Received-Signal-Strength Threshold

Optimization Using Gaussian Processes

Feng Yin, Yuxin Zhao, Fredrik Gunnarsson and Fredrik Gustafsson

Journal Article

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Feng Yin, Yuxin Zhao, Fredrik Gunnarsson and Fredrik Gustafsson, Received-Signal-Strength

Threshold Optimization Using Gaussian Processes, IEEE Transactions on Signal Processing,

2017. 65(8), pp. 2164-2177.

http://dx.doi.org/10.1109/TSP.2017.2655480

Postprint available at: Linköping University Electronic Press

http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-135065

1

Received-Signal-Strength Threshold OptimizationUsing Gaussian Processes

Feng Yin,Member, IEEE,Yuxin Zhao,Student Member, IEEE,Fredrik Gunnarsson,Senior Member, IEEE,Fredrik Gustafsson,Fellow, IEEE

Abstract—There is a big trend nowadays to use event-triggeredproximity report for indoor positioning. This paper presen tsa generic received-signal-strength (RSS) threshold optimizationframework for generating informative proximity reports. T heproposed framework contains five main building blocks, namelythe deployment information, RSS model, positioning metricselection, optimization process and management. Among others,we focus on Gaussian process regression (GPR) based RSS modelsand positioning metric computation. The optimal RSS thresholdis found through minimizing the best achievable localizationroot-mean-square-error formulated with the aid of fundamentallower bound analysis. Computational complexity is compared fordifferent RSS models and different fundamental lower bounds.The resulting optimal RSS threshold enables enhanced perfor-mance of new fashioned low-cost and low-complex proximityreport based positioning algorithms. The proposed frameworkis validated with real measurements collected in an office areawhere bluetooth-low-energy (BLE) beacons are deployed.

Index Terms—Gaussian process, indoor positioning, proximityreport, received-signal-strength, threshold optimization.

I. I NTRODUCTION

A. Background

Over the past few years, indoor localization and trackingusing wireless networks has received considerable attentiondue to the ever increasing demand on location-awarenessin various sectors. So far, most of the efforts have beenmade to improve the localization accuracy using advancedtechnologies, for instance statistical sensor fusion [3],ded-icated to optimally fuse different types of position-relatedmeasurements collected from indoor wireless infrastructures(for instance, cellular, wireless fidelity (Wi-Fi) and bluetoothlow-energy (BLE) nodes) and mobile devices. Due to the rapiddevelopment of the beaconing techniques, there is a big trendnowadays to use event-triggered proximity information fordeveloping new-fashioned, low-cost (e.g., less communicationoverhead, smaller database for storage, cheaper deploymentand maintenance) indoor positioning systems.

One way of obtaining a proximity report from the network isto compare an instantaneous RSS value with a tuned thresh-old Pth. A proximity report obtained in such way indicates

This work is funded by the European Union FP7 Marie Curie trainingprogramme on Tracking in Complex Sensor Systems (TRAX) withgrantnumber 607400. This work is an extension of our conference paper [1] and[2].

F. Yin and Y. Zhao and F. Gunnarsson are with Ericsson Research,Linkoping, SE-58330, Sweden. (E-mail: firstname.lastname@ericsson.com).

F. Gustafsson is with the Department of Electrical Engineering, Divisionof Automatic Control, Linkoping University, Linkoping,SE-58183, Sweden.(E-mail: fredrik@isy.liu.se).

whether or not a user equipment (UE) is in proximity of areference network node. Essentially, a proximity measurementcan be treated as a quantized RSS with merely two quantiza-tion levels [3, 4]. Unlike in the conventional paradigm, wherethe UE sends the measured RSS indication values periodicallyto the core network, a proximity report is triggered only whenthe UE’s status changes, for instance when the UE is crossinga border to another service region. Such proximity reportingscheme is beneficial in various ways. Among other benefits,the signaling between the UE and the core network can besignificantly reduced by sending much less frequently 1-bitproximity values instead of 6-8 bits RSS indication values.

In order to explain the concept of proximity based non-cooperative indoor positioning more clearly, we give an illus-trative example in Fig. 1, wherein we assume noise free RSSmeasurements and a simplistic propagation model with whichan RSS threshold corresponds to a circular coverage area inopen space. As we can see from the figure, the service areais divided into several small regions. An instantaneous RSSmeasurement being larger than a predefined threshold impliesthat the UE resides in the corresponding coverage area. Forinstance, the proximity vector, e.g.,[1, 1, 0], indicates that theUE is in the coverage area of the first and the second referencenodes, while outside of the coverage area of the third referencenode. The UE needs to upload the proximity vector only ifthere is status change in the proximity report, for instancefrom[1, 1, 0] to [0, 1, 0], when the UE moved from the marked placesome distance to the right. However, we note that in practicethe RSS measurements are subject to various types of noiseand we resort to a statistical framework for RSS thresholding.

B. Related Work and Our Contributions

In the literature, the proximity based positioning algorithmsare often called coarse grained algorithms or range-free algo-rithms. Since the publication of [4], a plethora of proximitybased positioning algorithms have been proposed, includingthe centroid algorithm [4], the approximate point in triangle(APIT) algorithm [5], the maximum-likelihood estimationbased algorithm [6], the ecolocation algorithm [7], and theiterative learning based algorithm [8], to mention a few. Themajority of the existing work considered large-scale coopera-tive sensor network localization subject to communicationcon-straints. To the best of our knowledge, RSS thresholding wasfirst considered for cooperative localization in [9]. Therein, asingle RSS threshold is optimized so as to limit the number ofthe neighboring senors. In our recent work [1], RSS threshold-ing was considered for non-cooperative, infrastructure-based

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indoor positioning, which can be regarded as a special caseof [9]. But the focus of [1] lies in the overall positioningperformance in a given service area and thorough treatment onthe measurement campaign, RSS modeling, model fitting andparameter calibration, signaling, and performance evaluationusing real data measured from a live network. In this work, weextend [1] to multiple RSS thresholds tuning. The performancemetric to be optimized is selected to be the overall positioningroot-mean-square-error (RMSE) represented in terms of theCramer-Rao bound or Barankin bound [2]. The former boundis suitable to benchmark estimation performance for medium-and large scale sensor networks, while the latter bound is moresuitable to benchmark small scale sensor networks. Moreover,we introduce advanced Gaussian process regression (GPR)based RSS models, perform detailed performance analyses andvalidate the results with more real data. Lastly, we incorporatethe derived fundamental lower bounds and the advanced GPRbased RSS models into RSS thresholds optimization. To give aquick overview, the proposed generic framework for selectinga set of reasonable RSS thresholds for proximity report basedpositioning is given in Figure 2.

C. Paper Organization and Notations

The remainder of this paper is organized as follows:Section II lists the prerequisites for performing the RSSthreshold optimization, including the deployment information,RSS modeling, and selection of an evaluation set of samplepositions. This section corresponds to the first step shown inFig. 2. Section III introduces fundamental lower bounds on theposition estimation. This section corresponds to the secondstep shown in Fig. 2. Section IV introduces a general RSSthreshold optimization procedure. This section corresponds tothe third step shown in Fig. 2. In Section V, we validate theproposed framework experimentally with real RSS measure-ments. Finally, Section VI concludes the paper.

