Localization in Wireless Networks

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Localization in Wireless Networks
David Madigan Rutgers University
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The Problem

• Estimate the physical location of a wireless
  terminal/user in an enterprise
• Radio wireless communication network,
  specifically, 802.11-based

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Example Applications

• Use the closest resource, e.g., printing to the
  closest printer
• Security 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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Physical Features Available for Use

• Received Signal Strength (RSS) from multiple
  access points
• Angles of arrival
• Time deltas of arrival
• Which access point (AP) you are associated with
• We use RSS and AP association
• RSS is the only reasonable estimate with current
  commercial hardware

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Known Properties of Signal Strength

• Signal strength at a location is known to vary as
  a log-normal distribution with some
  environment-dependent ?
• Variation caused by people, appliances, climate,
  etc.

Frequency (out of 1000)
Signal Strength (dB)

• The Physics signal strength (SS in dB) is
  known to decay with distance (d) as SS k1 k2
  log d

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Location Determination via Statistical Modeling

• Data collection is slow, expensive (profiling)
• Productization
• Either the access points or the wireless devices
  can gather the data
• Focus on predictive accuracy

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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.
  Microsofts RADAR system
• Take signal strength measures at many points in
  the site and build a multivariate regression
  model to predict location (e.g., Tirris group in
  Finland)
• Some work has utilized wall thickness and
  materials

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Bayesian Data Analysis
David Madigan
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Statistics
The subject of statistics concerns itself with
using data to make inferences and predictions
about the world Researchers assembled the vast
bulk of the statistical knowledge base prior to
the availability of significant computing Lots of
assumptions and brilliant mathematics took the
place of computing and led to useful and
widely-used tools Serious limits on the
applicability of many of these methods small
data sets, unrealistically simple models,
Produce hard-to-interpret outputs like p-values
and confidence intervals
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Bayesian Statistics
The Bayesian approach has deep historical roots
but required the algorithmic developments of the
late 1980s before it was of any use The old
sterile Bayesian-Frequentist debates are a thing
of the past Most data analysts take a pragmatic
point of view and use whatever is most useful
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Think about this
Denote q the probability that the next operation
in hospital A results in a death Use the data to
estimate (i.e., guess the value of) q
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Introduction
Classical approach treats ? as fixed and draws on
a repeated sampling principle Bayesian approach
regards ? as the realized value of a random
variable ?, with density f ?(?) (the
prior) This makes life easier because it is
clear that if we observe data Xx, then we need
to compute the conditional density of ? given Xx
(the posterior) The Bayesian critique focuses
on the legitimacy and desirability of
introducing the rv ? and of specifying its prior
distribution
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Bayes Theorem
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Bayes Theorem Example
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Bayes Theorem for Densities
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Hospital Example (0/27)
prior distribution
likelihood
posterior distribution
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Unreasonable prior distribution implies
unreasonable posterior distribution
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0.032
0.023
What to report? Mode? Mean? Median? Posterior
probability that theta exceeds 0.2? theta such
that Pr(theta gt theta) 0.05 theta such that
Pr(theta gt theta) 0.95
0.013
0.095
0.002
Posterior probability that theta is in
(0.002,0.095) is 90
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More formal treatment
Denote by qi the probability that the next
operation in Hospital i results in a death Assume
qi beta(a,b) Compute joint posterior
distribution for all the qi simultaneously
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Borrowing strength Shrinks estimate towards
common mean (7.4) Technical detail can use the
data to estimate a and b This is known as
empirical bayes
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Interpretations of Prior Distributions

• As frequency distributions
• As normative and objective representations of
  what is rational to believe about a parameter,
  usually in a state of ignorance
• As a subjective measure of what a particular
  individual, you, actually believes

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EVVE
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Bayesian Compromise between Data and Prior

• Posterior variance is on average smaller than the
  prior variance
• Reduction is the variance of posterior means over
  the distribution of possible data

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Conjugate priors
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Prediction

• Posterior Predictive Density of a future
  observation
• binomial example, n20, x12, a1, b1

?

y
y
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Prediction for Univariate Normal
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Prediction for Univariate Normal

• Posterior Predictive Distribution is Normal

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Prediction for a Poisson
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A More Challenging Example

• David Madigan
• Rutgers University
• Section 3.7 of Gelman et al.
• madigan_at_stat.rutgers.edu

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Bioassay Experiment

• Consider an animal experiment producing data of
  the form (xi,ni,yi), i1,k, where
• xi is the dose level
• given to ni animals
• of which yi had a positive outcome
• Reasonable model
• yi Bin(ni,?i) ), i1,k

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Here are the real data
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A Model

• Could model the four ?is separately but this
  ignores the dose information
• A typical dose-response models is this
• where logit(?i) log(?i /1- ?i)
• Typical Scientific Question In the light of the
  data, what is the probability that ? is bigger
  than 10?

