On the determination of the optimal parameters in the CAM model

In the field of complex systems, it is often possible to arrive at some simple stochastic or chaotic Low Order Models (LOMs) exploiting the time scale separation between leading modes of interest and fast fluctuations. These LOMs, although approximate, might provide interesting qualitative insights regarding some important aspects like the average time between two extreme events. Recently, the simplest example of a LOM with multiplicative noise, namely, a linear system with a linearly state dependent noise [also called correlated additive and multiplicative (CAM) model], has been considered as archetypal for numerous phenomena that present markedly non-Gaussian statistics. We show in this paper that the determination of the parameters of a CAM model from the (few) available data is far from trivial and that the actual most likely parameters might differ substantially from the ones determined directly from a (necessarily limited) short sequence of observations. We illustrate how this problem can be tackled, at least to the extent possible, using an approach that is based on Bayes' theorem. We shall focus on a CAM modeling the El Niño Southern Oscillation but the methodology can be extended to any phenomenon that can be described by a simplified LOM similar to the one examined here and where the available sequence of data is relatively short. We conclude that indeed a Bayesian approach can fix the problem.