Contemporary problems in modeling from time series

Mathematical modeling from discrete sequences of experimental data (time series) is an actively developing field in mathematical statistics and nonlinear dynamics. It started from approximation of a set of data points on a plane with a smooth curve, while currently such empiric models take the form of sophisticated differential and difference equations and are capable of describing even nonlinear oscillatory and wave phenomena. Practical applications of the empiric models are various ranging from future forecasts to technical and medical diagnostics. However, procedures for their construction are difficult to describe within a cohesive framework. In this article we give an overview of key problems in construction of dynamical models from chaotic series and contemporary approaches to their solution. Various practical situations are described system-atically according to the amount of a priori information about appropriate model structure, they are called «transparent», «gray», and «black boxes». We outline results of publications of many scientific groups in international and Russian journals during the period 1981-2005. To illustrate approaches, mainly original results of the authors and their colleagues are used.