‘Hybrid’ optimisation: a heuristic solution to the Markov-chain calibration problem

A Markov chain model for the natural course of succession through forest types was developed in a case study of mixed boreal forest located in Prioksko-Terrasnyi Biosphere Reserve, Near-Moscow Region, Russia. Stages of succession were specified in the conceptual scheme of succession transitions, and an original method of transition probability estimation, based upon expert data on the duration of the stages, generated a variety of transition matrices inheriting uncertainty both in the expert data and in the likelihood of alternative transitions in the scheme. To get rid of the uncertainty, the model has been calibrated to fit its observed state variables in terms of the relative area distribution among the specified forest types, the distribution becoming observable due to the application of a geographical information system (GIS) technology to forest inventory data. We formulate the calibration task as a multidimensional non-linear optimisation problem and find its formal approximate solution by means of a mathematical software routine. But the formal optimal solution has appeared to drive the model out of the expert-given intervals for stage durations, thus revealing disagreement between the model and observation data. Since the data may have errors, we have to look for a compromise in the model vs. data controversy. The formal solution, when combined with the original method of transition probability estimation, indicates the proper directions for a feasible choice of model parameters. Thus, the compromise solution (to the ‘hybrid’ optimisation problem) now consists in following those directions up (or down) to the nearest points on the expert-given intervals. Also, the formal solution has fixed the best likelihood ratios for the alternative transitions in the scheme, and the ‘hybrid’ inherits logically this advantage of the ‘parent’.