Throughout this paper, matrices are presented with up-percase letters and vectors with boldface lowercase letters.The operator[·]T stands for vector/matrix transpose and[·]−1

stands for the inverse of a non-singular square matrix. Theoperator tr(·) denotes the trace of a square matrix.‖ · ‖stands for the Euclidean norm of a vector and| · | denotes,depending on the context, either the cardinality of a set orthe determinant of a matrix. The operatorE(·) stands for thestatistical expectation. The operatorln(·) stands for the naturallogarithm and log(·) stands for the logarithm to base 10.Further,∇θ denotes the gradient operator.N (µ, σ2) denotesa Gaussian distribution with meanµ and varianceσ2. Theoperator erf(·) stands for the standard Gaussian error function.Lastly, we useIN , 1 and0 to denote an identity matrix of sizeN ×N , a vector of all 1s and a vector of all 0s, respectively.

The readers should not confuse the following notations:p

geographical position,P a collection of geographical posi-tions, p grid position specific for the online Gaussian processregression,P a set of grid positions,p test points for evaluat-ing the Barankin bound,PT transmit power,Pth a vector ofRSS thresholds.

#1

#2

#3[1, 1, 0]

[1, 0, 0]

[0, 1, 0]

[0, 0, 0]

[0, 1, 1]

[0, 0, 1]

d1

d2

d3

Fig. 1. Illustration of proximity report based indoor positioning. In thisexample, three reference nodes are deployed in a service area for positioningpurposes. For simplicity, we assume that the RSS measurements are noisefree and the underlying propagation model fits a linear log-distance model,cf. Section II-B. The coverage radius of each node,di, i = 1, 2, 3, is simply

determined bydi = d0 · 10

Pth,i−Ai

10Bi , where the notations will be explainedlater on. As we can see, the service area is divided into several small regions.An instantaneous RSS measurement being larger than a predefined thresholdimplies that the UE resides in the corresponding coverage area. For instance,the proximity vector, e.g.,[1, 1, 0], indicates that the UE is in the coveragearea of the first and the second reference nodes, while outside of the coveragearea of the third reference node. The UE needs to upload the proximity vectorif and only if there is at least one entry reversing the status, for instance from[1, 1, 0] to [0, 1, 0], when the UE moved from the marked place some distanceto the right.

Step 1: Prerequisites* Deployment Information

* RSS Modeling* Evaluation Set of Positions

Step 2: Fundamental Bound Analysis

Poptth

=optPthf(RSS model,deployment,Pth)

s.t. certain constraints

Step 3: RSS Thresholds Optimization

Fig. 2. Key steps of the proposed RSS thresholds optimization procedure.Herein,Popt

threpresents the optimized RSS thresholds. The connections are

the following. The fundamental bound analysis is based on deploymentinformation and RSS modeling. The outcome of the fundamental boundanalysis is the best achievable RMSE expression. Together with the evaluationset of positions, they are combined to evaluate the overall best achievablepositioning accuracy as a function of the rss threshold.

II. PREREQUISITES

Performing RSS threshold optimization in our work requiresnecessary deployment information, cf. Section II-A, a trainedRSS model, cf. Section II-B, and an evaluation set of samplepositions, cf. Section II-C.

A. Deployment

Throughout this paper, we restrict ourselves to indoor po-sitioning scenarios where a number ofN reference networknodes, such as cellular base stations, BLE beacons, Wi-Firouters, or a combination, are deployed. The reference networknodes are often placed rather uniformly in the surveillanceareaand mounted either on the ceiling or high on the wall to give apanoramic view. The geographical position and the transmis-sion power of each reference network node are assumed to be

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known a priori. In addition to a reference signal, a referencenetwork node may also broadcast information such as sensorID, position, transmit power, network configurations, as wellas a tiny amount of information about the event to be triggered.

B. RSS Model

A generic RSS model can be written as:

ri(p) = mi(p) + ei(p)︸ ︷︷ ︸

ri(p)

+ni, (1)

whereri(p) denotes the RSS observed at any positionp (threedimensional in general) by the UE from theith referencenetwork node,mi(p) denotes the received signal strength dueto path loss as a deterministic function ofp, ei(p) is a positiondependent noise term, andni is a position independent noiseterm. Herein, we useei(p) to represent the noise due to theshadowing effect, which is often position dependent. We usethe noise termni to account for the joint influence of theinterference from other devices, signal absorption from humanbodies, (unsuccessfully removed) small-scale fading, as wellas the background noise. We assumeni to be independent andidentically Gaussian distributed with zero mean and varianceσ2n,i. The sum ofmi(p) + ei(p) denoted byri(p) can be

understood as the clean RSS without noise. Model fitting canbe done either offline or online. We show three examples inthe sequel.

Example I: Offline Linear Least-Squares (LLS) Regres-sion. The underlying received signal strength due to path lossis represented by a deterministic, linear log-distance model asfollows:

mi(p) = Ai + 10Bi log

(di(p)

d0

)

, (2)

whereAi is the received signal strength measured at a ref-erence distanced0, Bi is the path loss exponent, anddi(p)is a short-hand notation of the Euclidean distance betweenp and theith reference network node’s position,pr,i, i.e.,di(p) , ||p − pr,i||. Furthermore, we assume that the shad-owing effect can be simply added to the varianceσ2

n,i of ni.The resulting RSS model can be seen as a special case of (1)and is widely known as log-distance path-loss model [10].

In the offline calibration phase, we are given a training dataset,Di = {(pi,j , ri(pi,j))| j = 1, 2, . . . ,M}. In this example,we perform the well known offline LLS fitting and obtain anestimate of the unknown parameters,θi = [Ai, Bi, σ

2n,i]

T . Itis well known thatθi can be obtained in closed form, see forinstance [3]. Herein and in the subsequent examples, we usethe calibrated parameters to replace the true parameter in theRSS prediction.

Finally, given a new position,p∗, we can predict that theobserved RSS measurement distributes as

ri(p∗) ∼ N (µi(p∗), σ2i (p∗)), (3)

where

µi(p∗) = mi(p∗) = Ai + 10Bi log

(di(p∗)

d0

)

, (4a)

σ2i (p∗) = σ2

n,i. (4b)

Example II: Offline Nonlinear Gaussian Process Regres-sion (GPR). We follow the same path loss function as givenin (2) in the first example. But the error term,ei(p), due tothe large-scale shadow fading, is represented by a zero-meanGaussian process

ei(p) ∼ GP(0, ki(p,p′)), (5)

where we follow the notation of a Gaussian processGP(·, ·)used in [11, Section 2.2]. The covariance (kernel) functionisselected to be

ki(p,p′) , E [ei(p)ei(p

′)] = σ2s,i exp

[−||p− p′||lc,i

]

, (6)

where σ2s,i accounts for the uncertainty introduced by the

shadow fading into the GP model andlc,i denotes the cor-relation distance. It is noted that in this model the noiseterms at two different positions, sayp andp′, are assumedto be spatially correlated according to the well-establishedGudmundson’s model [12]. Similar work but using differentcovariance functions can be found in [13]–[15].

Similar to the first example, we perform offline calibrationof the GPR based RSS model. We start with writing thelikelihood function of the observed RSS measurements asfollows:

p(ri(Pi); θi) ∼ N (mi(Pi),Ci(Pi,Pi)), (7)

with the following notations:

θi , [Ai, Bi, σ2s,i, lc,i, σ

2n,i]

T , (8a)

Pi , [pi,1,pi,2, . . . ,pi,M ], (8b)

ri(Pi) , [ri(pi,1), ri(pi,2), . . . , ri(pi,M )]T , (8c)

mi(Pi) , [mi(pi,1),mi(pi,2), . . . ,mi(pi,M )]T , (8d)

ki(p,Pi) , [ki(p,pi,1), ki(p,pi,2), . . . , ki(p,pi,M )]T ,(8e)

Ki(Pi,Pi) ,

ki(pi,1,Pi)ki(pi,2,Pi)

...ki(pi,M ,Pi)

, (8f)

Ci(Pi,Pi) , Ki(Pi,Pi) + σ2n,iIM . (8g)

The parameters inθi are usually unknown and need to becalibrated. A maximum-likelihood estimate (MLE),θi, isadopted here as an approximation of the underlying param-eters. Detailed derivations are given in Appendix A.

