Phase-randomisation in an integrated assessment model for Antarctic krill

An integrated model was configured to estimate eighteen different subsets of 119 total parameters in alternative configurations of a model assessing the status and productivity of Antarctic krill (Euphausia superba, hereafter krill). We fixed the parameters that were not estimated in any given configuration at pre-specified values. The model was fitted to survey and fisheries data for krill in Subarea 48.1, a statistical reporting area used by the Commission for the Conservation of Antarctic Marine Living Resources (CCAMLR). The number of estimated parameters was gradually increased across model configurations. The numbers of parameters estimated in configurations that were able to obtain an invertible Hessian matrix ranged from 48 to 107. Groups of estimated parameters in each configuration were activated in seven sequential stages (the "phases") in a series of at least 20 replicate trials for each configuration. The parameter phases were assigned and reassigned randomly in each trial until an invertible Hessian matrix was obtained, or until 3,000 phase assignments had been completed for the replicate without obtaining an invertible Hessian. Model configurations were evaluated in terms of the total objective value, the maximum gradient, the proportion of replicate trials that found the minimum observed negative log-likelihood, and the number of iterations required in each trial series before obtaining an invertible Hessian matrix. Configurations that estimated more parameters fitted the data better, but the order in which the parameters were estimated became more important in finding the best fit as the numbers of estimated parameters increased. Phase-randomized replicates in configurations estimating many parameters were more likely to estimate values representing local minima than the lowest negative log-likelihood. The best observed estimates for the base configuration were obtained in about half of the replicates. Configurations estimating more parameters than the base configuration fitted the data better but either did not produce invertible Hessian matrices or had high maximum gradients. Diagnostics were applied to Markov chain Monte Carlo sampling in the best model of each configuration that obtained an invertible Hessian matrix to test for convergence to the equilibrium distribution.