In this paper, a novel framework is proposed to enable a predictive deployment of unmanned aerial vehicles (UAVs) as temporary base stations (BSs) to complement ground cellular systems in face of downlink traffic overload. First, a novel learning approach, based on the weighted expectation maximization (WEM) algorithm, is proposed to estimate the user distribution and the downlink traffic demand. Next, to guarantee a truthful information exchange between the BS and UAVs, using the framework of contract theory, an offload contract is developed, and the sufficient and necessary conditions for having a feasible contract are analytically derived. Subsequently, an optimization problem is formulated to deploy an optimal UAV onto the hotspot area in a way that the utility of the overloaded BS is maximized. Simulation results show that the proposed WEM approach yields a prediction error of around 10%. Compared with the expectation maximization and k-mean approaches, the WEM method shows a significant advantage on the prediction accuracy, as the traffic load in the cellular system becomes spatially uneven. Furthermore, compared with two event-driven deployment schemes based on the closest-distance and maximal-energy metrics, the proposed predictive approach enables UAV operators to provide efficient communication service for hotspot users in terms of the downlink capacity, energy consumption and service delay. Simulation results also show that the proposed method significantly improves the revenues of both the BS and UAV networks, compared with two baseline schemes.