Gaussian Processes Online Observation Classification for RSSI-based Low-cost Indoor Positioning Systems

Transcription

1 RSSI (dbm) Gaussian Processes Online Observaion Classificaion for RSSI-based Low-cos Indoor Posiioning Sysems Maani Ghaffari Jadidi, Miesh Pael, and Jaime Valls Miro Absrac In his paper, we propose a real-ime classificaion scheme o cope wih noisy Radio Signal Srengh Indicaor (RSSI) measuremens uilized in indoor posiioning sysems. RSSI values are ofen convered o disances for posiion esimaion. However due o mulipahing and shadowing effecs, finding a unique sensor model using boh parameric and nonparameric mehods is highly challenging. We learn decision regions using he Gaussian Processes classificaion o accep measuremens ha are consisen wih he operaing sensor model. The proposed approach can perform online, does no rely on a paricular sensor model or parameers, and is robus o sensor failures. The experimenal resuls achieved using hardware show ha available posiioning algorihms can benefi from incorporaing he classifier ino heir measuremen model as a mea-sensor modeling echnique. I. INTRODUCTION The spreading of personal communicaion sysems ino many public and privae places, as well as he onse of new generaion of smarphones, has enabled he developmen of a vas number of indoor posiioning sysems based on sandard wireless communicaion echnologies [1], [2]. While indoor radio propagaion follows he same mechanisms as oudoor, shorer coverage range and greaer variabiliy of indoor environmens, e.g. he presence of ined meal in windows, make modeling he radio signal aenuaion significanly more challenging [3]. Furher, compared o oudoor scenarios, he number of Line-Of-Sigh (LOS) observaions are lower which means he common Friis free space model canno accuraely model he radio signal aenuaion. Therefore, for any indoor posiioning sysem ha relies on such models, he abiliy o differeniae LOS and Non-LOS (NLOS) observaions is beneficial. In his paper, we propose a probabilisic framework o explicily deec and sysemaically miigae NLOS radio signal observaions. The proposed approach is non-parameric, does no require a saisical characerizaion of waveforms, and can be incorporaed ino recursive Bayesian esimaion frameworks such as paricle filers as a mea-sensor modeling echnique. We use Gaussian process classificaion (GPC) for offline learning of decision regions based on dense disance and Radio Signal Srengh Indicaor (RSSI) measuremens, shown in Figure 1, and employ i in online scenarios using kd-ree srucures. Maani Ghaffari Jadidi and Jaime Valls Miro are wih Cenre for Auonomous Sysem, Faculy of Engineering and IT, Universiy of Technology Sydney, Ulimo, NSW 2007, Ausralia {maani.ghaffarijadidi, jaime.vallsmiro}@us.edu.au Miesh Pael is wih FX Palo Alo Laboraory Inc., Palo Alo, CA , USA miesh@fxpal.com -50 LOS NLOS log 10 (disance) (db) Fig. 1: The decision surface learned by a Gaussian process classifier using colleced LOS and NLOS measuremens. Each LOS/NLOS poin is averaged over 6 RSSI from 6 co-locaed BLE beacons wih he same ransmission power. The groundruh disances are compued using a laser range-finder sensor and an ieraive closes poin-based scan-maching echnique. A. Moivaion The main moivaion sems from he challenge faced in using Blueooh Low Energy (BLE) beacons for indoor posiioning. Assuming RSSI is he only quaniy available o he receiver, he common pracice repored in he lieraure is o conver he measured RSSI o disance. However, in our experience, under realisic condiions, radio signals are severely impaced due o shadowing and mulipahing effecs. These incidens are due o various facors such as he presence of people, he number of reflecive surfaces, and overall dynamics of he environmen. Therefore, a large number of spurious measuremens resuls in biased disance conversion and consequenly poor posiion esimaion performance. Through rejecing measuremens ha are no compaible wih he sensor model, we only add informaion o he esimaion process if i mainains is consisency. B. Conribuions The conribuions of his paper are wo folds. Firsly, we propose an online (adapive) echnique o model he BLE sensor so ha i can ackle he shadowing and mulipahing effecs of he signal. Secondly, we uilize he BLE sensor model in a posiion esimaion framework o localize a smarphone user or a robo in a given environmen. I should be noed ha by uilizing our approach, we eliminae he edious process of fingerprining he environmen o generae a radio map, raher we collec daa o model he BLE sensor which is a one ime process and has considerably lower overhead compared o fingerprining he environmen

