Domain-Constrained Semi-Supervised Mining of Tracking Models in Sensor Networks

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1 Doman-Constraned Sem-Supervsed Mnng of Trackng Models n Sensor Networks Rong Pan 1, Junhu Zhao 2, Vncent Wenchen Zheng 1, Jeffrey Junfeng Pan 1, Dou Shen 1, Snno Jaln Pan 1 and Qang Yang 1 1 Hong Kong Unversty of Scence & Technology {panrong,vncentz,panjf,dshen,snnopan, qyang}@cse.ust.hk 2 NEC Labs, Chna, Zhongguancun East Road, Bejng 10004, Chna zhaojunhu@research.nec.com.cn ABSTRACT Accurate localzaton of moble objects s a major research problem n sensor networks and an mportant data mnng applcaton. Specfcally, the localzaton problem s to determne the locaton of a clent devce accurately gven the rado sgnal strength values receved at the clent devce from multple beacon sensors or access ponts. Conventonal data mnng and machne learnng methods can be appled to solve ths problem. However, all of them requre large amounts of labeled tranng data, whch can be qute expensve. In ths paper, we propose a probablstc sem-supervsed learnng approach to reduce the calbraton effort and ncrease the trackng accuracy. Our method s based on sem-supervsed condtonal random felds whch can enhance the learned model from a small set of tranng data wth abundant unlabeled data effectvely. To make our method more effcent, we explot a Generalzed EM algorthm coupled wth doman constrants. We valdate our method through extensve experments n a real sensor network usng Crossbow MICA2 sensors. The results demonstrate the advantages of methods compared to other state-of-the-art objecttrackng algorthms. Categores and Subject Descrptors I.2.6 [Artfcal Intellgence]: Learnng; H.2. [Database Management]: Database Applcatons Data mnng General Terms Algorthms Keywords Localzaton, Calbraton, Trackng, Sensor Networks, EM, CRF 1. INTRODUCTION Recently, wreless sensor networks have attracted great nterests n several related research felds and ndustres. Many tasks such as context-aware computng [4] and envronmental montorng can be realzed wth the help of wreless sensor networks, whch offer Permsson to make dgtal or hard copes of all or part of ths work for personal or classroom use s granted wthout fee provded that copes are not made or dstrbuted for proft or commercal advantage and that copes bear ths notce and the full ctaton on the frst page. To copy otherwse, to republsh, to post on servers or to redstrbute to lsts, requres pror specfc permsson and/or a fee. KDD 07, August 12 15, 2007, San Jose, Calforna, USA. Copyrght 2007 ACM /07/000...$5.00. unque advantages of beng lghtweght, dstrbuted, envronmentaware and network-based. Object trackng, event-detecton and actvty recognton can now be realzed n sensor networks usng probablstc algorthms[7, 11]. It s a fundamental task for many of these applcatons to locate moble clent devces usng collected wreless sgnals (n terms of rado-sgnal-strength, RSS, values) from dfferent sensor nodes that act as beacons. In the past, some conventonal data mnng technologes have been appled for solvng the localzaton problem [7, 9]. Generally, some statstcal models are obtaned offlne whch can map sgnals to locatons. These models are then used onlne to predct the clent locatons based on the real-tme sgnal values. Among the past works, many researchers have developed rangng-based algorthms for localzng moble nodes n a wreless sensor network. One approach s multlateraton (e.g., [9]), consstng of two man steps. It frst transforms the sensor readngs nto a dstance measure. It then attempts to recover the coordnate locatons n terms of relatve dstance to the beacon nodes. Ths approach reles on an deal sgnal propagaton model and extensve hardware support. It suffers from low accuracy problem snce RSSs do not follow deal propagaton patterns, especally n complex envronments. In ths paper, we address ths problem usng a sem-supervsed statstcal relatonal learnng approach based on