M.Sc Thesis


M.Sc StudentNitai Stein
SubjectCooperative Estimation Via Altruism
DepartmentDepartment of Aerospace Engineering
Supervisor Full Professors Oshman Yaakov


Abstract

This thesis proposes a novel cooperative estimation approach. The scenario is that of a system comprising two subsystems that share information in order to generate a global estimate of a random parameter. Contrary to common cooperative estimation approaches, only the accomplishment of the global mission matters, so that each subsystem can relinquish the optimality of its own estimator. We call this cooperation strategy altruism, taking after the well-known phenomenon exhibited in nature. Two approaches for altruistic cooperation are presented. In the heterarchical cooperative estimation approach both estimators behave altruistically, sacrificing their own estimation performance for the purpose of improving global estimation performance.

In the hierarchical cooperative estimation approach one subsystem behaves egoistically (optimizing its own estimation performance) whereas the other behaves altruistically.


The thesis shows that using the two subsystems egoistically, even if cooperating in terms of information sharing, is inferior to the proposed altruistic approaches. The choice between the two altruistic approaches is up to the decision of the system designer. A more daring designer would choose the heterarchical approach, which is globally optimal but requires the sacrifice of estimation performance of both estimators; a more conservative designer would prefer the hierarchical approach, which yields a sub-optimal solution, but leaves the first estimator optimal.


To define the two approaches as proper mathematical optimization problems, a global cost function is defined. The cost function takes a form similar to the mean squared error (MSE), and, in fact, constitutes a generalization of it. This cost function is common to both of the proposed approaches. However, in each approach different constraints are applied, and, hence, despite the common cost function, the two approaches yield two separate optimization problems.


For each problem, equations that yield candidate minimizers of the cost function are derived. Finding explicit solutions to these equations is generally very hard. It is proved that among those candidates, there necessarily exist global minimizers of the cost function. It is shown that these optimal solutions yield centroidal Voronoi tessellations (CVT). In terms of estimation, this means that the optimal altruistic estimators are optimal (in the minimum mean squared error sense) inside their Voronoi regions.


When the underlying distribution is Gaussian, explicit expressions are obtained for the estimators in both approaches. Implementation requires only the calculation of the largest eigenvalue of the parameter conditional covariance matrix and its corresponding eigenvector.


To demonstrate the superiority of the proposed altruistic approaches over the egoistic approach, the achieved costs in the Gaussian case are compared to the minimum mean squared error. It is shown that the gain from using altruism depends on the dynamic range of the spectrum of the parameter conditional covariance matrix, and on the parameter dimension. Numerical examples are used to show the behavior of the cost functions in scalar and 2D Gaussian cases, and to demonstrate the validity of the expressions for optimal estimators.