In order to give a training data driven RSS model that takesinto account all noise sources, we compute according to [11]the Gaussian posterior probability of an observed RSS valueat a new positionp∗ by

p(ri(p∗)|Di; θi) ∼ N(µi(p∗), σ

2i (p∗)

), (9)

where

µi(p∗) = kTi (p∗)C

−1i (ri −mi) +mi(p∗) (10a)

σ2i (p∗) = σ2

n,i + σ2s,i − kT

i (p∗)C−1i ki(p∗). (10b)

Note that in (10a) and (10b),ri, mi, ki(p∗), andCi are shortfor ri(Pi), mi(Pi), ki(p∗,Pi), andCi(Pi,Pi).

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The above introduced GPR will be called full GPR in thesequel for the reason that it processes the training data (ofsizeM ) in a batch manner. The corresponding computationalcomplexity scales asO(M3) and the memory requirementscales asO(M2). Some sparse GPR methods have beenproposed to reduce the computational complexity toO(s2M)and the memory requirement toO(sM), wheres (s ≪ M ) isthe size of a set of inducing input points that are sub-sampledsmartly from the original training dataset. A unifying viewof the state-of-the-art sparse GPR methods has been given in[16]. Next, we apply the recent recursive Gaussian processregression method [17] for deriving our online RSS model.

Example III: Online Gaussian Process Regression.Thenotations, if not re-defined, will follow those given for thefull GPR method. For simplicity and easier comparison withthe full GPR, let us imagine that the training data arrives oneby one as time evolves, namely we have a new data point{pi,t, ri(pi,t)}, at each time instancet = 1, 2, . . . ,M .

Similar to the sparse GPR methods, we introduce a setof s grids P = [p1, p2, . . . , ps], representing the desiredtracking area. Herein, we simply assume that all the ref-erence nodes use the sameP. The corresponding RSSobservations at these grids are latent variablesri(P) ,

[ri(p1), ri(p2), . . . , ri(ps)]T . Furthermore, we denoteDg,i ,

{ri(P), P}. For notational brevity in the sequel,ri is shortfor ri(P), and its mean and covariance matrix are denotedby mi and Ki, respectively. Our first aim is to compute theposterior distribution ofri at any time instancet (t ≥ 1)given the training dataDi,1:t , {Pi,1:t, ri(Pi,1:t)}. Herein,the noisy RSS observations are stacked intori(Pi,1:t) ,

[ri(pi,1), ri(pi,2), . . . , ri(pi,t)]T , and the locations are stacked

into Pi,1:t , [pi,1,pi,2, . . . ,pi,t].The main steps of the recursive GPR [17] based RSS model

are summarized as follows:

1) Initialization: Set initial mean vectorµgi,0 , mi and the

covariance matrixCgi,0 , Ki. Compute the inverse of

Ki and store it for later use.2) Recursive Processing: For eacht = 1, 2, . . . ,M , do the

following computations:

Ji,t = ki(pi,t, P)K−1i , (11a)

µpi,t = mi(pi,t) + Ji,t

(µ

gi,t−1 − mi

), (11b)

σ2,pi,t = σ2

s,i + Ji,t

(C

gi,t−1 − Ki

)JTi,t, (11c)

Gi,t =1

σ2n,i +

(σpi,t

)2Cgi,t−1J

Ti,t, (11d)

µgi,t = µ

gi,t−1 + Gi,t

(ri(pi,t)− µp

i,t

), (11e)

Cgi,t = C

gi,t−1 − Gi,tJi,tC

gi,t−1. (11f)

After the recursive processing of the online GPR, cf.(11a)–(11f), we have

p(ri|P,Di) ∼ N(

µgi,M ,Cg

i,M

)

. (12)

3) Prediction: At the end of the training phase, i.e.,t = Massumed in this specific example, the posterior distribu-tion of ri(p∗) at a novel input positionp∗, given Di

andDg,i, can be approximated by

p(ri(p∗)|p∗, P,Di) ≈ N (ri(p∗)|µi(p∗), σ2i (p∗)),

(13)where

µi(p∗)= kTi (p∗)K

−1i (µg

i,M − mi) +mi(p∗), (14a)

σ2i (p∗)=σ2

s,i+σ2n,i+kT

i (p∗)K−1i

(

Cgi,MK−1

i −Is

)

ki(p∗).

(14b)Herein,ki(p∗) is short forki(p∗, P).

Note that the detailed derivations of (11a)–(11f) can befound in [17] and the derivations of (14a) and (14b) are givenin Appendix B. It is easy to verify that the computationalcomplexity scales asO(s3) for evaluatingK−1

i only once inthe initialization step,O(s2) for µ

gi,t and C

gi,t at any time

instancet in the recursive processing step. As compared to thefull GPR method, online GPR method is able to reduce theoverall computational complexity fromO(M3) to O(s2M)with s ≪ M . Although the online GPR method providesthe same overall computational complexity as some sparseGPR methods, it can process new data online with lesscomputational effort. More precisely, when we have a newinput/observation pair{pi,M+1, ri(pi,M+1)} at time M + 1after the training phase, it requires onlyO(s2) complexity tocomputeµg

i,M+1 andCgi,M+1.

In the above RSS modeling procedure, we assumed thatan estimate of the true parameterθi is known prior to theonline GPR. This can be the case when some historical/expertknowledge is available or a small set of the training datacan be used to train the parameters like we did for the fullGP in the second example. Alternatively, [18] demonstratedthat these parameters can be learned online as training datacome in. To summarize, online GPR model is more flexible touse and adaptive to new arrival data. While if the underlyingRSS model is time invariant and the computational cost issecondary, full GPR using all available measurements for bothhyper-parameter optimization and prediction will theoreticallygive the best modeling results.

C. Evaluation Set of Sample Positions

Apart from the deployment information and the RSS model,we need also an evaluation set of sample positions for whichlocalization accuracy will be evaluated. An evaluation setcanbe selected for instance to contain uniform grids or a pluralityof trajectories that cover the area where positioning is ofinterest.

D. Concluding Remarks

This section introduces and compares several RSS models.A good RSS model is able to give more information aboutthe UE position in a proximity measurements, which is es-sentially a hard-thresholded RSS measurement. The Gaussiandistributive profile of the RSS makes the computation of aninformation metric tractable. More details will be given atbeginning of Section III.

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III. F UNDAMENTAL LOWER BOUNDS ONPOSITION

ESTIMATION

This section aims to derive two fundamental lower boundson the proximity report based position estimation mean-square-error (MSE). These are the Cramer-Rao bound (CRB)and the Barankin bound (BB), respectively. It is known thatCRB falls into the category of small error bounds, while BBfalls into the category of large error bounds. Small/Large-errorbound is intended for small/large estimation error [19]. Herein,we assume that all regularity conditions for computing thesetwo bounds are fulfilled.

