Using Cramer-Rao-Lower-Bound to Reduce Complexity

of Localization in Wireless Sensor Networks

Dominik Lieckfeldt, Dirk Timmermann

Department of Computer Science and Electrical EngineeringInstitute of Applied Microelectronics and Computer Engineering

University of Rostock

dominik.lieckfeldt@uni-rostock.de

Outline

1. Introduction

2. Goal

3. Localization in wireless sensor networks Overview Cramer-Rao-Lower-Bound Complexity and energy consumption

4. Characterizing Potential Benefits

5. Conclusions / Outlook

6. Literature

Using CRLB to Reduce Complexity of Localization in WSNs 2

Introduction

• Wireless Sensor Network (WSN): Random deployment of large

number of tiny devices Communication via radio

frequencies Sense parameters of

environment

• Applications Forest fire Volcanic activity Precision farming Flood protection

Using CRLB to Reduce Complexity of Localization in WSNs 3

• Location of sensed information important parameter in WSNs

Introduction – Localization Example

4Using CRLB to Reduce Complexity of Localization in WSNs

Parameters: m … Number of beacons n … Number of

unknowns N=m+n … Total number

of nodes

Beacon

Unknown

Error ellipse

Goal of this Work

• Investigate potential impact and applicability of adapting and scaling localization accuracy to: Activity Importance Energy level Other parameters (context)

• Obey fundamental trade-off between:accuracy <-> complexity

• Benefits: Decreased communication Prolonged lifetime of WSN

Using CRLB to Reduce Complexity of Localization in WSNs 5

Localization in WSN

• Possible approaches Lateration (typically used) Angulation Proximity

• Lateration Use received signal strength

(RSS) to estimate distances : RSS ~ 1/d²

Idea: – Estimate distances to beacons– Solve non-linear system of

equations

Using CRLB to Reduce Complexity of Localization in WSNs 6

24,1

241

241

23,1

231

231

22,1

221

221

)()(

)()(

)()(

dyyxx

dyyxx

dyyxx

23

4

12,1d

3,1d

4,1d

i,jd

Localization in WSN

• Measurements of RSS are disturbed: Interference Noise

• How accurate can estimates of position be? Cramer-Rao-Lower-Bound (CRLB) poses lower bound on

variance of any unbiased estimator

Using CRLB to Reduce Complexity of Localization in WSNs 7

22

1

2 1

2

2,1

2,1

,,1

2

2,1

rss )~()~(E1

yyxx

dd

dd

d

bCRLB

N

i

N

ij ji

jiji

N

ij

… Path loss coefficient… standard deviation of

RSS measurements… true parameter… estimated parameter

pn

rss

Distance

Geometry

2

rss

p

10ln

10

n

b

x

x~

Cramer-Rao-Lower-Bound

Using CRLB to Reduce Complexity of Localization in WSNs 8

CRLBCRLB

Error model of RSS

measurements

Error model of RSS

measurements

Number of beacons

Number of beacons GeometryGeometry

Lower bound on variance of

position error

Lower bound on variance of

position errorRSS [dBm]

Pro

ba

bili

ty

-5 0 2.5-2.5 5 x

Cramer-Rao-Lower-Bound

-5 0 2.5-2.5 5 x

-5 0 2.5-2.5 5

0

0.2

0.4

x

Pro

bab

ilit

y

• Example 1 dimension True position at x=0 Disturbed position estimates Probability density of position estimates Standard deviation or root mean square error

more intuitive than variance

Using CRLB to Reduce Complexity of Localization in WSNs 9

22rss )ˆ()ˆ(E yyxxCRLB

2

Cramer-Rao-Lower-Bound – An Example

• 2 beacons, 1 unknown

Using CRLB to Reduce Complexity of Localization in WSNs 10

0 0.1 0.2 0.3 0.40

50

100

150

200

250

300

Distance/ [rad]

Inc

rea

se

of

rss [

%]

0 0.1 0.2 0.3 0.40

50

100

150

200

250

300

Distance/ [rad]

Inc

rea

se

of

rss [

%]

4/

Complexity of Localization

• Complexity depends on: Dimensionality (2D/3D) Number of Beacons Number of nodes with unknown

position

Using CRLB to Reduce Complexity of Localization in WSNs 11

Energy Consumption and Localization

• Communication Two-way communication beacon <->

unknown Main contribution to total energy

consumption

• Calculation Simplest case: Energy spend ~ number

of beacons

Using CRLB to Reduce Complexity of Localization in WSNs 12

Ene

rgy

Number of beacons

Reducing Complexity of Localization in WSNs

• How to reduce Complexity? Constrain number of beacons used Idea:

Select those beacons first that contribute most to localization

accuracy!

