Non-linear And Non-normal Filter Based On Monte-carlo Technique

Non-Linear and Non-Normal Filter Based on Monte-Carlo Technique Hisashi TANIZAKI Kobe University, Kobe 657, JAPAN Abstract: A non-linear and/or non-normal filter is proposed in this paper. Generating random draws of the state vector directly from the filtering density, the filtering estimate is obtained, which gives us a recursive algorithm. There, we do not evaluate any integration included in the density-based filtering algorithm such as the numerical integration procedure and the MonteCarlo integration approach. The Monte-Carlo experiments indicate that the proposed non-linear and non-normal filter shows a good performance. Key Words: Non-Linear, Non-Normal, Filtering, Random Draws, Rejection Sampling. 1 Introduction A non-linear and non-normal filtering algorithm is proposed in this paper. Given random draws of the state vector which are directly generated from the filtering density, the filtering estimate is recursively obtained. There, we do not evaluate any integration included in the density-based filtering algorithm. We have numerous density-based filtering algorithms. Kitagawa [6] and Kramer and Sorenson [8] proposed the numerical integration procedure. Tanizaki [11], Tanizaki and Mariano [12] and Mariano and Tanizaki [9] utilized the Monte-Carlo integration with importance sampling to derive non-linear and non-normal filtering algorithm. Moreover, 1 Carlin, Polson and Stoffer [2] suggested the Monte-Carlo integration procedure with Gibbs sampling.1 Thus, in this paper, a Monte-Carlo procedure of filtering algorithm using the simulation technique is proposed, where we utilize the random draws only. The procedure improves over the other non-linear filters developed in the past from the following three points, i.e., computational burden, simplicity of computer programming and no ad hoc assumptions. The numerical procedure proposed in Kitagawa [6] and Kramer and Sorenson [8] has the disadvantages: (i) Location of nodes has to be set by a researcher. (ii) We have to derive the densities from measurement and transition equations by hand. (iii) Computational burden increases more than proportionally as the dimension of the state variable is high. The problems of the Monte-Carlo integration procedure with importance sampling developed by Tanizaki [11], Tanizaki and Mariano [12] and Mariano and Tanizaki [9] are: (i) The importance density has to be appropriately chosen by a researcher. (ii) We need to derive the densities from measurement and transition equations by hand. Use of Monte-Carlo integration with Gibbs sampler (Carlin, Polson and Stoffer [2]) also has some problems: (i) We need to assume the distributions of nuisance parameters. (ii) The Gibbs sampler leads to a great amount of data storage. (iii) This procedure also takes a lot of time computationally. The Monte-Carlo procedure proposed in Tanizaki and Mariano [13] has the following disadvantages: (i) The Monte-Carlo approach requires a large amount of data storage. (ii) Precision of the state-variable estimate is poor. 1 The non-linear and non-normal filter which Carlin, Polson and Stoffer [2] proposed is not a recursive algorithm. For the Gibbs sampler, see Geman and Geman [4]. 2 An alternative procedure which we propose utilizes the Monte-Carlo technique. By generating random draws directly fr