On-Line Probabilistic Classification with Particle Filters Pedro Højen-Sørensen, Nando de Freitas,...
-
Upload
sherman-henderson -
Category
Documents
-
view
212 -
download
0
Transcript of On-Line Probabilistic Classification with Particle Filters Pedro Højen-Sørensen, Nando de Freitas,...
![Page 1: On-Line Probabilistic Classification with Particle Filters Pedro Højen-Sørensen, Nando de Freitas, and Torgen Fog, Proceedings of the IEEE International.](https://reader035.fdocuments.us/reader035/viewer/2022072006/56649cfa5503460f949cbea3/html5/thumbnails/1.jpg)
On-Line Probabilistic On-Line Probabilistic Classification with Particle Classification with Particle FiltersFilters
Pedro Højen-Sørensen, Nando de Freitas, and Torgen Fog, Proceedings of the IEEE International Workshop on Neural Networks for Si
gnal Processing (NNSP2000), 2000 (to appear)
Cho, Dong-Yeon
![Page 2: On-Line Probabilistic Classification with Particle Filters Pedro Højen-Sørensen, Nando de Freitas, and Torgen Fog, Proceedings of the IEEE International.](https://reader035.fdocuments.us/reader035/viewer/2022072006/56649cfa5503460f949cbea3/html5/thumbnails/2.jpg)
IntroductionIntroduction
Sequential Classification Problems Condition monitoring and real-time decision systems
Monitoring patients, fault detection problem
Particle filter provide an efficient and elegant probabilistic solution to this problem.
It becomes possible to compute the probabilities of class membership when the classes overlap and evolve with time.
This classification framework applied to any type of classifier, but for demonstration purposes, multi-layer perceptrons (MLPs) are used.
![Page 3: On-Line Probabilistic Classification with Particle Filters Pedro Højen-Sørensen, Nando de Freitas, and Torgen Fog, Proceedings of the IEEE International.](https://reader035.fdocuments.us/reader035/viewer/2022072006/56649cfa5503460f949cbea3/html5/thumbnails/3.jpg)
Model SpecificationModel Specification
Markov, Nonlinear, State Space Representation Transition model: p(t|t–1)
t RRn corresponds to the parameters (weights) of a neural network f(xt, t)
The parameters are assumed to follow a random walk t = t–1 + ut.
The process noise could be Gaussian ut ~ N(0, t2In
)
Observation model: p(yt|xt,t) xt RRnx denotes the input data at time t.
yt {0,1}ny represents the output class labels.
The likelihood of the observations should be given by the following binomial (Bernoulli) distribution
tt ytt
yttttt ffp 1)),(1(),(),|( θθθ xxxy
![Page 4: On-Line Probabilistic Classification with Particle Filters Pedro Højen-Sørensen, Nando de Freitas, and Torgen Fog, Proceedings of the IEEE International.](https://reader035.fdocuments.us/reader035/viewer/2022072006/56649cfa5503460f949cbea3/html5/thumbnails/4.jpg)
Estimation Objectives Our goal will be to approximate the posterior distributi
on p(0: t|d1: t) and one of its marginals, the filtering density p(t|d1: t), where d1: t = {x1:t, y1:t}
By computing the filtering density recursively, we do not need to keep track of the complete history of the parameters.
![Page 5: On-Line Probabilistic Classification with Particle Filters Pedro Højen-Sørensen, Nando de Freitas, and Torgen Fog, Proceedings of the IEEE International.](https://reader035.fdocuments.us/reader035/viewer/2022072006/56649cfa5503460f949cbea3/html5/thumbnails/5.jpg)
Particle FilteringParticle Filtering
![Page 6: On-Line Probabilistic Classification with Particle Filters Pedro Højen-Sørensen, Nando de Freitas, and Torgen Fog, Proceedings of the IEEE International.](https://reader035.fdocuments.us/reader035/viewer/2022072006/56649cfa5503460f949cbea3/html5/thumbnails/6.jpg)
Generic Particle Filter for Generic Particle Filter for ClassificationClassification
![Page 7: On-Line Probabilistic Classification with Particle Filters Pedro Højen-Sørensen, Nando de Freitas, and Torgen Fog, Proceedings of the IEEE International.](https://reader035.fdocuments.us/reader035/viewer/2022072006/56649cfa5503460f949cbea3/html5/thumbnails/7.jpg)
Bayesian Importance Sampling Step Importance functions
Recursive formulas
Transition prior p(t|t–1) is used as importance distribution for the MLPs.
t
kktktt qqp
11:1:10:1:0 ),|()()|( θθθθ dd
),|(
)|(),,|(
1:0:1
1:01:1
ttt
ttttttt q
ppw
θθ
θθθ
d
dxy
),|( tttt pw θxy
![Page 8: On-Line Probabilistic Classification with Particle Filters Pedro Højen-Sørensen, Nando de Freitas, and Torgen Fog, Proceedings of the IEEE International.](https://reader035.fdocuments.us/reader035/viewer/2022072006/56649cfa5503460f949cbea3/html5/thumbnails/8.jpg)
Selection Step EE(Ni) = Nwt
(i)
MCMC step A skewed importance weights distribution
Many particles have no children, whereas others have a large number of children.
![Page 9: On-Line Probabilistic Classification with Particle Filters Pedro Højen-Sørensen, Nando de Freitas, and Torgen Fog, Proceedings of the IEEE International.](https://reader035.fdocuments.us/reader035/viewer/2022072006/56649cfa5503460f949cbea3/html5/thumbnails/9.jpg)
A Simple Classification A Simple Classification ExampleExample Experimental Setup
An MLP with 4 hidden logistic functions and an output logistic function
N = 200, t = 0.2 (0 = 10)
![Page 10: On-Line Probabilistic Classification with Particle Filters Pedro Højen-Sørensen, Nando de Freitas, and Torgen Fog, Proceedings of the IEEE International.](https://reader035.fdocuments.us/reader035/viewer/2022072006/56649cfa5503460f949cbea3/html5/thumbnails/10.jpg)
Results
![Page 11: On-Line Probabilistic Classification with Particle Filters Pedro Højen-Sørensen, Nando de Freitas, and Torgen Fog, Proceedings of the IEEE International.](https://reader035.fdocuments.us/reader035/viewer/2022072006/56649cfa5503460f949cbea3/html5/thumbnails/11.jpg)
An Application to Fault An Application to Fault DetectionDetection Monitoring the exhaust valve condition in a
marine diesel engine The main goal
Detection of the leakage before the engine performance becomes unacceptable or irreversible damage occurs.
![Page 12: On-Line Probabilistic Classification with Particle Filters Pedro Højen-Sørensen, Nando de Freitas, and Torgen Fog, Proceedings of the IEEE International.](https://reader035.fdocuments.us/reader035/viewer/2022072006/56649cfa5503460f949cbea3/html5/thumbnails/12.jpg)
Experimental Setup and Results An MLP with 2 hidden unit and 5 input nodes (PCA is
used for dimensionality reduction.) 500 particles
![Page 13: On-Line Probabilistic Classification with Particle Filters Pedro Højen-Sørensen, Nando de Freitas, and Torgen Fog, Proceedings of the IEEE International.](https://reader035.fdocuments.us/reader035/viewer/2022072006/56649cfa5503460f949cbea3/html5/thumbnails/13.jpg)
ConclusionsConclusions
We presented a novel on-line classification scheme and demonstrated it on two problems. We believe this strategy has great potential and that it
needs to be further tested on other types of parametric classifiers and classification domains.