2 C. Noaion Probabiliies and probabiliy densiies are no disinguished in general. Marices are capialized in bold, such as in X, and vecors are in lower case bold ype, such as in x. Vecors are column-wise and 1: n means inegers from 1 o n. Random variables, such as X, and heir realizaions, x, are someimes denoed inerchangeably. x [i] denoes a reference o he i-h elemen of he variable. An alphabe such as X denoes a se. A reference o a es se quaniy is shown by x. Finally, E[ ] and V[ ] denoe he expeced value and variance of a random variable, respecively. D. Ouline In he following secion, we presen he relaed work. In Secion III, he problem formulaion, and required preliminaries are explained. We presen deails of sensor modeling and analysis in Secion IV. The posiioning algorihm is explained in Secion V. We presen he experimenal resuls in Secion VI and Secion VII concludes he paper. II. RELATED WORK The idea of inegraing non-parameric models ino Bayesian filering is no new. In [4], he sysem dynamics and observaion models in exended and unscened Kalman filers and Paricle Filers (PFs) are appropriaely replaced by Gaussian Processes (GPs). In comparison o parameric models, upon he availabiliy of sufficien raining daa, resuls show improvemen in racking accuracy. Machine learning echniques are also exensively considered for indoor localizaion sysems. In locaion fingerprining approach, kernel mehods in he form of Suppor Vecor Machines (SVMs) and GPs frameworks have become he sandard way of indoor posiioning [5] [8]. However, hese approaches require he edious process of mapping he RSSI values in differen locaions in he environmen, prior o he experimen which is disinc from he online approach we use in his work. Furhermore, he likelihood map is non-adapive and does no ake ino accoun dynamic of he environmen. An imporan par of online RSSI-based posiioning sysems is he radio signal pah-loss model [9]. Such models are usually based on Friis free space model and are only valid if here is a direc and collision-free pah beween ransmier and receiver, wih no reflecion and refracion due o nearby obsacles, and in he far-field of ransmiing anenna [3], [10]. A key challenge here is o be able o idenify and miigae NLOS observaions [11] [16]. To he bes of our knowledge, approaches in [15], [16] are concepually he closes o his work. In [15], he problem of range error miigaion using SVM and GP regression is sudied. The approach uses l 2 and l 1 -minimizaion and characerizes he ranging error based on a se of feaures exraced from he received waveform. In [16], a se of saisical feaures are exraced from he received signal; a classifier discards he NLOS measuremens, and he disance o he ransmier is esimaed using regression echniques. In his work, we do no rely on feaure exracion from he received signal, he receiver has only access o he received RSSI (unlike [15]), and he classificaion oupu is incorporaed ino he probabilisic posiioning framework for sequenial esimaions. In paricular, insead of discarding measuremens we use a probabilisic mixure measuremen model. The echnique in [17] uses he floor plan o associae mulipah componens of he propagaed radio signal o he surrounding geomery. An environmen survey prior o he experimen is required as well as more sophisicaed hardware for daa collecion. In [18], indoor channel models for a wider range of frequencies o mee 5G 5h generaion wireless sysems requiremens are sudied. The probabiliy of LOS observaions is modeled using exponenial decays as a funcion of disance. However, i is menioned ha high variabiliy exiss beween differen deploymens and openness of he area. I is clear ha using purely disance resuls in a passive model and canno cope wih online radio signal variaions. The proposed soluion in his work is a nonparameric represenaion of LOS probabiliies using disance and RSSI and akes he spaial correlaion of radio signal