condtonal random felds (Sem-CRF). We assume that a moble sensor node moves n a sensor network envronment. The RSS of the moble node can be receved by several sensors n the network, whch are then forwarded to a processor for trackng. It can also happens n the way that all sensors n the network are sendng sgnals to the moble sensor node, whch performs the localzaton tself. In ether case, we have a sequence of mult-dmensonal vectors that corresponds to a trace. Each vector along the trace can be labeled wth a physcal locaton coordnate, or unlabelled. Ths paper makes the followng contrbutons. Frst, we dentfy and solve a major bottleneck n the applcaton of data mnng technologes n sensor networks. Second, we present a novel semsupervsed learnng method for moble-node trackng and localzaton by utlzng both labeled and unlabelled RSS trace data. Thrd, we ntroduce doman-drven heurstcs for reducng the complexty of the learnng procedure, whch greatly mprove the scalablty of the statstcal models. Fnally, we valdate the proposed methods through the experments over a real sensor network. 2. THE Sem-CRF ALGORITHM FOR TRACKING AND LOCALIZATION 2.1 Problem Statement Consder a two-dmensonal trackng problem. Our objectve s 1023

2 to determne the locaton of a moble object y t =(u t,v t) C as t moves n a sequence, gven the observed sgnal vectors x t. Fgure 3 shows an example of the floor n one of our expermental test beds, whch conssts of N =beacon nodes and one moble unknown node. The localzaton problem can be converted to a supervsed classfcaton problem f we had suffcent labeled data for each locaton. However, when the labeled data are nsuffcent at each locaton, we wsh to make the best use of some partally labelled or totally unlabeled RSS sequences as well n our predcton. Now let us formally ntroduce the notaton of tranng data. In our study, the tranng data consst of a set of fully labelled sequences D f = {(X 1,Y 1),...,(X M,Y M )}, and a set of partally labeled (or totally unlabeled) sequences D p = {(X M+1,Y M+1),...,(X M+L,Y M+L)},whereX s a sequence of sgnal vectors x 1,...,x m, =1,...,M + L,andY s a sequence of correspondng locatons y 1,...,y m, =1,...,M + L. Some values of y j are unknown for M +1 M + L. A moble robot can be employed to collect these unlabelled data by smply wanderng around. 2.2 Lnear-chan CRF The CRF Model In ths paper, we propose a statstcal relatonal learnng approach usng CRF to explot the relatonshp between RSS readngs at two neghborng tme ponts n terms of ther correspondng physcal dstance. As a moble object moves around, a sequence of RSS values can be receved, wth each correspondng to a certan locaton. Ths process can be modeled by an 1-D Lnear-chan CRF as ntroduced n the followng secton, where the states correspond to the locaton labels and the nputs or observatons correspond to the RSS readngs. Lnear-chan CRF models have been wdely used to model sequental data. These models can be roughly vewed as condtonally-traned fnte state machnes [5]. A lnear-chan CRF, as shown n Fgure 1, defnes a dstrbuton over state sequence y = y 1,y 2,...,y T gven an nput sequence x = x 1,x 2,...,x T by makng a frst-order Markov assumpton on states, where T s Unobserved Y Observed X y 1 y 2 x 1 x 2 Fgure 1: Dagram of Lnear-chan CRF the length of the sequence. These Markov assumptons mply that the dstrbuton over sequences factorzes n terms of potental functons Φ t(y t 1,y t, x) as: p(y x) = Q t y T x T Φt(yt 1,yt, x), (1) Z(x) where the partton functon Z(x) s the normalzaton constant that makes the probablty of all state sequences sum to one. It s defned as follows: Z(x) = X Y Φ t(y t 1,y t, x). (2) y t The potental functons Φ t(y t 1,y t, x) can be nterpreted as the cost of makng a transton from state y t 1 to state y t at tme t, smlar to a transton probablty n an HMM. Computng the partton functon Z(x) requres summng over the exponentally many possble state sequences y. By explotng the Markov