Applying Rprop Neural Network for the Prediction of the Mobile Station Location

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1 Sensors 0,, ; do:0.3390/s OPE ACCESS sensors ISS Communcaton Applyng Rprop eural etwork for the Predcton of the Moble Staton Locaton Chen-Sheng Chen, * and Jum-Mng Ln Department of Informaton Management, Tanan Unversty of Technology, o. 59, Jhongjheng Rd., YongKang Dst., Tanan Cty 700, Tawan Department of Communcaton Engneerng, Chung-Hua Unversty, o. 707, Sec., WuFu Rd., Hsnchu, Tawan; E-Mal: jmln@chu.edu.tw * Author to whom correspondence should be addressed; E-Mal: t0043@mal.tut.edu.tw; Tel.: ext. 5038; Fax: Receved: 5 January 0; n revsed form: 7 March 0 / Accepted: Aprl 0 / Publshed: 8 Aprl 0 Abstract: Wreless locaton s the functon used to determne the moble staton (MS) locaton n a wreless cellular communcatons system. When t s very hard for the surroundng base statons (BSs) to detect a MS or the measurements contan large errors n non-lne-of-sght (LOS) envronments, then one need to ntegrate all avalable heterogeneous measurements to ncrease the locaton accuracy. In ths paper we propose a novel algorthm that combnes both tme of arrval (TOA) and angle of arrval (AOA) measurements to estmate the MS n LOS envronments. The proposed algorthm utlzes the ntersectons of two crcles and two lnes, based on the most reslent back-propagaton (Rprop) neural network learnng technque, to gve locaton estmaton of the MS. The tradtonal Taylor seres algorthm (TSA) and the hybrd lnes of poston algorthm (HLOP) have convergence problems, and even f the measurements are farly accurate, the performance of these algorthms depends hghly on the relatve poston of the MS and BSs. Dfferent LOS models were used to evaluate the proposed methods. umercal results demonstrate that the proposed algorthms can not only preserve the convergence soluton, but obtan precse locaton estmatons, even n severe LOS condtons, partcularly when the geometrc relatonshp of the BSs relatve to the MS s poor. Keywords: tme of arrval (TOA); angle of arrval (AOA); back-propagaton nseural network (BP); reslent back-propagaton (Rprop)

2 Sensors 0, 408. Introducton The problem of poston determnaton of a moble user n a wreless network has been studed extensvely n recent years. It has receved sgnfcant attenton and varous locaton dentfcaton technologes have been proposed n the past few years. A recent report and order ssued by the U. S. Federal Communcatons Commsson (FCC) n July 996 requres that all wreless servce provders provde the locaton nformaton to emergency 9 (E-9) publc safety servces. The separate accuracy requrements of the E-9 mandate were set for network-based technologes: wthn 5 meters for 67 percent of calls, and wthn 300 meters for 95 percent of the calls. To date, satsfyng the FCC accuracy requrement s very dffcult. Most papers and ther algorthms could not acheve ths goal. The varous technques proposed nclude sgnal strength (SS), angle of arrval (AOA), tme of arrval (TOA) and tme dfference of arrval (TDOA). The sgnal strength scheme uses a known mathematcal model to descrbe the path loss attenuaton wth dstance. A fuzzy logc technque wth a geometrcal soluton was appled to calculate range estmates through sgnal strength measurements []. The AOA scheme estmates the sgnal drecton of arrval [], to derve the moble staton (MS) locaton, by usng ether a drectve antenna, or an antenna array leadng to multple lnes of poston. The TOA locaton scheme measures the propagaton tme for a rado wave to travel between the MS and a base staton (BS). The TDOA scheme determnes the poston of MS by examnng just the dfference n tme from MS to multple BSs, rather than the absolute arrval tme. Dfferent potental applcatons of wreless locaton servces have been well developed, ncludng the E-9 subscrber safety servces, locaton-based bllng, fleet management and ntellgent transportaton system (ITS) [3]. One crtcal problem n wreless locaton systems s the non-lne-of-sght (LOS) propagaton effect. A common requrement for hgh locaton accuracy s the presence of a lne-of-sght (LOS) path between the MS and each partcpatng BS. Due to the sgnal reflecton or dffracton between MS and BSs, LOS errors can sgnfcantly mpact wreless locaton performance. Extensve research on LOS effect mtgaton for locaton estmaton have been carred out n the past few years. Snce n LOS the delay has a hgher varance than under LOS condtons, [4] proposed a decson framework to detect LOS BSs va tme seres of estmates. An algorthm was proposed n [5] for TOA systems to mtgate LOS effects by applyng weghts whch are nversely proportonal to ther resduals for all possble BS combnatons, then the LOS BS wth a larger resdual has a lower effect on the MS locaton estmaton. Smlar resdual schemes were proposed for both AOA systems n [6] and TDOA systems n [7]. The algorthms descrbed n [5-7] perform well provded there are many avalable BSs beng LOS wth the MS n the system. Otherwse, they cannot mprove the locaton accuracy. Based on the LOS stuaton and how much a pror knowledge of the LOS error s avalable, dfferent LOS dentfcaton and correcton algorthms for moble user locaton are proposed [8]. Another major concern that affects the choce of locaton scheme to deploy n cellular communcaton systems s hearablty [9]. Here, hearablty s the ablty to receve sgnals from a suffcent number of BSs smultaneously at a suffcent power level [0]. It s dffcult for the MS to receve sgnals from neghborng BSs n cellular communcaton systems, and the hearablty s poor due to near-far effect and multple access nterference. [] ndcated the respectve sgnal strength thresholds clearly show that the coverage n rural areas s much less than n urban areas. The lkelhood of fndng three BSs wth receved sgnal strength ndcaton stronger than 00 db s