propagaion ino accoun. III. PROBLEM FORMULATION AND PRELIMINARIES We now define he problems we sudy in his paper and hen briefly explain he required preliminaries o solve hese problems. Le M = {m [j] j = 1 : n m } be a se of known and fully observable feaures whose elemens, m [j] R 3, represen BLE beacons locaions. The robo has a receiver ha can only receive he RSSI of a broadcased signal. Le S Z be he se of possible RSSI measuremens a ime. The observaion consiss of an n s -uple random variable (S [1],..., S [ns] ) whose elemens can ake values s [k] S, k {1 : n s }. We denoe he robo posiion up o ime by x 0: {x 0,..., x } where x R 3. Given he se of known BLE beacons and noisy observaions, we wish o solve he following problems. Problem 1 (Measuremen model): Le Z R 0 be he se of possible range measuremens a ime ha is calculaed hrough a nonlinear mapping s z. The measuremen model p(z x ) is a condiional probabiliy disribuion ha represens he likelihood of range measuremens. Find he mapping from signal o range measuremens and he likelihood funcion ha describes he measuremen noise. Problem 2 (Posiioning): Le z 1: {z 1,..., z } be a sequence of range measuremens up o ime. Le x be a Markov process of iniial disribuion p(x 0 ) and ransiion equaion p(x x 1 ). Given p(z x ), esimae recursively in ime he poserior disribuion p(x 0: z 1: ). In he firs problem, we ry o characerize he received signal and hrough an appropriae model ransform i o a range measuremen. Furhermore, we need o find a likelihood funcion ha describes he measuremen noise. The second problem can be seen as a range-only selflocalizaion problem. For simpliciy, since he map is known, i is eliminaed from condiional probabiliies erms. We now express he main assumpions we use o solve he defined problems.

3 Assumpion 1 (Consan ransmission power): The ransmission power of all beacons during posiioning experimens remain fixed. Since a differen ransmission power leads o a differen signal propagaion behavior, i.e. a shorer or a longer range, his assumpion guaranees ha he sensor model complies wih he employed beacons. Assumpion 2 (Known daa associaion): Each beacon has a unique hardware idenifier ha is available o he receiver device. This assumpion is usually saisfied in pracice as each beacon has a unique MAC-address ha broadcass i ogeher wih he RSSI. Finally, we assume ha he only available informaion o he receiver is he RSSI, his is he common case for exising wireless rouers and BLE beacons. However, if he ime difference of arrival (ransmission ime) be available o he receiver device, he posiion esimaion accuracy can be improved. A. Blueooh low energy echnology Blueooh Low Energy [19] proocol was devised in I operaes in he 2.4 GHz license-free band and hence shares he same indoor propagaion characerisics as 2.4 GHz WiFi ransceivers. Unlike WiFi, BLE uses 40 channels each wih a widh of 2 MHz [20]. B. Gaussian processes classificaion Supervised classificaion is he problem of learning inpuoupu mappings from a raining daase for discree oupus (class labels). Gaussian process classificaion [21] is a nonparameric Bayesian echnique ha uses saisical inference o learn dependencies beween poins in a daase. The problem in his paper is a binary classificaion. We define a raining se D {(x [i], y [i] ) i = 1 : n o } of dimension d which consiss of a d-dimensional inpu vecor x and a class label y { 1, +1} for n o observaions. In GPC, he inference is performed in wo seps; firs compuing he predicive disribuion of he laen variable corresponding o a query case, f D, x N (E[f ], V[f ]), and hen a probabilisic predicion, p(y = +1 D, x ), using a sigmoid funcion. The non-gaussian likelihood and he choice of he sigmoid funcion can make he inference analyically inracable. Hence, approximae echniques such as Expecaion Propagaion (EP) [22] needs o be used. The vecor of hyperparameers (parameers of he covariance and mean funcions), θ, can be opimized by maximizing he log of he marginal likelihood funcion, log p(y X, θ), where X is he d n design marix of aggregaed inpu vecors x, and y = [y [1],..., y [n] ] T. The GPC model implemened in his work uses a consan mean funcion, squared exponenial covariance funcion wih auomaic relevance deerminaion as described in [23], whereas