A. Preliminaries

A proximity report is obtained by hard thresholding an RSSmeasurement, concretely,

ci =

{

0, ri(p) ≤ Pth,i

1, ri(p) > Pth,i

, (15)

whereci is introduced here to denote the proximity informa-tion obtained through comparing a thresholdPth,i with theinstantaneous RSS value,ri, measured atp = [x, y, z]T bythe UE via communicating with theith reference networknode. We setci equal to ‘1’/‘0’when the UE is in/beyondproximity of the ith reference network node. We furtherdefine a vector of RSS measurements received by the UEfrom all reference network nodes asr = [r1, r2, . . . , rN ]T

and define the corresponding vector of proximity reports asc = [c1, c2, . . . , cN ]T .

By taking advantages of the Gaussian distributive profile of(3) for the log-distance model or (9) for the full GPR modelor (13) for the online GPR model, we can easily derive theprobability of observingci at p, i.e.,

Pr{ci;p} =

G(

Pth,i−µi(p)σi(p)

)

, ci = 0

1−G(

Pth,i−µi(p)σi(p)

)

, ci = 1, (16)

where G(·) denotes the Gaussian cumulative distributionfunction (CDF). For notational brevity, we defineGi(p) ,

G(

Pth,i−µi(p)σi(p)

)

. Similarly, we use Pr{c;p} to denote theprobability of observing vectorc at p and due to the assump-tion that the measurements are mutually independent, we havePr{c;p} =

∏Ni=1 Pr{ci;p}. Lastly, we note that in (16)µi(p)

and σi(p) follow the expressions given in (4a) and (4b) forthe linear log-distance model, or (10a) and (10b) for the fullGPR model, or (14a) and (14b) for the online GPR model.

B. Proximity Report Based Position Estimator

We define a proximity based position estimator byp(c). Adirect measure of the estimation error—the mean-square-error(MSE), is very often used for comparing different parameterestimators. The MSE of any proximity based position estima-tor can be expressed as

MSE(p(c)) , EPr{c;p}[

(p(c)− p) (p(c)− p)T]

= b(p)bT (p) + Σ(p), (17)

where the bias,b(p), and the covariance matrix,Σ(p), aredefined, respectively, by

b(p) , EPr{c;p}[p(c)]− p, (18a)

ρ(p) , EPr{c;p}[p(c)] = b(p) + p, (18b)

Σ(p) , EPr{c;p}[

(p(c)−ρ(p)) (p(c)−ρ(p))T]

. (18c)

C. Fundamental Lower Bounds

It is known from the literature, see for instance [19], thatfundamental lower bounds can be placed on the MSE of aparameter estimator by

MSE(p(c)) � b(p)bT (p) + Υ (p)Γ (p)−1Υ (p)T , (19)

where the information matrix,Γ (p), and the translation ma-trix, Υ (p), are defined as follows:

Γ (p) , EPr{c;p}[η(p)η(p)T

], (20)

Υ (p) , EPr{c;p}[(p(c) − ρ(p))η(p)T

], (21)

whereη(p) is some function ofp. Next, we show two wellknown lower bounds that adopt differentη(p).

Cramer-Rao Bound (CRB): When we let

η(p) =

∂∂p

Pr{c;p}Pr{c;p} , (22)

the translation matrix,Υ (p), defined in (21) can be easilyproven to become

Υ (p) =

(∂ρT (p)

∂p

)T

, (23)

and the information matrix,Γ (p), defined in (20) becomes thewell known Fisher information matrix (FIM), namely,

Γ (p) = EPr{c;p}

∂∂p

Pr{c;p}(

∂∂p

Pr{c;p})T

Pr2{c;p}

. (24)

The FIM can be further expressed, due to the independenceassumption on the measurements, as

Γ (p) =

N∑

i=1

EPr{ci;p}

∂∂p

Pr{ci;p}(

∂∂p

Pr{ci;p})T

Pr2{ci;p}

,

N∑

i=1

fi,xx fi,xy fi,xz

fi,yx fi,yy fi,yz

fi,zx fi,zy fi,zz

, (25)

where for any combination ofm,n ∈ {x, y, z},

fi,mn=∑

ci∈{0,1}

∂∂m

Pr{ci;p} · ∂∂n

Pr{ci;p}Pr{ci;p}

=

(∂Gi(p)

∂m· ∂Gi(p)

∂n

)

·(

1

Gi(p)+

1

1−Gi(p)

)

. (26)

The first-order derivatives ofGi(p) for all the three RSSmodels are given in Appendix C. HavingGi(p) as well asits derivatives, we can easily evaluate (25) to get the FIM.

6

Finally, inserting the results ofΥ (p) andΓ (p) into the right-hand-side of (19) gives the CRB.

Barankin Bound (BB): In this case, as will be shown lateron,η(p) and eventuallyΥ (p) andΓ (p) are also functions ofan additional set of test points. This set of test points, denotedas pj , j = 1, 2, . . . ,Mb, must be preselected according toa given UE positionp. This set of test points decides thetightness of the Barankin bound to be derived below.

For the BB, we set

η(p)=

[Pr{c; p1}Pr{c;p} ,

Pr{c; p2}Pr{c;p} , . . . ,

Pr{c; pMb}

Pr{c;p}

]T

−1T . (27)

The translation matrix can be derived based on the results in[19] to be

Υ (p)=[ρ(p1)−ρ(p),ρ(p2)−ρ(p), . . . ,ρ(pMb)−ρ(p)] ,

(28)and the information matrixΓ (p) becomes the Barankin in-formation matrix (of dimensionMb ×Mb), with the (j, k)thentry given by

Γjk(p, pj , pk) = EPr{c;p}

[Pr{c; pj}Pr{c;p}

Pr{c; pk}Pr{c;p}

]

− 1, (29)

=

N∏

i=1

∑

ci∈{0,1}

Pr{ci; pj}Pr{ci; pk}Pr{ci;p}

− 1,

(30)

where the second equation is owing to the independenceassumption on the proximity reports.

In order to give a tight lower bound, the test points needto be carefully selected. The optimal set of test points can befound as[p

opt1 , popt

2 , . . . , poptMb

]= max

p1,...,pMb

Υ (p)Γ (p)−1Υ (p)T . (31)

Unfortunately, this optimization problem is cumbersome inourcase. A much easier way of providing a set of sub-optimal testpoints is to minimize the diagonal elements of the Barankinmatrix, Γjj(p, pj), namely,

ps-optj = argmin

pj

Γjj(p, pj), ∀j = 1, 2, . . . ,Mb. (32)

According to (30), we have

Γjj(p, pj) =

N∏

i=1

∑

ci∈{0,1}

Pr2{ci; pj}Pr{ci;p}

− 1. (33)

We adopt the following procedure to find the test points:

1) Generate a set ofK (with K ≫ Mb) candidate testpoints, pk, uniformly and rather densely in the servicearea.

2) For each candidate test point, evaluateΓkk, k =1, 2, . . . ,K, according to (33) and store them.

3) Find amongΓkk, k = 1, 2, . . . ,K, all the local minimaΓj′j′ and record their indexesj′.

4) Set a selection threshold,ε, search for allj ∈ {1, ..., j′−1, j′ +1, ...,K} that satisfyΓjj − Γj′j′ < ε and usepj

as the final test points. Repeat this for allj′.

The test points selected in the above way should well representthe location ambiguities that a position estimation algorithmgets stuck at more easily. Another big advantage of the BBlies in that the map information can be easily incorporatedinto the test point selection procedure. Lastly, we take theoptimized test pointsps-opt

j , j = 1, 2, . . . ,Mb into (28) and (30)to evaluateΥ (p) andΓ (p), respectively. Similarly, insertingthe results ofΥ (p) and Γ (p) into the right-hand-side of(19) gives the BB. It was shown in [20, Theorem I] that theBarankin bound is an increasing function of the number of testpoints and can converge to certain value when the regularityconditions are fulfilled. We have confirmed this experimentallyin [2].

