Channel-aware Distributed Classification in ... - Semantic Scholar

Channel-aware Distributed Classification in Wireless Sensor Networks Using Binary Local Decisions Mohammad Fanaei, Matthew C. Valenti, Natalia A. Schmid, and Vinodkrishnan Kulathumani Wireless Communications Research Lab (WCRL) Department of Computer Science and Electrical Engineering West Virginia University, Morgantown, WV, USA 2011 SPIE Defense, Security, and Sensing Signal Processing, Sensor Fusion, and Target Recognition XX April 27, 2011 M. Fanaei et al.

Wireless CommunicationsAprilResearch 27, 2011 1/ Lab 22

Distributed Classification ( in WSN Using Binary Local Decisions

Outline 1 Introduction, Motivation, and Problem Statement 2 System Model of Our Distributed Classification WSN 3 Fusion Rule Derivation 4 Numerical Analysis 5 Conclusions

M. Fanaei et al.



Introduction, Motivation, and Problem Statement


M. Fanaei et al.




Definitions

WSNs and Distributed Detection and Classification (DDC)

2 1

Wireless sensor network (WSN): Large number of densely-deployed spatially-distributed autonomous sensors with limited capabilities that cooperate with each other to achieve a common goal. M. Fanaei et al.




Definitions

WSNs and Distributed Detection and Classification (DDC)

2 1

Main constraints: Limited bandwidth and energy resources. Limited processing capability of local sensors.

M. Fanaei et al.




Problem and Solutions

Problem Statement and Possible Solutions General Problem of DDC in WSNs: Distributed local sensors observe the conditions of their surrounding environment and process their local observations. The processed data is sent to a fusion center (FC). The FC makes the ultimate global decision.

Possible Approaches: There are M hypotheses to be classified. Multi-class classification by each sensor and sending it with dlog2 M e bits. Multi-class classification by each sensor and sending it with less bits than dlog2 M e (for example, binary decision transmission). Binary classification by each sensor and sending binary decisions.

M. Fanaei et al.




Problem and Solutions

Problem Statement and Possible Solutions General Problem of DDC in WSNs: Distributed local sensors observe the conditions of their surrounding environment and process their local observations. The processed data is sent to a fusion center (FC). The FC makes the ultimate global decision.

Possible Approaches: There are M hypotheses to be classified. Multi-class classification by each sensor and sending it with dlog2 M e bits. Multi-class classification by each sensor and sending it with less bits than dlog2 M e (for example, binary decision transmission). Binary classification by each sensor and sending binary decisions.

M. Fanaei et al.



System Model of Our Distributed Classification WSN


M. Fanaei et al.




K

K

System Model

System Model of Our Distributed Classification WSN I n1 x1 H0 H1 H2 . . . HM-1

Sensor 1

y1

u1 n2

x2

uK

nK

….......

Sensor K

y2

u2 ….......

…...

……. xK

Sensor 2

Fusion Center

u0

yK

M : The number of independent and mutually exclusive hypotheses, M ≥ 2, with known prior probabilities of pj = P [H = Hj ]. H0 is the null/rejection hypothesis and its existence means that none of the other M − 1 hypotheses has occurred. M. Fanaei et al.




System Model

System Model of Our Distributed Classification WSN II Each non-null hypothesis Hj is associated with a known influence field, Aj .

2 1

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System Model

System Model of Our Distributed Classification WSN II Each non-null hypothesis Hj is associated with a known influence field, Aj . Spatial region in the surrounding of Hj in which it can be sensed using some specific modality. Each influence field is characterized by its area and considered flat over its area.

Entire influence field of the underlying hypothesis is inside the sensing area. The center of the influence field of the underlying hypothesis is known or has been reliably estimated

M. Fanaei et al.




System Model

System Model of Our Distributed Classification WSN II If the sensors are distributed uniformly within the sensing area, the average number of sensors that can be placed in the influence field of A hypothesis Hj will be Kj = b Sj Kc. K0 = 0

K1 = 3

K2 = 21 2

1

M. Fanaei et al.




System Model

System Model of Our Distributed Classification WSN II Assuming known center of underlying influence field, for each sensor i the set of hypotheses is divided into two disjoint subsets: Ci1 and Ci0 . 1 CA = {H1 , H2 }

0 CA = {H0 }

2 1

M. Fanaei et al.




System Model

System Model of Our Distributed Classification WSN II Assuming known center of underlying influence field, for each sensor i the set of hypotheses is divided into two disjoint subsets: Ci1 and Ci0 . 1 CB = {H2 }

0 CB = {H0 , H1 }

2 1

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System Model

System Model of Our Distributed Classification WSN III n1 x1

y1

u1 n2

x2

uK

nK

….......

