### Sequential Monte Carlo Methods

Filtering is the problem of sequentially estimating the states
(parameters or hidden variables) of a system as a set of
observations become available on-line. To solve it, one begins by
modeling the evolution of the system and the noise in the
measurements. The resulting models typically exhibit complex
nonlinearities and non-Gaussian distributions, thus precluding
analytical solution.

A popular solution strategy for the general filtering problem
is to use sequential Monte Carlo methods, also known as Particle
Filters (Doucet: 1998, Doucet, de Freitas and Gordon: 2000, Gordon,
Salmond and Smith: 1993). These methods allow for a complete
representation of the posterior distribution of the states, so that
any statistical estimates can be easily computed. They can therefore,
deal with any nonlinearities or distributions.

**Links**
- Particle Filter Group.
- Sequential Monte Carlo Methods Homepage.

### Stochastic Reachability Analysis with application to Aircraft Conflict Detection

The primary concern of all advanced air traffic management systems
(ATMSs) is to guarantee safety of air travel. Safety is typically
quantified in terms of numbers of conflicts, i.e., situations
where two aircraft come closer than a certain distance from one
another. The safety distance is encoded by means of a minimum
allowed horizontal separation and a minimum vertical separation.
Currently, for en-route airspace the minimum horizontal separation
is 5 nautical miles (nmi), while close to airports it is reduced to
3 nmi. The minimum vertical separation is 1000 feet (ft).

To prevent conflicts, ATMS resorts to a two stage process. In the
first stage conflict detection is performed: the positions of the
aircraft in the future are predicted based on their current
positions and flight plans, and they are compared so as to detect
potential situations of conflict. Once a potential conflict has
been detected, the trajectories of the aircraft involved in the
conflict are re-planned in the conflict resolution stage.

**Links**
- HYBRIDGE Web-site.
- Center-TRACON Automation System.