In this paper we introduce a class of granular evolving system modeling approach within the framework of interval analysis. Our aim is to present an interval-based learning algorithm which develops both, granular and singular approximations of nonlinear nonstationary functions using singular data. The algorithm is capable of incrementally creating/adapting both model parameters and structure. These are key features in nonlinear systems modeling. In addition, interval analysis provides rigorous bounds on approximation errors, rounding errors, and on uncertainties in data propagated during computations. The learning algorithm is simple and particularly suited to process stream of data in real time. In this paper we focus on the foundations of the approach and on the details of the learning algorithm. An application concerning economic time series forecasting illustrates the usefulness and efficiency of the approach.