Sensor K

y2

u2 ….......

xK

Sensor 2 …...

…….

H0 H1 H2 . . . HM-1

Sensor 1

Fusion Center

u0

yK

K: The number of distributed local sensors. x = [X1 , X2 , . . . , XK ]: Vector of independent (given any specific underlying hypothesis) local sensor observations: p(x|Hj ) = QK i=1 p(Xi |Hj ). M. Fanaei et al.




System Model


y1

u1 n2

x2

y2

u2

uK

Sensor K

nK

….......

….......

xK

Sensor 2 …...

…….

H0 H1 H2 . . . HM-1

Sensor 1

Fusion Center

u0

yK

Conditional local observation of sensor i, given hypothesis Hj : ( vi , Hj ∈ Ci0 Xi |Hj = s + vi , Hj ∈ Ci1 with i.i.d. vi ∼ N (0, σo2 ). M. Fanaei et al.




System Model


y1

u1 n2

x2

uK

nK

….......

Sensor K

y2

u2 ….......

xK

Sensor 2 …...

…….

H0 H1 H2 . . . HM-1

Sensor 1

Fusion Center

u0

yK

Ui , 1 ≤ i ≤ K: Binary local decision ( based on following rule: 0, Xi < βi Ui = γi (Xi ) = 1, Xi > βi Sensors are able to distinguish only the occurrence or not occurrence of the M − 1 non-null hypotheses. M. Fanaei et al.




System Model

System Model of Our Distributed Classification WSN III n1 x1 H0 H1 H2 . . . HM-1

Sensor 1

y1

u1 n2

x2

uK

nK

….......

Sensor K

y2

u2 ….......

…...

……. xK

Sensor 2

Fusion Center

u0

yK

Ui ’s are sent to the fusion center through parallel transmission channels. y = [Y1 , Y2 , . . . , YK ]: The output of the parallel AWGN channel received by the fusion center. Yi = Ui + ni where i.i.d. ni ∼ N (0, σn2 ).

U0 = γ0 (y) ∈ {0, 1, . . . , M −1}: The final decision at the fusion center. M. Fanaei et al.



Fusion Rule Derivation


M. Fanaei et al.

Wireless Communications AprilResearch 27, 2011 10 / Lab 22



Fusion Center’s Decision Metric S0 =

PK

i=1 Yi :

Fusion center’s decision metric.

S0 =

K X

Ui + n0 , n0 ∼ N (0, Kσn2 )

i=1

S0 tends to have larger values if the influence field of the underlying hypothesis is larger. Influence fields of different hypotheses should have enough separation.

Γ = {α1 , α2 , . . . , αM −1 }: The set of decision thresholds based on which the fusion center can classify the underlying hypothesis using S0 . The fusion rule will be as follows: (α0 = −∞, αM = ∞) U0 = j if and only if αj ≤ S0 < αj+1 , j = 0, 1, . . . , M − 1

M. Fanaei et al.




Overview of Fusion Rule Derivation I

Find conditional pdf of sensor decision, Ui , given hypothesis Hj . fUi |Hj (ui |Hj )

M. Fanaei et al.

( vi , if Hj ∈ Ci0 Xi |Hj = s + vi , if Hj ∈ Ci1 vi ∼ N (0, σo2 ) AND ( 0, if xi < βi ui = γi (xi ) = 1, if xi > βi





Find conditional pdf of sensor decision, Ui , given hypothesis Hj . fUi |Hj (ui |Hj ) Find conditional pdf of received signal at fusion center, Yi , given decision at local sensor, Ui . fYi |Ui (yi |ui )

M. Fanaei et al.

( vi , if Hj ∈ Ci0 Xi |Hj = s + vi , if Hj ∈ Ci1 vi ∼ N (0, σo2 ) AND ( 0, if xi < βi ui = γi (xi ) = 1, if xi > βi yi = ui + ni , ni ∼ N (0, σn2 )






( vi , if Hj ∈ Ci0 Xi |Hj = s + vi , if Hj ∈ Ci1 vi ∼ N (0, σo2 ) AND ( 0, if xi < βi ui = γi (xi ) = 1, if xi > βi yi = ui + ni , ni ∼ N (0, σn2 )

fYi |Hj = fYi |Ui × fUi |Hj M. Fanaei et al.




