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Distributed Projection Approximation Subspace Tracking Based on Consensus Propagation Carolina Reyes, Thibault Hilaire, Christoph F. Mecklenbruker creyes@nt.tuwien.ac.at, thibault.hilaire@lip6.fr, cfm@nt.tuwien.ac.at Institute of

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Distributed Projection Approximation Subspace Tracking Based on Consensus Propagation

Carolina Reyes, Thibault Hilaire, Christoph F. Mecklenbräuker

creyes@nt.tuwien.ac.at, thibault.hilaire@lip6.fr, cfm@nt.tuwien.ac.at Institute of Communications and Radio Frequency Engineering Vienna University of Technology

CAMSAP December 16, 2009

INSTITUT FÜR NACHRICHTENTECHNIK UND HOCHFREQUENZTECHNIK

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Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions

Outline

1

Introduction

2

PAST-Consensus Propagation Algorithm

3

Simulation Results

4

Summary and Conclusions

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Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions

Problem Statement Finite capacity of communication channel. Bit rate constraints. Sensor network architectures are structured in a centralized/small distributed fashion. Average data collected from the whole network is more important than individual node data.

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Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions

Applications Industrial, building, home system automation. Monitoring (concentrations of chemicals in hydrology, agriculture, pollution control, prediction of avalanches and land slides). Healthcare sensor implantation in human bodies.

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Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions

Projection Approximation Subspace Tracking Algorithm Mathematical model Let x(t) ∈ CN be the data vector observed at time t, with r narrow-band signal waves impinging on an array of N sensors x(t) = A(ω(t))s(t)+v(t), A =       1 1 1 ejω1 ejω2 ejωr . . . . . . . . . e(N−1)jω1 e(n−1)jω2 e(n−1)jωr       , s(t) =     s1(t) . . . sr(t)    

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Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions

Projection Approximation Subspace Tracking Algorithm Image source: Euclidean Subspace, Wikipedia

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Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions

Projection Approximation Subspace Tracking Algorithm Cost function Bin Yang [1] proposed to minimize the following cost function J′(W(t)) = t

i=1 βt−i

x(i) − W(t)y(i)

  • 2,

by the the approximation y(i) = WH(i − 1)x(i). [1] B. Yang, “Projection Approximation Subspace Tracking”, IEEE

  • Trans. Sig. Proc., vol. 43, no. 1, pp. 95-107, 1995.

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Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions

What is Projection Approximation Subspace Tracking? Requires O(nr) operations per update. n: Input vector dimension. r: Desired number of eigencomponents. t: Number of snapshots. Constrained to r < n < t.

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Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions

Consensus Propagation

−5 5 10 15 20 25 30 35 −5 5 10 15 20 25 30 35 10 11 16 17 18 23

Figure: Sensor network with neighborhood N17 for radius 9

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Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions

Consensus Propagation

Figure: Sensor network with neighborhood N17 for radius 9

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Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions

Consensus Propagation

Figure: Sensor network with neighborhood N17 for radius 9

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Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions

Consensus Propagation Algorithm for n := 1, 2, . . . , N do Input {yj(t − 1), wj}j∈Nn are the pairs sent to node n in step t − 1 yn(t) =

  • j∈Nn

yj(t − 1)wj

  • /
  • j∈Nn

wj

  • Broadcast the pair {yn(t), wn} to all nodes in Nn

Output: yn(t) is the estimation of the average in step t at node n endfor

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Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions

Consensus Propagation Algorithm for n := 1, 2, . . . , N do Input {yj(t − 1), wj}j∈Nn are the pairs sent to node n in step t − 1 yn(t) =

  • j∈Nn

yj(t − 1)wj

  • /
  • j∈Nn

wj

  • endfor

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Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions

Consensus Propagation Algorithm for n := 1, 2, . . . , N do Input {yj(t − 1), wj}j∈Nn are the pairs sent to node n in step t − 1 yn(t) =

  • j∈Nn

yj(t − 1)wj

  • /
  • j∈Nn

wj

  • Broadcast the pair {yn(t), wn} to all nodes in Nn,

endfor

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Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions

Consensus Propagation Algorithm for n := 1, 2, . . . , N do Input {yj(t − 1), wj}j∈Nn are the pairs sent to node n in step t − 1 yn(t) =

  • j∈Nn

yj(t − 1)wj

  • /
  • j∈Nn

wj

  • Broadcast the pair {yn(t), wn} to all nodes in Nn, wn = 1 /
  • |Nn|

endfor

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Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions

Consensus Propagation Algorithm for n := 1, 2, . . . , N do Input {yj(t − 1), wj}j∈Nn are the pairs sent to node n in step t − 1 yn(t) =

  • j∈Nn

yj(t − 1)wj

  • /
  • j∈Nn

wj

  • Broadcast the pair {yn(t), wn} to all nodes in Nn, wn = 1 /
  • |Nn|

endfor

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Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions

Consensus Propagation Algorithm for n := 1, 2, . . . , N do Input {yj(t − 1), wj}j∈Nn are the pairs sent to node n in step t − 1 yn(t) =

  • j∈Nn

yj(t − 1)wj

  • /
  • j∈Nn

wj

  • Broadcast the pair {yn(t), wn} to all nodes in Nn, wn = 1 /
  • |Nn|

Output: yn(t) is the estimation of the average in step t at node n endfor

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Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions

Consensus Propagation Algorithm for n := 1, 2, . . . , N do Input {yj(t − 1), wj}j∈Nn are the pairs sent to node n in step t − 1 yn(t) =

  • j∈Nn

yj(t − 1)wj

  • /
  • j∈Nn

wj

  • Broadcast the pair {yn(t), wn} to all nodes in Nn, wn = 1 /
  • |Nn|

Output: yn(t) is the estimation of the average in step t at node n endfor

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Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions

PAST-Consensus Propagation Algorithm Initialize: β, P1(0), . . . , PN(0), W1(0), . . . , WN(0) for t := 1, 2, . . . do for n := 1, 2, . . . , N do Input: xn(t) aggregate xn(t)=Snx(t − 1) from all nodes∈Nn yn(t) = WH

n (t − 1)xn(t)

locally average yn(t) hn(t) = Pn(t − 1)yn(t) gn(t) = hn(t)/[β + yH

n (t)hn(t)]

Pn(t) = 1

β(Pn(t − 1) − gn(t)hH n (t))

en(t) = Dn(x(t) − Wn(t − 1)yn(t)) Wn(t) = Wn(t − 1) + en(t)gH

n (t)

broadcast {xn(t), yn(t), wn} to all nodes∈Nn end end

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Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions

PAST-Consensus Propagation Algorithm Initialize: β, P1(0), . . . , PN(0), W1(0), . . . , WN(0) for t := 1, 2, . . . do for n := 1, 2, . . . , N do Input: xn(t) end end

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Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions

PAST-Consensus Propagation Algorithm Initialize: β, P1(0), . . . , PN(0), W1(0), . . . , WN(0) for t := 1, 2, . . . do for n := 1, 2, . . . , N do Input: xn(t) aggregate xn(t)=Snx(t − 1) from all nodes∈Nn yn(t) = WH

n (t − 1)xn(t)

locally average yn(t) end end

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Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions

PAST-Consensus Propagation Algorithm Initialize: β, P1(0), . . . , PN(0), W1(0), . . . , WN(0) for t := 1, 2, . . . do for n := 1, 2, . . . , N do Input: xn(t) aggregate xn(t)=Snx(t − 1) from all nodes∈Nn yn(t) = WH

n (t − 1)xn(t)

locally average yn(t) hn(t) = Pn(t − 1)yn(t) gn(t) = hn(t)/[β + yH

n (t)hn(t)]

Pn(t) = 1

β(Pn(t − 1) − gn(t)hH n (t))

en(t) = Dn(x(t) − Wn(t − 1)yn(t)) Wn(t) = Wn(t − 1) + en(t)gH

n (t)

end end

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Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions

PAST-Consensus Propagation Algorithm Initialize: β, P1(0), . . . , PN(0), W1(0), . . . , WN(0) for t := 1, 2, . . . do for n := 1, 2, . . . , N do Input: xn(t) aggregate xn(t)=Snx(t − 1) from all nodes∈Nn yn(t) = WH

n (t − 1)xn(t)

locally average yn(t) hn(t) = Pn(t − 1)yn(t) gn(t) = hn(t)/[β + yH

n (t)hn(t)]

Pn(t) = 1

β(Pn(t − 1) − gn(t)hH n (t))

en(t) = Dn(x(t) − Wn(t − 1)yn(t)) Wn(t) = Wn(t − 1) + en(t)gH

n (t)

broadcast {xn(t), yn(t), wn} to all nodes∈Nn end end

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Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions

