The paper presents a new approach to restoration characteristics randomized models under small amounts of input and output data. This approach proceeds from involving randomized static and dynamic models and estimating the probabilistic characteristics of their parameters. We consider static and dynamic models described by Volterra polynomials. The procedures of robust parametric and non-parametric estimation are constructed by exploiting the entropy concept based on the generalized informational Boltzmann’s and Fermi’s entropies.

We propose a new method of randomized forecasting (RF-method), which operates with models described by systems of linear ordinary differential equations with random parameters. The RF-method is based on entropy-robust estimation of the probability density functions (PDFs) of model parameters and measurement noises. The entropy-optimal estimator uses a limited amount of data. The method of randomized forecasting is applied to World population prediction. Ensembles of entropy-optimal prognostic trajectories of World population and their probability characteristics are generated. We show potential preferences of the proposed method in comparison with existing methods.

A method was proposed for approximate solution of the system of nonlinear algebraic equations and inequalities by computer-aided generation of the sequence of residues of this system that were calculated through the collections of random vectors generated at each algorithmic step. It is based on the batch iterations using the simple Monte Carlo trials. The almost sure convergence with an exponential rate of this sequence to the global minimum of residue was proved. For the finite number of iterations, the probabilistic estimates of the deviation of the residue value from its global minimum were established. The method can be used for approximate solution of systems of equations and inequalities with algorithmically defined functions satisfying the Hölder condition.

We propose a new forecasting procedure that includes randomized hierarchical dynamic regression models with random parameters, measurement noises and random input. We developed the technology of entropy-randomized machine learning, which includes the estimation of characteristics of a dynamic regression model and its testing by generating ensembles of predicted trajectories through the sampling of the entropy-optimal probability density functions of the model parameters and measurement noises. The density functions are determined at the learning stage by solving the constrained maximization problem of an information entropy functional subject to the empirical balances with real data. The proposed procedure is applied to the randomized forecasting of the daily electrical load in a regional power system. We construct a two-layer dynamic model of the daily electrical load. One of the layers describes the dependence of electrical load on ambient temperature while the other simulates the stochastic quasi-fluctuating temperature dynamics.

Randomized machine learning focuses on problems with considerable uncertainty in data and models. Machine learning algorithms are formulated in terms of a functional entropy-linear programming problem. We adapt these algorithms to forecasting problems on an example of the evolution of thermokarst lakes area in permafrost zones. Thermokarst lakes generate methane, a greenhouse gas affecting climate change. We propose randomized machine learning procedures using dynamic regression models with random parameters and retrospective data (climatic parameters and remote sensing of the Earth’s surface). The randomized machine learning algorithm developed below estimates the probability density functions of model parameters and measurement noises. Randomized forecasting is implemented as algorithms transforming the optimal distributions into the corresponding random sequences (sampling algorithms). The randomized forecasting procedures and technologies are trained, tested, and then applied to forecast the evolution of thermokarst lakes area in Western Siberia.