Overview of Fusion Rule Derivation I Find conditional pdf of sensor decision, Ui , given hypothesis Hj . fUi |Hj (ui |Hj ) Find conditional pdf of received signal at fusion center, Yi , given decision at local sensor, Ui . fYi |Ui (yi |ui )

ΦYi |Hj (ν) = E eνYi Hj Z ∞ = eνyi fYi |Hj (yi ) dyi −∞ n o = L fYi |Hj (yi |Hj ) −ν

Definition

fYi |Hj = fYi |Ui × fUi |Hj

M. Fanaei et al.

L|−ν Find conditional moment generating function (MGF) of Yi , given Hj .




Overview of Fusion Rule Derivation I Find conditional pdf of sensor decision, Ui , given hypothesis Hj . fUi |Hj (ui |Hj )

ΦS0 |Hj (ν) =

K Y

ΦYi |Hj (ν)

i=1

Find conditional pdf of received signal at fusion center, Yi , given decision at local sensor, Ui . fYi |Ui (yi |ui )


M. Fanaei et al.

Find conditional MGF of S0 , given Hj .





Overview of Fusion Rule Derivation I Find conditional pdf of sensor decision, Ui , given hypothesis Hj . fUi |Hj (ui |Hj ) Find conditional pdf of received signal at fusion center, Yi , given decision at local sensor, Ui . fYi |Ui (yi |ui )


M. Fanaei et al.

Find conditional pdf of S0 , given Hj . L−1 |−ν Find conditional MGF of S0 , given Hj .







fYi |Hj = fYi |Ui × fUi |Hj M. Fanaei et al.

Find conditional posterior pdf of Hj , given S0 .

Find conditional pdf of S0 , given Hj . L−1 |−ν Find conditional MGF of S0 , given Hj .





Overview of Fusion Rule Derivation II Conditional posterior pdf of Hj , given S0 : fHj |S0 (Hj |s0 ) =

pj

K X

p 2πKσn2

`=0

"

(s0 − `)2 a` exp − 2Kσn2

#

Each coefficient a` is a specific function of a subset of βi ’s, i.e. local sensor decision thresholds. Minimum error probability Bayesian decision rule: Hˆj = arg max fHj |S0 (Hj |s0 ) j∈{0,1,...,M −1}

The intersection of fHj |S0 (Hj |x), j = 0, 1, . . . , M − 1, determines the optimum decision region thresholds at the fusion center.

M. Fanaei et al.




Overview of Fusion Rule Derivation II Conditional posterior pdf of Hj , given S0 : fHj |S0 (Hj |s0 ) =

pj

K X

p 2πKσn2

`=0

"

(s0 − `)2 a` exp − 2Kσn2

#

Each coefficient a` is a specific function of a subset of βi ’s, i.e. local sensor decision thresholds. Minimum error probability Bayesian decision rule: Hˆj = arg max fHj |S0 (Hj |s0 ) j∈{0,1,...,M −1}

The intersection of fHj |S0 (Hj |x), j = 0, 1, . . . , M − 1, determines the optimum decision region thresholds at the fusion center.

M. Fanaei et al.




Local vs Global Optimization

Locally Optimal vs. Globally Optimal Local Thresholds Based on Bayesian decision theory, locally optimal decision rule of sensor i for a binary decision making is ui =1 P Hj ∈ Ci1 |xi ≷ P Hj ∈ Ci0 |xi ui =0

ui =1

xi ≷ βi,Local ui =0

s σ2 = + O ln 2 s

! P Hj ∈ Ci0 P Hj ∈ Ci1

2. βi,Local depends only on the variance of observation noise, σO

Globally optimal local decision thresholds depend on variances of both 2 and σ 2 . It is optimized through observation noise and channel noise, σO n brute force search over all possible discrete values for βi ’s.

