Nonstationarity in Multifactor Models of Discrete Jump Processes, Memory, and Application to Cloud Modeling

Because of the mathematical and numerical limitations, standard statistical methods known from the literature are not applicable to inferring jump processes under exogenous influence. Such processes can be considered, for example, in the atmosphere (transitions between different cloud types) and in the ocean (phase transitions between water and ice). Reasons for these intrinsic limitations of standard methods are investigated and a method for the inference of discrete microscopic jump models based on macroscopic ensemble observations is presented. It significantly extends the recently developed methods of nonstationary Markov model parameterization (which are constrained to a single exogenous factor and to direct individual observations of the jump process realizations). The main advantage of the new method is the possibility of inference from indirect ensemble observations with multiple exogenous factors. Moreover, this method allows for a new possibility to test whether the available time series is best described via stationary or nonstationary and Markovian (with memory) or independent (without memory) processes. It also allows estimation of the relative significance of the exogenous factors and their impact on the jump probabilities. The new framework provides a unified toolkit for data analysis of jump processes with the same level of detail now possible for standard continuous state space tools. The resulting numerical algorithm is applied to analysis of the total relative cloud cover data in the midlatitudes and in the tropics under the influence of some meteorologically relevant local and global exogenous factors.