3 Sensors 0, 409 only 35 percent n rural areas, whereas t s about 84 percent n urban areas. It s dffcult for an MS to detect three or more BSs for locaton purposes n rural areas. It s evdent that the hearablty n an IS-95 system s extremely poor []. The number of BSs that can be heard by a MS s only one f the MS s near ts servng BS. Two or three BSs can be heard only at the edge of a cell. The nsuffcent number of avalable BSs lmts the locaton-based servces and mpedes the mplementaton of locaton systems. Due to poor hearablty, t s reasonable to consder the hybrd methods by ntegratng two or more schemes. Comparng tme-based methods wth angle-based categores, both have ther own advantages and lmtatons. In general, the AOA (angle-based) scheme requres only two BSs for locaton estmaton, but t s necessary to deploy antenna array at BS for the AOA to work properly. In the case of tme-based methods such as TOA and TDOA schemes, they requre at least three BSs to be precsely located for a -D locaton estmaton [3], nevertheless, they offer better accuracy than those of angle-based AOA schemes. The hybrd algorthm n [4] combnng a TDOA technque wth an addtonal AOA at the servng BS, can offer more accurate estmaton n small error condtons. A hybrd TOA/AOA algorthm n [5], based on a nonlnear constraned optmzaton, can locate the MS even n LOS envronments for the case of three partcpatng BSs. A hybrd range/range dfference algorthm was used to estmate MS locaton n a global system for moble communcatons (GSM) when only two BSs are avalable and the MS s located at the center of mass of the servng cell [6]. There s always an ambguty n the MS locaton f only two TOA measurements are used. In order to resolve the ambguty when an MS can be heard by only two BSs, both TOA and AOA nformaton are requred. Hence, we have proposed the hybrd TOA/AOA postonng methods for such hearablty-constraned condtons n [7]. The MS s located at the varous ntersectons of two crcles and two lnes, because the LOS error always appears as a postve bas n the TOA measurements. Snce the LOS range errors are always postve, TOA measurements are greater than the true values. Therefore the true MS locaton should be nsde the regon enclosed by the overlap of the two crcles. The above ntersectons are defned as feasble ntersectons. By usng two AOA measurements to elmnate the least lkely feasble ntersecton, the geometrcal postonng methods are based on the weghted sum of the remanng feasble ntersectons enclosed by two TOA crcles and two AOA lne. These methods wth dfferent weghtng algorthms can effectvely elmnate the LOS errors and provde more accurate postonng. Artfcal neural network (A) s an nformaton processng method nspred by the bologcal nervous system, whch can approxmate nonlnear functons based on data sets. The system employs a set of actvaton functons and nput-output of sample patterns that do not requre a pror selecton of a mathematcal model. The back-propagaton neural network (BP) s currently the most representatve learnng algorthm n A [8], and has been successfully appled to a wde range of scentfc areas, especally n applcatons nvolvng forecastng, mage processng, pattern recognton and sgnal processng, and many others. BP contnuously adjusts a set of weghts of nputs and ther correspondng outputs for the connectons n the network to produce a mappng from nput vectors to output vectors. It s an teratve algorthm usng the gradent steepest descent method to mnmze the error between the network actual output and desred output. Ths s partcularly useful for those problems wth an unknown optmal algorthm. The major drawbacks of the tradtonal BP are ther slow learnng process and have a tendency to be trapped nto a local mnmum.

4 Sensors 0, 40 Reslent back-propagaton (Rprop) s the best algorthm n terms of convergence speed, accuracy as well as robustness wth respect to the tranng parameters [9]. The Rprop s a local adaptve learnng algorthm, the basc dea s to elmnate the harmful nfluence due to the weght step sze of the partal dervatve. Comparng to the back-propagaton algorthm, the Rprop converges faster and needs less tranng. In ths paper, we propose the algorthm based on Rprop to estmate MS locaton f both TOA and AOA measurements are smultaneously avalable from two BSs. We proposed two types for nput data collecton. The frst type (dvded type) establshes dfferent nput data subsets accordng to the number of the remanng feasble ntersectons. The second type (composte type) s to make the remanng feasble ntersectons n order. Durng the tranng perod, the Rprop s traned to establsh the nonlnear relatonshp between the remanng feasble ntersectons and the MS locaton. The remanng feasble ntersectons become the nput data after tranng, and are passed through the traned Rprop to estmate the MS locaton. Smulaton results show that the proposed algorthm always performs consstently better than Taylor seres algorthm (TSA) [0,], hybrd lnes of poston algorthm (HLOP) [], and even the geometrcal postonng methods proposed by us n [7]. Although t requres tranng, our algorthm are satsfed the FCC standard of accuracy n most of the LOS error models. For both TSA and HLOP, the MS locaton accuracy can be crtcally affected by the relatve geometry between BSs and MS. The proposed algorthm performs equally well for any MS locaton whle TSA may not converge when the MS s located on the straght lne passng through two of the BSs, and HLOP would produce a large locaton error when the measured angle s close to 90 or 70. The remander of ths paper s organzed as follows. In Secton, we descrbe the MS postonng methods by usng TSA and HLOP. The geometrcal postonng methods are revewed n Secton 3. Secton 4 brefly descrbes BP and Rprop methods. In Secton 5, we propose the algorthm based on Rprop to determne the poston of the MS. ext, Secton 6 compares the performance of the proposed algorthm wth the other methods through smulaton result. Fnally, Secton 7 draws conclusons.. Taylor Seres Algorthm (TSA) and Hybrd Lnes of Poston Algorthm (HLOP) If both the TOA and AOA measurements are accurate, then only one BS s requred to locate the MS [7]. In realty, both TOA and AOA measurements contan errors due to LOS propagaton. Thus more than one BS s requred for MS locaton of reasonable accuracy. Takng nto account the constrant on hearablty, the number of BSs avalable for estmatng MS locaton s lmted to two n ths paper. However, each BS has both TOA and AOA measurement capabltes. Let t denote the propagaton tme from the MS to BS, =,. The dstances between BS and MS can be expressed as: r c t ( x X ) ( y Y ) () where (x, y) and (X, Y ) are the locatons of MS and BS, respectvely. c s the propagaton speed of the sgnals. If θ s the angle between MS and BS, wth respect to a reference drecton (x-axs), BS s located at (X, Y ) = (0, 0), BS s located at (X, Y ) = (X, 0), and MS s located at (x, y), as shown n Fgure, then θ can be obtaned as:

