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1
Sparse Event Detection in Wireless Sensor Networks using Compressive
SensingJia Meng, Husheng Li, and Zhu Han
the 43rd Annual Conference on Information Sciences and Systems (CISS), 2009
2
OutlineIntroductionSystem ModelCompressive Sensing AlgorithmSimulation Results and AnalysisConclusions
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IntroductionThe dogma of signal processing maintains
that a signal must be sampled at a Nyguist rate at least twice its bandwidth in order to be represented without error
In practice, we often compress the data soon after sensing, trading off signal representation complexity (bits) for some error(consider JPEG image compression in digital cameras, for example)
Clearly, this is wasteful of valuable sensing/sampling resources
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IntroductionIn this paper, we investigate how to employ
compressive sensing in wireless sensor networksSpecifically, we target on two problems of wireless
sensor networks1. The number of events is much less compared to
the number of all sources2. Different events may happen simultaneously and
cause interference to detect them individuallyTo overcome the above two problems, we propose
a sparse event detection scheme in wireless sensor networks by employing compressive sensing
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System ModelThere are a total of N sources randomly
located in a fieldThose source randomly generate the events to
be measuredWe denote K as the number of events that the
sources generateK is a random number, and is much smaller than
NWe denote as the event vector, in which
each component has a binary value, i.e.,Obviously X is a sparse vector since
1NX
0,1nX K N
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System ModelIn the system, there are M active monitoring
sensors trying to capture these eventsThere are two challenges for those
monitoring sensors1. All those events happen simultaneously
As a result, the received signals are interfering with each other
2. The received signal is deteriorated by propagation loss and thermal noise
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System ModelThe received signal vector can be written as
is the thermal noise vector whose component is independent and has zero mean and variance of
is the channel response matrix whose component can be written as
is the distance from the source to the sensing device
is the propagation loss factor is the Raleigh fading modeled as complex Gaussian
Noise with zero mean and unit variance
1 1 1M M N N MY G X 1M
2M NG
/ 2
, , ,m n m n m nG d h
,m nd thmthn
,m nh
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System ModelNotice that the number of events, the number
of sensors, and total number of sources have the following relation
Consequently, the received signal vector Y is an condensed representation of the event
Event vector Y has aliasing of vector X, due to the low sampling rate M
K M N
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Compressive Sensing AlgorithmProblem Formulation and AnalysisBayesian Detection
1. Model Specification2. Marginal Likelihood Maximization3. Heuristic using Prior Information
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Problem Formulation and AnalysisDefinition : Restricted Isometry Property
(RIP)For any vector V sharing the same K nonzero entries as X, if
for some , , then the matrix G preserves the information of the K-sparse signal.
It has been proved that if G is an i.i.d. Gaussian matrix or random ±1 entry matrix, then the K-sparse signal is compressible with high probability if
2
21 1GV
V
0
log /M cK N K N
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Problem Formulation and AnalysisSince M < N there are infinite number of
satisfy
The problem is to find the sparse reconstructed signal
The above optimization is called the l1-magic in the literatureThe complexity is
X̂ˆY GX
1ˆ
ˆ ˆargminY GX
X X
3O N
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Bayesian DetectionConsidering the fact that the components of
X are either 0 or 1we adopt the Bayesian compressive sensing
[12–14], which is fully probabilistic and introducing a set of hyper-parameters
[12] M. E. Tipping, “Sparse Bayesian learning and the relevance vector machine”, Journal of Machine Learning Research, vol. 1, p.p. 211-244, Sept. 2001.[13] M. E. Tipping and A. C. Faul, “Fast marginal likelihood maximisation for sparse Bayesian models”, in Proceedings of the Ninth International Workshop on Artificial Intelligence and Statistics, Key West, FL, Jan 3-6.[14] S. Ji, Y. Xue and L. Carin, “Bayesian compressive sensing”, IEEE Trans. Signal Processing, vol. 56, no. 6, June 2008.
