In statistics

Statistics

Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

, particle filters, also known as Sequential Monte Carlo method

Monte Carlo method

Monte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in computer simulations of physical and mathematical systems...

s (SMC), are sophisticated model estimation

Estimation

Estimation is the calculated approximation of a result which is usable even if input data may be incomplete or uncertain.In statistics,*estimation theory and estimator, for topics involving inferences about probability distributions...

 techniques based on simulation

Simulation

Simulation is the imitation of some real thing available, state of affairs, or process. The act of simulating something generally entails representing certain key characteristics or behaviours of a selected physical or abstract system....

. Particle filters have important applications in econometrics

Econometrics

Econometrics has been defined as "the application of mathematics and statistical methods to economic data" and described as the branch of economics "that aims to give empirical content to economic relations." More precisely, it is "the quantitative analysis of actual economic phenomena based on...

, and in other fields.

Particle filters are usually used to estimate Bayesian models in which the latent variable

Latent variable

In statistics, latent variables , are variables that are not directly observed but are rather inferred from other variables that are observed . Mathematical models that aim to explain observed variables in terms of latent variables are called latent variable models...

s are connected in a Markov chain

Markov chain

A Markov chain, named after Andrey Markov, is a mathematical system that undergoes transitions from one state to another, between a finite or countable number of possible states. It is a random process characterized as memoryless: the next state depends only on the current state and not on the...

 — similar to a hidden Markov model

Hidden Markov model

A hidden Markov model is a statistical Markov model in which the system being modeled is assumed to be a Markov process with unobserved states. An HMM can be considered as the simplest dynamic Bayesian network. The mathematics behind the HMM was developed by L. E...

 (HMM), but typically where the state space of the latent variables is continuous rather than discrete, and not sufficiently restricted to make exact inference tractable (as, for example, in a linear dynamical system

Linear dynamical system

Linear dynamical systems are a special type of dynamical system where the equation governing the system's evolution is linear. While dynamical systems in general do not have closed-form solutions, linear dynamical systems can be solved exactly, and they have a rich set of mathematical properties...

, where the state space of the latent variables is restricted to Gaussian distributions and hence exact inference can be done efficiently using a Kalman filter

Kalman filter

In statistics, the Kalman filter is a mathematical method named after Rudolf E. Kálmán. Its purpose is to use measurements observed over time, containing noise and other inaccuracies, and produce values that tend to be closer to the true values of the measurements and their associated calculated...

).