5 Sensors 0, 4 y Y tan ( ) x X () TSA [0,] and HLOP [] methods are commonly used to estmate the MS locaton, whch are brefly descrbed n ths secton... Taylor Seres Algorthm (TSA) TOA and AOA measurements are used as nputs to the Taylor seres poston estmator. Let (x, y) s the MS locaton and (x v, y v ) s the ntally estmated poston, let x = x v + δ v and y = y v + δ v. The MS locaton s obtaned by lnearzng the TOA and AOA equatons through the use of a Taylor seres expanson and retanng second-order terms, we have: where r x A,, y v ( xv X ) ( yv Y ), A z (3) r rv r rv r r z, and,, v x x v, y y v x v, y v v, y, tan v Y v,,. x X x x v, y v y x v, y v The least-squares (LS) soluton to the estmaton problem s gven by: v T T ( A A) A z (4) It requres a proper ntal poston guess close to the true soluton and can acheve hgh accuracy. Ths method s recursve and the computatonal overhead s ntensve n the teraton. Due to the ntal guess of the MS locaton s not accurate enough, the convergence of the teratve process s not assured [0,]. In addton, to avod the dvergent problems n the smulatons, the true MS locaton s used as an ntal poston... Hybrd Lnes of Poston Algorthm (HLOP) Ths scheme makes use of the orgnal nonlnear range equatons to produce lnear lnes of poston (LOP), rather than crcular LOP, to locate the MS. The method takes the advantage of smpler computaton of MS locaton. The detals of the lnear LOP approach can be acqured by usng the TOA measurements as n [3], and the hybrd lnear LOP algorthm wth AOA measurement n []. Combnng the lnear LOP and two AOA lnes, the MS locaton can be determned by: Gl h (5) where x l denotes the MS locaton, y X 0 r r X G tan and tan h 0. X tan

6 Sensors 0, 4 Agan, the LS soluton to Equaton (5) s gven by: 3. Geometrcal Postonng Methods T T l ( G G) G h (6) From the vewpont of geometrc approach, the TOA value measured at any BS can be used to form a crcle centered at the BS. The MS poston s then gven by the ntersecton of the crcles from multple TOA measurements. Smlarly, a sngle AOA measurement constrans the MS along a lne. Each of the followng equatons descrbes a crcle for TOA, a lne for AOA, as shown n Fgure : Crcle : x y r (7) Crcle : x X y (8) r Lne : tan x y 0 (9) Lne : tan x y tan X (0) Fgure. Geometrc layout of the two crcles and two lnes. If there s no error or even no nose at all, the crcles and lnes wll ntersect at only one pont. However, ths s usually not the case n practce where the LOS effect exsts. LOS propagaton s qute common and t serously degrades locaton accuracy. The ntersectons of two TOA crcles and two AOA lnes wll be spread over a regon, whch wll be offset from the true MS locaton. Because of the fact that LOS effect always ncreases the propagaton delay, the measured TOA estmated are always greater than the true values due to the excess path length. The true MS locaton must le n the regon of overlap of the two crcles. As mentoned earler, the ntersectng ponts that are wthn ths are defned as feasble ntersectons. Hence, the feasble ntersectons must satsfy the followng nequaltes smultaneously: x y r () x X y () r

7 Sensors 0, 43 The most drect method s to utlze these feasble ntersectons of the crcles and lnes to estmate the MS locaton. To acheve hgh accuracy of MS locaton wth less complexty, we have proposed a class of geometrcal postonng methods n [7] and outlned as follows. 3.. Averagng Method By usng two AOA measurements, the least lkely ntersecton s frst elmnated. The MS locaton s obtaned to calculate the average value of all the remanng AOA measurements wth feasble ntersectons. Step. Fnd all the feasble ntersectons of the two crcles and two lnes. Step. Assume F and F are the ntersectons of the two crcles as shown n Fgure, F s consdered to be the least lkely ntersecton f 0 < θ, θ < 80, and F s consdered to be the least lkely ntersecton f 80 < θ, θ < 360. Delete the least lkely ntersecton from the set of feasble ntersectons and there wll be remanng feasble ntersectons. Step 3. The MS locaton s estmated by averagng these remanng feasble ntersectons, where: x x and y y (3) 3.. Dstance-Weghted Method However, not all the remanng feasble ntersectons can always provde nformaton of the same value for locaton estmaton. In ths method, the weghts are nversely proportonal to the squared value of the dstance between the remanng feasble ntersectons and the average MS locaton. Steps 3 are the same as those of the averagng method. Step 4. Calculate the dstance d between each remanng feasble ntersecton (x, y ) and the average locaton : d ( x x ) ( y y ), (4) Step 5. Set the weght for the th remanng feasble ntersecton to (x d, y d ) s determned by: d x x d and y d d d d y. Then the MS locaton (5) One can see n the averagng method and dstance-weghted method, all the remanng feasble ntersectons wll affect the MS locaton estmaton. In the followng we also propose two methods of sort averagng and sort-weghted, whch can be appled wthout consderng the nfluence of feasble ntersectons for too far away from the average MS locaton.

8 Sensors 0, Sort Averagng Method Steps 4 are the same as those of the dstance-weghted method. Step 5. Rank the dstances d n ncreasng order and re-label the remanng feasble ntersectons n ths order. Step 6. The MS locaton x s estmated by the mean of the frst M remanng feasble ntersectons: M M x M M, y M M y ( M 0.5* ) (6) 3.4. Sort-Weghted Method Steps 5 are the same as those of the sort averagng method. Step 6. The MS locaton s estmated by a weghted average of the frst M remanng feasble ntersectons wth weght = : 3.5. Threshold Method x M d M d x, y M d M d y ( M 0.5* ) (7) The weght of ths method s based on how close the remanng feasble ntersectons are. Those feasble ntersectons that are closer to one another are assgned wth greater weghts. In other words, those ntersectons that are n close proxmty wll be assgned wth greater weghts. Steps and are the same as those of the averagng method. Step 3. Calculate the dstance d mn, m, n, between any par of feasble ntersectons. Step 4. Select a threshold value D thr as the average of all the dstances d mn. Step 5. Set the ntal weght I k, k, to be zero for all remanng feasble ntersectons. If d D, then I I and I I for m, n. mn thr m m n n Step 6. The MS locaton (x t, y t ) s estmated by: x t I x and I y t I y I (8) 4. The Tradtonal BP Algorthm and the Rprop Algorthm 4.. The Tradtonal BP Algorthm In ths secton, we descrbe the methodology based on artfcal neural network (A). It s a technque that models the learnng procedures of a human bran, and employs a set of actvaton

