M.Sc Student | Nitai Stein |
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Subject | Cooperative Estimation Via Altruism |

Department | Department of Aerospace Engineering |

Supervisor | Full Professor Oshman Yaakov |

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.