Statistical Manifold

We prove that the information geometry's Frobenius manifold is a symplectic manifold having Poisson structures. By proving this statement, a bridge is created between the theories developed by Vinberg, Souriau and Koszul and the Frobenius manifold approach. Until now these approaches seemed disconnected.

In this paper, a novel statistical manifold algorithm is proposed for position estimation of sensor nodes in a wireless network, making full use of distance information available among unknown nodes and simultaneous localization of multiple unknown nodes. To begin, a ranging model including the distance information among unknown nodes is established. With the reparameterization of the natural parameter and natural statistic, the solution problem of the ranging model is transformed into a parameter estimation problem of the curved exponential family. Then, a natural gradient method is adopted to deal with the parameter estimation problem of the curved exponential family. To ensure the convergence of the proposed algorithm, a particle swarm optimization method is utilized to obtain initial values of the unknown nodes. Experimental results indicate that the proposed algorithm can improve the positioning accuracy, compared with the traditional algorithm.