9 Sensors 0, 45 functons, ether nonlnear or lnear, thus one doesn t requre a pror selecton of a mathematcal model. Further, ths method has been proved to be very useful for varous applcatons. One of the most nfluental developments n A was the nventon of the BP, whch provdes advantages of non-lnear problem solvng ablty. BP s a mult-layered, feed-forward archtecture wth supervsed learnng method for computer learnng and modelng. A supervsed feed-forward neural network can not only learn from the tranng data to dscover patterns representng the nput and output varables, but approxmate many problems wth hgh accuracy. In a supervsed learnng approach, a set of nput varables s used for whch the correspondng output varables are known. Generally speakng, the BP archtecture comprses one nput layer, one output layer, wth one or a number of hdden layers n between them. Although a network wth multple hdden layers s possble, a sngle layer s suffcent to model arbtrarly complex nonlnear functons. Wth proper selecton of archtecture, t s capable of approxmatng most problems wth hgh accuracy and generalzaton ablty. The nput layer receves nformaton from the external sources and passes ths nformaton to the network for processng. The hdden layer determnes the mappng relatonshps between neurons are stored as weghts of connectng lnks. When the nput and output varables are related nonlnearly, the hdden layer can extract hgher level features and facltate generalzaton. The output from the output layer s the predcton of the net for the correspondng nput. The structure of BP chosen for the present problem s shown n Fgure. Each layer conssts of several neurons and the layers are nterconnected by sets of correlaton weghts. A standard A comprses numerous smple processng unts called neurons. Each node s connected to other neurons through drected connectng lnks; each neuron s a processng unt that contans an actvaton functon and an assocated weght. The actve functon s mathematcal formula and used to transform the output such that t falls wthn an acceptable range. In ths paper, the actvaton functons of hdden layer and output layer are hyperbolc tangent sgmod functon and lnear transfer functon. A weght returns a mathematcal value for the relatve strength of connectons to transfer data from one layer to another layer. BP estmate relaton between nput and output of sample patterns by updatng teratvely the weghts n the network so as to mnmze the dfference between the actual output vectors and the desred output vectors. The back propagaton learnng algorthm s composed of ntalzaton, a forward pass, and a backward pass. The weghts and bases n the network are ntalzed to small random numbers. Once these parameters have been ntalzed, the network s ready for tranng. A tranng pattern conssts of a set of the nput vectors and the correspondng output vectors. In the begnnng, a set of tranng patterns are fed to the nput layer of the network. The forward pass starts from the nput layer, the net nputs of the neurons are multpled wth correspondng weghts, then summated, and transferred to the hdden layer. The actvated sgnals are outputted from the hdden layer, and are passed forward to the output layer. Fnally, the output of BP s generated. Subsequently n the backward pass, the error between actual output and desred output s calculated. The error functon s defned as the mean squared sum of dfferences between the actual output vector T k and the desred output vector O k : k ( T k O k ) (9)

10 Sensors 0, 46 Fgure. A fully connected multlayer feed-forward network wth one hdden layer. The error sgnal at the output layer s propagated backward to the nput layer through the hdden layer n the network. Back-propagaton s so named because the error dervatves are calculated n the opposte drecton of sgnal propagaton. In the tranng process, the gradent descent method calculates and adjusts the weght of the network to mnmze the error. In the weght updatng algorthm, the dervatve of the error wth respect to the weght was frst negated then multpled by a small constant β known as the learnng rate, as expressed n the followng equaton: w ( t) j ( t) w (0) j The negatve sgn ndcates that the new weghtng vector s movng n a drecton opposte to that of the gradent. In the learnng process of neural network, the learnng rate affects the speed of convergence. The tranng process may lead to an oscllatory state f a learnng rate s too fast, on the other hand, the convergence speed may suffer f the learnng rate s too slow. The tranng process may not converge n the case of ether a too hgh or too low value for the learnng rate β. To accelerate the convergence, a momentum α can be added to the learnng procedures [4]: w () ( t) ( t) ( t) j wj wj In Equaton () α s between 0 and. When α = 0, a weght change s completely dependent on the value of gradent. When α =, the amount of new weght change s set to that of the last weght change and the gradent s smply gnored. The weghts are adjusted to make the actual output move closer to the desred output and to obtan the fnal outputs. Ths process s repeated untl the error s less than a pre-specfed level for each of the tranng data ponts, or a large number of tranng teratons have already been run. In summary, the flow chart of tranng procedure of BP s n Fgure 3 and the steps are lsted as follows.

11 Sensors 0, 47 Set the number of the layer and the number of neurons n each layer: () Set β, α and ntal values of the weghts, and the bases n the network are ntalzed to small random numbers. () Gvng nput and output vectors. (3) Compute the output values of each layer and unt n a feed-forward drecton. () Calculate the output for the jth hdden neuron. () Calculate the output for the kth output neuron. (4) Calculate the error functon at the output neuron. (5) Compute the deltas for each of the precedng layers by back propagatng the errors. () Calculate error for the kth output neuron. () Calculate error for the jth hdden neuron. (6) Update all weghts and bases (7) Repeat steps 3-7 untl the teraton has fnshed or the algorthm s convergent. Fgure 3. The flow chart of the calculaton procedure for BP.

