A smoothing algorithm for estimating stochastic, continuous time model parameters and its application to a simple climate model

Even after careful calibration, the output of deterministic models of environmental systems usually still show systematic deviations from measured data. To analyse possible causes of these discrepancies, we make selected model parameters time variable by treating them as continuous time stochastic processes. This extends an approach that was proposed earlier using discrete time stochastic processes. We present a Markov chain Monte Carlo algorithm for Bayesian estimation of such parameters jointly with the other, constant, parameters of the model. The algorithm consists of Gibbs sampling between constant and time varying parameters by using a Metropolis-Hastings algorithm for each parameter type. For the time varying parameter, we split the overall time period into consecutive intervals of random length, over each of which we use a conditional Ornstein-Uhlenbeck process with fixed end points as the proposal distribution in a Metropolis-Hastings algorithm. The hyperparameters of the stochastic process are selected by using a cross-validation criterion which maximizes a pseudolikelihood value, for which we have derived a computationally efficient estimator. We tested our algorithm by using a simple climate model. The results show that the algorithm behaves well, is computationally tractable and improves the fit of the model to the data when applied to an additional time-dependent forcing component. However, this additional forcing term is too large to be a reasonable correction of estimated forcing and it alters the posterior distribution of the other, time constant parameters to unrealistic values. This difficulty, and the impossibility of achieving a good simulation when making other parameters time dependent, indicates a more fundamental, structural deficit of the climate model. This is probably related to the poor resolution of the ocean in the model. Our study demonstrates the technical feasibility of the smoothing technique but also the need for a careful interpretation of the results. © 2009 Royal Statistical Society.