Formal Definitions of Conservative PDFs

Under ideal conditions, the probability density function (PDF) of a random variable, such as a sensor measurement, would be well known and amenable to computation and communication tasks. However, this is often not the case, so the user looks for some other PDF that approximates the true but intractable PDF. Conservativeness is a commonly sought property of this approximating PDF, especially in distributed or unstructured data systems where the data being fused may contain un-known correlations. Roughly, a conservative approximation is one that overestimates the uncertainty of a system. While prior work has introduced some definitions of conservativeness, these definitions either apply only to normal distributions or violate some of the intuitive appeal of (Gaussian) conservative definitions. This work provides a general and intuitive definition of conservativeness that is applicable to any probability distribution, including multi-modal and uniform distributions. Unfortunately, we show that this \emph{strong} definition of conservative cannot be used to evaluate data fusion techniques. Therefore, we also describe a weaker definition of conservative and show it is preserved through common data fusion methods such as the linear and log-linear opinion pool, and homogeneous functionals. In addition, we show that after fusion, weak conservativeness is preserved by Bayesian updates. These strong and weak definitions of conservativeness can help design and evaluate potential correlation-agnostic data fusion techniques

Unifying Consensus and Covariance Intersection for Efficient Distributed State Estimation over Unreliable Networks

This thesis studies the problem of recursive distributed state estimation over unreliable networks. The main contribution is to fuse the independent and dependent information separately. Local estimators communicate directly only with their immediate neighbors and nothing is assumed about the structure of the communication network, specifically it need not be connected at all times. The proposed estimator is a Hybrid one that fuses independent and dependent (or correlated) information using a distributed averaging and iterative conservative fusion rule respectively. It will be discussed how the hybrid method can improve estimators's performance and make it robust to network failures. The content of the thesis is divided in two main parts. In the first part I study how this idea is applied to the case of dynamical systems with continuous state and Gaussian noise. I establish bounds for estimation performance and show that my method produces unbiased conservative estimates that are better than Iterative Covariance Intersection (ICI). I will test the proposed algorithm on an atmospheric dispersion problem, a random linear system estimation and finally a target tracking problem. In the second part, I will discuss how the hybrid method can be applied to distributed estimation on a Hidden Markov Model. I will discuss the notion of conservativeness for general probability distributions and use the appropriate cost function to achieve improvement similar to the first part. The performance of the proposed method is evaluated in a multi-agent tracking problem and a high dimensional HMM and it is shown that its performance surpasses the competing algorithms