12 Sensors 0, Rprop Algorthm Compared to the tradtonal BP algorthm, the Rprop algorthm can provde faster tranng and rate of convergence, and has the capablty to escape from local mnma. The Rprop s known to be very robust wth respect to ther nternal parameters and therefore regarded as one of the best frst-order learnng methods among the A algorthms. Rprop s a frst-order algorthm and ts tme and memory requrement scale s lnear wth the number of parameters to optmze. The Rprop algorthm s probably the most easly adjustable learnng rule, slght varatons of the values of parameters can not affect the convergence tme. The actvaton functon of the hdden and output layers s treated as lnear transfer functon. Rprop s easy to mplement and the hardware mplementaton s descrbed n [5]. Comparng to back-propagaton, one of the advantages of Rprop algorthm s that the magntude of the partal dervatve does not affect weght update and t depends only on the sgns of the partal dervatve. Thus t allows for faster convergence than the back-propagaton can do. Rprop performs a drect adaptaton of the weghtng step based on local gradent nformaton. A crucal dfference to the prevously developed adaptaton technques s that the adaptaton effort won t be blurred by the gradent behavor. The man dea of Rprop s to reduce the potental spurous effect of the partal dervatve on weght-updates by retanng only the sgn of the dervatve as an ndcaton of the drecton n whch the error functon wll be changed by the weght-update. We ntroduce an ndvdual update-value Δ j (t) for each weght, whch solely determnes the sze of the weght-update. Ths adaptve update-value evolves durng the learnng process based on ts local sght on the error functon, accordng to the followng learnng rule [9]: ( t) j ( ( t) j t) j ( t) j,,, f w f wj else ( t) j ( t) ( w ( w t) j t) j 0 0 () where 0 < η < < η +. We can smply descrbe the adaptaton rule as follows: Whenever the partal dervatve of the error functon ψ wth respect to the correspondng weght w j changes ts sgn, t ndcates that the value of last update was too bg and the algorthm has jumped over a local mnmum. The update-value Δ j s decreased by a factor η. If the dervatve retans ts sgn, the update-value s slghtly ncreased by the factor η + n order to accelerate convergence n shallow regons. Once the update-value for each weght s adapted, the weght-update tself follows a very smple rule: f the dervatve s postve (ncreasng error), the weght s decreased by ts update-value, f the dervatve s negatve, the update-value s added to the weght: ( t) ( t) j, f 0 wj ( t) ( t) ( t) wj j, f 0 (3) wj 0, else

13 Sensors 0, 49 There s one excepton to the rule above. If the partal dervatve changes sgn,.e., the prevous step was too large and the mnmum was mssed, the prevous weght-update s reverted: ( t) ( t) ( t) ( t) w j wj, f 0 w w (4) Due to that backtrackng weght step, the dervatve s supposed to change ts sgn once agan n the followng step. In order to avod a double punshment of the update value, there should be no adaptaton of the update value n the succeedng step. In practce ths can be done by settng n the Δ j update-rule above. 5. Proposed Locaton Algorthm Based on Rprop To mprove the accuracy of MS locaton, we proposed the employment of Rprop, a supervsed learnng neural network to obtan an approxmaton of MS locaton. The remanng feasble ntersectons are fed to the nput layer, and MS locaton s the only one varable n the output layer. Gven a number of known nput-output tranng patterns, the Rprop models are traned contnuously and deployed to adjust the weghts wth one hdden layer. A traned Rprop s to mnmze the dfference between the actual MS locaton and the desred MS locaton. The network has the followng nput-output mappng: Input: V remanng feasble ntersectons (V =,,, 6) Output: desred MS locaton The number of the remanng feasble ntersectons depends on the geometrc relatonshp of the two TOA crcles and two AOA lnes. In ths case, the number of the remanng feasble ntersectons s between and 6. Every measurement wll result n one nput data. Fgure 4 llustrates the structure used n Rprop MS locaton forecastng model of a three-layered network. Two types of nput layer for tranng purpose are dentfed and explaned n detal as follows: Type (Dvded Type): Accordng to the number of the remanng feasble ntersectons, the frst type establshes dfferent nput data subsets respectvely. For each measurement, we collect the V remanng feasble ntersectons and put them nto the V-th nput data subsets separately. There are sx data subsets n ths nput layer for tranng purpose, and the measurement number of each subset won t be dentcal. From smulaton results, the measurement number of 4 remanng feasble ntersectons s the maxmum, whle the measurement number of remanng feasble ntersectons s the mnmum. The detaled steps are as follows: () Collect the V remanng feasble ntersectons of two TOA crcles and two AOA lnes. (V =,,, 6). () If the number of remanng feasble ntersecton s V, then placng these V ponts n the V-th subset. The V remanng feasble ntersectons are belongng to the correspondng V-th nput data subsets separately. (3) The 6 nput data subsets wth varous measurement numbers are traned accordng to Rprop. j j

14 Sensors 0, 40 Type (Composte Type): The second type s a collecton of the V remanng feasble ntersectons n order. Regardless of the number of remanng feasble ntersectons n each measurement, we wll only establsh one nput data set. The summaton of all the measurement number for the 6 subsets s equal to the number of all measurements. The detaled steps are as follows: () Collect the V remanng feasble ntersectons for each measurement and expand to sx ones n a data set. () The method to expand the remanng feasble ntersectons to 6 ones durng each measurement s as follows. () If the number of remanng feasble ntersectons s V, replcate them by (6/V) tmes. (V =,, 3) () If the number of remanng feasble ntersectons s 4, take the average value of these 4 ponts and treat t as the ffth pont. By ths manner the 6th pont s the average of the 5 prevous numbers. () If 5 remanng feasble ntersectons are collected, take an average of these 5 ponts as the 6th one. (3) After expanson, placng the 6 remanng feasble ntersectons n the nput data set for tranng purposes. Fgure 4. Structure of the predcton models for (a) Type (Dvded Type) and (b) Type (Composte Type). (a)

