Bayesian Graphical Models for Location Determination

Transcription

1 ayesian Graphical Models for Location Determination David Madigan Rutgers University & Avaya Labs joint work with Wen-ua Ju, P. Krishnan, and A.. Krishnakumar at Avaya Labs Research and Richard P. Martin and Eiman Elnahrawy at Rutgers C

2 The Problem Estimate the physical location of a wireless terminal/user in an enterprise Radio wireless communication network, specifically, based

3 Example Applications Use the closest resource, e.g., printing to the closest printer ecurity: in/out of a building Emergency 911 services Privileges based on security regions (e.g., in a manufacturing plant) Equipment location (e.g., in a hospital) Mobile robotics Museum information systems

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5 Physical Features Available for Use Received ignal trength (R) from multiple access points Angles of arrival Time deltas of arrival Which access point (AP) you are associated with We use R and AP association R is the only reasonable estimate with current commercial hardware

6 Known Properties of ignal trength ignal strength at a location is known to vary as a log-normal distribution with some environment-dependent σ Variation caused by people, appliances, climate, etc. 3 Frequency (out of 1) ignal trength (d) The Physics: signal strength (; in d) is known to decay with distance (d) as = k 1 + k 2 log d

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8 Location Determination via tatistical Modeling Data collection is slow, expensive ( profiling ) Productization Either the access points or the wireless devices can gather the data Focus on predictive accuracy

9 Prior Work discrete, 3-D, etc. Take signal strength measures at many points in the site and do a closest match to these points in signal strength vector space. [e.g. Microsoft s RADAR system] Take signal strength measures at many points in the site and build a multivariate regression model to predict location (e.g., Tirri s group in Finland) ome work has utilized wall thickness and materials

10 Krishnan et al. Results Infocom 24 moothed signal map per access point + nearest neighbor

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12 Probabilistic Graphical Models X D 1 D 2 D Graphical model = picture of some conditional independence assumptions For example, D 1 is conditionally independent of D 3 given X

13 Markov Properties for Acyclic Directed Graphs (ayesian Networks) (Global) separates A from in G an(a,,) m A (Local) α nd(α)\pa(α) pa (α) (Factorization) f(x) = Π f(x v x pa(v) ) } equivalent X D 1 D 2 D 3 p(x,d 1,D 2,D 3, 1, 2 ) =p(x) p(d 1 X) p(d 2 X) p(d 3 X) p( 1 D 1,D 2 ) p( 2 D 2 ) 1 2

14 Monte Carlo Methods and Graphical Models X imple Monte Carlo: ample in turn from p(x), p(d 1 X), p(d 2 X), p(d 3 X), p( 1 D 1,D 2 ), and p( 2 D 2 ) D 1 D 2 D 3 Gibbs ampling: ample in turn from 1 2 p(x D 1,D 2,D 3, 1, 2 ) p(d 1 X, D 2,D 3, 1, 2 ) p( 2 X, D 1,D 2,D 3, 1 )

15 Full Conditionals from the Graphical Model p(d 1 X,D 2,D 3, 1, 2 ) X p(x, D 1,D 2,D 3, 1, 2 ) D 1 D 2 D 3 = p(x) p(d 1 X) p(d 2 X) p(d 3 X) p( 1 D 1,D 2 ) p( 2 D 2 ) 1 2 p(d 1 X) p( 1 D 1,D 2 ) UG/WinUG automates this via adaptive rejection sampling and slice sampling

16 Full Conditionals from the Graphical Model X D 1 D 2 D 3 Incorporating Data, etc. uppose the D s were observed. Then sample from: p(x D 1,D 2,D 3, 1, 2 ) p( 1 X, D 1,D 2,D 3, 2 ) p( 2 X, D 1,D 2,D 3, 1 ) 1 2

17 Full Conditionals from the Graphical Model X D 1 D 2 D 3 Incorporating Data, etc. uppose the D s were observed. Then sample from: p(x D 1,D 2,D 3, 1, 2 ) p( 1 X, D 1,D 2,D 3, 2 ) p( 2 X, D 1,D 2,D 3, 1 ) 1 2 θ ayesian Analysis. Treat parameters the same as everything else.

