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Research ArticleIWKNN An Effective Bluetooth Positioning MethodBased on Isomap and WKNN

Qi Wang Yingying Feng Xiangde Zhang Yanrui Sun and Xiaojun Lu

Department of Mathematics Northeastern University Shenyang Liaoning Province China

Correspondence should be addressed to Xiaojun Lu luxiaojun0625sinacom

Received 2 July 2016 Revised 21 September 2016 Accepted 29 September 2016

Academic Editor Ruay-Shiung Chang

Copyright copy 2016 Qi Wang et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Recently Bluetooth-based indoor positioning has become a hot research topic However the instability of Bluetooth RSSI (ReceivedSignal Strength Indicator) promotes a huge challenge in localization accuracy To improve the localization accuracy this papermeasures the distance of RSSI vectors on their low-dimensional manifold and proposes a novel positioning method IWKNN(Isomap-basedWeighted119870-Nearest Neighbor)The proposed method firstly uses Isomap to generate low-dimensional embeddingfor RSSI vectors Then the distance of two given RSSI vectors is measured by Euclidean distance of their low-dimensionalembeddings Finally the position is calculated by WKNN Experiment indicates that the proposed approach is more robust andaccurate

1 Introduction

Positioning is a basic requirement in peoplersquos daily life It isthe basis of LBS (Location-Based Service) [1 2] Generallypositioning problems could be classified as outdoor andindoor The outdoor positioning mainly uses GPS (GlobalPositioning System) GPRS (General Packet Radio Service)and so forth while the indoor positioning mostly utilizesshort range signals such as Wi-Fi (Wireless-Fidelity) [3 4]and Bluetooth [5ndash7] And the testing position is determinedbased on the received signalsThese positioning technologieshave been widely used in different kinds of applications

Beacon is a commonly used Bluetooth signal sourceFigure 1 shows a picture of beacon It is small and low-powerBluetooth dissipation equipment It can work for even twoyears just with a fastener battery So beacon can be placedwithout too many restrictions having broad prospects ofapplication

This paper aims to improve the accuracy of Bluetooth-based indoor positioning when beacons are preplaced atsome given places in a room A smart device is used to receivethese signals Then the position of device can be calculatedbased on received Bluetooth RSSI data

Classical positioning methods could be classified as twotypes One is function-based model This kind of method is

mostly developed based on signal propagation function [8 9]for example trilateration algorithm [10 11] and IoT (Internetof Things) [12] This kind of model requires an estimatedsignal propagation function If the positions of emitters aregiven then any testing location could be calculated basedon RSSI data and signal propagation function This kind ofmethod does not need to store the vast RSSI data of referencepositions The localization accuracy relies on the measure-ment error of signal propagation function The less the erroris the more accurate the localization is Because the signalintensity of beacon is instable the error of signal propagationmodel would be relatively large So it is inaccurate to localizea Bluetooth receiver by this kind of method

The other kind of method could be described as classifi-cation algorithm which is built up based on position finger-print Figure 2 illustrates general positioning process basedon fingerprint Lemic et al [13] and Jaffre et al [14] determinethe final position by KNN (119870-Nearest Neighbor) They findthe nearest119870 reference positions to calculate unknown posi-tion according to Euclidean distance of RSSI vectors Caso etal [15] develop WKNN (Weighted 119870-Nearest Neighbor) for3D positioning by weighting each nearest neighbor Besidesthis Shin et al [16] propose EWKNN (Enhanced WeightedK-Nearest Neighbor) by making parameter 119870 variable toimprove the positioning performance

Hindawi Publishing CorporationMobile Information SystemsVolume 2016 Article ID 8765874 11 pageshttpdxdoiorg10115520168765874