he error funcion likelihood (probi regression), and EP echnique for approximae inference is done using he open source Gaussian process (GP) library in [21]. C. Paricle filers In he problem of localizaion using RSSI, he observaion space is nonlinear, and he poserior densiy is ofen mulimodal. Paricle filers are a non-parameric implemenaion of he Bayes filer ha are suiable for racking and localizaion problems where dealing wih global uncerainy is crucial [24] [26]. In his work, we use Sample Imporance Resampling (SIR) filer embedded wih he sysemaic resampling algorihm. To deec he degeneracy and perform resampling, we compue he effecive sample size which corresponds o he reciprocal of he sum of squares of paricle weighs. IV. SENSOR MODELING AND ANALYSIS In his secion, we ackle he firs problem. To model he mapping from he signal o he measuremen space, i.e. RSSI o range, we use Friis free space model [3], [10] in which he signal aenuaion is proporional o he logarihm of he disance. This model can characerize radio signals propagaion in LOS scenarios; however, in NLOS and he presence of cluer, i may perform poorly which negaively affecs he posiioning algorihm. We firs describe experimenal daa collecion rounds, followed by how we use he experimenal daa o esimae he parameers of he pahloss model and rain he GP classifier. A. Daa collecion rounds We employ a robo equipped wih an Inerial Measuremen Uni (IMU) and a laser range-finer o localize using laser odomery. We use his resul as a proxy for groundruh o esimae disances o he BLE beacons a known locaions. We empirically found ha he effecive range of BLE beacons o define a meaningful relaion beween RSSI and disance is abou 10 m, which is consisen wih he available lieraure [1]. Hence, all daa collecion rounds for sensor modeling are performed along a 10 m range o capure he main rend of daa. In Round I, RSSIs are colleced in LOS and NLOS scenarios. The NLOS is creaed arificially by blocking he LOS using furniure such as chairs. In Round II, on a differen working day, we colleced anoher LOS daase. The colleced daa from Round I and II are illusraed in Figure 2. B. Pah-loss model parameers esimaion The signal propagaion in an indoor environmen is a complex physical phenomenon, and i is ofen no possible o find a unique model o characerize i. However, he simplified free space pah-loss model can capure he essence of signal propagaion. The model depends on, a X in dbm, which capures he ransmission power, anenna characerisics and he average channel aenuaion, he received power, p RSSI in dbm, he pah-loss exponen γ, and a reference disance, d 0 in m, for he anenna far-field. The model can be expressed as follows. p RSSI = a X + 10γ log 10 ( z d 0 ) + ɛ (1)

4 RSSI (dbm) RSSI (dbm) RSSI (dbm) RSSI (dbm) log 10 (disance) (db) (a) -90 log 10 (disance) (db) (b) LOS measuremens Pah-loss model -90 log 10 (disance) (db) (c) -75 log 10 (disance) (db) Fig. 2: Raw RSSI measuremens are from 6 co-locaed BLE beacons colleced along 10 m range for (a) Round I: LOS (12680 poins) and (b) Round I: NLOS (9380 poins). The NLOS measuremens have lower signal srengh due o shadowing and non-consrucive mulipahing effecs. (c) shows Round II: LOS (10640 poins), and (d) shows pah-loss model parameer esimaion using he maximum likelihood and a Gaussian noise model. The poins indicae he median of measuremens from all 6 beacons wih a similar ime-samp, i.e wihin en milliseconds. (d) where ɛ is he received signal power noise and assumed o have an independen and idenically disribued (i.i.d.) Gaussian disribuion, ɛ N (0, σ 2 ). The hree model parameers a X, γ, and d 0 can hen be esimaed using he nonlinear leas squares parameer esimaion echnique, i.e. maximum likelihood esimaion wih a Gaussian noise assumpion. Figure 2d shows he model wih parameers esimaed using he Round II daase. Remark 1: From Equaion (1), i is clear ha if p RSSI, in dbm, follows a normal disribuion, hen he received power, in Wa, follows a log-normal disribuion. Therefore, we can assume ha he disance follows a log-normal disribuion as well. In pracice, we calculae he range z from a known value of p RSSI. C. GP classifier raining and validaion To increase he diversiy of raining daa, we use NLOS observaions from Round I and LOS