Several large-scale projects including FamilySearch, Ancestry, BALSAC (University of Quebec), and others have gathered incredible amounts of genealogical data ranging from millions to billions of individuals. To study the structure of this data, we propose a model that generates a genealogical network based on real-world genealogical data using two key features: (i) geodesic distance between couples prior to union and (ii) the number of children per couple. The distribution of the distance to a couples' nearest common ancestor in an observed community captures the global scale at which biological cycles form in the underlying genealogical network. Similarly, the number of children per couple captures the local structure given by the degree distribution in the genealogical network. Constructing imitation data which approximates a real-world network's structure and growth rate is desirable for use in generalizable machine learning models. This model, which we refer to as the Target Model, provides a foundation for further work in predicting family network growth and structure.

This dissertation presents solutions to two open problems in estimation theory. The first is a tractable analytical solution for problems in multi-target filtering which are too complex to solve using traditional techniques. The second explores a new approach to the nonlinear filtering problem for a general class of models. The approach to the multi-target filtering problem which involves jointly estimating a random process of the number of targets and their state, developed using the probability hypothesis density (PHD) filter alleviates the intractability of the problem by avoiding explicit data association. Moreover, the notion of linear jump Markov systems is generalized to the multiple target case to accommodate births, deaths and switching dynamics to derive a closed form solution to the PHD recursion for this so-called linear Gaussian jump Markov multi-target model. The proposed solution is general enough to accommodate a broad class of practical problems which are deemed intractable using traditional techniques. Based on this closed form solution, an efficient method is developed for tracking multiple maneuvering targets that switch between multiple models without the need for gating, track initiation and termination, or clustering for extracting state estimates. The approach to the nonlinear filtering problem explores the framework of the virtual linear fractional transformation (LFT) model which localizes the nonlinearity to the feedback with a simple and sparse structure. The LFT is an exact representation for any differentiable nonlinear mapping and therefore amenable to a general class of problems. An alternative analytical approximation method is presented which avoids linearization of the state space model. The uncorrelated structure of the feedback connection gives of the state space model. The uncorrelated structure of the feedback connection gives better second-order moment approximation of the nonlinearly mapped variables. By arranging the unscented transform in the feedback, the prediction and estimation steps are derived in closed form. The proposed filters for the discrete-time model and continuous-time dynamics with sampled-data measurements respectively are shown to be robust under highly nonlinear and uncertain conditions where standard analytical approximation based filters diverge. Moreover, the LFT based filters are efficient for online implementation. In addition, the LFT framework is applied to extend the closed form solution of the PHD recursion to the nonlinear jump Markov multi-target model.

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