2 Mobile Information Systems

Figure 1 A picture of beacon

Cheng et al [17] introduce SVM (Support VectorMachine) to classify the RSSI vectors Mu et al [18] classifythe RSSI data by ANN (Artificial Neural Network) Zhe [19]develops a positioning method based on Bayes method Heuses more flexible models to estimate distributions of beaconsignals Fras et al [20] proposed a newmethod that combinesBayes and WKNN Alfakih et al [21] develop a positioningalgorithm based on GMM (Gaussian Mixture Model) Theycalculate the probability that the testing RSSI belongs to eachreference RSSI dataThe final location is calculated by sum ofweighted coordinates of reference locations Different fromBayes method GMM is to calculate the probability based onGaussian Mixture function

Kim et al [22] and Thuong et al [23] estimate thepositioning performance for KNN algorithm Jaffre et al [14]compare KNN and Bayes positioning method with severaldifferent distance measurements and value 119870 Experimentalresult states that KNN and Euclidean distance can get thesmallest mean positioning error when 119870 = 4 Zhang etal [24] use beacons as emitter and compare the positioningperformance of ANN SVM and improved KNN The paperreports that the improved KNN algorithm has the highestpositioning accuracy

Because of the instability of Bluetooth received RSSIvectors have a large variety This means that RSSI vectors offaraway positions may have smaller Euclidean distances thanthat of neighborhood It leads to choosing a faraway referenceposition as candidate location sometimes This enlarges thelocalization error of Euclidean distance based methods suchas KNN and WKNN So just using Euclidean distance tomeasure similarity of RSSI vectors is not accurate Bluetoothpositioning requires a more robust distance measurementmethod

Manifold learning [25] finds the low-dimensional embed-ding for high-dimensional data Researchers have pro-posed different algorithms for manifold learning such asMDS (Multidimensional Scaling) [26] LLE (Locally LinearEmbedding) [27] and Isomap [28] The geodesic distance isa useful distance metric of Isomap in manifold learning Itcan be used to measure the similarity of high-dimensional

vectors Figure 3 illustrates the comparison of Euclidian andgeodesic distances

We consider Bluetooth indoor positioning as a high-dimensional data matching problem Calculate low-dimen-sional embeddings for the training and testing RSSI dataUse the Euclidian distance of low-dimensional embeddingsto measure the approaching extent of two given RSSI vectorsThe final location is calculated by WKNN Figure 4 demon-strates the flow chart of proposed method

Figure 5 illustrates the comparison of positioning resultsof WKNN and proposed IWKNN Figure 5(a) gives a local-ization illustration by WKNN based on Euclidian distancewhile Figure 5(b) shows a positioning process by proposedIWKNN In Figure 5 the red rectangles are real testingpositions The pink dots are selected 119870 nearest trainingpositions (119870 = 4) The green hollow dots are obtainedfinal position Figure 5 shows that the proposed method canselect more proper nearest positions than WKNN achievinga better localization

This paper is organized as follows Section 2 introducesthe mathematical model of indoor positioning problem Sec-tion 3 illustrates the proposed localization algorithmbased onIsomap andWKNN Section 4 shows the experimental resultand analysis Section 5 concludes the whole paper

2 The Mathematical Model ofBluetooth Positioning

This section presents the mathematical model of Bluetoothpositioning problem

Here let us consider a general Bluetooth positioningproblem We establish a rectangular coordinate system in aregion 119898 beacons are distributed evenly in this area Thena cell phone in this region would receive and measure RSSIsfrom all beacons They can be grouped as a RSSI vector TheBluetooth positioning problem is to determine the coordinateof cell phone in the rectangular coordinate system

Suppose there are total 119899 reference positions Administra-tor should record Bluetooth RSSI at each reference positionfor a period of time When a user is standing at some place inthis area he can measure the Bluetooth RSSI by cell phone orother equipmentThen the coordinate of person can be calcu-lated based on the newly measured and recorded RSSI data

Let 119909119894 represent the coordinate of 119894th reference location119909119894 = [119886119894 119887119894] 119894 = 1 119899 rssi119905119894119895 is the RSSI received fromthe 119895th beacon at the 119894th reference position at time 119905 120572119905119894 isa vector composed of RSSI on the 119894th reference location fromdifferent beacons at time 119905 120572119905119894 = [rssi1199051198941 rssi1199051198942 rssi119905119894119898]119879119894 = 1 119899 Here 119898 is the number of beacons 119899 is thenumber of reference locations