logit(?i) a b xi
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Note

• Can flip between log odds and probability
• if log(?i /1- ?i) ? ?xi
• then (?i /1- ?i) exp(? ?xi)
• then ?i exp(? ?xi) - ?i exp(? ?xi)
• then ?i ?i exp(? ?xi) exp(? ?xi)
• then ?i (1 exp(? ?xi)) exp(? ?xi)
• so ?i exp(? ?xi)/ (1 exp(? ?xi))
• logit-1(? ?xi)

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Bayesian Analysis
For now, use a flat (improper) prior
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Localization in Wireless Networks
David Madigan Rutgers University
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Krishnan et al. Results
Infocom 2004

• Smoothed signal map per access point nearest
  neighbor

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Probabilistic Graphical Models
X

• Graphical model picture of some conditional
  independence assumptions
• For example, D1 is conditionally independent of
  D3 given X

D1
D2
D3
S1
S2
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Conditional Independence
X Y Z
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Markov Properties for Acyclic Directed
Graphs (Bayesian Networks)
(Global) S separates A from B in Gan(A,B,S)m ? A
B S (Local) a nd(a)\pa(a) pa (a)

equivalent
(Factorization) f(x) ? f(xv xpa(v) )
X
p(X,D1,D2,D3,S1,S2) p(X) p(D1X) p(D2X)
p(D3X) p(S1D1,D2) p(S2D2)
?
D1
D2
D3
S1
S2
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Monte Carlo Methods and Graphical Models
Simple Monte Carlo Sample in turn from
X
p(X), p(D1X), p(D2X), p(D3X), p(S1D1,D2), and
p(S2D2)
D1
D3
D2
Gibbs Sampling Sample in turn from
S1
S2
p(X D1,D2,D3,S1,S2)
p(D1 X, D2,D3,S1,S2)

p(S2 X, D1,D2,D3,S1)
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Full Conditionals from the Graphical Model
p(D1 X,D2,D3,S1,S2)
X
? p(X, D1,D2,D3,S1,S2)
D1
D3
p(X) p(D1X) p(D2X) p(D3X) p(S1D1,D2) p(S2D2)
D2

S1
S2

• ? p(D1X) p(S1D1,D2)

BUGS/WinBUGS automates this via adaptive
rejection sampling and slice sampling
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Full Conditionals from the Graphical Model
Incorporating Data, etc. Suppose the Ds were
observed. Then sample from
X
p(X D1,D2,D3,S1,S2)
p(S1 X, D1,D2,D3, S2)
D1
D3
D2
p(S2 X, D1,D2,D3,S1)
S1
S2
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Full Conditionals from the Graphical Model
Incorporating Data, etc. Suppose the Ds were
observed. Then sample from
X
p(X D1,D2,D3,S1,S2)
p(S1 X, D1,D2,D3, S2)
D1
D3
D2
p(S2 X, D1,D2,D3,S1)
S1
S2
Bayesian Analysis. Treat parameters the same as
everything else.
q
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Bayesian Graphical Model Approach
Y
X
S1
S2
S3
S4
S5
average
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Y1
X1
Y2
X2
S11
S12
S13
S14
S15
Yn
Xn
S21
S22
S23
S24
S25
b50
b40
b20
b10
b30
b51
b41
b21
b31
b11
Sn1
Sn2
Sn3
Sn4
Sn5
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Plate Notation
Yi
Xi
Dij
Sij
i1,,n
b1j
b0j
j1,,5
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Hierarchical Model
Y
X
S1
S2
S3
S4
S5
b1
b2
b3
b4
b5
b
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Hierarchical Model
Yi
Xi
Dij
Sij
i1,,n
b1j
b0j
j1,,5
m0
t0
t1
m1
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Pros and Cons

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

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What if we had no locations in the training data?
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Zero Profiling?

• Simple sniffing devices can gather signal
  strength vectors from available WiFi devices
• Can do this repeatedly
• Locations of the Access Points

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Why does this work?

• Prior knowledge about distance-signal strength
• Prior knowledge that access points behave
  similarly
• Estimating several locations simultaneously

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Corridor Effects
Y
X
C1
C2
C3
C4
C5
S1
S2
S3
S4
S5
b1
b2
b3
b4
b5
b
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Results for N20, no locations
corridor main effect
corridor -distance interaction
average error
0 0 20.8 0 1 16.7 1 0 17.8 1 1 17.3
with mildly informative prior on the distance
main effect
corridor main effect
corridor -distance interaction
average error
0 0 16.3 0 1 14.7 1 0 15.8 1 1 15.9
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Discussion

• Informative priors
• Convenience and flexibility of the graphical
  modeling framework
• Censoring (30 of the signal strength
  measurements)
• Repeated measurements normal error model
• Tracking
• Machine learning-style experimentation is clumsy
  with perl-WinBUGS

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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 2000 based on Rappaport 1992