observaions from Round II. The oal number of raw observaions aken from 6 BLE beacons is abou 20, 000. We compue he median of he observaions wihin en milliseconds o reduce he effec of ouliers and improve he accuracy of he raining se, leading o abou 2000 poins. We hen downsample daa o abou 1000 poins o keep he compuaional aspec of GPC manageable. Each raining poin inpu consiss of a 2-dimensional vecor concaenaed from he RSSI observaion and he corresponding groundruh range. The arge labels are se o +1 and 1 for LOS and NLOS, respecively. Figure 1 shows he inferred probabiliy surface in which he higher probabiliies correspond o LOS observaions. Noe ha in pracice one does no have access o he groundruh disance. Insead, he esimaed disance o a beacon ogeher wih he RSSI observaion are he inpu. To employ he classifier online, he resuls are sored in a kd-ree daa srucure wih an appropriae resoluion. We evaluae he performance of he classifier using he Receiver Operaing Characerisic curve (ROC) and he area under he ROC (AUC) [27]. The raw measuremens wihou any filering are used o conduc wo ess. Firs, we use all observaions from Round I NLOS and Round II LOS. In he second es, we use all observaions from Round I and II which conain abou 32, 000 poins. Figure 3 illusraes he ROC analysis resuls where he AUC indicaes he average performance of he classifier on each es se. V. POSITIONING ALGORITHM We now formulae a measuremen model ha embeds he classifier ino he Bayesian filering algorihm. Le C [i] be a Bernoulli random variable whose realizaion a ime indicaes LOS probabiliy for he i-h paricle. Wihou loss of generaliy, he join probabilisic measuremen model can be defined as follows. p(z, c [i] x [i] ) = p(c [i] z, x [i] )p(z x [i] ) (2) The condiional probabiliy p(z x [i] ) is he so-called likelihood funcion of he Bayesian filering and in he radiional for he i-h paricle. Therefore, he join probabiliy of he range measuremen and LOS can be seen as a new likelihood funcion. However, his model is only valid if he measuremen is SIR filer reurns an imporance weigh w [i]

5 True posiive rae AUC = AUC = False posiive rae Fig. 3: The receiver operaing characerisic curve and he area under he curve for he rained GP classifier. The classifier is validaed using he LOS and NLOS measuremens colleced on Round I and II. The average performance of he classifier on he larger es se is lower, LOS. The classifier can heoreically deec he NLOS when p(c [i] z, x [i] ) p los, where p los is a hreshold for LOS deecion and can be se using he ROC analysis performed earlier [27]. As such, in he absence of any prior knowledge abou he environmen, we rea NLOS measuremens as random wih a consan probabiliy p rand. Consequenly, he measuremen funcion can be wrien as: p(z, c [i] x [i] ) = { [i] p(c z, x [i] p rand )p(z x [i] ) if p(c [i] z, x [i] ) > p los oherwise (3) To query he probabiliy p(c [i] z, x [i] ) from he classifier, we use he raw RSSI observaion and esimaed disance o he corresponding beacon as h(x [i] ) (x [i] m [j] ) T (x [i] m [j] ) (4) The formulaed probabilisic measuremen model incorporaes he developed classifier ino he SIR filer framework. As we will see laer, one can only use p(z x ) o compuing he filering disribuion of he robo posiion, e.g. using a normal or a log-normal disribuion, however, he join measuremen model improves he confidence abou he correcness of he model-measuremen relaion. As i is assumed here is no inerocepive sensor available, we do no have any knowledge regarding he ransiion probabiliy model p(x +1 x ). Le he sae vecor be x [i] = [x [i,1] ẋ [i,1] x [i,2] ẋ [i,2] ] T, where ẋ [i] denoes he he i- h paricle s velociy a ime. Assuming a consan velociy moion model, he sae equaion becomes: 1 s 0 0 x [i] +1 = F x[i] + u, F = (5) s where s is he sampling ime, u N (0, Q), and Q is a diagonal moion noise covariance marix. Noe ha he receiver heigh insalled on he robo is fixed as he robo TABLE I: Parameers used in he posiioning experimens. Parameer Symbol Value Compared SIR paricle filer varians: Gaussian PFG - Gaussian wih classifier PFG-C - Lognormal PFL - Lognormal wih classifier PFL-C - Pah-loss model parameers: Aenuaed ransmission power a X The pah-loss exponen γ 1.72 Reference disance d m Measuremen model: Classifier