assumpton, however, Z(x) (as well as the node margnal p(y t x) and the Vterb labelng) can be calculated effcently by varants of the standard dynamc programmng algorthms used n HMM. We assume that potentals factorze themselves accordng to a set of features f k that are gven and fxed, so that! Φ(y t 1,y t, x) =exp λ k f k (y t 1,y t, x,t). (3) X k The model parameters are a set of real weghts Λ = {λ k }, one for each feature (to be defned below). The feature functons can descrbe any aspect of a transton from y t 1 to y t as well as y t and the global characterstcs of x. For example, f k may have value 1 when the dstance between y t 1 and y t s smaller than 50cm Parameter Estmaton The parameters Λ can be estmated through a maxmum lkelhood procedure usng the tranng data. That s, we can estmate them by maxmzng the condtonal log-lkelhood of the labeled sequences n the tranng data Ψ={(X 1,Y 1),...,(X M,Y M ))}, whch s defned as: MX L(Λ) = log(p (Y X ;Λ)), (4) =1 where M s the number of sequences. As dscussed n Sutton et al. n [10], L(Λ) s concave n lght of the convexty of the knd of functons g(x) =log P exp x Inference Gven the condtonal probablty of the state sequence defned by a CRF n Equaton (1) and the parameters Λ, the most probable labelng sequence can be obtaned as Y =argmaxp (Y X;Λ), (5) Y whch can be effcently calculated usng the Vterb algorthm []. The margnal probablty of states at each poston n the sequence can be computed by a dynamc programmng nference procedure smlar to the forward-backward procedure for HMM [3]. We can defne the forward values α t(y x) by settng α 1(y x) equal to the probablty of startng wth state y and then terate as follows: α t(y x) = X y α t 1(y x)exp `Λ t(y,y,x), (6) where Λ t(y,y,x) s defned by: Λ t(y,y,x) = X k Then Z(x) equals to P y αt(y x). β (y x) can be defned smlarly. λ k f k (y t 1 = y,y t = y, x) (7) The backward values β t(y x) = X y β t+1(y x)exp `Λ t(y, y, x). () After that, we calculate the margnal probablty of each locaton gven the observed sgnal sequence: P (y t = g x) = αt(g x) βt(g x). (9) Z(x) So far, we have ntroduced a lnear-chan CRF model for unknown moble-node trackng. We can see that f k (y t 1,y t, x) (n 1024

3 Equaton (3)) s an arbtrary feature functon over the entre observed sequences and the states at postons t and t 1. In our problem, the locatons are two-dmensonal contnuous values. The number of possble locatons are nfnte large. Therefore, t s extremely dffcult to compute the feature of two arbtrary locatons. Fortunately, the trackng area s known n advance usually. One soluton s to dscretze a 2-D locaton space nto grds. For nstance, n a 5m 4m area, we can dvde t nto 10 grds wth each grd beng 50 50cm 2. Ths example s shown n Fgure 2. In ths way, we can convert the known locatons nto such grds. In the test phase, f a moble object s located at grd g, we can use the coordnates of the center pont n g to represent the locaton of the moble object. After lmtng the locaton space, t s possble to use the lnear-chan CRF approach for trackng problem. However, a major ssue s how to determne the sze of the grd. Ths problem can be solved n two ways. Frst, the sze often s determned by the nature of the problem tself, whch s decded by the precson requrement posed by applcaton users. Another approach s to study the problem emprcally, as we wll do n the expermental secton. g 1 g 2 g 24 g 35 g 46 g g 9 Fgure 2: A demo of reducton of locatons to grds Incorporatng Doman Constrants After reducng locatons to grds, we can specalze the feature functons for each possble transton among dfferent grds. That s, we can defne f k (y t 1 = g, y t = h, x) by f (g,h) (t 1,t,x). However, the number of the transton feature functons, as well as the correspondng parameters, reaches n 2 (n s the number of grds), whch can be qute large. For nstance, n the above example n Fgure 2, n = 0, then n the CRF learnng, we need to estmate 6400 parameters for the potental f k (y