Joint Localiza,on Algorithms for Network Topology Ambiguity Reduc,on

Transcription

1 Ss. Cyril and Methodius University in Skopje T Faculty of Electrical Engineering and Information Technologies (FEEIT) Institute of Telecommunications (ITK) Joint Localiza,on Algorithms for Network Topology Ambiguity Reduc,on Marko Angjelicinoski, Daniel Denkovski, Vladimir Atanasovski and Liljana Gavrilovska {markoang; danield; vladimir; liljana}@feit.ukim.edu.mk 4 th COST IC0902 Workshop Rome, Italy, October 9-11, 2013

2 Outline IntroducPon Problem definipon Joint Maximum Likelihood (JML) Concluding remarks 2

3 IntroducPon TransmiXer localizapon is an important aspect in commercial, public safety and military applicapons of current and future wireless networks Numerous approaches possible Received Signal Strength (RSS)- based localizapon is a viable localizapon solupon due to the: Inherent presence of the RSS extracpon feature in all radio devices Sufficient precision for a variety of pracpcal applicapons However, there are associated challenges in the process as the operapng environment of wireless networks is very hosple + the radio environmental informapon (e.g. wireless channel model parameters) and network configurapon informapon (sensor posipons) might be unreliable or absent This presentapon analyzes the network topology ambiguity problem and proposes novel localizapon algorithms that improve the transmixer localizapon performance while reducing the network topology ambiguity 3

4 Outline IntroducPon Problem definipon Joint Maximum Likelihood (JML) Concluding remarks 4

5 General network setup Anchors Agent position estimate Single transmicer (agent) with unknown posipon Set N, N =n of measuring sensors (anchors) with known posipons The anchors measure the RSS of the signal broadcasted by the transmixer Agent The transmixer posipon espmated through (unbiased) es,ma,on using the RSS observapons Using appropriate path loss model* The localiza,on error quanpfied through the covariance matrix Agent localization covariance * R. K. Marn, and R. Thomas, Algorithms and bounds for esmang locaon, direconality, and environmental parameters of primary spectrum users, IEEE Transacon on Wireless Communicaons (11), pp , November 2009.

6 Anchor posipon ambiguity Anchors position estimates The posipon informapon of some anchors is obtained through previous es,ma,on The anchor posipon es,mates are and they sapsfy Ambiguous and ofen very unreliable network topology informapon The ambiguous anchor posipon informapon can cause severe deteriora,on of the localizapon algorithms performance New localiza,on algorithms for scenarios with ambiguous topologies needed Agent localization covariance increases

7 Outline IntroducPon Problem definipon Joint Maximum Likelihood (JML) Concluding remarks 7

8 Joint localizapon algorithms The ambiguity problem can be alleviated with localizapon algorithms that jointly espmate the agent s and the anchors exact posi,ons* Improve the transmicer localiza,on performance Reduce the network topology ambiguity The joint localizapon relies on the assumppon that the erroneous informapon about the anchor posipons is obtained by some previous espmapon Modeled as a random process parameterized with respect to the exact anchor posipons This work presents a general joint RSS- based localiza,on framework developed using non- Bayesian es,ma,on formalism* The unknown agent s and ambiguous anchors posipons are regarded as determinis,c parameters the technically correct espmapon approach for scenarios with imprecise anchor posi,on informa,on * M. Angjelicinoski, D. Denkovski, V. Atanasovski, and L. Gavrilovska, SPEAR: Source Posion Esmaon for Ambiguity Reducon, IEEE Communicaon LeXers (submixed)

9 Joint localizapon algorithms: Non- Bayesian formalism Ambiguity regions Ki Two subsets of anchors: Subset V, V =v - > anchors with certain i.e. precisely known posipon (Base StaPons or APs) Subset U, U =u - > anchors with ambiguous posipons The data vector is The unknown parameters vector is determinispc Assump,ons: PropagaPon model: log- distance path loss model in log- normal uncorrelated shadowing Anchor posipon espmates: i.i.d. Gaussian* * J. Hemmes, D. Thain, and C. Poellabauer, Cooperave Localizaon in GPS- Limited Urban Environments, First Internaonal Conference ADHOCNETS 2009, Ontario, CA, 2009, pp

10 Joint Maximum Likelihood The main task of the RSS- based joint localizapon framework is to es,mate θ based on the informa,on contained in X* Employing Maximum Likelihood (ML) approach* results in the Joint Maximum Likelihood (JML) localizapon algorithm where p(x;θ) is the joint probability density func,on of the data vector X When, then and the JML becomes the Legacy ML (LML) localizapon algorithm * S. M. Kay, Fundamentals of Stascal Signal Processing: Esmaon Theory, Prence- Hall, 199.

11 JML performance evaluapon (1/4) à transmixer localizapon Simulated area Transmit power Simula,on parameters 20mX20m 0dBm Path loss exponent 2.5 Reference distance TransmiXer posipon Shadowing std. deviapon 1m Number of anchors Number of ambiguous anchors Anchor ambiguity std. deviapon (9,9)m 1:1: db (U=N) 1:3: m Monte- Carlo trials (L) 000 Circular normal y [m] ID: ID:5 ID:3 ID:2 ID:4 ID:9 Tx:1(9,9) ID:7 ID:1 ID: ID: x [m] scenario GOAL: Compare the Root Mean Squared Error (RMSE) performance of the JML with the performance of the LML in terms of the transmicer localiza,on capabili,es 11

12 JML performance evaluapon (2/4) à transmixer localizapon The RMSE gain increases as the inipal anchor posi,on ambiguity increases (relapve to the average network internodes distances) 9 The joint localizapon framework suitable for scenarios with large ini,al topology ambigui,es RMSE [m] 7 5 LML(Δ=4m) LML(Δ=7m) 4 LML(Δ=m) JML(Δ=4m) The joint agent/ambiguous 3 anchor localizapon results JML(Δ=7m) in RMSE gain for the JML(Δ=m) transmicer localiza,on σ [db] The JML performs better than the LML reliable transmitter localization is possible in scenarios with ambiguous anchor positions

