In recent years, due to significant progress in sensing, communication, and embedded-system technologies, mobile sensor networks have been exploited in monitoring and predicting environmental fields (e.g., temperature, salinity, pH, or biomass of harmful algal blooms). To deal with practical situations, phenomenological and statistical modeling techniques shall be used to make inferences from observations. However, such statistical models need to be carefully tailored such that they can be practical and usable for mobile sensor networks with limited resources. In this dissertation, we consider the problem of using mobile sensor networks to estimate and predict environmental fields modeled by spatio-temporal Gaussian processes. In the first part of the dissertation, we first present robotic sensors that learn a spatio-temporal Gaussian process and move in order to improve the quality of the estimated covariance function. For a given covariance function, we then theoretically justify the usage of truncated observations for Gaussian process regression for mobile sensor networks with limited resources. We propose both centralized and distributed navigation strategies for resource-limited mobile sensing agents to move in order to reduce prediction error variances at points of interest. Next, we formulate a fully Bayesian approach for spatio-temporal Gaussian process regression such that multifactorial effects of observations, measurement noise, and prior distributions of hyperparameters are all correctly incorporated in the posterior predictive distribution. To cope with computational complexity, we design sequential Bayesian prediction algorithms in which exact predictive distributions can be computed in constant time as the number of observations increases. Under this formulation, we provide an adaptive sampling strategy for mobile sensors, using the maximum a posteriori (MAP) estimation to minimize the prediction error variances.In the second part of the dissertation, we address the issue of computational complexity by exploiting the sparsity of the precision matrix used in a Gaussian Markov random field (GMRF). The main advantages of using GMRFs are: (1) the computational efficiency due to the sparse structure of the precision matrix, and (2) the scalability as the number of measurements increases. We first propose a new class of Gaussian processes that builds on a GMRF with respect to a proximity graph over the surveillance region, and provide scalable inference algorithms to compute predictive statistics. We then consider a discretized spatial field that is modeled by a GMRF with unknown hyperparameters. From a Bayesian perspective, we design a sequential prediction algorithm to exactly compute the predictive inference of the random field. An adaptive sampling strategy is also designed for mobile sensing agents to find the most informative locations in taking future measurements in order to minimize the prediction error and the uncertainty in the estimated hyperparameters simultaneously.
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