t 1,y t, x). Although we can stll estmate the values of the parameters wth large n, t wll certanly ncrease the computatonal cost and run the rsk of overfttng. What s worse, learnng CRF wth more parameters requres more tranng data, whch wll ncrease the labellng costs. In addton, we also need to trade off the complexty of the model and ts generalzaton capablty. If we ncrease the grd sze to reduce the computatonal cost, we wll sacrfce the estmaton accuracy. In ths paper, we ncorporate the doman constrants n the data mnng process to reduce the number of parameters that need be learned. In partcular, we note that a moble object n a sensor network typcally moves around n the same way, such that the lkelhood of transtng between two neghborng ponts are roughly same. The lkelhood of travelng between two dstant ponts wll also be roughly the same, although the value wll be much smaller. Such a doman constrant s supported by our experments. g 0 To ncorporate the doman constrants mentoned above, we use a so-called parameter tyng technque that s desgned for speech recognton [6] to combne smlar parameters. Our assumpton s that the characterstcs of two transtons wth the same dstance are alke. Intutvely, n Fgure 2, we observe that the transtons g 1 g 2 and g g 9 should happen wth smlar frequences as they both transt by one grd n terms of Eucldean dstance. Smlarly, the transtons g 24 g 35 and g 35 g 46 should happen smlarly as the ther Eucldean dstances are both 2 grds. From ths observaton, we can te the parameters of the transton feature functons so long as they have the same transton dstance, whch s defned as follows: The value of the transton feature functon f k (y t 1,y t, x) equals one f and only f ds(y t 1,y t)=k, where the ds defnes the dstance between the two ponts. As expected, the number of parameters s greatly reduced by usng ths constrant. 2.3 The Sem-CRF Algorthm In ths secton, we ntroduce how to ncorporate sequences whose labels are fully or partally observed n the parameter estmaton of CRF. An effcent method for parameter estmaton wth ncomplete data can be derved by the extenson of EM algorthm [2]. In ths paper, we use a Generalzed Expectaton Maxmzaton (GEM) algorthm to learn the parameters Λ of CRF wth both fully and partally observed data [1]. In the GEM algorthm, the probablstc optmzaton problem s dvded nto two-step teratons. The unobserved data are estmated n the E-step wth the parameters obtaned n the last teraton and the parameters of CRF are optmzed n the M-step. We frst compute the log lkelhood of Equaton (4) wth expectaton over the unobserved data as follows: L(Λ; Λ t ) = P M =1 log P (Y X;Λ)+ P M+L P P (Y (u) X,Y (o) ;Λ t )logp(y (u) = P M =M+1 Y (u) P=1 M+L =M+1,Y (o) Pt Λt(y,y,X ) P M+L =1 log Z(X P )+ P (Y (u) X,Y (o) ;Λ t ) P t Λt(y,y,X ) Y (u) In ths equaton, Y (u) s the unobserved locatons of the -th sequence, Y (o) s the observed counterpart, and Λ t(y,y,x )= X k λ kf k (y,y,x ), Z(X )= X y exp Xt Λt(yt 1,yt,X). Smlar to Equaton (4), L(Λ; Λ t ) s also concave. We can use the same method to optmze t. The only problem left s how to nfer for partally observed sequences. We need to change Equatons (6) and () for some cases. If y t = j s observed, we drectly assgn 1 to α t(y = j x) and β t(y = j x) and assgn 0 to the other values of α t and β t.ify t s not observed, we follow Equatons (6) and (). The new nference formulae are summarzed n Equatons (10) to (12). >< 0 y t j α t(y = j x) = 1 y t = j >: P y α t 1(y x)exp (Λ t(y,y,x)) y t unseen (10) X ;Λ) 1025

