using belief propagation to counter correlated reports...

Using Belief Propagation to Counter

Correlated Reports in Cooperative

Spectrum Sensing

Mihir Laghate and Danijela Cabric11th December 2014

D. Markovic / Slide 2

Outline

Motivation & Prior Work

Bayesian Network Model

Identifying Source of Correlation

Spectrum Sensing Using Loopy Belief Propagation

Conclusions

2


Correlated Reports in Spectrum Sensing

Cooperation utilizes diversity in fading channels to improve sensing

Correlations in reports caused by:

Correlated fading channels

Common interferers

Colluding users

Correlation reduces sensing accuracy!

3

PrimaryUser

Secondary Users

Fusion Center

Reports( )s t

( )ku t

( )ky t

Our goals:

Define mathematical model describing both phenomena

Identify sets of users who are correlated

Identify source of correlation

Sense spectrum occupancy by loopy belief propagation

Sensing results

( ), ( ) {0,( }, 1)k ks t u t y t


System Model

Channel from PU to SUs may have unknown correlations

SUs may be honest or malicious. If malicious,

– they can be independently malicious or colluding

– attack strategy is statistical and unknown

Binary random variables

4

1 1

2 2

( ) ( )

( ) ( )0

y t u t

y t u tP

Primary User

Secondary Users

Fusion Center

ReportsCollusion

Channel

Channel

CorrelatedChannels

( )s t

( )ku t ( )ky tSensing Results

( ), ( ) {0,( }, 1)k ks t u t y t

Statistical attacks:


Prior Work: Correlated Channels

5

Primary User

Secondary Users

Fusion Center

Reports

Channel

Channel

Correlated Channels

( )s t


Increased correlation limits sensing accuracy (Mishra et al. 2006)

Sensing when given locations of SUs and using (Gudmundson1991)’s model for correlations due to shadow fading:

– (Min et al. 2011) cluster SUs based on distance

– (Xue et al. 2014) weight reports according to correlation


Prior Work: Collaborative Malicious SUs

6

Primary User

Secondary Users

Fusion Center

ReportsCollusion

Channel

Channel( )s t


Collusion can increase false alarm and misdetection probability when majority rule is used (Yu et al. 2013)

(Rawat et al. 2011) sense spectrum by a M-out-of-N rule when colluding SUs attack by sending the same report

Channel

Channel


Novelty of our Work

7

System

Relaxing Assumptions in System Information:

No location information of SUs

Channel correlation structure is unknown

Approach:

Single framework for two sources of correlation: fading channels and collusion

Model causal nature of system, i.e., conditional distributions of sensing results and attack strategies


Outline





Conclusions

8


Bayesian Network Model 1/2

9

System

Model causal nature of system, i.e., conditional distributions of sensing results and attack strategies

Bayesian Network

Model


Bayesian Network Model 2/2

10

Factorization of joint distribution

MAP & ML estimators require learning this factorization

Learning factorization learning structure of model identifying sets of colluding SUs and

SUs with correlated channels

1 4 1 4

31 2 4

3 4 3 41 1 2 2

, ,..., , ,...,

|, | |

, | ,| |

P Ps u u y y s

P P u s Pu u s u s

P P P y y u uy u y u

Sensing Results

SU Reports


Outline





Conclusions

11


Learning Structure of Model

Existence of latent variables with 2 neighbors is not learnable:(Pearl 1988)

– Sensing results:

– Spatial correlation nodes:

12

1,2X

s

3X

4X

1u

2u

3u

4u 3,4

A

1y

2y

3y

4y

Ground Truth

Spatial Correlation

Sensing Result

Attack Strategy

SU Reports

1 2 3 4, , ,u u u u

3 4,X X

Observable Variables

Latent Variables


Source of Correlation is Unidentifiable

Collusion indistinguishable from channel correlation!

13

s

1y

2y

3y

4y

Ground Truth

Correlations SU Reports

Learnable Structure

1,2X

1u 2u

3X

4X

3u

4u

3,4A


Sets of Correlated SUs

We can identify sets of SUs with correlated reports

– Proposed method in paper based on principle of additive information distances on trees from (Choi et al 2011)

14

s

1y

2y

3y

4y

Ground Truth


Learnable Structure

1 1,2

2 3,4


1r

2r



We can identify sets of SUs with correlated reports

– Proposed method in paper based on principle of additive information distances on trees from (Choi et al 2011)

For m-th set, is an unknown distribution which is required for spectrum sensing

15

s

1y

2y

3y

4y

Ground Truth


Learnable Structure

1 1,2

2 3,4


( ) | ( )mm

y t s tr P

1r

2r


Outline





Conclusions

16


Sensing by Loopy Belief Propagation

17

Iterative message passing algorithm to estimate posterior distributions of

and .