D. Computational Complexity

Before showing the results, let us first recap the followingnotations:N is the number of reference network nodes,Mthe number of RSS measurements collected at each node,sthe number of grids for the online GP regression,K the totalnumber of candidate test points andMb the final candidatetest points for computing the Barankin bound.

The computation of the CRB requires the first order deriva-tives of Gi(p). From our derivations in Appendix C, thecomputational complexity scales asO(N) for the linear RSSmodel, while scales asO(NM2) for the full GPR model andscales asO(Ns2) for the online GPR model. The computationof the BB does not require the first order derivatives ofGi(p) as in the CRB computation, but it requires a set ofcarefully selected test points. It is easy to derive from theabove subsection that the computational complexity scalesasO(NK) + O(M3

b ). One practical way to suppress the highcomplexity is to makeK andMb relatively small at the costof reduced accuracy of the Barankin bound.

E. Discussions on Bias

In the previous subsection, we derived two fundamentallower bounds, namely the CRB and the BB, on the MSEof any proximity based position estimator. Obviously, it ismuch simpler to compute the two bounds for unbiased positionestimator due to the fact thatb(p) = 0, ∀p in the parameterspace. Two different methods might be used to derive anunbiased, proximity report based position estimator. In thefirst method, an unbiased estimator might be obtained as aby-product when deriving the BB in a more complicatedmanner according to [21]. In the second method, we mightuse dithering techniques to generate dither noise and add itto the proximity measurement before running the maximum-likelihood estimation. We have shown some interesting resultsrecently in [22], [23]. In the sequel, we will focus on unbiasedposition estimator. However, our method also applies for anybiased estimator if the biasb(p) is of known form, whichneeds to be obtained either analytically or replaced by themaximum bias tolerance planned for each position in the worstcase.

IV. RSS THRESHOLDOPTIMIZATION

In Section II and Section III, we listed the prerequisites andinformation criteria that serve as elementary building blocks

7

for RSS threshold optimization. In this section, we aim to givea generic RSS thresholding procedure, followed by discussionson the implementation aspect.

We summarize the results obtained from Section II asfollows:

• We obtained the sensor deployment information, includ-ing for instance the floor plan, reference network nodepositions, transmit power, etc.

• We obtained a calibrated RSS model for each referencenetwork nodei, i = 1, 2, . . . , N , specified by the modelparameters:

1) θi = [Ai, Bi, σ2i ]

T in the linear log-distance model,cf. Example I given in Section II-B; or

2) θi = [Ai, Bi, σ2s,i, lc,i, σ

2n,i] in the nonlinear GPR

model, cf.Example II and Example III given inSection II-B.

• We obtained an evaluation set,X ∗, which contains knownsample positionsp∗

i , [x∗i , y

∗i , z

∗i ]

T that might be con-sidered to be of varied importance quantified by theweighting factorsw∗

i , i = 1, 2, . . . , |X ∗|. The weightingfactors are positive and sum up to one.

Next, we tune the thresholdsPth = [Pth,1, . . . , Pth,N ] oftheN reference nodes. The workings are as follows:

1) For every sample positionp∗i ∈ X ∗, evaluate some

localization accuracy metricf(p∗i ,Pth). Take weighted

average by

f(Pth) =

|X ∗|∑

i=1

w∗i f(p

∗i ,Pth), (34)

where f(Pth) is short for an overall accuracy metricfunction that is dependent of deployment and RSSmodel, andf(p∗

i ,Pth) is short for an accuracy metricfunction that is dependent of the evaluation sampleposition, RSS model and deployment.

2) SolvePoptth = optimize

Pth

f(Pth), subject to certain con-

straints onPth.Lastly, devices that newly appear in the deployment area

will be configured with the optimized reporting threshold,P

optth . The proximity reports can be used directly for position-

ing purposes, see for instance proximity report based particlefiltering in [24], or serve as basis for analyzing user behaviors,such as dwell time and motion patterns. RSS threshold opti-mization is supported by an architecture, and we characterizethe architecture by describing the logical entities performingdifferent steps in the procedure. Furthermore, different nodecandidates are discussed for the logical entities.

Figure 3 shows the signaling chart of our proposed RSSthresholding procedure, where most of the key steps areperformed by acomputation entity. It can also be so thatthe offline processing is performed in aconfiguration entity,and the online processing in a separatefusion entity. Anotherpossibility is that the calibration efforts are performed in adedicatedcalibration entity. These logical entities can be im-plemented separately or jointly in a mobile device, a referencenetwork node, or some other network node. The necessarycommunication in the two phases between a device and logical

ReferenceNetwork Nodes

UserEquipment

ComputationEntity

Offline

Online

Positioning

Reference Signal

RSS Measurements

RSS Modeling

ThresholdOptimization

Threshold Configuration

Reference Signal

RSS Measurement Evaluation

Proximity Report

Fig. 3. Signaling chart of the proposed RSS thresholding procedure.

entity may be via a link with a reference network node, viasome other communication link, or internally in the device.

Next, we provide more details about the implementationof the proposed RSS thresholding procedure, especially theoptimization part.

• There exist many choices on the localization accuracymetric, for instance, the mean-square-error (MSE) andthe localization outage probability of a specific proximitybased position estimator, fundamental lower bounds onthe MSE such as the Cramer-Rao bound (CRB) and theBarankin bound (BB). Bayesian type bounds [25] canalso be used if the evaluation set contains a multiple oftrajectories that follow certain motion model.

• We consider in the remainder of this paper the bestachievable localization RMSE of unbiased, proximityreport based position estimator, which is formulated interms of the CRB or the BB derived in Section III. Moreprecisely, we have

f(p∗i ,Pth) ,

√

tr(CRB(p∗i ,Pth)) (35)

orf(p∗

i ,Pth) ,√

tr(BB(p∗i ,Pth)), (36)

where both the CRB and the BB on the estimation ofp∗i are computed in a snapshot manner for every sample

position in the evaluation set.• Practical constraints on the RSS thresholds may take

into account the fact that the RSS thresholds,Pth,i,∀i = 1, 2, ..., N , are integer valued (due to quantization)and contained in an interval, say[Pmin

th,i, Pmaxth,i ], due to the

sensor reading limits.• Originally in [1], we set identical RSS thresholdsPth,1 =

Pth,2 = . . . = Pth,N = Pth, test every possible RSS

8

threshold candidate, and choose the one that minimizesthe cost function. While this method does not applyto multiple RSS thresholds optimization due to the ex-ponentially increased computational complexity as wasexplained in [1]. To sidestep this complex combinatorialproblem, we first relax the integer constraint and resortto numerical optimization method to solve for an locallyoptimal set of RSS thresholds,Popt

th , from the minimiza-tion problem. The result is then rounded to the nearestfeasible integer.

• The cost function that minimizes the best achievablelocalization RMSE in the above example is not ensuredto be convex in terms of the RSS thresholds. A numericaloptimization method may stuck at local optimum.

• For the case that some reference nodes malfunction,principally, we need to redo RSS thresholding for theremaining nodes. While for the case that more referencenodes are to be added in a calibrated network, we need toconsider where to place them in the first place. Intuitively,it is more meaningful to install them in the area where thelocalization accuracy is relatively poor. Besides, we needto collect RSS measurements and train the RSS model pa-rameters for those newly installed reference nodes beforewe can redo RSS thresholds optimization. For large-scalesensor network, however, this is prohibitively expensive.In order to maintain similar performance with reasonablecomputation time, we can redo the optimization only forthe ‘neighbors’ of the removed nodes in the former caseor jointly for the newly added nodes and their ’neighbors’in the latter case.