If the RSSI data are separately received at time 119905 =1199051 1199052 119905119897 then the matrix of RSSI data for 119894th referencelocations could be written as

[1205721199051119894 1205721199052119894 120572119905119897119894 ] = (rssi11990511198941 rssi11990521198941 sdot sdot sdot rssi1199051198971198941rssi11990511198942 rssi11990521198942 sdot sdot sdot rssi1199051198971198942 d

rssi1199051119894119898 rssi1199052119894119898 sdot sdot sdot rssi119905119897119894119898

) (1)

Mobile Information Systems 3

Receiver PreprocessingDatabase

Positioning algorithm

Obtaining testing vector

Beacon1

Beaconm

[a b]

x1

x2

xn

120572t1

120572t2

120572tn

Figure 2 Positioning process based on fingerprint

0 2 4 6 8 10 12 14 16

Geodesic

Euclidean minus100

minus80

minus60

minus40

65

43

21

0

Figure 3 The difference of geodesic distance and Euclidean distance

Let 120572119894 be the fingerprint of the 119894th reference location then120572119894 = [rssi1198941 rssi1198942 rssi119894119898]119879= 1119897 [[ 119897sum119895=1rssi1199051198951198941 119897sum119895=1rssi1199051198951198942 119897sum119895=1rssi1199051198951198942]]

119879 (2)

Here 119894 = 1 2 119899For any unknown testing position = [119886 119887] the received

RSSI vector 120572119909 at could be written as120572119909 = [rssi1199091 rssi1199092 rssi119909119898]119879 (3)

Then the localization problem could be expressed by thefollowing formula[ ] = 119879 (1205721 1205722 120572119899 120572119909) (4)

Here 119879 is the positioning function to be determined whichprojects a RSSI vector to coordinate of position This prob-lem is generally solved by positioning fingerprint methodsThese kinds of methods are mostly developed based on theEuclidean distance of RSSI vectors such as KNN [13 14]WKNN [15] and EWKNN [16] This paper develops a per-formance enhanced positioningmethod which employs low-dimensional embedding distance tomeasure the similarity ofRSSI vectors and use WKNN to determine unknown testinglocation

3 Proposed Method

This section presents the proposed positioning methodIWKNN Firstly the low-dimensional embeddings of ref-erence and testing RSSI vectors are calculated by IsomapThen the unknown testing position is generated accordingto distances of low-dimensional embeddings by WKNN


Creating the database

Get the distance matrix DG by Floyd

Compute a low-dimensional embedding B

Estimate location by WKNN based on B

xi 120572ti

Testing point 120572x

[a b]

Training phase

Testing phase

Generate undirected weighted graph G

Process the data and designa high-dimensionalmanifold A

Figure 4 The block diagram of the proposed algorithm

2

4

62 4 6 8 10 12 14 16

50

100

150

200

(a) Positioning by WKNN (119870 = 4)

2 4 6 8 10 12 14 16

6080

100120140160180200220

2

3

4

5

6

(b) Positioning by proposed IWKNN (119870 = 4)

Figure 5 Positioning by WKNN and proposed IWKNN

31 Overview As an important method in manifold learn-ing Isomap can find the low-dimensional embedding forhigh-dimensional manifold It can keep the neighborhoodstructure of high-dimensionalmanifold in a low-dimensionalspace

Let 119860 be a set which is composed of (119899 + 1) RSSI vectors119860 = 120572119894 120572119909 | 119894 = 1 2 119899 (5)119860 sub 119877119898Suppose 119861 is a low-dimensional embedding of 119860 119861 =120573119894 120573119909 | 119894 = 1 119899 where 119861 sub 1198771198981015840 and 1198981015840 lt 119898Let 119891 be a smooth embedding projection from 119861 to 119860119891 119861 rarr 119860 satisfying 120572119894 = 119891(120573119894) (for all 119894 = 1 2 119899) and120572119909 = 119891(120573119909)

