A New Class of Particle Filters for Random Dynamic Systems with Unknown Statistics
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A New Class of Particle Filters for Random Dynamic Systems with Unknown Statistics Joaqu´ın M´ıguez Departamento de Electr´onica e Sistemas, Universidade da Coru˜na, Facultade de Inform´atica, Campus de Elvi˜na s/n, 15071 A Coru˜na, Spain Email: [email protected]
´ Monica F. Bugallo Department of Electrical and Computer Engineering, State University of New York at Stony Brook, Stony Brook, NY 11794-2350, USA Email: [email protected]
Petar M. Djuri´c Department of Electrical and Computer Engineering, State University of New York at Stony Brook, Stony Brook, NY 11794-2350, USA Email: [email protected] Received 4 May 2003; Revised 29 January 2004 In recent years, particle filtering has become a powerful tool for tracking signals and time-varying parameters of random dynamic systems. These methods require a mathematical representation of the dynamics of the system evolution, together with assumptions of probabilistic models. In this paper, we present a new class of particle filtering methods that do not assume explicit mathematical forms of the probability distributions of the noise in the system. As a consequence, the proposed techniques are simpler, more robust, and more flexible than standard particle filters. Apart from the theoretical development of specific methods in the new class, we provide computer simulation results that demonstrate the performance of the algorithms in the problem of autonomous positioning of a vehicle in a 2-dimensional space. Keywords and phrases: particle filtering, dynamic systems, online estimation, stochastic optimization.
1.
INTRODUCTION
Many problems in signal processing can be stated in terms of the estimation of an unobserved discrete-time random signal in a dynamic system of the form xt = fx (xt−1 ) + ut , yt = f y (xt ) + vt ,
t = 1, 2, . . . , t = 1, 2, . . . ,
(1) (2)
where (a) xt ∈ RLx is the signal of interest, which represents the system state at time t; (b) fx : RLx → Ix ⊆ RLx is a (possibly nonlinear) state transition function; (c) ut ∈ RLx is the state perturbation or system noise at time t; (d) yt ∈ RL y is the vector of observations collected at time t, which depends on the system state;
(e) f y : RLx → I y ⊆ RL y is a (possibly nonlinear) transformation of the state; (f) vt ∈ RL y is the observation noise vector at time t, assumed statistically independent from the system noise ut . Equation (1) describes the dynamics of the system state vector and, hence, it is usually termed state equation or system equation, whereas (2) is commonly referred to as observation equation or measurement equation. It is convenient to distinguish the structure of the dynamic system defined by the functions fx and f y from the associated probabilistic model, which depends on the probability distribution of the noise signals and the a priori distribution of the state, that is, the statistics of x0 . We denote the a priori probability density function (pdf) of a random signal s as p(s). If the signal s is statistically dependent on some observation z, then we write the conditional
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