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19 ayesian Graphical Model Approach X Y D 1 D 2 D 3 D 4 D 5 average X, Y ~ unif D i ( X, Y ) = distance to the ith access 2 i ~ N( bi + bi 1 log Di,! i ), i = 1,...,5 point

20 X 1 Y 1 X 2 Y 2 D 11 D 12 D 13 D 14 D 15 D 21 D 22 D 23 D 24 D X n Y n b 1 b 2 b 3 b 4 b 5 D n1 D n2 D n3 D n4 D n5 b 11 b 21 b 31 b 41 b 51 n1 n2 n3 n4 n5

21 Plate Notation X i Y i D ij ij i=1,,n b j b 1j j=1,,5

22 Leave-one-out error (feet) imple ayes ierarchical ayes moothnn Training ample ize

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24 moothnn () versus ayesian () Model, Error in Feet N= N= N= N= N= N=253 N=5 N=1 N=2 N=5 N= CA Down Data CA Down Data CA Down Data CA Down Data N=5 CA Up Data N=1 CA Up Data N=2 CA Up Data N=55

25 ierarchical Model X Y D 1 D 2 D 3 D 4 D β 1 β 2 β 3 β 4 β 5 β

26 ierarchical Model X i Y i D ij ij i=1,,n b j b 1j j=1,,5 µ τ µ 1 τ 1

27 imple ayesian () versus ierarchical ayesian () Model, Error in Feet N=5 N=1 N=2 N=5 N=1 N= N=5 N=1 N=2 N=5 N= CA Down Data CA Down Data CA Down Data CA Down Data N=5 CA Up Data N=1 CA Up Data N=2 CA Up Data N=55

28 Pros and Cons ayesian model produces a predictive distribution for location MCMC can be slow Difficult to automate MCMC (convergence issues) Perl-WinUG (perl selects training and test data, writes the WinUG code, calls WinUG, parses the output file)

29 What if we had no locations in the training data? Leave-one-out error (feet) imple ayes - No Locations ierarchical ayes - No Locations moothnn- With Locations Training ample ize

30 Results with No Locations: imple (), ierarchical (), Error in Feet N=5 N=1 N=2 N=5 N=1 N= N=5 N=1 N=2 N=5 N= CA Down Data CA Down Data CA Down Data CA Down Data N=5 CA Up Data N=1 CA Up Data N=2 CA Up Data N=56

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32 Zero Profiling? imple sniffing devices can gather signal strength vectors from available WiFi devices Can do this repeatedly Locations of the Access Points

33 Why does this work? Prior knowledge about distance-signal strength Prior knowledge that access points behave similarly Estimating several locations simultaneously

34 Corridor Effects X Y D 1 D 2 D 3 D 4 D 5 C 1 C 2 C 3 C 4 C β 1 β 2 β 3 β 4 β 5 β

35 Results for N=2, no locations corridor main effect corridor -distance interaction average error with mildly informative prior on the distance main effect corridor main effect corridor -distance interaction average error

36 Corridor Effect: None (N), Main (M), Interaction (I), oth (), Error in Feet N=5 N=1 N=2 N=5 N=1 N= N M I N M I N M I N M I N M I N M I N=5 N=1 N=2 N=5 N= CA Down Data CA Down Data CA Down Data CA Down Data N M I N M I N M I N M I N M I N=5 N=1 N=2 N= CA Up Data CA Up Data CA Up Data N M I N M I N M I N M I

37 Discussion Informative priors Convenience and flexibility of the graphical modeling framework Censoring (3% of the signal strength measurements) Repeated measurements & normal error model Tracking Machine learning-style experimentation is clumsy with perl-winug

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39 Prior Work Use physical characteristics of signal strength propagation and build a model augmented with a wall attenuation factor Needs detailed (wall) map of the building; model portability needs to be determined [RADAR; INFOCOM 2] based on [Rappaport 1992]