13 JML performance evaluapon (3/4) à anchor ambiguity reducpon GOAL: Compare the Root Mean Squared Error (RMSE) performance of the JML in terms of reducing the ini,al network topology ambiguity 1 1 ID:3 ID:7 ID: ID:9 y [m] ID: ID:2 Tx:1(9,9) ID:1 IniPal Anchor PosiPon Ambiguity 4 2 ID:5 ID:4 ID: x [m] 13

14 JML performance evaluapon (4/4) à anchor ambiguity reducpon RMSE [m] JML (sensid:9, Δ=7m).5 JML (sensid:, Δ=7m) Initial Anchor Position Ambiguity σ [db] The JML improves the reliability of the anchor position estimates reduced network topology ambiguity

15 Outline IntroducPon Problem definipon Joint Maximum Likelihood (JML) Concluding remarks 15

16 Concluding remarks In challenging environments, the informapon about the sensors posipons might be unreliable the problem can be alleviated using joint localizapon algorithms This work introduced general RSS- based localizapon framework using non- Bayesian espmapon formalism that: Improves the transmicer localiza,on performances Reduces the sensors posi,onal ambiguity The JML is introduced as a typical representapve of the proposed framework The JML performance evaluapon proves that substanpal transmixer localizapon improvements and network ambiguity reducpon gains can be axained via joint localizapon 1

17 THANK YOU FOR YOUR KIND ATTENTION! QUESTIONS? hxp://wingroup.feit.ukim.edu.mk 17

18 Backup slides 1

19 TheorePcal performance bounds The Cramer- Rao Lower Bound (CRLB) [1] bounds the covariance matrix of any unbiased, non- Bayesian espmator of θ via the inverse of the Fisher Informa,on Matrix (FIM) Objec,ve: prove convergence of the derived bounds by invespgapng the asympto,c performance of the JML The JML is expected to achieve the CRLB in terms of RMSE for small environmental variability and small anchor posipon errors (i.e. in the small varia,on/error region) RMSE: simulapon [1] S. M. Kay, Fundamentals of StaPsPcal Signal Processing: EsPmaPon Theory, PrenPce- Hall, 199. PosiPon Error Bound (PEB): theorepcal 19

20 Bound convergence (1/3) RMSE[m] à transmixer localizapon High radio environmental variability region RMSE PEB > 0 5 σ [db] 12 4 RMSE - PEB 0 0 Small variapons region RMSE - PEB 0 (approaches 2 zero) The JML 2 σ achieves [db] the Δ [m] Δ [m] σ [db] derived bound SPEB[m] Large inipal network ambiguity RMSE - PEB < 0-2 Typical behavior for biased espmapon 2 large inipal Δ ambiguipes [m] introduce bias in the espmapon process and JML performs bexer than expected in terms of RMSE

21 Bound convergence (2/3) RMSE[m] à anchor localizapon (ID:2) 15 5 σ [db] 12 RMSE - PEB Small variapons region 4 RMSE 0 - PEB 0 (approaches 0 zero) The JML achieves the derived bound 2 σ [db] Δ [m] Δ [m] σ [db] The Bound converges even for large environmental variability SPEB[m] y [m] ID: ID:5 ID:3 ID:2 ID: x [m] ID:9 Tx:1(9,9) ID:7 ID:1-4 Large inipal network ambiguity RMSE - PEB < The espmapon is biased and the bound does Δ not [m] converge NoPce that ID:2 is very close to the transmicer ID: ID:

22 Bound convergence (3/3) à anchor localizapon (average) RMSE - PEB 12 0 RMSE[m] σ [db] 4 2 σ [db] 4 SPEB[m] Δ [m] Δ [m] σ [db] Averaging over sensors increases the region of convergence of the bound and reduces the bias The bound for far anchors converges in Δ larger [m] region compared with close anchors

23 Joint Maximum Likelihood The main task of the RSS- based joint localizapon framework is to es,mate θ based on the informa,on contained in X [1] Employing Maximum Likelihood (ML) approach [1], results in the Joint Maximum Likelihood (JML) localizapon algorithm where p(x;θ) is the joint probability density func,on of the data vector X Employing the path loss and the anchor posipon espmate distribupon assump,ons, the log- likelihood funcpon L(θ X)=lnp(X;θ) results in When, then and the JML becomes the Legacy ML (LML) localizapon algorithm [1] S. M. Kay, Fundamentals of StaPsPcal Signal Processing: EsPmaPon Theory, PrenPce- Hall,

24 Performance bounds (1/2) The Cramer- Rao Lower Bound (CRLB) [1] bounds the performance of any unbiased, non- Bayesian espmator of θ via the inverse of the Fisher Informa,on Matrix (FIM) The informa,on inequality provides the CRLB espmapon bound for θ [1] The informapon inequality bounds the covariance matrix of any unbiased espmator of determinispc parameters with the inverse of the FIM calculated for the respecpve parameters Fisher InformaPon Matrix for the parameter vector θ QuanPfies the informa,on about the unknown parameter θ vector contained in the observed data X [1] S. M. Kay, Fundamentals of StaPsPcal Signal Processing: EsPmaPon Theory, PrenPce- Hall,

25 Performance bounds (2/2) The Equivalent FIM (EFIM) can be calculated as Schur s complement of the FIM The posiponal informapon can be studied separately for each node in the network The CRLB for each network node (transmixer and sensors) is obtained by employing the EFIM expressions in the informapon inequality 25