PAST-Consensus Propagation Algorithm Initialize: β, P1(0), . . . , PN(0), W1(0), . . . , WN(0) for t := 1, 2, . . . do for n := 1, 2, . . . , N do Input: xn(t) aggregate xn(t)=Snx(t − 1) from all nodes∈Nn yn(t) = WH

n (t − 1)xn(t)

locally average yn(t) hn(t) = Pn(t − 1)yn(t) gn(t) = hn(t)/[β + yH

n (t)hn(t)]

Pn(t) = 1

β(Pn(t − 1) − gn(t)hH n (t))

en(t) = Dn(x(t) − Wn(t − 1)yn(t)) Wn(t) = Wn(t − 1) + en(t)gH

n (t)

broadcast {xn(t), yn(t), wn} to all nodes∈Nn end end

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Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions

Simulation Parameters Parameter Variable Value Number of nodes N 36 Number of incoming signals r 1 Frequency= cos(DOA) ωr(t) 0.1

  • Max. number of snapshots

tmax 1000 Forgetting factor β 0.97 Transmission radius 9 Topology Planar array SNR

  • 20dB to 20dB

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Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions

Performance Evaluation of the RMSE for (r = 1), constant

−20 −15 −10 −5 5 10 15 20 10

−4

10

−3

10

−2

10

−1

10 SNR [dB] Root Mean Square Error RMSE for PAST, n=17 RMSE for d−PAST, n=17 RMSE for global PAST, n=17 RMSE average over N=36

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Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions

Simulation Parameters Parameter Variable Value Number of nodes N 36 Number of incoming signals r 2 Frequencies= cos(DOA) ωr(t) 0.5:-0.5, -0.5:0.5

  • Max. number of snapshots

tmax 1000 Forgetting factor β 0.97 Transmission radius 9 Topology Planar array SNR 3dB

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Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions

Consensus Propagation

200 400 600 800 1000 −1 −0.5 0.5 1 time freq= cos(DOA) freq true avg freq estimated

Centralized PAST result for whole sensor array (N = 36, r = 2)

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Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions

Consensus Propagation

200 400 600 800 1000 −1 −0.5 0.5 1 time freq=cos(DOA) freq true avg freq estimated

PAST result neighborhood N17 (N = 6, r = 2)

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Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions

Consensus Propagation

200 400 600 800 1000 −1 −0.5 0.5 1 time freq=cos(DOA) freq true avg freq estimated

Distributed PAST-Consensus result for sensor No. 17 (r = 2)

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Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions

Summary Locally average the vector yn(t) in n with information from Nn. n broadcasts its local observation xn(t), the locally filtered r-dimensional vector yn(t), and a weight wn. yn(t) contains information from the updated signal subspace at t − 1 as well as new observation data xn(t).

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Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions

Preliminary Conclusions Signal subspace tracking can be implemented without a centralised fusion center. Current RMSE performance shows benefits, but also plenty of room for improvement.

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Introduction PAST-Consensus Propagation Algorithm Simulation Results Summary and Conclusions

Next Steps How to select suitable weights? Only do consensus propagation on yn(t)? Or also on Wn(t)? Alternative approach based on distributed RLS

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Root Mean Square Error for N = 1to18

−20 −18 −16 −14 −12 −10 −8 −6 −4 −2 10

−2

10

−1

10 SNR [dB] Root Mean Square Error RMSE for PAST, n=17 RMSE for d−PAST, n=17 RMSE for global PAST, n=17 n=1 n=2 n=3 n=4 n=5 n=6 n=7 n=8 n=9 n=10 n=11 n=12 n=13 n=14 n=15 n=16 n=18

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Root Mean Square Error for N = 19to36

−20 −18 −16 −14 −12 −10 −8 −6 −4 −2 10

−1

10 SNR [dB] Root Mean Square Error RMSE for PAST, n=17 RMSE for d−PAST, n=17 RMSE for global PAST, n=17 n=19 n=20 n=21 n=22 n=23 n=24 n=25 n=26 n=27 n=28 n=29 n=30 n=31 n=32 n=33 n=34 n=35 n=36

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RMSE Definition RMSE =

  • 1

901

1000

  • t=100

|ω1(t) − ˆ ω1(t)|2 (1) Average RMSE = 1 36

  • 1

901

1000

  • t=100

|ω1(t) − ˆ ω1(t)|2 (2)

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