15 Sensors 0, 4 Fgure 4. Cont. (b) The tranng data s dfferent from the data that uses to estmate the MS locaton. That s, the tranng nput-output patterns s no longer be used after tranng s down. In real applcaton, we collect the remanng feasble ntersectons and the desred MS locaton to tran the neural network pror to the practcal use. After the tranng, then the remanng feasble ntersectons as nput data (wth the MS locatons be unknown) can not only pass through the traned Rprop more quckly, but estmate the better approprate MS locaton. Whenever we start to fnd the postons, the remanng feasble ntersectons can be used as the traned nput, as we expect ths model can estmate MS locatons quckly and precsely. In addton, we have found that when there are only 00 peces of nput-output patterns as tranng data, the proposed algorthm stll work better than the other methods. Therefore, the concluson s that the proposed algorthm can be appled n practcal stuatons. 6. Smulatons Results In ths secton for farly comparson wth varous methods we apply the computer smulatons to demonstrate the performance of the proposed algorthm. In the smulatons, the BSs are respectvely located at (0, 0) and (,000 m, 0). Each smulaton s performed by 0,000 ndependent runs, and the MS locaton s chosen randomly accordng to a unform dstrbuton wthn the rectangular area formed by the ponts I, J, K and L, as shown n Fgure. To valdate the effectveness of the proposed algorthm, the remanng feasble ntersectons and the desred MS locaton are collected for comparson. The proposed hybrd TOA/AOA algorthm employs Rprop for MS locaton estmaton and performance evaluaton. Regardng the LOS effects, three propagaton models are adopted, namely, the unformly dstrbuted nose model [5], the based unform random model [], as well as the dstance-dependent model [5].

16 Sensors 0, 4 The frst LOS propagaton model s based on the unformly dstrbuted nose model [5], n whch the TOA measurement error s assumed to be unformly dstrbuted over (0, U ), for =, where U s the upper bound and the AOA measurement errors are assumed to be f = w 5 and f = w 0, where w s a unformly dstrbuted varable over [, ] [6]. Before applyng Rprop to estmate MS locaton, the parameter must be set n advance, such as the numbers of hdden neurons, and tranng teratons (epochs). The parameter settngs for network archtectures must be determned carefully to avod constructng a worse network model; otherwse, they may ncrease the computatonal cost and produce worse results. Tral-and-error methods are used to determne the parameter settngs for network archtectures. We attempted to fnd the optmal parameter as well as mantan good performance at the same tme. Sngle hdden layer s the most wdely used one among varous learnng methods for neural networks. It s well enough to model arbtrarly complex nonlnear functons [7]. Therefore, the number of hdden layers s set at one. To examne how close the forecast to the real MS locaton, the root-mean-square (RMS) error s employed to evaluate the performance of the proposed algorthm. If the network outputs are relatvely close to the real MS locaton, RMS error wll have small values. Fgure 5 shows the RMS error of convergence versus the ncreased number of epochs. One can see the Rprop wth one hdden layer can map the remanng feasble ntersectons to MS locaton. At the begnnng of the tranng perod, the error s reduced rapdly. When the number of epochs s above,000, t wll not gve further performance mprovement. Some general rules to determne the number of hdden neurons are: () 0.5 (p + q), () p, () p +, (v) 3p +, where p and q are nput neurons and output neurons, respectvely [8]. The RMS errors wth varous numbers of hdden-layer neurons are lsted n Fgure 6 for comparson. One can see the RMS error converged to the same mnmum value for varous hdden layer neurons. The accuracy of MS locaton s hardly affected by the numbers of hdden-layer neurons. The number of neurons n the hdden layer s set to be because of the satsfactory predcton performance. In order to avod ncreasng the computaton load, hdden neurons and,000 tranng teratons are used n the smulatons. If BS s the servng BS of MS, ts TOA and AOA measurements should be more accurate. The varables of ths model are chosen as follows: U = 00 m, U = 400 m, τ = 5 and τ = 0. The proposed algorthm based on Rprop algorthm produce more accurate estmatons of MS locaton than those based on BP wth a learnng rate of 0.0, as shown n Fgure 7. We can see that BP-based wth,000 epochs has the worst performance and Rprop-based wth,000 epochs has the best performance. Even Rprop-based wth 00 epochs perform better than BP-based wth,000 epochs. The number of epochs of Rprop s less than the tradtonal BP. So Rprop method can offer much faster convergence and requre much less convergence tme. The results show that Rprop algorthm can accurately estmate MS locaton. The superor performance for the Rprop algorthm has been demonstrated by comparng the RMS errors. One can see that Rprop algorthm produces more accurate estmatons of MS locaton than tradtonal BP algorthm. Therefore, we select Rprop algorthm to estmate MS locaton n ths paper.

17 RMS RMS Sensors 0, Fgure 5. RMS errors reducton accordng to the number of epochs. Type (upper bound=00 m) Type (upper bound=300 m) Type (upper bound=400 m) Type (upper bound=500 m) Type (upper bound=600 m) Type (upper bound=700 m) Type (upper bound=00 m) Type (upper bound=300 m) Type (upper bound=400 m) Type (upper bound=500 m) Type (upper bound=600 m) Type (upper bound=700 m) Epochs Fgure 6. The RMS errors wth varous neurons numbers of hdden layer Type (upper bound=00 m) Type (upper bound=300 m) 0 Type (upper bound=400 m) Type (upper bound=500 m) 00 Type (upper bound=600 m) Type (upper bound=700 m) Type (upper bound=00 m) 90 Type (upper bound=300 m) Type (upper bound=400 m) 80 Type (upper bound=500 m) Type (upper bound=600 m) Type (upper bound=700 m) * ( p + q ) p * p + 3 * p + Hdden neurons

