Maximum Likelihood pqEDMD Identification

The ordinary least squares (OLS) regression for linear system identification might give biased results when noise affects some explicative variables. As OLS is at the core of the extended dynamic mode decomposition algorithm, it is interesting to pay attention to alternative methods, such as maximum likelihood estimation (MLE), to deal with the identification problem. This study explores this direction, discusses the question of defining the probability distribution of the observable functions, and illustrates the performance of the algorithm with two case studies. The first one shows a successful application of MLE to a simple reaction network, while the second, more complex example based on the Duffing equation highlights the method limitation in relation with the empirical construction of the probability distribution of the observables.