V. EXPERIMENTAL VALIDATION

In Section IV, we have shown the whole procedure ofoptimizing RSS threshold(s) for enhanced overall localizationperformance. In what follows, we will validate our idea exper-imentally using real RSS measurements collected in an indoorbluetooth-low-energy (BLE) network. There is a big trendnowadays to use BLE networks for Internet of Things (IoT)due to the low power consumption and efficient monitoringcapabilities of the BLE devices [26]. The optimized RSSthresholds not only lead to novel proximity based position-ing methods but also enable user equipment (UE) assistedmeasurement to support handover, for instance, in WidebandCode Division Multiple Access (WCDMA) and Long-TermEvolution (LTE) systems [27], [28].

A. Sensor Deployment, Measurement Campaign and ModelFitting

Our experiments were conducted in a typical office envi-ronment at Ericsson Research, Linkoping, Sweden. In totalN = 12 BLE beacons are placed uniformly in the area.The floor plan as well as the known beacon positions areillustrated using a local two-dimensional (2-D) coordinatesystem in Fig. 4. The BLE beacons serve as transmitters andbroadcast data packages regularly. The transmit power is setto PT = −58 dBm identically for all BLE beacons. Moderatescale measurement campaign was conducted during normal

0 5 10 15 20 25 30 35 40 45 50 550

5

10

15

20

25

30BLE BeaconsGrid PointsEvaluation Points

#12

#1

#11

#2

#3

#8

#5

#10

#4

#7

#9

#6

x-position, in meter

y-p

ositi

on,

inm

ete

r

Fig. 4. Illustration of the floor plan where in total 12 BLE beacons aredeployed, the preselected grids for online GPR modeling andthe preselectedevaluation set of test positions for RSS threshold optimization.

TABLE I3-D POSITIONS(UNIT IN METER) OF 12 BLE BEACONS, THE AMOUNT OF

RSSMEASUREMENTS(UNIT IN SAMPLES) COLLECTED BY EACH BLEBEACON AS THE OUTCOME OF THE MEASUREMENT CAMPAIGN AND THE

CALIBRATED LINEAR LOG-DISTANCE MODEL PARAMETERS,A,B, σ (UNITIN dB).

ID Position (x, y, z) # RSS Data Parameters (A, B, σ)# 1 (2.27, 24.26, 2.60) 1583 (-57.6, -2.5, 6.9)# 2 (16.47, 20.34, 2.35) 2898 (-68.3, -1.8, 7.2)# 3 (14.83, 11.48, 0.71) 2675 (-68.0, -1.9, 6.5)# 4 (30.89, 20.36, 2.35) 3845 (-60.8, -2.3, 7.5)# 5 (29.13, 14.90, 2.54) 4295 (-60.3, -2.4, 6.9)# 6 (49.00, 20.34, 2.35) 1179 (-66.5, -1.9, 6.7)# 7 (37.74, 4.10, 2.60) 2438 (-55.7, -2.6, 6.5)# 8 (18.82, 4.00, 2.60) 642 (-73.0, -1.2, 5.9)# 9 (45.51, 13.30, 2.25) 3014 (-61.9, -2.0, 7.0)# 10 (29.66, 4.10, 2.60) 3024 (-57.6, -2.6, 6.1)# 11 (5.32, 14.81, 2.35) 955 (-60.6, -2.6, 7.2)# 12 (8.00, 20.32, 2.60) 1666 (-63.7, -1.9, 6.5)

work hours. Throughout the measurement campaign, the UEwalked along various predefined tracks and measured a totalnumber of28214 RSS measurements from the BLE beacons.For simpler Gaussian process regression model, the trackpositions are assumed to be precisely known.1 The obtainedRSS measurements were eventually uploaded to a computationentity via Wi-Fi for RSS model fitting and threshold opti-mization. More details about the sensor deployment and theoutcome of the measurement campaign are given in Table I.

We consider three RSS models, namely the linear log-distance model, the nonlinear full GPR model and the onlineGPR model, respectively. The linear model fitting follows thesteps given in the first example of Section II and the calibratedmodel parameters are given in Table I. These parameters serveas our prior knowledge when choosing the initial guessesof the full GPR model parameters. The nonlinear full GPRmodel is learned with 600 RSS measurements randomlysampled from the full dataset. Herein, we only choose 600measurements so that too long overall simulation time can beavoided for full GPR model calibration and RSS thresholds op-

1The track positions were essentially obtained from an app-based position-ing algorithm developed by SenionLab and are supposed to be subject to zeromean Gaussian error with covariance matrixΣp = σ2

pI2, where the standarddeviation (std) is identical for bothx- andy-direction and varies moderatelyfrom 10−4 to 2.6 meter for different positions.

9

(a)

(b)

Fig. 5. Illustration of the training data set (600 samples marked by black+) and the calibrated full GPR model for the 4th BLE beacon: (a)depictsthe posterior mean of (9) and (b) depicts the posterior variance of (9).

timization. We show both the posterior mean and the posteriorcovariance for the fourth BLE beacon in Fig. 5. In contrast,due to the significantly reduced computational complexity,weuse the full dataset to train the online GPR model and useit in the RSS threshold optimization. Herein, a number of 39grids are predefined in the online GPR model and they aredepicted in Fig. 4. In Fig. 6, we show similar results for theonline GPR model. When comparing Fig. 5 with Fig. 6, it isobserved that posterior mean map of the online GPR modelis richer in context than that of the full GPR model. Thismight be due to larger number of measurements used in themodeling. However, the posterior covariance is accurate onlyat the predefined grids and needs to be interpolated at otherpoints. Hence, if the evaluation set contains a lot of samplepositions that are far from the grid points, we will probablysuffer from higher approximation error.

Lastly, we note that due to space limitation, we are unableto show the posterior mean and covariance for other beacons.The results for other BLE beacons are similar. It is clear fromFig. 5 that the energy pattern is not isotropic. To the left sideof the fourth BLE beacon is concrete wall and the energy ishighly absorbed. While to its right side, the energy is muchless absorbed owing to the wooden wall. These explain whywe did not see a uniform, isotropic energy pattern. This is theadvantage of the GP model over empirical models as the dataspeak for the model.

(a)

(b)

Fig. 6. Illustration of the training data set (3845 samples marked by black+) and the calibrated online GPR model for the 4th BLE beacon: (a) depictsthe posterior mean of (13) and (b) depicts the posterior variance of (13).

B. RSS Threshold Optimization

Before performing RSS threshold optimization, we firstgenerate a setX ∗ of 100 evaluation positions,p∗

i , uniformlyin the deployment area, as shown in Fig. 4.2 The weightingfactors are set equally asw∗

i = 1/|X ∗| for all evaluationpositions inX ∗.

As the localization accuracy metric, we compute the bestachievable RMSE either in terms of the CRB or the BB.Herein, we assume that thez-component of all sample po-sitions is fixed to 1.3 meter and knowna priori. Thus, weonly concern about position estimation inx- andy-directions.As a consequence,f(p∗

i ,Pth) boils down tof([x∗i , y

∗i ],Pth).

Our focus is to optimize a RSS threshold individually for eachBLE beacon and compare the overall localization performancewith that led by a global RSS threshold. We use the MATLABfunctionfminconto solve the multi-variate minimization prob-lem with the cost function given in (34) and the constraints−100 dBm≤ Pth,i ≤ −70 dBm, i = 1, 2, . . . , 12. Afterwards,we round the optimized RSS thresholds,P

optth to the nearest

integers. The starting point is selected to bePth,1 = Pth,2 =. . . = Pth,12 = −90 dBm. The number of test points,Mb, forcomputing the BB is set to 25 to save computational time.