Using CRLB to Reduce Complexity of Localization in WSNs 13

Related Work

• Impact of geometry not considered

• No local rule which prevents insignificant beacons from broadcasting their position

Using CRLB to Reduce Complexity of Localization in WSNs 14

Beacon Placement

Beacon Placement

Weighting range measurements

Weighting range measurements

Simulate localization error

Simulate localization error

Variance/Distance[LZZ06, CPI06, BRT06]Variance/Distance[LZZ06, CPI06, BRT06]

Detect outliers[OLT04, PCB00]Detect outliers[OLT04, PCB00]

Choose nearest k beacons[CTL05]

Choose nearest k beacons[CTL05]

Characterizing Potential Benefits

• Simulations using Matlab

• Aim: Proof of Concept Determine how likely it is that

constraining the number of beacons is possible without increasing CRLB significantly

Using CRLB to Reduce Complexity of Localization in WSNs 15

Characterizing Potential Benefits

• Simulation setup: Random deployment of m beacons and 1

unknown

Using CRLB to Reduce Complexity of Localization in WSNs 16

For every deployment calculate:– – k=m: consider all beacons– k<m: consider all combinations

of subsets of beacons determine ratio

mkk

miki ,,1;,,1)()(

k

m

1)(/)()( )()()( mkk iii

5 7 9 11 133

10

20

30

40

50

60

70

80

90

100

k

P(C

RL

Bo

k ) [

%]

5 7 9 11 133

10

20

30

40

50

60

70

80

90

100

k

P(C

RL

Bo

k ) [

%]

optimalrandom

Characterizing Potential Benefits

Using CRLB to Reduce Complexity of Localization in WSNs 17

• Potential of approach m=13 beacons Event: “CRLBok“ (equals 5% increase)05,0)()( ki

Potentially highest savingsin terms of energy and communication effort

Conclusion / Outlook

• Preliminary study based on CRLB Considers strong impact of geometry on

localization accuracy

• Selection of subsets of beacons for localization is feasible in terms of: Prolonging lifetime of sensor network Decreasing communication

• Outlook Determine/investigate local rules for

selecting subset of beacons

Using CRLB to Reduce Complexity of Localization in WSNs 18

Literature

[BHE01] Nirupama Bulusu, John Heidemann, and Deborah Estrin. Adaptive beacon placement. In ICDCS '01: Proceedings of the The 21st International Conference on Distributed Computing Systems, pages 489–503, Washington, DC, USA, 2001. IEEE Computer Society.

[BRT06] Jan Blumenthal, Frank Reichenbach, and Dirk Timmermann. Minimal transmission power vs. signal strength as distance estimation for localization in wireless sensor networks. In 3rd IEEE International Workshop on Wireless Ad-hoc and Sensor Networks, pages 761–766, Juni 2006. New York, USA.

[CPI06] Jose A. Costa, Neal Patwari, and Alfred O. Hero III. Distributed weighted-multidimensional scaling for node localization in sensor networks. ACM Transactions on Sensor Networks, 2(1):39–64, February 2006.

[CTL05] King-Yip Cheng, Vincent Tam, and King-Shan Lui. Improving aps with anchor selection in anisotropic sensor networks. Joint International Conference on Autonomic and Autonomous Systems and International Conference on Networking and Services, page 49, 2005.

[LZZ06] Juan Liu, Ying Zhang, and Feng Zhao. Robust distributed node localization with error management. In MobiHoc '06: Proceedings of the seventh ACM international symposium on Mobile ad hoc networking and computing, pages 250–261, New York, NY, USA, 2006. ACM Press.

[OLT04] E. Olson, J. J. Leonard, and S. Teller. Robust range-only beacon localization. In Proceedings of Autonomous Underwater Vehicles, 2004.

[PCB00] Nissanka B. Priyantha, Anit Chakraborty, and Hari Balakrishnan. The cricket location-support system. In 6th ACM International Conference on Mobile Computing and Networking (ACM MOBICOM), 2000.

[PIP+03] N. Patwari, A. III, M. Perkins, N. Correal, and R. O'Dea. Relative location estimation in wireless sensor networks. In IEEE TRANSACTIONS ON SIGNAL PROCESSING, volume 51, pages 2137–2148, August 2003.

[SHS01] Andreas Savvides, Chih-Chieh Han, and Mani B. Strivastava. Dynamic fine-grained localization in ad-hoc networks of sensors. Pages 166–179, 2001.

Using CRLB to Reduce Complexity of Localization in WSNs 19

Questions?

Thank you for your Attention!

Introduction – Localization Example

Using CRLB to Reduce Complexity of Localization in WSNs 21

20 30 40 50 60 70 800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

X: 50Y: 0.954

n (Number of connections to any node)

P(n

)

• Example Scenario: N=10000 nodes with 10% beacons Area: (1000x1000)m

• Start-up phase: Transmission range is chosen to provide

connection to at least 3 beacons Minimum transmission power Initial localization of nodes in range of at

least 3 beacons

• In refinement phase: Every node has connections to 50 other

nodes -> need to select subset of beacons for

localization

0 20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Transmission radius [m]

P(m

ore

than

2 b

eaco

ns in

ran

ge)

X: 45Y: 0.9526