hreshold p los 0.4 Gaussian; sandard deviaion σ n 3 m Gaussian; random probabiliy p rand 0.1 Lognormal; sandard deviaion σ ln 0.4 dbm Lognormal; random probabiliy p rand (d 0 σ ln 2π) 1 Moion model: Posiion sandard deviaion σ u 0.1 m Velociy sandard deviaion σ v 0.05 m / sec Paricle filer: Number of paricles n p 100 Resampling hreshold n hr 20 BLE Beacon Parameer: Transmission Power T x +4 dbm Broadcasing Frequency B f 10 Hz operaes on an even floor. VI. EXPERIMENTAL RESULTS To validae he proposed measuremen modeling using he GP classifier, we evaluae our approach on an indoor posiioning algorihm using BLE beacons. The daase is colleced during working hours in an office space and he robo is moved wih a moderae speed of 0.2 m / sec on average. In he following, we explain he experimenal seup and resuls as well as a discussion on he limiaions of his work and compuaional complexiy analysis of he proposed algorihm. A. Experimenal seup and evaluaion crieria Tradiionally, Cramér-Rao Lower Bound (CRLB) has been developed and used for sysem designs and evaluaions, since i can predic he achievable performance before building he sysem [25], [28]. We uilized CRLB o approximae he heoreical lower bound for he mean-squared error. We define he efficiency, η, of a sysem using CRLB and he Roo Mean Squared Error (RMSE) as follows. CRLB η = 100 (6) RMSE The explanaions of he compared echniques and used parameers are provided in Table I. We compare he resuls for indoor posiioning using he SIR Paricle Filer (PF) wih Gaussian (PFG) and log-normal (PFL) likelihood funcions, and wih and wihou incorporaing he classifier, PFG-C and PFL-C, respecively. To deec he degeneracy, we calculae he effecive sample size, n eff = ( n p i=1 w[i] ) 1, and perform resampling when n eff < n hr ; where n p is he number of paricles and n hr is a hreshold 1 < n hr < n p. All he resuls presened in his paper use n p = 100 and = 20, and he robo posiion is esimaed using he n hr

6 TABLE II: Comparison of indoor posiioning algorihms using paricle filering wih and wihou incorporaing he online classifier on Daase I and II. The resuls are averaged over 100 runs; mean ± sandard error. PFG PFG-C PFL PFL-C CRLB (m) RMSE (m) 8.08 ± ± ± ± 0.05 η (%) 5.76 ± ± ± ± 0.04 Time (sec) 10.6 ± ± ± ± 0.29 weighed average of all paricles posiions. Moreover, he ransmission power T x of all beacons is +4 dbm. Fig. 5: The empirical cumulaive disribuion funcions of he four compared echniques. Fig. 4: The indoor posiioning resuls in an office environmen populaed wih BLE beacons. For clariy, The esimaed rajecories are ploed by skipping 50 ime seps beween any wo successive posiions. B. Indoor posiioning resuls The daase is colleced in a research office pariioned ino separae office cabins and consiss of radiional office furniure. The daa is colleced using a TurleBo equipped wih an IMU sensor and a laser range-finder which are used for groundruh pose esimaion. The beacons signals are recorded using a smarphone Android app. The daase is colleced by maneuvering he robo over a disance of 70 m in an office space of m 2, as shown in Figure 4. The mehods are implemened using Robo Operaing Sysem (ROS) [29] and resuls for indoor posiioning are processed using MATLAB. The nominal sampling rae is BLE beacons is 10 Hz; however, in pracice, we experienced a sampling rae of 7 Hz, on average, for he enire daase. Figure 5 shows he empirical cumulaive disribuion funcion (CDF) of he four compared echniques. The empirical CDF is an unbiased esimae of he populaion CDF and is a consisen esimaor of he rue CDF. Each curve illusraes he median of 100 CDF from 100 independen runs. The PFG-C demonsraes he bes performance by he localizaion error of abou 2 m. Noe ha faser rise from zero o one along he verical axis is a desirable oucome. The saisical summary of he resuls is depiced in Figure 6. As an example, he esimaed rajecory using PFG-C and PFL-C are also illusraed in Figure 4. The proposed classifier has a desirable effec on he robo posiion esimaion where he robo posiion has fewer flucuaions. Fig. 6: The saisics from indoor posiioning resuls using paricle filering wih normal and log-normal noise disribuions. The incorporaion of he classifier ino he sensor model leads o a more accurae