Recently, there has been a growing interest in wireless sensor networks due to the advanced embedded system and network technologies. Their applications include, but are not limited to, environment monitoring, building comfort control, traffic control, manufacturing and plant automation, and surveillance systems. The conventional inverse problem approach based on physical transport models is computationally prohibitive for resource-constrained, multi-agent systems. In contrast, Gaussian process and Gaussian Markov models have been widely used to draw statistical inference from geostatistical and environmental data. However, the statistical models need to be carefully tailored such that they can be practical and usable for mobile sensor networks with limited resources. In addition, reducing localization uncertainty in low-cost mobile sensors is very challenging. Thus, a fundamental problem in various applications is to correctly fuse the spatially collected data and estimate the process of interest under localization uncertainty.Motivated by the aforementioned issues, in this dissertation, we consider the problem of using mobile sensor networks to estimate and predict environmental fields modeled by spatio-temporal Gaussian processes and Gaussian Markov random fields in the presence of localization uncertainty.In the first part of this dissertation, we formulate Gaussian process regression with observations under the localization uncertainty. In our formulation, effects of measurement noise, localization uncertainty, and prior distributions are all correctly incorporated in the posterior predictive statistics. The analytically intractable posterior predictive statistics are proposed to be approximated by two techniques, viz., Monte Carlo sampling and Laplace's method. In addition, the localization problem is studied in this part, when the position of the robot is estimated by a maximum likelihood estimation (MLE) using vision data. We transform the high dimensional vision data to a set of uncorrelated feature candidates. A multivariate GP regression with unknown hyperparameters is formulated to connect the set of selected features to their corresponding sampling positions. In order to decrease computational load and increase the accuracy of localization, a feature reduction approach is developed. Therefore, a subset of the features is selected to minimize the localization error using cross-validation methods.In the second part of the dissertation, we consider the problem of predicting a spatial (spatio-temporal) random field using sequential noisy observations collected by robotic sensors. The random field of interest is modeled by a Gaussian Markov random field (GMRF) instead of Gaussian process. In this way, we proposed iteratively updated predictive inferences. We derive the exact Bayesian solution to the problem of computing the predictive inference of the random field, taking into account uncertain hyperparameters, measurement noise, and uncertain localization in a fully Bayesian point of view. We show that the exact solution is not scalable as the number of observations increases. To cope with this exponentially increasing complexity, we propose scalable approximations with a controllable tradeoff between approximation error and complexity to the exact solution. Finally, we derive an approximate Bayesian solution to the problem of the simultaneously localization and computing the predictive inferences, taking into account observations, uncertain hyperparameters, measurement noise, kinematics of robotic sensors, and uncertain localization.
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