We depict the multiple RSS thresholds optimized individ-ually for each BLE beacon versus the global RSS threshold

2In contrast to [1], we used here a smaller number of evaluation positionsthat well represent the deployment area. The final results donot differ much,but the computation cost can be reduced significantly, especially for the BBbased RSS thresholds optimization.

10

1 2 3 4 5 6 7 8 9 10 11 12-90

-85

-80

-75multiple Pth + linear modelsingle Pth + linear modelmultiple Pth + full GPR modelsingle Pth + full GPR modelmultiple Pth + online GPR modelsingle Pth + online GPR model

sensor index

RS

Sth

resh

olds

,in

dBm

(a)

1 2 3 4 5 6 7 8 9 10 11 12-90

-85

-80

-75multiple Pth + linear modelsingle Pth + linear modelmultiple Pth + full GPR modelsingle Pth + full GPR modelmultiple Pth + online GPR modelsingle Pth + online GPR model

sensor index

RS

Sth

resh

olds

,in

dBm

(b)

Fig. 7. Illustration of multiple RSS thresholds optimized individually foreach BLE beacon versus the global RSS threshold optimized for all BLEbeacons: (a) CRB based (b) BB based.

optimized for all beacons based on the CRB in Fig. 7(a) andbased on the BB in Fig. 7(b). To give better a vision, we didnot use the rounded values in these figures. Interestingly, theaverage of the multiple RSS thresholdsP opt

th , 1N

∑Ni=1 P

optth,i

is very close to the global RSS threshold in both cases. Theglobal RSS threshold does not differ much for different com-binations of RSS model and lower bound. The improvementbecomes more obvious when we decrease the number of BLEbeacons in the network. For instance, when we merely useBeacon 5, 6, 12, the difference will be around one meter.In these figures, it is also obvious that advanced GPR basedRSS models help improve the positioning accuracy about twometers. The positioning performance is comparable for the twoGPR models, but the computational time of the online GPRmodel is about 50 times faster than that of the full GPR model.Furthermore, we show the overall best achievable RMSEfor different combinations of RSS model and lower boundin Fig. 8. Only modest performance improvement has beenobtained by using the RSS threshold optimized individuallyfor each beacon. However, much more computational time has

Linear full GPR online GPR5.5

6

6.5

7

7.5

8

8.5

multiple Pth + linear modelsingle Pth + linear modelmultiple Pth + full GPR modelsingle Pth + full GPR modelmultiple Pth + online GPR modelsingle Pth + online GPR model

RSS model

Ove

rall

RM

SE

,in

me

ter

(a)

Linear full GPR online GPR4.5

5

5.5

6

6.5

7

7.5

8

8.5

multiple Pth + linear modelsingle Pth + linear modelmultiple Pth + full GPR modelsingle Pth + full GPR modelmultiple Pth + online GPR modelsingle Pth + online GPR model

RSS model

Ove

rall

RM

SE

,in

me

ter

(b)

Fig. 8. Illustration of the overall best achievable RMSE using multiple RSSthresholds optimized individually for each BLE beacon versus the global RSSthreshold optimized for all BLE beacons (a) CRB based (b) BB based.

also been observed when optimizing multiple RSS thresholds.The RMSE led by the BB based RSS thresholds is higher thanthe CRB based counterparts. Similar to the conclusion drawnin [2], the BB is less optimistic about the mean-square-errorof proximity based position estimator than the CRB.

Next, we aim to show the improvement made when usingan optimal RSS threshold as compared to a suboptimal one.Herein, we simply consider the global RSS threshold forthree different RSS models. In Fig. 9, we depict the overallbest achievable localization RMSE as a function of the RSSthresholdPth, which ranges from -100 dBm to -70 dBm withan increment 1 dBm. As is shown in the figure, the minimumis achieved at -83 dBm, -84 dBm, and -84 dBm respectivelyfor the three different RSS models. The optimal thresholdscoincide with those given in Fig. 7(a) if they are rounded tothe nearest integer therein. The corresponding RMSE valuescoincide with those shown in Fig. 8(a) for the CRB case. It isnot surprising to see the convex profile of the RMSE curvesin all cases. The reason is that too large or too small thresholdgives very little information about an unknown location. We

11

-100 -95 -90 -85 -80 -75 -700

10

20

-100 -95 -90 -85 -80 -75 -700

10

20

-100 -95 -90 -85 -80 -75 -700

10

20Ove

rall

RM

SE

,in

me

ter

RSS threshold candidates, in dBm

Fig. 9. Overall best achievable localization RMSE (in termsof CRB) versusthreshold candidates for the linear log-distance model in subfigure-I, nonlinearfull GPR model in subfigure-II, and nonlinear online GPR model in subfigureIII, respectively.

have explained this with a toy example in the introduction. Aformal proof of the convexity of our cost function might beinteresting for our future work.

Lastly, we demonstrate a target tracking example using thetrained RSS thresholds. More precisely, particle filteringhasbeen applied to obtain the position estimates for a specifictrajectory as shown in Figure 10. The trajectory is selectedfrom one of the52 predefined tracks. It is noted that with thefull GPR-based RSS measurement model, the best positioningperformance has been achieved. While with the full GPR-based proximity measurement model, less accurate positionestimates have been obtained. Nevertheless, the two corridorson this floor can be successfully distinguished in all cases.This is important for determining pedestrian flow required invarious indoor applications. To compare the statistical per-formance of the different measurement models and differentthresholding algorithms, the cumulative distribution function(CDF) of the estimation errors are compared in Figure 11. Itis clear that the median positioning error for RSS model isaround 1.3 meter. For proximity model with global threshold,the median error is approximate 2.3 meter and for proximitymodel with optimized thresholds from CRB and BB, there isaround 0.3 meter improvement. With the optimized thresholdsfrom BB, the position accuracy has been improved for mostof the cases, while for around20% of the cases large positionerrors have been observed.

VI. CONCLUSIONS

We have introduced a generic framework of RSS thresholdoptimization for indoor sensor networks. With the aid of thisframework, we can obtain the most informative proximityreport by means of optimizing the overall localization accuracyin a deployment area. Two pivotal building elements of thiswork are the RSS model and the fundamental lower bound.We have tested both the full GPR based model and anonline GPR based model. The computational complexity oftraining these two models scales asO(M3) and O(s2M),

[m]0 10 20 30 40 50 60

[m]

0

5

10

15

20

25

30

35

Ground truth

Position estimates: RSS

Position estimates: Proximity with optimized thresholds from CRB

Position estimates: Proximity with optimized thresholds from BB

Position estimates: Proximity with global threshold

Fig. 10. Illustration of the estimated positions of an exemplary track.

Position error in meter0 1 2 3 4 5 6 7 8 9

CD

F

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

RMSE: RSS RMSE: Proximity Global thresholdRMSE: Proximity with optimized thresholds from CRB RMSE: Proximity with optimized thresholds from BB

Fig. 11. CDF of the positioning error of an exemplary track.

respectively. As our localization accuracy metric, the bestachievable positioning RMSE is expressed either in terms ofthe Cramer-Rao Bound or the Barankin Bound. The compu-tational complexity of the CRB scalesO(NM2) for the fullGPR model andO(s2N) for the online GPR model. The com-putational complexity of the BB scales asO(NK)+O(M3

b ).Large number of test points for computing the BB is hencecomputationally prohibitive. We tested different combinationsof RSS model and lower bound in various experiments withreal data collected from a live BLE network deployed atEricsson Research. We compared the performance of usingdifferent thresholds for different reference nodes and using asingle threshold for all reference nodes. The results confirmedthat the former modality outperforms the latter, but for largescale sensor network the improvement can be negligible.