13
Maximum Likelihood Estimation (MLE)假設有五個袋子,各袋中都有無限量的餅乾 ( 櫻桃口味
或檸檬口味 ) ,已知五個袋子中兩種口味的比例分別是1. 櫻桃 100%2. 櫻桃 75% + 檸檬 25% 3. 櫻桃 50% + 檸檬 50% 4. 櫻桃 25% + 檸檬 75% 5. 檸檬 100%
從同一個袋子中連續拿到 2 個檸檬餅乾,那麼這個袋子最有可能是上述五個的哪一個? Ans : 5
0
0.252
0.502
0.752
1
14
Maximum a posteriori (MAP)假設有五個袋子,各袋中都有無限量的餅乾 ( 櫻桃口味
或檸檬口味 ) ,已知五個袋子中兩種口味的比例分別是1. 櫻桃 100% ( 拿到的機率 0.1)2. 櫻桃 75% + 檸檬 25% ( 拿到的機率 0.2)3. 櫻桃 50% + 檸檬 50% ( 拿到的機率 0.4)4. 櫻桃 25% + 檸檬 75% ( 拿到的機率 0.2)5. 檸檬 100% ( 拿到的機率 0.1)
從同一個袋子中連續拿到 2 個檸檬餅乾,那麼這個袋子最有可能是上述五個的哪一個? Ans : 4
0.1 × 0=0
0.2 × 0.252 =0.0125
0.4 × 0.502=0.1
0.2 × 0.752 =0.1125
0.1 × 1=0.1
|| i ii
p x p xp x
p
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Model SpecificationThe noise in the system is composed of
propagation loss with zero mean and varianceThe probability density function can be
approximated as Gaussian distribution as
Due to the assumption of independence of , he likelihood of the complete data set can be written as
2
21
| 0,M
ii
p N
nY
/ 2 22 22
1| , 2 exp
2
Mp Y X Y GX
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Model SpecificationThe real distribution of X is Bernoulli
distributionHowever, the close form solution in our
problem is hard to be obtainedInstead, we assume a zero-mean Gaussian
prior distribution over the signal X
where is a vector of N independent hyper-parameters
1
1
2/ 2 1/ 2
1
| | 0,
2 exp2
N
n nn
NN n n
nn
p X N X
x
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Model SpecificationGiven , the posterior parameter distribution
conditioned over the signal is given by combining the likelihood and prior with Bayes’ rule
which is a Gaussian distribution ith covariance and mean of
2
2
2
| , || , ,
| ,
p Y X p Xp X Y
p Y
,N
12
2
1, ,
T
T
n
A G G
G Y
A diag
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Marginal Likelihood MaximizationThe sparse Bayesian model is formulated as
the local maximization with respect to of the marginal likelihood, or equivalently its logarithm
with
2
2
1
log | ,
log | , |
1log 2 log
2T
L p Y
p Y X p X dX
M C Y C Y
2 1 TC I GA G
19
Marginal Likelihood MaximizationA point estimate for the parameters is
then obtained by evaluating (11) with , giving a posterior mean approximator
However, marginal likelihoods are generally difficult to compute, i.e., values of and which maximize cannot be obtained in closed form
For the updating of , differentiate (12), and then equate it to 0. After rearranging, we have
MPMP
MPGX G
2 L
2
inewi
i
20
Marginal Likelihood Maximization
where is the posterior mean signal from (11), and is defined as
with being the diagonal element of the posterior signal covariance from (10) computed with current and values
For the variance , differentiation leads to re-estimate
2
inewi
i
i thii
1ii iiN
iiNthi
2
22
2new
i i
Y G
M
21
Heuristic using Prior InformationAfter the reconstruction of , if the algorithm
converges to wrong results, there are two possible situations
1. The algorithm can converge to either around 0 and 1, but with the wrong position for the sparse events
could not be easily distinguished
2. have values deviating from 0 or 1 easy to find the error using threshold
methods
X̂
X̂
23
Simulation Results and AnalysisThere are a total of N = 256 events randomly
located within 500m-by-500m areaThe M wireless sensors are also randomly
located within this areaThe minimal distance between a event and a
sensor is 5mThe propagation loss factor is 3The transmitted power is normalized to 1 and
the thermal noise is 10-12
The number of random events is K which is a small number
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Simulation Results and AnalysisIllustration of Correct Detection
Illustration of Incorrect Detection