canonical examples. Other Markov chain Monte Carlo simulation methods are also briefly described, and comments are made on the advantages of sample-based approaches for Bayesian inference summaries.
Markov chain Monte Carlo (MCMC) simulation methods are being used increasingly in statistical computation to explore and estimate features of likelihood surfaces and Bayesian posterior distributions. This paper presents simple conditions which ensure the convergence of two widely used versions of MCMC, the Gibbs sampler and Metropolis-Hastings algorithms.
The Poisson model for rainfall occurrences in which storm intensity and duration are represented by two independent random variables is extended to consider intensity and duration as bivariate random variables each with a marginal exponential distribution. A numerical optimization method using annual maxima is adopted for parameters estimation. Comparison is made with the results of a numerical procedure which uses the Gumbel distribution as an approximation to the probability distribution of the extremes of the bivariate exponential model. A case study is presented using data from 18 raingauge stations in northern Italy. For rainfall durations of practical interest the theoretically derived relationships between probabilities and intensities compare favourably with observed relationships.
Several Markov chain methods are available for sampling from a posterior distribution. Two important examples are the Gibbs sampler and the Metropolis algorithm. In addition, several strategies are available for constructing hybrid algorithms. This paper outlines some of the basic methods and strategies and discusses some related theoretical and practical issues. On the theoretical side, results from the theory of general state space Markov chains can be used to obtain convergence rates, laws of large numbers and central limit theorems for estimates obtained from Markov chain methods. These theoretical results can be used to guide the construction of more efficient algorithms. For the practical use of Markov chain methods, standard simulation methodology provides several variance reduction techniques and also give guidance on the choice of sample size and allocation.
The spatial structure of the depth of rainfall from a stationary storm event is investigated by using point process techniques. Cells are assumed to be stationary and to be distributed in space either independently according to a Poisson process, or with clustering according to a Neyman-Scott scheme. Total storm rainfall at the centre of each cell is a random variable and rainfall is distributed around the centre in a way specified by a spread function that may incorporate random parameters. The mean, variance and covariance structure of the precipitation depth at a point are obtained for different spread functions. For exponentially distributed centre depth and a spread function having quadratically exponential decay, the total storm depth at any point in the field is shown to have a gamma distribution. The probability of zero rainfall at a point is investigated, as is the stochastic variability of model parameters from storm to storm. Data from the Upper Rio Guaire basin in Venezuela are used in illustration.
Markov chain Monte Carlo methods for Bayesian computation have until recently been restricted to problems where the joint
distribution of all variables has a density with respect to some fixed standard underlying measure. They have therefore not
been available for application to Bayesian model determination, where the dimensionality of the parameter vector is typically
not fixed. This paper proposes a new framework for the construction of reversible Markov chain samplers that jump between
parameter subspaces of differing dimensionality, which is flexible and entirely constructive. It should therefore have wide
applicability in model determination problems. The methodology is illustrated with applications to multiple change-point analysis
in one and two dimensions, and to a Bayesian comparison of binomial experiments.