4 >< 0 y t j β t(y = j x) = 1 y t = j >: P y β t+1(y x)exp (Λ t(y,y,x)) y t unseen (11) P (y t = k x) = αt(y = k x) βt(y = k x) Z x. (12) We now summarze the Sem-CRF learnng algorthm n Table 1. There are several ways of parameter ntalzaton. The common one s to randomly assgn them values from 0 to 1. To speed up the convergence, we use an alternatve that prelmnarly estmates parameters wth labeled data. As to the number of teratons, we wll dscuss t n the expermental secton. Table 1: The tranng algorthm for CRF wth both fully and partally observed data. Algorthm Sem-CRF Input: The fully and partally observed data D f, D p Output: The parameters Λ of CRF 1 Intalze parameters Λ 0 of CRF. Λ t =Λ 0. whle log-lkelhood has not converged or the max number of teratons s not reached, do % ====== E-step ====== Compute the expectatons of the all unobserved locatons, by Equatons (10) to (12). % ====== M-step ====== Optmze Λ usng L-BFGS. Λ t =Λ. endwhle return Λ Camera-1 2 Camera Moble Node 7 Projected Plane Real Plane Camera-3 5 Camera-2 Beacon Node Fgure 3: Expermental Test-Bed 4 MICA2 3. EXPERIMENTAL EVALUATION MICA2Dot Sony AIBO LEGO Mndstorms 3.1 Expermental Setup We test the effectveness and robustness of our locaton trackng algorthm for moble sensor nodes n a sensor network based on the RSS sgnals. Our experments are performed n the Pervasve Computng Laboratory (Fgure 3) n the Department of Computer Scence and Engneerng at Hong Kong Unversty of Scence and Technology. The room s large enough for us to set up an expermental test-bed of 5.0 meters by 4.0 meters. In Fgure 3, P 1P 3 = P 4P 6 =5.0m and P 1P 4 = P 3P 6 =4.0m. There are three man components of our setup: Wreless Sensor Networks. We use CrossBow MICA2 and MICA2Dot to construct a wreless sensor network. We program these sensor nodes to broadcast and detect beacon frames perodcally so that they could measure the RSS of each other. Moble Robots. We try dfferent knds of robots that can run freely around the floor at dfferent speeds, such as a Sony AIBO dogs, LEGO Mndstorms and off-the-shelf toy cars. Fgure 3 shows that a sensor node s attached on top of a toy car whch can be remotely controlled by rado at the speed of 0.4 m/s. A Camera Array s used to record the ground truth locatons of the moble robots for our tranng and test data. We use two performance measurements to evaluate the orgnal CRF and the CRF model usng parameter tyng (denoted by CRF- PT) localzaton algorthms. The frst metrc s the mean errordstance values between estmated and true locatons. The second measurement s the accuracy n percentage. Gven an error-dstance threshold θ, the accuracy rate s the probablty that the dstance between the estmated and true locatons s smaller than θ. Two more baselnes n our experments nclude (1) Logstc Regresson (LR), (2) Support Vector Regresson (SVR). We control a moble robot to run and stop around the test area (Fgure 3) for collectng data wth samplng nterval 0.5s. The data set formed a trace of length about 600m wth 3, 000 examples. For every experment below, we randomly select a subset of the data as fully observed tranng data, a subset of data as partally observed tranng data by randomly removng the locatons assocated wth them, and evaluate the performance on the rest. To reduce the statstcal varablty, we repeated the experments for 30 tmes and reported the average results. 3.2 Convergence of Sem-CRF One queston about the the Sem-CRF algorthm s the convergence of the EM teratons. In ths experment, we use 10 fully labelled and 50 partally labelled sequental data to tran the CRF, where the length of each sequence s 5 and only one node s labelled n the partally labelled data. Fgure 4 shows the convergence rate of Sem-CRF. We can see that about 4 teratons are enough. In the experments of ths paper, the maxmum number of teratons of the Sem-CRF s set to 10. Summaton of Log Lkelhooh Number of Iteratons Fgure 4: Convergence rate of Sem-CRF 1026