Here the low-dimensional embedding is calculated byIsomap The distance of RSSI vectors is measured by Euclid-ian distance of their low-dimensional embeddings Finallythe unknown testing position is calculated based on low-dimensional distances by WKNN

32 Geodesic Distance of High-Dimensional Data Assume allthe elements in 119860 are from the same manifold Ideally anyelement in 119860 can be linearly expressed by its neighborhoodpointsWe connect each point in119860 to its119872 nearest neighbors(119872 is a predetermined parameter) Then an undirectedweighted graph is constructedThus a distance matrix of thisgraph could be generated by Euclidean distance The edgeweight for given two points 120572119894 and 120572119895 is 119889119864(119894 119895) = 120572119894 minus 120572119895

According to Isomap algorithm [28] the shortest distanceof any two high-dimensional RSSI vectors is used as geodesic


7m

18m

(a) The positions of beacons

7m

18m

(b) The reference positions

Figure 6 The layout of beacons and reference positions

distance It can be calculated by Floydrsquos algorithm [28] whichis shown in119863119866 = 119889119866 (119894 119895)119896= 119889119864 (119894 119895) 119896 = 0

min (119889119866 (119894 119895)119896minus1 119889119866 (119894 119896)119896minus1 + 119889119866 (119896 119895)119896minus1) 119896 ge 1 (6)

Here 119896 = 0 119899 + 133 Low-Dimensional Embedding Then a low-dimensionalembedding could be calculated as follows [28]120591 (119863) = 119867119878119867minus2 (7)

where 119878119894119895 = (1198632119894119895) 119867 = 119868 minus (1(119899 + 1))119864119864119879 119868 is a unit matrixand 119864 = [1 1 1]119879

Let 1205821 1205822 1205821198981015840 be the largest 1198981015840 eigenvalues of 120591(119863)(in descending order) and their corresponding eigenvectorsbe 1199061 1199062 1199061198981015840 Then all of the vectors in set 119861 could beobtained by [28][1205731 1205732 120573119899 120573119909]= [radic12058211199061 radic12058221199062 radic12058211989810158401199061198981015840]119879 (8)

Then the low-dimensional embedding of 119860 = 1205721 1205722 120572119899 120572119909 is 119861 = 1205731 1205732 120573119899 12057311990934 Distance of Low-Dimensional Embedding The traininglow-dimensional fingerprint of the 119894th reference point is[1205731 1205732 120573119899] The testing low-dimensional vector is 120573119909The distance of 120573119894 (119894 = 1 2 119899) and 120573119909 is measured byEuclidian distance 119889119894 = 1003817100381710038171003817120573119894 minus 1205731199091003817100381710038171003817 (9)

35 Positioning by WKNN According to WKNN [15]let 1198891199011 1198891199012 119889119901119870 stand for the 119870 minimum values in1198891 1198892 119889119899 Then the reference coordinates [119886119901119895 119887119901119895] 119895 =1 119870 are selected as candidate reference locations Finally

the coordinate of testing position [ ] is estimated by thefollowing formula [15][ ] = 1sum119870119895=1 119908119895 119870sum119895=1119908119895 [119886119901119895 119887119901119895] (10)

where 119908119895 = 1119889119901119895 + 119888 (11)

Here 119888 is a relative small positive number

4 Experimental Result and Analysis

41 Experimental Details The experiment is carried out in alaboratory of Northeastern University The laboratory roomis 7times18m2 A 6times18m2 subarea is used as experimental area

In our experiment beacons are used to emit Bluetoothsignals whose parameters were adjusted referencing [5ndash7]The emitting interval is set with 2HZ and power of minus8 dbSamsung Galaxy S3 is used to receive Bluetooth signals at afrequency of 1Hz

Figure 6 shows the positions of beacons and referencelocations The hollow dots in Figure 6(a) show the layout of30 beacons They are placed evenly on the ceiling with meandistance of 2 meters All the beacons are recorded with MACaddress and positions The solid dots in Figure 6(b) showthe layout of 24 reference positions with mean distance of 2meters on the ground

