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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

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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.

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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

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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

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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

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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

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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

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Heuristic using Prior Information

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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 AnalysisProposed method l1-magic

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Simulation Results and AnalysisIllustration of Correct Detection

Illustration of Incorrect Detection

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Simulation Results and AnalysisHeuristic Improvement

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Simulation Results and AnalysisNoise Effect

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ConclusionsPropose a compressive sensing method for

sparse event detection in wireless sensor networks

Formulate the problem and propose solutionsIntroduced a fully probabilistic Bayesian

framework which helps dramatically reduce the sampling rate