5 3.3 Sem-CRF vs. Baselnes In the followng experments, we fx the tranng data sze at 550, and tune the rato of the labelled data from 0.33 to 1. In Fgure 5, we show the mean error performance of the four algorthms descrbed above. As can be seen, Sem-CRF consstently outperforms the other algorthms n terms of mean error dstance, whle CRF beats the remanng two baselnes. One mportant reason s they both effectvely leverage the sequental nformaton of the moble node. Moreover, as Sem-CRF can also learn from the unlabelled data, t gans much better performance when there are a lot of such data wth a small porton of labelled ones. We lst some more nformaton of these experments ncludng the accuracy performance n Table 2. Mean Error (cm) LR SVR CRF GEM CRF proporton of labelled nstances Fgure 5: Vary the rato of tranng set sze Table 2: Performance of the tested approaches Approach Mean(cm) Accuracy at 100cm Sem-CRF % CRF % SVR % LR % 3.4 Impact of Grd Szes The grd sze may affect the performance of the localzaton algorthms. In ths experment, we fx the rato of labelled data at 5% and vary the sde length of the grds from 10cm to 100cm. Fgure 6 shows the expermental results of CRF and Sem-CRF. From the fgure we can see that when the grd sze ranges from 20cm to 50cm, the performance of both the two methods s less senstve than that wth the grd sze of 10cm and 100cm. Mean Error (cm) CRF GEM CRF Grd Sze (cm) Fgure 6: Vary the grd sze (rato of labelled data s 5%.) 4. CONCLUSION AND FUTURE WORKS We have presented a new approach to reducng the calbraton effort when trackng moble sensor nodes n a wreless sensor network. Our approach made extensve use of the sequental nformaton of movng sensor s trajectory. These sequences provded unlabelled examples whch can be used to tran CRF together wth the manually labelled RSS values. We ntroduce a Sem-CRF model to utlze such partally labelled data. By usng parameter tyng technques we sgnfcantly mprove the performance of Sem-CRF algorthm whle reducng calbraton effort. A sensor network was set up based on Crossbow MICA2 and MICA2Dot nodes whch are used as both beacon and moble nodes. Expermental results showed that the proposed method could acheve a better performance wth relatvely fewer number of labelled examples. In the future, we plan to contnue to test the Sem-CRF based framework n a large scale sensor network. We are also nterested n ntroducng dfferent factors, such as changng tme and space, to see how the knowledge learned n one settng can be appled to another. 5. REFERENCES [1] H. L. Cheu,, S. W. Lee, and P. K. Lesle. Actvty recognton from physologcal data usng condtonal random felds, January [2] A. P. Dempster, N. M. Lard, and D. Rubn. Maxmum lkelhood from ncomplete data va the em algorthm. Journal of the Royal Statstcal Socety, 1(39):1 3, [3] J. Lafferty, A. McCallum, and F. Perera. Condtonal random felds: Probablstc models for segmentng and labelng sequence data. In Proc. 1th Internatonal Conf. on Machne Learnng, pages Morgan Kaufmann, San Francsco, CA, [4] J. Lester, T. Choudhury, N. Kern, G. Borrello, and B. Hannaford. A hybrd dscrmnatve/generatve approach for modelng human actvtes. In IJCAI, pages , [5] A. McCallum. Effcently nducng features of condtonal random felds. In C. Meek and U. Kjærulff, edtors, UAI, pages Morgan Kaufmann, [6] C. Neukrchen, D. Wllett, and G. Rgoll. Soft State-Tyng for HMM-Based Speech Recognton. In 5th Internatonal Conference on Spoken Language Processsng (ICSLP), pages , Sydney, 199. [7] X. Nguyen, M. I. Jordan, and B. Snopol. A kernel-based learnng approach to ad hoc sensor network localzaton. ACM Transactons on Sensor Networks, 1(1): , [] L. R. Rabner. A tutoral on hdden markov models and selected applcatons n speech recognton. Proceedngs of the IEEE, 77(2):257 26, 199. [9] A. Savvdes, C. Han, and M. B. Strvastava. Dynamc fne-graned localzaton n ad-hoc networks of sensors. In Proceedngs of the 7th Annual Internatonal Conference on Moble Computng and Networkng, pages , Rome, Italy, [10] C. Sutton and A. McCallum. An ntroducton to condtonal random felds for relatonal learnng. In L. Getoor and B. Taskar, edtors, Introducton to Statstcal Relatonal Learnng. MIT Press, [11] J. Yn, X. Cha, and Q. Yang. Hgh-level goal recognton n a wreless LAN. In Proceedngs of the Nneteenth Natonal Conference on Artfcal Intellgence, pages 57 54, San Jose, CA, USA, July