To build a usable training fingerprint RSSI data isrecorded for 200 seconds at each reference position AndKalman filter [29] is used to preprocess RSSI data

Considering that a person may move fast with a cellphone we just use one vector of RSSI data for positioning Inthis paper 74 testing positions are used in our experimentThese positions are distributed evenly with one-meter dis-tance in the 6 times 18m2 experimental area

42 Selection of 119872 and 1198981015840 In Isomap algorithm 119872 and1198981015840 are important parameters which are the count of nearestneighbors and dimension of low-dimensional embedding Toselect a proper 119872 and 1198981015840 some experiments are carried outfirstly


Table 1 The positioning errors with different 119872119872 Max error (m) Mean error (m) Min error (m) Std error (m)3 4991 1668 0026 09194 5389 1657 0230 10065 5086 1620 0093 09526 3851 1796 0473 09077 4112 1723 0326 08288 4278 1910 0335 09539 4317 1936 0027 105610 3400 1865 0078 081811 5187 2037 0424 109412 5806 1983 0389 114813 4290 1942 0377 100914 5388 1938 0292 097115 3814 1672 0278 079116 3815 1592 0303 077517 3528 1635 0334 072618 4516 1732 0297 084119 3602 1635 0331 086420 3605 1516 0001 078221 3600 1618 0117 077622 4601 1655 0350 097623 4590 1622 0345 093424 4902 1633 0345 0982

Table 2 The mean positioning error with different 11989810158401198981015840119872 15 16 17 18 19 20 212 1778 1772 1957 1807 1895 2004 19203 1856 1710 1811 1735 1680 1801 17124 1770 1744 1687 1663 1601 1573 15295 1672 1592 1635 1732 1635 1516 16186 1625 1523 1712 1736 1657 1597 16177 1741 1600 1705 1703 1761 1675 16098 1751 1585 1703 1783 1614 1731 15949 1770 1695 1718 1843 1639 1730 159310 1778 1686 1666 1825 1672 1682 160611 1785 1676 1718 1823 1749 1695 155912 1764 1618 1722 1776 1665 1646 158413 1742 1577 1695 1761 1630 1635 159214 1736 1633 1697 1755 1653 1675 161515 1745 1638 1674 1741 1678 1702 1606

Table 1 shows the positioning errors with different 119872As can be seen from the table when 119872 is between 15 and21 the maximum and mean positioning errors are relativelysmall and stable According to our experiment the proposedmethod achieves minimummean error when 119872 = 20

Table 2 shows the mean positioning error with different119872 and1198981015840119872 is assigned between 15 and 211198981015840 varies from 2

to 15 In this table we can find that when1198981015840 is between 4 and6 the mean positioning error is relatively small

According to these experiments we set119872 = 20 and1198981015840 =5 in this paper

43 The Effectiveness of Isomap The effectiveness of Isomapis firstly examined in our Bluetooth localization experimentWe compare the performance of WKNN and EWKNN [16]and their Isomap enhanced versions WKNN is separatelytested with 119870 = 3 4 9 And the threshold of EWKNNis mean of distance vector

Figure 7 shows the error distribution of different meth-ods In Figure 7 the first-column figures illustrate theerror distributions of original methods The second columnshows the error distribution of Isomap enhanced methodsThe third column shows the differences of original andIsomap enhanced methods From Figure 7 we can findthat the error distribution of developed Isomap enhancedmethods (IWKNN and IEWKNN) is more concentrated inthe low error region than original methods (WKNN andEWKNN)

Figure 8 illustrates the cumulative probability of errordistribution for differentmethodsThe curves in these figuresdemonstrate that the Isomap enhanced methods (IWKNNand IEWKNN) achieve better performance than originalmethods (WKNN and EWKNN)

