Aerosol model selection and uncertainty modelling by adaptive MCMC technique

M. Laine, J. Tamminen
2008 Atmospheric Chemistry and Physics  
Introduction Conclusions References Tables Figures ◭ ◮ ◭ ◮ Back Close Full Screen / Esc Printer-friendly Version Interactive Discussion Atmos. Abstract We apply Bayesian model selection techniques on the statistical inversion problem of the GOMOS instrument. The motif is to study which type of aerosol model best fits the data and to show how the uncertainty of the aerosol model can be included in the error estimates. The competing models consist of various formulations, each having different 5
more » ... nknown parameter vectors. We have developed an Adaptive Automatic Reversible Jump Markov chain Monte Carlo method (AARJ) for sampling values from the posterior distributions of the unknowns of the models. The algorithm is easy to implement and can readily be employed for model selection as well as for model averaging, to properly take into account the uncertainty of the modelling. 10 15 and natural framework to consider uncertainty in the estimated values as well as the model uncertainty. In many cases, classical approximative estimation methods can be seen as special cases of some more general Bayesian analyses, see for example Kaipio and Somersalo (2004) . In Bayesian inference, the uncertainty of the estimated value is a primary target of the 20 investigation. Whenever computationally possible, the result of the analysis is the full multi-dimensional posterior probability density of the unknowns. The approach allow the study of many kinds of uncertainties, including uncertainty in the model itself. Prior information from different sources can be pooled and incorporated statistically correctly and the correlation structure of the unknowns can be fully explored. Practical tools for 25 applying Bayesian inference to modelling problems are provided by the Markov chain 10792 Abstract Introduction Conclusions References Tables Figures ◭ ◮ ◭ ◮ Back Close Full Screen / Esc Printer-friendly Version Interactive Discussion Monte Carlo (MCMC) methods. MCMC is a common title for algorithms that simulate values from a probability distribution known only up to a normalizing constant. A typical case of such a task is to find the posterior distribution of the unknown parameters of a geophysical model. For application examples and more details on applying Bayesian MCMC methods in geophysical research see, for example, Tamminen and Kyrl (2001); 5 Tamminen (2004); Haario et al. (2004). In this article the Bayesian model selection and averaging is applied to the GOMOS (ESA 2007) aerosol model selection problem. GOMOS (Global Ozone Monitoring by Occultation of Stars) is an instrument on board the Envisat satellite that uses stellar occultation to measure the atmosphere Abstract ACPD Abstract Introduction Conclusions References Tables Figures ◭ ◮ ◭ ◮ Back Close Full Screen / Esc Printer-friendly Version Interactive Discussion Abstract ACPD Abstract Introduction Conclusions References Tables Figures ◭ ◮ ◭ ◮ Back Close Full Screen / Esc Printer-friendly Version Interactive Discussion Abstract 20 Abstract Introduction Conclusions References Tables Figures ◭ ◮ ◭ ◮ Back Close Full Screen / Esc Printer-friendly Version Interactive Discussion Abstract 25 DRAM adaptation (Haario et al., 2006) adds a new component to the AM method that is called Delayed Rejection (Abstract Introduction Conclusions References Tables Figures ◭ ◮ ◭ ◮ Back Close Full Screen / Esc Printer-friendly Version Interactive Discussion chain is Markovian and that the so called reversibility condition holds. This means that all the standard MH distributional convergence statements hold. In the DRAM method the DR algorithm is used together with several different adaptive Gaussian proposals. This helps the algorithm in two ways. Firstly, it enhances the adaptation by providing accepted values that make the adaptation start earlier. Secondly, it allows the sampler 5 to work better for non Gaussian targets and with non linear correlations between the components. The ergodicity of the DRAM method is proven by Haario et al. (2006). A new feature presented in this article is the combination of the DRAM and AM adaptations with the automatic RJMCMC. The practical application presented is the aerosol model selection in the GOMOS inversion. We want to note that Hastie (2005) 10 has also suggested a combination of adaptation and automatic RJMCMC of Green. The adaptation method (so called Adaptive Acceptance Probability, AAP) used in his work is, however, different from the adaptation employed here. We regard our AARJ method to be more general and easily applicable to high dimensional nonlinear models typical in geophysical problems. 15 3.3 The AARJ algorithm Here we present a schema for the algorithm for AARJ, an Adaptive Automatic Reversible Jump MCMC for model selection and model averagingl problems with a fixed number of models M 1 , . . . , M k . 3.3.1 The algorithm 20 1. Run separate adaptive MCMC chains using the DRAM method for all the proposed models. Collect the mean vectors µ (i ) and the Cholesky factors R (i ) of the covariance matrices of the chains, i =1, . . . , k. 2. Run automatic RJMCMC using the target approximations calculated in step (1). 3. If the current model is kept, use the standard random walk MH with Gaussian Abstract ACPD Abstract Introduction Conclusions References Tables Figures ◭ ◮ ◭ ◮ Back Close Full Screen / Esc Printer-friendly Version Interactive Discussion
doi:10.5194/acp-8-7697-2008 fatcat:fr5jscbsqjgmlf56r26rjvrcby