Particle filter

Other name of Particle filter is sequential Monte Carlo method. It is referred to as a technique which is used for estimation purposes. One can see its importance in applications of econometrics. These are used in the estimation of Bayesian models. Bayesian model is a model in which you will find latent variables. These latent variables found in the model are connected by markov chain. Bayesian model is very much similar to a model known as hidden markov model.

The difference between the models is that the latent variables state space is continuous rather than being discrete as in Bayesian model. Particle filter have a certain aim. Its aim is that it helps in determining the sequence of the hidden parameters. The sequence determined by it is based on the data into observation. The value of data is 1 divided by k. The value of k starts from 0. The parameter that has been estimated using Bayesian model is a follower of posterior distribution.

There are various kinds of filters. One is auxiliary particle filter. It is an algorithm that is introduced by Pitt and shepherd. It was introduced in the year 1999. It is referred to as a sequential Monte Carlo algorithm. This algorithm is called as a sampling method that is used for making an approximation for a distribution. The distribution in this case makes use of a structure that is temporal. The distribution is represented in the form of particles.

There is a formula for this kind of distribution. It is determined as probability of x divided by z. here x is referred to as an unobserved state that is observed at a particular time t. z is considered to be the observation sequence. This sequence starts from time 0 and ends till time t. Another category of filters is kalman filters. It has a distribution formula that is based on the assumption of two models. Those two models are known as transition model and multivariate model. Particle filter is a very general form. Few of its assumptions are based upon the models discussed here i.e. transition model and sensor model.

One cannot do sampling directly from the density known as true posterior density. It is impossible to do such a thing. Sampling can be easily done through a distribution known as proposal distribution. If state space assumptions are used then the estimation of the weights called as importance weights can be done in a recursive manner. The term recursive means anything that can be repeated again and again. There is a term related with sampling. It is called sampling-importance resampling.

If there are some unequal weighted particles, then it can be sampled again into a new set of equal weighted particles. This method has been introduced by Gordon, salmond and smith. After this method was introduced, Gordon proved it mathematically with the help of mathematical calculations. Particles that are drawn into a model are drawn with the help of proposal distribution. This distribution is used to do the evaluation of importance weights. These filters can be improved if new observations are proposed.