Inputs:

Reports for

Sets of correlated SUs Forms factor graph

Priors for spectrum occupancy and

Output:

Estimate of which gives:

( )s t {1,..., }( ) | ( )Ky t s tP

{1,..., }( )Ky t 1,...,t T

( )s t

( ) | ( )m m

y t s tP r

{1,..., }( ) | ( )Ks t y tP

{1,

0... }

,1,( ) | ( )ˆ( ) arg max K

s

s t y ts t P

m

(1)s

(2)s

1r

2r

3r

1,1

1

1y

3,1

3

1y

1,2

1

2y

2,2

2

2y

3,3

3

2y

2,1

2

1y

Set

Set

Set

Variable nodes

t =2

t =1

Factor nodes

1

2

3


Priors for Loopy Belief Propagation

Prior for spectrum occupancy is uniform on {0, 1}

Assumes no knowledge of PU traffic

Prior for

Perfect honest SU would have

Prior defined as distance from the perfect honest SUFor example, for an independent SU:

18

( ) | ( )m m

y t s tP r

( )s t

1 if ( ) ( )

( ) | ( )0 otherwise

k

k

y t s tP y t s t

1l

1 1

1 (0,0) 0 (1,0)4

0 (0,1) 1 (1,)

)(

1kp r

r r

r rr

( ) 0s t ( ) 1s t


Simulations

Proposed algorithm is compared with a “naïve” BP algorithm (Pennaet al. 2012) which assumes that all SUs are uncorrelated and may be malicious

System

Pairs of colluding SUs

Remaining uncorrelated SUs

Identical sensing statistics for individual SU:detection probability 0.95, false alarm probability 0.05.

19


Simulations: Increasing # Uncorrelated SUs

20

System: 1 pair of colluding SUs, 9 message passing iterations

Probability of Detection Probability of False Alarm

Proposed algorithm is compared with a “naïve” BP algorithm (Penna et al. 2012)which assumes that all SUs are uncorrelated and may be malicious

Detection probability improved while false alarm probability unchanged

Gain in detection probability reduces with increasing no. of uncorrelated SUs


Simulations: Increasing # Correlated Pairs

21

System: 8 SUs, T=12 time slots, 9 message passing iterations

Probability of Detection Probability of False Alarm

For all sensing algorithms, detection probability reduces and false alarm probability increases with increase in number of colluding pairs

Detection probability of proposed algorithm suffers less than naïve algorithm

False alarm probability is comparable to naïve algorithm


Conclusions

Modeled channel correlations and colluding statistical attacks as a Bayesian network

Proved that identifying source of correlation in reports is not possible

Proposed loopy belief propagation algorithm to sense spectrum while learning the correlations in the SU reports

– Improved detection probability

– Gains increase with number of colluding pairs

22


Selected References

23

(Mishra et al. 2006) S. M. Mishra, A. Sahai, and R. W. Brodersen, “Cooperative Sensing among Cognitive Radios,” in IEEE International Conference on Communications, 2006, vol. 4.(Gudmundson 1991) M. Gudmundson, “Correlation model for shadow fading in mobile radio systems,” Electronics Letters, vol. 27, no. 23, Nov. 1991.(Min et al. 2011) A. W. Min, K. G. Shin, and X. Hu, “Secure Cooperative Sensing in IEEE 802.22 WRANs Using Shadow Fading Correlation,” IEEE Trans. on Mobile Computing, Oct. 2011.(Xue et al. 2014) D. Xue, E. Ekici, and M. C. Vuran, “Cooperative Spectrum Sensing in Cognitive Radio Networks Using Multidimensional Correlations,”IEEE Transactions on Wireless Communications, vol. 13, no. 4, Apr. 2014.(Yu et al. 2013) C.-K. Yu, M. Laghate, A. H. Sayed, and D. Cabric, “On the effects of colluded statistical attacks in cooperative spectrum sensing,” in IEEE SPAWC, 2013.(Rawat et al. 2011) A. S. Rawat, P. Anand, H. Chen, and P. K. Varshney, “Collaborative Spectrum Sensing in the Presence of Byzantine Attacks in Cognitive Radio Networks,” IEEE Transactions on Signal Processing, vol. 59, no. 2, Feb. 2011.(Pearl 1988) J. Pearl, Probabilistic reasoning in intelligent systems: networks of plausible inference. San Francisco, CA, USA: Morgan Kaufmann Publishers Inc., 1988.(Choi et al. 2011) M. J. Choi, V. Y. F. Tan, A. Anandkumar, and A. S. Willsky, “Learning Latent Tree Graphical Models,” Journal of Machine Learning Research, vol. 12, Jul. 2011.(Penna et al. 2012) F. Penna, Y. Sun, L. Dolecek, and D. Cabric, “Detecting and Counteracting Statistical Attacks in Cooperative Spectrum Sensing,” IEEE Transactions on Signal Processing, vol. 60, no. 4, Apr. 2012.

Thank you!

Questions?

Backup Slides


Loopy Belief Propagation Algorithm

Messages repeatedly sent along edges in both directions

Are joint distributions of variables

26

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2s

1r

2r

3r

1,1

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1y

3,1

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Variable nodes

Factor nodes

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Variable node to factor node

Factor node to variable node

Factor nodes weigh by received reports


Beliefs and Estimator

Beliefs estimate posterior marginal of variables

Estimator:

27

1s

2s

1r

2r

3r

1,1

1

1y

3,1

3

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1,2

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Trajectory of beliefs in system with 8 SUs and T=5 time slots


Attack Strategies

All SUs within one colluding set are statistically equivalent

Attack strategy operates on aggregated reports, e.g., sum

Define for

28

if '

1o.w.m

m

jj

jj

m

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[0.5,1] {0,1, | |, }m

using belief propagation to counter correlated reports...

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