Entropy-randomized estimation of nonlinear dynamical model parameters on observation of dependent process

A. Yu. Popkov ; Yu. A. Dubnov ; Yu. S. Popkov

Čelâbinskij fiziko-matematičeskij žurnal, Tome 9 (2024) no. 1, pp. 144-159

Résumé

The work is devoted to the development of the method of randomized machine learning in the direction of estimating dynamic models of related processes using real data, one of which is considered as the main one, and the other as a dependent one. The model of the main process in this concept is implemented by a dynamic model based on differential equations with parameters, which in turn are implemented by a static model in a different time scale. Randomized machine learning is a new theory at the intersection of data science, machine learning and data mining, based on the concept of entropy to estimate the probabilistic characteristics of model parameters. Such characteristics are the probability distributions of the corresponding objects, the estimates of which are the distributions implemented by the probability density functions or discrete distributions. Achieving this goal becomes possible thanks to the idea of moving from models with deterministic parameters to models with random parameters and, additionally, measured at the output with random noise, which achieves the consideration of the stochastic nature, which is obviously present in any natural phenomenon. As a demonstration of the proposed method, we consider the problem of predicting the total number of infected people based on the SIR dynamic epidemiological model, in which one of the parameters is considered as the state of the associated process implemented by the static model. Its evaluation is carried out according to the observations of the main process, and forecasting is carried out using the model of the associated one. Conducted experiment using real data on COVID-19 cases in Germany shows the efficiency of the proposed approach. The forecast obtained by the classical least squares method leads to an underestimation of the model output compared to the real observed data, while the proposed approach demonstrates more flexibility and potentially allows you to obtain forecasts that are more adequate to the real data, which confirms its effectiveness and adequacy under conditions a small amount of data with a high level of uncertainty.

Détail

Keywords: randomized machine learning, entropy, entropy estimation, forecasting, randomized forecasting.