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Bayesian RAIM/Pesonen
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Bayesian RAIM/Pesonen
Bayesian Receiver Autonomous Integrity Monitoring Technique Henri - - PowerPoint PPT Presentation
Bayesian Receiver Autonomous Integrity Monitoring Technique Henri - - PowerPoint PPT Presentation
1 Bayesian Receiver Autonomous Integrity Monitoring Technique Henri Pesonen and Robert Pich Tampere University of Technology Department of Mathematics ION 2009 International Technical Meeting January 27, Anaheim, CA no model is right,
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Bayesian RAIM/Pesonen
Motivation
Practically all GNSS-receivers implement RAIM in some form Detect & exclude faulty observations Decide of position estimate is reliable Almost all RAIM methods are based on statistical hypothesis tests Criticized for convoluted approach Alternative approach is Bayesian model comparison
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Bayesian RAIM/Pesonen
Traditional RAIM/FDE
Classic RAIM/FDE is based on frequentist hypothesis testing Seek to reject hypothesis based on how unlikely it is RAIM/FDE as a two-stage procedure: global test local test
H1 H0 χ2
1−α,n−p
T
H0,i H1,i
α0 2 α0 2
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Bayesian RAIM/Pesonen
Frequentist hypothesis testing
Choose test statistic Choose significance level Find critical region Type I and II errors Assume we have correct models Test statistic realizes in critical region Two possible reasons: 1) H0 wrong 2) sample is rare Assume sample is not rare ➔ H0 is wrong “reject the null hypothesis with significance level α”
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Bayesian RAIM/Pesonen
Bayesian model comparison
Hypotheses (models) can be compared directly using Bayesian theory Probability of a model given data “probability that the null hypothesis is true is P” Sometimes it is easier to think in terms of odds “What are the odds that the data is from one model?” M P (Mi | D) = P (D | Mi) P (Mi) P (D)
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Bayesian RAIM/Pesonen
Bayesian model comparison
The posterior odds are the prior odds multiplied by the Bayes factor. Bayes factor is the ratio of evidences of models given the data Decisions can be made based on the odds
Oi j = P (Mi | D) P
- M j | D
= P (Mi) P
- M j
× P (D | Mi) P
- D | M j
- Oij
log10 Oij Probability for Mi against Mj [1, 3.2) [0, 0.5) Not worth more than barely a mention [3.2, 10) [0.5, 1) Substantial [10, 31.6) [1, 1.5) Strong [31.6, 100) [1.5, 2) Very strong [100, ∞) [2, ∞) Decisive
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Bayesian RAIM/Pesonen
RAIM technique employing Bayesian model comparison theory
Build statistical models for the observations (at most one failure at a time) Prior distribution for the bias (in addition to state)
M0 : y = H0x0 + v Mi : y = H0x0 + biei + v = [H0, ei]
- Hi
x0 bi
- xi
+v, i = 1, . . ., n,
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Bayesian RAIM/Pesonen
The Bayes factor Prior
Analytical formulation using normal distributions
Bi j = ci c j × exp(gi(zi) − g j(zj)).
× 1 , i = 0
1
- σ2
b(eT i S −1ei+ 1 σ2 b
)
, i 0 c∗
i =
g∗
i (zi) = 1
2 , i = 0
((y−Hiµi)T S −1ei)2 eT
i S −1ei+ 1 σ2 b
, i 0
g S = HPx0HT + R,
zi = y − Hiµi
- Pi0
= P (n − 1 channels are clean and 1 is contaminated) P (n channels are clean) = P (channel is clean)n−1 P (channel is contaminated) P (channel is clean)n = (1 − )n−1 (1 − )n =
- 1 −
(16)
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Bayesian RAIM/Pesonen
Position solution is the posterior distribution given the most plausible model
Most probable model is the model with the best odds against the ‘null’ model Position solution From posterior we can compute measures of quality How probable that the error is at most r ?
k = arg max
i
Oi0
p(xk|y, Mk)
Use as a prior at next time step
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Bayesian RAIM/Pesonen
Bayesian Receiver Autonomous Integrity Monitoring Method
Condition for correct biased observation to be identified Best odds against the ‘null’ model Check if the odds for the best model are good enough
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Bayesian RAIM/Pesonen
Simulations
Observation noise Every twenty epochs, one channel generates blunder
- bservations from
Satellites generated uniformly on Tracks simulated using constant velocity model
- :G98GR<#
+,*3"># C3 ,0#-:G9!5
6<#
9G< >#C3'#"$('77!('"#4')'#5'0')$('.#60 $057'#OY8GS98GSPZOY8GS98GSPZO8GS98GS[8GRP# We compare the detection performance of RAIM/FDE and BRAIM
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Bayesian RAIM/Pesonen
Simulations (large variance priors)
8/1-& n !;
<&
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&
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K%*/0!9$7,&,*/!
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Bayesian RAIM/Pesonen
Simulations (small variance priors)
8/1-& n !;
<&
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<=6)(0&>?&@AB& C1.2"2#&@AB& !1"7D.(&@AB& H*%%$7)!9$7,&,*/!
I;! 3C! J!
K%*/0!9$7,&,*/!
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!
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Bayesian RAIM/Pesonen
GPS-data test
BRAIM was tested using GPS-observations Artificial test was carried out by dropping good quality
- bservations and by using the observation with the
worst CNR
100 200 300 400 500 600 700 800 20 40 60 80 Kalman BRAIM
||error|| Epoch
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Bayesian RAIM/Pesonen
Summary
Bayesian model comparison can be used as an alternative to traditional hypothesis test based-RAIM methods Frequentists: “reject the null hypothesis with significance level α” Bayesians: “probability that the null hypothesis is true is P” BRAIM has Low computational complexity (under certain conditions) No restrictions on number of faulty observations (models) Natural expansion to time series Position domain investigations needed
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Bayesian RAIM/Pesonen