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In this paper, we consider mobile sensor networks that use spatiotemporal Gaussian processes to predict a wide range of spatiotemporal physical phenomena. Nonparametric Gaussian process regression that is based on truncated observations is proposed for mobile sensor networks with limited memory and computational power. We first provide a theoretical foundation of Gaussian process regression with truncated observations. In particular, we demonstrate that prediction using all observations can be well approximated by prediction using truncated observations under certain conditions. Inspired by the analysis, we then propose a centralized navigation strategy for mobile sensor networks to move in order to reduce prediction error variances at points of interest. For the case in which each agent has a limited communication range, we propose a distributed navigation strategy. Particularly, we demonstrate that mobile sensing agents with the distributed navigation strategy produce an emergent, swarming-like, collective behavior for communication connectivity and are coordinated to improve the quality of the collective prediction capability.