APPENDIX A

The maximum-likelihood estimate of the GPR model pa-rameters,θi, can be obtained by maximizing the Gaussianprior likelihood function, cf.(7), with respect toθi, which isequivalent to

argminθi

g(θi) , (ri −mi)TC−1

i (ri −mi) + ln |Ci|. (37)

For clarity, we note that the optimization variables are in themean vectormi and covariance matrixCi. More precisely,

12

Ai andBi are in the mean vectormi, while σ2s,i, σ

2n,i andlc,i

are in the covariance matrixCi.Various existing numerical methods can be adopted to

solve this minimization problem, such as the limited-memoryBFGS (LBFGS) quasi-Newton method [11] and the conjugategradient (CG) method. Herein, we adopt the former methodwhich requires the first-order derivatives of the cost function,g(θi). Equation (37) is essentially convex with respect to theparameters. Similar proof can be found in [29, Section 4.8.2].Due to space limitation, we only give the results as follows:

∂g(θi)

∂Ai

= ai(C−1

i + (C−1i )T

)(ri −mi) (38a)

∂g(θi)

∂Bi

= bi

(C−1

i + (C−1i )T

)(ri −mi) (38b)

∂g(θi)

∂σ2s,i

= tr

{[

C−1i −

(C−1

i (ri −mi))(·)T

] ∂Ci

∂σ2s,i

}

(38c)

∂g(θi)

∂lc,i= tr

{[

C−1i −

(C−1

i (ri −mi))(·)T

]∂Ci

∂lc,i

}

(38d)

∂g(θi)

∂σ2n,i

= tr

{[

C−1i −

(C−1

i (ri −mi))(·)T

] ∂Ci

∂σ2n,i

}

, (38e)

where

ai ,∂(ri −mi)

T

∂Ai

= −1T , (39a)

bi ,∂(ri −mi)

T

∂Bi

= −10 · [log di,1, . . . , log di,M ] ,

(39b)[

∂Ci

∂σ2s,i

]

j,k

=

{

1, j = k

exp[−||pi,j−pi,k||

lc,i

]

, j 6= k(39c)

[∂Ci

∂lc,i

]

j,k

=

{0, j = k

σ2s,i exp

[−||pi,j−pi,k||

lc,i

]||pi,j−pi,k||

l2c,i, j 6= k

(39d)

∂Ci

∂σ2n,i

= IM . (39e)

Here we use(A)(·)T to denote(A)(A)T for brevity.

APPENDIX B

Imagine thatDg,i = {P, ri} is also a training data setdespite thatri is latent. Given a novel input,p∗, the posteriordistribution of observing a noisy RSSri(p∗), givenDg,i, canbe easily obtained as follows:

p(ri(p∗)|p∗,Dg,i) ∼ N(

µpi,g,

(σpi,g

)2)

, (40)

where

µpi,g = ki(p∗, P)T K−1

i (ri − mi) +mi(p∗) (41a)

σ2,pi,g = σ2

s,i + σ2n,i − ki(p∗, P)T K−1

i ki(p∗, P). (41b)

The posterior distribution ofri(p∗), givenDi and P, can becomputed analytically via the following marginalization:

p(ri(p∗)|p∗,Di, P)=

∫

p(ri|P,Di)p(ri(p∗)|p∗,Dg,i,Di)dri,

(42)and approximated with reduced computational complexity, likein [30], by

p(ri(p∗)|p∗,Di, P)≈∫

p(ri|P,Di)p(ri(p∗)|p∗,Dg)dri.

(43)Since bothp(ri(p∗)|p∗,Dg,i) and p(ri|P,Di) are Gaussiandistributed, applying Lemma A.1 in [31] yields eventually(14a) and (14b).

APPENDIX C

Log-distance Model: The first-order derivatives ofGi(p)can be derived for anym ∈ {x, y, z} as follows:

∂Gi(p)

∂m≡ ∂

∂m

{1

2erf

(Pth,i−µi(p)√

2σi

)}

=−10Bi√2πσi ln 10

exp

[(Pth,i−µi(p))

2

−2σ2i

]m−mr,i

||p−pr,i||2.

(44)

Inserting the above result into (26) and performing somealgebraic manipulations yields

fi,mn = γi ·(m−mr,i)(n− nr,i)

||p− pr,i||4, (45)

whereγi is a variable in terms of the calibrated RSS modelparameters and the RSS threshold, more precisely,

γi =200B2

i

πσ2i ln

2(10)·exp

[−(Pth,i−µi(p))

2

σ2

i

]

1− erf2(

Pth,i−µi(p)√2σi

) . (46)

Full GPR model: The first-order derivatives can be derivedfor the full GPR model with a bit more efforts as follows:

∂Gi(p)

∂m≡ ∂

∂m

{1

2erf

(Pth,i − µi(p)√

2σi(p)

)}

= γi ·[

∂ui(p)∂m

vi(p) − ∂vi(p)∂m

ui(p)

v2i (p)

]

, (47)

where

γi =1√πexp

[

− (Pth,i − µi(p))2

2σ2i (p)

]

, (48a)

ui(p) = Pth − µi(p), (48b)

vi(p) =√2σi(p), (48c)

∂ui(p)

∂m= −∂µi(p)

∂m, (48d)

∂vi(p)

∂m=

1√

2σ2i (p)

∂σ2i (p)

∂m. (48e)

13

In this caseµi(p) and σ2i (p) are given in (10a) and (10b).

Furthermore, their derivatives are calculated as follows:

∂µi(p)

∂m=

∂ki(p,Pi)

∂mC−1

i (yi −mi) +10Bi

ln 10

m−mr,i

||p− pr,i||2,

(49a)

∂σ2i (p)

∂m= −∂ki(p,Pi)

∂m

(

C−1i +

(C−1

i

)T)

ki(p,Pi),

(49b)

where ∂ki(p,Pi)∂m

is a vector of sizeM × 1 with the jth entrygiven by[∂ki(p,Pi)

∂m

]

j

=−σ2

s,i

lc,iexp

[−||p− pi,j ||lc,i

](m−mi,j)

||p− pi,j ||.

(50)Having the results (10a), (10b), (49a) and (49b), we can theneasily evaluateui(p), vi(p),

∂ui(p)∂m

, ∂vi(p)∂m

via (48), andeventually the derivative ofGi(p) via (47).

Online GPR model: The derivations are similar to thoseshown above for the full GPR model. In this caseµi(p) andσ2i (p) are given in (14a) and (14b). Similarly, their derivatives

are calculated as follows:

∂µi(p)

∂m=

∂ki(p, P)

∂mK−1

i (µgi,M − mi) +

10Bi

ln 10

m−mr,i

||p− pr,i||2(51a)

∂σ2i (p)

∂m= −∂ki(p, P)

∂m

(

K−1i +

(K−1

i

)T)

ki(p, P), (51b)

where ∂ki(p,P)∂m

is a vector of sizes × 1 with the jth entrygiven by[∂ki(p, P)

∂m

]

j

=−σ2

s,i

lc,iexp

[−||p− pj ||lc,i

](m−mj)

||p− pj ||. (52)

Having the results (14a), (14b), (51a) and (51b), we can theneasily evaluateui(p), vi(p),

∂ui(p)∂m

, ∂vi(p)∂m

via (48), andeventually the derivative ofGi(p) via (47).

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