On the determination of the optimal parameters in the CAM model

Abstract

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.

Large Ensemble Particle Filter for Spatial Climate Reconstructions Using a Linear Inverse Model

“…In all cases, the SIR‐LIMs clearly outperform the benchmark experiments only driven by external forcing (noDA experiment), even though the amplitude of extreme short‐lived cooling events caused by strong volcanic eruptions are underestimated. However, the use of a LIM as a dynamical model still limits the full potential of PFs and future work with more sophisticated models with non‐Gaussian and even additive noise (Bianucci & Mannella, 2021; Martinez‐Villalobos et al., 2019) and/or with other non‐linear forecast models (Nadiga, 2021) could help improve the reconstructions of extreme anomalies.…”

A mechanism for the skew of ensemble forecasts

“…To address this, we used the conditional moments (variance, skew, and excess kurtosis) in our numerical ensembles to estimate the conditional parameters E , g , and b , assuming that they are related to the conditional moments in the same way as the stationary parameters are related to the stationary moments in Equations A4a–A4c. Figure 4 shows the parameters estimated from the control ensemble for three initial conditions, x o = 1, 3, and 5, as a function of forecast lead time τ —in practice, we suggest fitting the conditional pdfs using a more robust technique, like the maximum likelihood method or a Bayesian procedure (Bianucci and Mannella, 2021) than the “method of moments” used here. The conditional parameters estimated from the control ensemble were then specified in Equation (A2a) to determine candidate conditional SGS pdfs.…”