M. Fanaei et al.




Local vs Global Optimization

Locally Optimal vs. Globally Optimal Local Thresholds Based on Bayesian decision theory, locally optimal decision rule of sensor i for a binary decision making is ui =1 P Hj ∈ Ci1 |xi ≷ P Hj ∈ Ci0 |xi ui =0

ui =1

xi ≷ βi,Local ui =0

s σ2 = + O ln 2 s

! P Hj ∈ Ci0 P Hj ∈ Ci1

2. βi,Local depends only on the variance of observation noise, σO

Globally optimal local decision thresholds depend on variances of both 2 and σ 2 . It is optimized through observation noise and channel noise, σO n brute force search over all possible discrete values for βi ’s.

M. Fanaei et al.



Numerical Analysis


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Numerical Analysis

Environment Description

1

WSN Environment Description K = 15, M = 3 (H0 , H1 , and H2 ). P [H0 ] = 0.6, P [H1 ]0.3, and P [H2 ] = 0.1. Total area of the observation environment = 15. A1 = 5, and A2 = 15. Assuming uniform distribution of the sensors:

1

K1 = 5 and K2 = 15.

There are two non-null hypotheses. Local sensors are divided into two disjoint groups: K1 = 5 sensors that can be inside the influence field of either of the non-null hypotheses. Other K − K1 = 10 sensors that can only be inside the influence field of hypothesis H2 .

1

2

2

Pc : Average optimized probability of correct classification at the fusion center. M. Fanaei et al.



Numerical Analysis

Pc vs SNRO

Effect of Observation SNR on Classification Performance

Average Probabbility of Correct Classification at Fusion Center (P P ) C

1

0.95

0.9

0.85

0.8 SNRC=15 dB (Globally Optimum) SNRC=15 dB (Locally Optimum) SNRC=10 dB (Globally Optimum) SNRC=10 dB (Locally Optimum) SNRC=55 dB (Globally Optimum) SNRC SNRC=5 dB (Locally Optimum) SNRC=0 dB (Globally Optimum) SNRC=0 dB (Locally Optimum)

0.75

07 0.7

0.65

-2.5

M. Fanaei et al.

0

2.5

5 7.5 Observation SNR (SNRO) in dB

10

12.5

15



Numerical Analysis

Pc vs SNRC

Effect of Channel SNR on Classification Performance


1

0.95

0.9

0.85

0.8 SNRO=15 dB (Globally Optimum) SNRO=15 dB (Locally Optimum) SNRO=10 dB (Globally Optimum) SNRO=10 dB (Locally Optimum) SNRO=5 dB (Globally Optimum) SNRO=5 dB (Locally Optimum) SNRO=0 dB (Globally Optimum) SNRO=0 dB (Locally Optimum)

0.75

07 0.7

0.65

-2.5

M. Fanaei et al.

0

2.5

5 7.5 Channel SNR (SNRC) in dB

10

12.5

15



Numerical Analysis

Pc vs K

Effect of Number of Sensors on Classification Performance


1

0.95

0.9

SNRO=5 dB, SNRC=5 dB (Globally Optimum) SNRO 5 dB SNRO=5 dB, SNRC SNRC=5 5 dB (L (Locally ll O Optimum) ti ) SNRO=5 dB, SNRC=0 dB (Globally Optimum) SNRO=5 dB, SNRC=0 dB (Locally Optimum) SNRO=0 dB, SNRC=5 dB (Globally Optimum) SNRO=0 dB, SNRC=5 dB (Locally Optimum) SNRO=0 dB, dB SNRC=0 dB (Globally Optimum) SNRO=0 dB, SNRC=0 dB (Locally Optimum)

0.85

0.8

0.75

07 0.7

0.65

5

M. Fanaei et al.

10

15 K

20

25



Conclusions


M. Fanaei et al.



Conclusions

Summary and Future Work

Summary and Future Work Summary The problem of multi-class classification using local binary decisions was summarized. A model for a simple WSN deployed as a multi-class classification system was introduced. Detailed analysis of the proposed classification WSN was presented. Numerical performance of the proposed classification system was presented in a typical scenario. Future Work Considering asynchronous communication between the local sensors and fusion center. Considering fading channels between the local sensors and fusion center. Considering more complex influence fields of the underlying hypothesis. Simulation of the proposed distributed classification system using actual data. M. Fanaei et al.



Conclusions

Questions

Thank You Very Much for Your Attention. Questions?

M. Fanaei et al.



Channel-aware Distributed Classification in ... - Semantic Scholar

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