Computer Science, Software Engineering & Robotics Workshop, FGCU, April 27-28, 2012

Fault Prediction with Particle Filters

by

David Hatfield

mentors: Dr. D. Kern & Dr. J. Zalewski

Computer Science, Software Engineering & Robotics Workshop, FGCU, April 27-28, 2012

The Particle Filter is a sequential Monte Carlo algorithm used to estimate the true state of a system given a series of measurements (which are corrupted by error) taken periodically over time.

What Is a Particle Filter?

Computer Science, Software Engineering & Robotics Workshop, FGCU, April 27-28, 2012

The Monte Carlo method is an algorithm to conduct computations by random sampling of data to assess results statistically.

Example of computing area under a curve:http://chc60.fgcu.edu/EN/HistoryDetail.aspx?c=6

What Is a Monte Carlo Method?

Computer Science, Software Engineering & Robotics Workshop, FGCU, April 27-28, 2012

Essential Steps in the Algorithm

Computer Science, Software Engineering & Robotics Workshop, FGCU, April 27-28, 2012

p(xk|Dk-1) denotes the Probability Density Function of the state vector given all the measurements up to time k – 1 (denoted by Dk-1).

The following are given by Bayes theorem:Prior distribution:Likelihood function:Posterior distribution:

p (x k∣Dk−1)=∫ p (x k∣x k−1) p(x k−1∣Dk−1)d k−1

p ( y k∣D k−1)=∫ p ( y k∣x k ) p( x k∣D k−1)d k

p ( yk∣D k)=p( y k∣xk ) p( xk∣Dk−1)

p ( yk∣Dk−1)

Recursive Bayesian Solution

Computer Science, Software Engineering & Robotics Workshop, FGCU, April 27-28, 2012

Predicting Future StatesAt time k, the pdf of the state at time k + p

may be calculated as follows:

The set of values the state vector may take may be classified as either normal or faulty states. Once the pdf for a future time is obtained, the probability of a fault occurring may may be calculated by integrating the pdf over the set of all faulty states.

p (x k + p∣D k)=∫ p (x k∣D k )[ ∏j−k + 1

k+ p

p( x j∣x j−1)] dx k : k+ p−1

Computer Science, Software Engineering & Robotics Workshop, FGCU, April 27-28, 2012

Robotics: Localization, navigation, and tracking as well as fault detection, prediction, and diagnosis.

Image and Audio Enhancement: Reduction of noise in image and audio data.

Economics and Finance: Estimation of latent variables in Econometrics.

Selected Applications