18 CDF CDF Sensors 0, 44 Fgure 7. Comparson of average MS locaton based on BP and Rprop BP - Type wth 000 epochs BP - Type wth 000 epochs Rprop - Type wth 000 epochs Rprop - Type wth 000 epochs Rprop - Type wth 00 epochs Rprop - Type wth 00 epochs Locaton error (m) Fgure 8 shows cumulatve dstrbuton functons (CDFs) of the locaton error for dfferent methods based on unformly dstrbuted nose model. To check for convergence, the ntal guess of MS locaton n TSA s chosen to be the true soluton n our smulatons. Smulatons demonstrated that at least three teratons are requred for TSA to converge. We can see that TSA and HLOP offer the worst performance, and the proposed algorthm has good ablty to make more accurate estmatons of MS locaton. The dvded type wth sx nput data subsets for tranng performs better than the composte type. Thus the proposed algorthm can always yeld better performance, compared wth TSA, HLOP and the other geometrcal postonng methods. Fgure 8. Comparson of error CDFs when LOS errors are modeled as the upper bound Rprop - Type (Dvded Type) Rprop - Type (Composte Type) Averagng Dstance-weghted Sort-averagng Sort-weghted Threshold HLOP TSA Locaton error (m)

19 RMS Error Sensors 0, 45 Based on the proposed neural network structure stated above, the Rprop can be appled to estmate the locaton of MS for every nput data. Fgure 9 shows how the average locaton error s affected by the upper bound on LOS error. The upper bound of LOS error for BS s 00 m and those of other BSs range from 00 m to 700 m. After the tranng perod, the superor performance for the proposed algorthm can be proved by comparng the RMS error of MS locaton. In general, as the upper bound on LOS error ncreases, then the average magntudes of the LOS errors wll also ncrease, whch leads to less accurate locaton estmaton. ote that the proposed algorthm can deal wth large errors more effectvely than the other methods. For most of the upper bound on LOS errors, the average locaton errors for TSA and HLOP are larger than at least twce that for the proposed algorthm. The senstvty of the proposed algorthm wth respect to the LOS effect s much less than those of TSA and HLOP methods. Fgure 9. Performance comparson of the locaton estmaton methods when the upper bound s used to model the LOS Averagng Dstance-weghted Sort-averagng Sort-weghted Threshold HLOP TSA Rprop - Type (Dvded Type) Rprop - Type (Composte Type) Upper bound on LOS range error (m) The second LOS propagaton model s based on the dstance-dependent LOS error model [5]. The LOS range error for the th range s taken to be ξ = χ R, where χ s a proportonal constant and R s the true range between -th BS and MS [5]. It makes ntutve sense to vew LOS errors as beng proportonal to the dstance traveled by the sgnal. The AOA measurement error s assumed to be f = w τ, for =, [6]. The varables are chosen as follows: χ = 0.3, χ = 0., τ =.5 and τ = 5 Fgure 0 shows the CDF of the average locaton error of dfferent algorthms wth dstance-dependent LOS error. Compared wth the other methods, the accuracy of MS locaton was consderably mproved wth the proposed algorthm. We can see that TSA has the worst performance, and the proposed algorthm wth dvded type has the best performance, followed by the proposed algorthm wth composte type.

20 CDF CDF Sensors 0, 46 Fgure 0. CDFs of the locaton error wth dstance-dependent LOS error Rprop - Type (Dvded Type) 0.4 Rprop - Type (Composte Type) Averagng 0.3 Dstance-weghted Sort-averagng 0. Sort-weghted Threshold 0. HLOP TSA Locaton error (m) The thrd LOS propagaton model s based on a based unform random varable [], n whch the measured error of TOA between MS and BS s assumed to be γ = ρ + w μ, for =, where ρ and μ are constants. Smlarly, the measured error of AOA, s modeled as a = b + w c, for =, where b and c are constants. The error varables for the two BSs are chosen as follows: ρ = 50 m, ρ = 50 m, μ = μ = 00 m, b =.5, b = 3 and c = c = 5. The resultng CDF curves of the locaton error are as shown n Fgure. The dvded type wth sx nput data subsets provdes better locaton estmaton than the composte type. TSA and HLOP lead to less accurate results under ths condton, and the proposed algorthm can stll gve better MS locaton estmate. Fgure. Comparson of locaton error CDFs wth based unform random error Rprop - Type (Dvded Type) 0.4 Rprop - Type (Composte Type) Averagng 0.3 Dstance-weghted Sort-averagng 0. Sort-weghted Threshold 0. HLOP TSA Locaton error (m)

21 y CDF Sensors 0, 47 Fgure shows CDFs of the MS locaton error for all methods based on a based unform random varable model. The proposed algorthm wth,000 epochs s slghtly better than that wth 00 epochs. The smulaton results show that the postonng precson of the proposed algorthm wth only 00 epochs stll yeld superor performance when compared wth TSA, HLOP and the other geometrcal postonng methods. Our proposed algorthm wth 00 epochs stll enhances the performance of MS locaton estmaton effectvely. Fgure. CDFs of the locaton error of the other dfferent methods and the proposed algorthm wth,000 and 00 epochs Averagng Dstance-weghted Sort-averagng Sort-weghted Threshold HLOP TSA Rprop - Type wth 000 epochs Rprop - Type wth 000 epochs Rprop - Type wth 00 epochs Rprop - Type wth 00 epochs Locaton error (m) Fgure 3. Dstrbuton of the dvergence ponts for TSA x

22 y Sensors 0, 48 Fgure 4. Dstrbuton of the dvergence ponts for HLOP x When the MS s close to the condton of beng algned wth the two BSs, TSA may not converge. HLOP can result n large locaton errors when the measured angle approaches 90 or 70. We defne the dvergence pont when the RMS error s above 3,000 m. The dstrbutons of the dvergence ponts of TSA and HLOP are shown n Fgures 3 and 4, respectvely. The dvergence probabltes for dfferent LOS errors are between 0.3% and 4.66% [7]. The dvergence ponts of TSA and HLOP are not used to calculate the RMS errors and CDF s n our smulatons. ote that the proposed algorthms do not have such dvergence problem for ths stuaton. TSA and HLOP won t be any dvergent problems n the case of more than two BSs avalable to use. o matter what LOS propagaton model s consdered, the smulaton results show that the proposed algorthms for Rprop can gve very accurate results n the MS locaton estmaton after the tranng perod. 7. Conclusons Ths paper proposes novel Rprop-based algorthm to obtan approxmate MS locaton. We combne both TOA and AOA measurements to estmate the MS locaton under the condton that the MS s heard by only two BSs. The key ssue s to apply Rprop to model the relatonshp of the remanng feasble ntersectons and MS locaton. After tranng, the proposed algorthm can reduce the effects of LOS errors and mprove MS locaton performance. One the other hand, the tradtonal methods of TSA and HLOP may not converge when the MS/BSs have an undesrable geometrc layout. The postonng accuracy of the proposed algorthms s hardly affected by the relatve poston between the MS and BSs. Smulaton results show that the convergence performance of the proposed algorthms are qute well and provdes the capabltes to explctly reduce the effects of LOS errors. In summary, the proposed algorthm can always yeld better performance than TSA, HLOP and the geometrcal postonng methods for dfferent levels of LOS errors.