locaion and scale esimaion. The classifier makes he posiioning algorihm more robus o noisy observaions and ouliers, improving he overall reliabiliy of he sysem (Figure 5). This is, in paricular, appealing for he case of he normal likelihood. From he physical naure of he radio signal propagaion, he ranging bias is always posiive. Therefore, a symmeric disribuion such as he Gaussian likelihood performs poorly in characerizing he noise. However, depending on he parameers, here are insances ha he normal and log-normal disribuions behave similarly. Neverheless, he classifier improves he esimaion performance for boh ypes of noise models. Table II shows he numerical comparison beween differen algorihms from 100 independen runs. The CRLB value for normal and log-normal disribuions is inherenly differen as he noise variance for he former is in meers and he laer in dbm. Thus, one should compare he efficiency of mehods wih a similar likelihood funcion. However, we can compare all algorihms using RMSE. Overall, PFG- C and PFL-C show beer performance compared o heir corresponding algorihms ha do no use he classifier. C. Discussion The main limiaion of he proposed online classificaion echnique is ha he RSSI range varies according o he BLE beacon TP. Therefore, using beacons ha have differen TPs as compared o hose used for raining he classifier

7 will resul in lower performance. Furhermore, he classifier canno improve he sensor model if i is no (a leas empirically) compaible wih he underlying physical naure of he RF signal propagaion. Therefore, i can only ac as a proxy for consisen observaion selecion which can deec and miigae desrucive mulipahing, shadowing, or sensor failures, i.e. weak baeries or hardware failures. Finally, o our experience, collecion of NLOS daa is of grea imporance. If NLOS daa has a subsanial overlap wih LOS daa, hen he performance of he rained classifier will decrease dramaically. In small environmens, his effec can be undersood from consrucive mulipahing or parially blockage of LOS during daa collecion. D. Compuaional complexiy For n o observaions, he approximae inference using EP scales as O(n 3 o) which is performed offline. The kd-ree srucure is suiable for efficien search in low-dimensional spaces, such as he case in his work. For n p paricles and n z neares neighbor queries, he algorihm scales as O(n p n z log n q ), where n q is he number of sored query poins, and usually n z n p. VII. CONCLUSION AND FUTURE WORK We sudied he problem of indoor posiioning using BLE beacons. We developed an online classificaion sraegy o improve he consisency of received measuremens wih he employed sensor model. Our experimenal resuls under realisic condiions show promising improvemens and he proposed classifier can be used as a mea-sensor modeling echnique o cope wih spurious measuremens. The proposed mehod is paricularly simpler and more scalable han he popular fingerprining echnique as he raining phase is in he sensor space insead of spaial coordinaes of an environmen. The fuure work includes furher sudies and improvemen of he sensor model in he presence of semi-dynamic obsacles. Inegraion of incremenal moion measuremens such as IMUs can also improve he accuracy of posiion racking. Moreover, increasing he sampling rae can provide a beer efficiency hrough a higher flow of informaion ino he esimaion process. Lasly, he simulaneous esimaion of he robo (receiver) and BLE beacons posiions is an ineresing avenue o follow. ACKNOWLEDGEMENT This work has been suppored by Yahoo Research under he Faculy Research and Engagemen Program (FREP) an academic oureach iniiaive. The auhors would also like o hank FX Palo Alo Laboraories Inc., for sharing Blueooh Low Energy daase colleced by hem o validae our proposed algorihm. REFERENCES [1] H. Liu, H. Darabi, P. Banerjee, and J. Liu, Survey of wireless indoor posiioning echniques and sysems, IEEE Trans. on Sysems, Man, and Cyberneics, Par C (Applicaions and Reviews), vol. 37, no. 6, pp , [2] G. Yanying, A. Lo, and I. Niemegeers, A survey of indoor posiioning sysems for wireless personal neworks, Comm. Surveys and Tuorials, vol. 11, no. 1, pp , [3] T. S. Rappapor, Wireless communicaions: principles and pracice. 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