| The problem of filtering is an important content in the study of the information science,and it has been used in many applications such as radar,communication,navigation and sonar.The probability and statistics methods are the main ideas to tackle the issue.The existing filtering methods are trying to improve the performance by the numerical technique,but have seldom considered the geometric property in the procedure of filtering.As the filtering problem becomes more complicated,the usual filtering methods may be difficult to satisfy the requirement of filtering precision due to poor information.How to make full use of the probability distribution itself and the probabilistic distribution characteristics and information in the procedure to support the applications in the complex scenarios becomes the new study idea.Information geometry is the best tool that make the exploitation and exploration for the geometric structure and information.In this thesis,we will consider the filtering problem in the viewpoint of the information geometry,and give some graceful inspiration for proposing filtering algorithms.Firstly,we use Bayesian filtering to provide the unify framework for the filtering problems.The statistical manifold is constructed by the joint probability of the state and the measurement based on Bayesian principle.Bridging by the joint probability,we convert the Bayesian filtering into the estimation on the statistical manifold,and the information geometric optimization technique is used to seek the optimal estimation.With the natural gradient descent method,we derive the natural gradient descent filtering method.Then the Kalman filter,the extended Kalman filter and the iterated extended Kalman filter are been proving as the special formulation of natural gradient descent filtering.Same as the definition in the information filtering,the natural gradient descent filtering can induce the information filtering,the extended information filtering and the iterated information filtering.Thus,we can unify the state space,the information space and the probability space by utilizing the information geometry.Secondly,we use the determination points technique to describe the state and the measurement approximately.Then,we can use the natural gradient descent filtering method to induce the iterated unscented Kalman filter and iterated cubature Kalman filtering in the state space,and iterated unscented information filtering and iterated cubature information filtering in the information space correspondingly.Thirdly,we use the random points to describe the state and the measurement approximately in the Bayesian filtering directly.The non-Gaussian characteristics have been preserved in the filtering.Based on the framework of the ensemble Kalman filter,the individual random points are been updated by the natural gradient descent filtering.This improvement achieves the better performance comparing with the original ensemble Kalman filter.Besides,we use the Gaussian sum approximation to hold the non-Gaussian characteristics,and propose the Gaussian sum ensemble Kalman filter.Finally,we use the monte Carlo technique to approximate the Bayesian filtering.For the two existing methods,namely sequential monte carlo method and Markov chain monte carlo(MCMC)method,we built the bridge between the natural gradient and Riemannian manifold MCMC method.With the relationship between natural gradient and the Kalmantype filter,we can embed the Kalman-type filter into the Riemannian manifold MCMC method.The Kalman-type filter is used to grasp the local characteristics in the filtering procedure,and the Riemannian manifold MCMC is utilized hold the whole procedure.This combination can achieve the fast computation and better performance.In this thesis,we use the information geometry to unify the filtering in the state space,information space and the probability space.With the study being explored in depth,the filtering methods can benefit from the information geometry with more inspirations and instructions. |