23 Sensors 0, 49 References. Song, H.L. Automatc vehcle locaton n cellular systems. IEEE Trans. Veh. Technol. 007, 43, Rappaport, T.S.; Reed, J.H.; Woerner, B.D. Poston locaton usng wreless communcatons on hghways of the future. IEEE Commun. Mag. 996, 34, Caffery, J.J.; Stuber, G. Overvew of radolocaton n CDMA cellular systems. IEEE Commun. Mag. 998, 36, Borras, J.; Hatrack, P.; Mandayam,.B. Decson theoretc framework for LOS dentfcaton. In Proceedngs of 48th IEEE Vehcular Technology Conference, Ottawa, Canada, 8 May 998; pp Chen, P.C. A nonlne-of-sght error mtgaton algorthm n locaton estmaton. In Proceedngs of IEEE Wreless Communcatons and etworkng Conference, ew Orleans, LA, USA, 4 September 999; pp Xong, L. A selectve model to suppress LOS sgnals n angle-of-arrval (AOA) locaton estmaton. In Proceedngs of The nth IEEE Internatonal Symposum on Personal, Indoor and Moble Rado Communcatons, Boston, MA, USA, 8 September 998; pp Cong, L.; Zhuang, W. on-lne-of-sght error mtgaton n TDOA moble locaton. Proc. IEEE Globecom 0,, Cong, L.; Zhuang, W. on-lne-of-sght error mtgaton n moble locaton. IEEE Trans. Wrel. Commun. 005, 4, Yap, J.H.; Ghaher-r, S.; Tafazoll, R. Accuracy and hearablty of moble postonng n GSM and CDMA networks. In Proceedngs of Thrd Internatonal Conference on 3G Moble Communcaton Technologes, London, UK, 8 0 May 00; pp Reed, J.H.; Krzman, K.J.; Woerner, B.D.; Rappaport, T.S. An overvew of the challenges and progress n meetng the E-9 requrement for locaton servce. IEEE Commun. Mag. 998, 36, Reed, J.H.; Krzman, K.J.; Woerner, B.D.; Rappaport, T.S. An overvew of the challenges and progress n meetng the E-9 requrement for locaton servce. IEEE Commun. Mag. 998, 36, Ma, C. Integraton of GPS and cellular networks to mprove wreless locaton performance. In Proceedngs of IO GPS/GSS 003, Portland, OR, USA, 9 September 003; pp Caffery, J.; Stuber, G. Subscrber locaton n CDMA cellular networks. IEEE Trans. Veh. Technol. 998, 47, Cong, L.; Zhuang, W. Hybrd TDOA/AOA moble user locaton for wdeband CDMA cellular systems. IEEE Trans. Wrel. Commun. 00,, Venkatraman, S.; Caffery, J.; You, H.R. A novel TOA locaton algorthm usng LOS range estmaton for LOS envronments. IEEE Trans. Veh. Technol. 004, 53, Sprto, M.A. Moble staton locaton wth heterogeneous data. In Proceedngs of 5nd Vehcular Technology Conference, IEEE VTS-Fall VTC 000, Boston, MA, USA, 4 8 September 000; pp

24 Sensors 0, Chen, C.S.; Su, S.L.; Huang, Y.F. Hybrd TOA/AOA geometrcal postonng schemes for moble locaton. IEICE Trans. Commun. 009, E9-B, Rumelhart, D.E.; Hnton, G.E.; Wllams, R.J. Learnng representatons by back-propagatng errors. ature 996, 33, Redmller, M.; Braun, H. A drect adaptve method for faster backpropagaton learnng: The RPROP algorthm. In Proceedngs of IEEE Internatonal Conference on eural etworks, San Francsco, CA, USA, 8 March Aprl 993; pp Foy, W. Poston-locaton solutons by Taylor seres estmaton. IEEE Trans. Aerosp. Electron. Syst. 976, AES-, Torrer, D. Statstcal theory of passve locaton systems. IEEE Trans. Aerosp. Electron. Syst. 984, AES-0, Venkatraman, S.; Caffery, J. Hybrd TOA/AOA technques for moble locaton n non-lne-of-sght envronments. In Proceedngs of IEEE Wreless Communcatons and etworkng Conference, Atlanta, GA, USA, 5 March 004; pp Caffery, J. A new approach to the geometry of TOA locaton. In Proceedngs of 5nd Vehcular Technology Conference, IEEE VTS-Fall VTC 000, Boston, MA, USA, 4 8 September 000; pp Jacobs, R.A. Increased rates of convergence through learnng rate adaptaton. eural etw. 988,, Patnak, L.M.; Rajan, K. Target detecton through mage processng and reslent propagaton algorthms. eurocomputng 000, 35, Chen, C.L.; Feng, K.T. An effcent geometry-constraned locaton estmaton algorthm for LOS envronments. In Proceedngs of Internatonal Conference on Wreless etworks, Communcatons and Moble Computng, Mau, HI, USA, 3 6 June 005; pp Kohzada,.; Boyd, M.S. A comparson of artfcal neural network and tme seres models for forecastng commodty prces. eurocomputng 996, 0, Venkatachalan, A.R.; Sohl, J.E. An ntellgent model selecton and forecastng system. J. Forecast. 999, 8, by the authors; lcensee MDPI, Basel, Swtzerland. Ths artcle s an open access artcle dstrbuted under the terms and condtons of the Creatve Commons Attrbuton lcense (

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