Finally the performance of all tested methods is given inTable 3 According to this table Isomap almost improves allof positioning indicators clearlyThe best values of importantindicators are given in bold font The minimum of maxerror is 4994 which is obtained by IWKNN with 119870 = 9The minimum of mean error is 1463 which is obtained byIWKNN with 119870 = 4 The minimum of min error is 00005which is obtained by IWKNNwith119870 = 3 and IEWKNNTheminimum of std error is 0674 which is obtained by IWKNNwith119870 = 7Themaximumof ldquoratio of error less than 2metersrdquois 7896 which is obtained by IWKNN with 119870 = 5 Thistable supports the effectiveness of proposed Isomap enhancedpositioning methods

According to Table 3 we suggest 119870 = 4 or 5 for smallmean positioning error and large ldquoratio of error less than 2metersrdquo

44 Comparison with Different Methods In order to test theperformance of proposed method more clearly we comparethe proposed IWKNN (119870 = 4) with six different positioningmethods

Figure 9 compares the performances of different meth-ods From this table we can see that Bayes method hasthe best performance in the low error part and the pro-posed IWKNN performs best in almost all the remainingpart

Table 4 illustrates the positioning error of differentmethods According to this table GMM Bayes andWKNN-Bayesrsquos max errors are better than other methods But theimproved Bayes method proposed by Fras et al [20] cangenerate small max mean and std errors The proposedIWKNN has the minimal mean error and maximal ldquoratio oferror less than 2 metersrdquo


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(a) WKNN (K = 3) (b) IWKNN (K = 3) (c) (b)-(a)

(d) WKNN (K = 4)(e) IWKNN (K = 4) (f) (e)-(d)

(g) WKNN (K = 5) (h) IWKNN (K = 5) (i) (h)-(g)

(j) WKNN (K = 6) (k) IWKNN (K = 6) (l) (k)-(j)

(m) WKNN (K = 7) (n) IWKNN (K = 7) (o) (n)-(m)

(p) WKNN (K = 8) (q) IWKNN (K = 8) (r) (q)-(p)

(s) WKNN (K = 9) (t) IWKNN (K = 9) (u) (t)-(s)

(v) EWKNN (w) IEWKNN (Isomap EWKN-N) (x) (w)-(v)

Figure 7 Error distributions of original and Isomap enhanced methods


0

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bilit

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(a) WKNN and IWKNN (119870 = 3)

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Figure 8 Continued


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(h) EWKNN and IEWKNN

Figure 8 Cumulative error distribution of original and Isomap enhanced methods

Table 3 The positioning error of original and Isomap enhanced methods

Method Max error (m) Mean error (m) Min error (m) Std error (m) le2m ()WKNN (119870 = 3) 7000 1505 00008 0772 7399IWKNN (119870 = 3) 5046 1481 00005 0758 7558WKNN (119870 = 4) 6558 1508 0001 0732 7449IWKNN (119870 = 4) 7110 1463 0002 0710 7605WKNN (119870 = 5) 6803 1541 0014 0712 7612IWKNN (119870 = 5) 6162 1478 0036 0693 7896WKNN (119870 = 6) 6045 1581 0002 0707 7145IWKNN (119870 = 6) 5216 1512 0011 0680 7708WKNN (119870 = 7) 5774 1631 0023 0703 6918IWKNN (119870 = 7) 5273 1558 0041 0674 7427WKNN (119870 = 8) 5135 1689 0023 0721 6550IWKNN (119870 = 8) 5138 1604 0066 0683 7077WKNN (119870 = 9) 5447 1749 0019 0739 6151IWKNN (119870 = 9) 4994 1660 0032 0703 6714EWKNN 7000 1502 0002 0756 7426IEWKNN 5046 1477 00005 0743 7604

Table 4 The positioning error of different methods

Method Max error (m) Mean error (m) Min error (m) Std error (m) le2m ()Triangulation 11443 2592 00288 1616 4180IoT [12] 7576 3077 00331 1482 2772GMM [21] 12880 3506 0 1965 6503Bayes 14318 2391 0 1611 4446WKNN-Bayes [20] 8062 2031 0 1116 5342WKNN 6558 1508 0001 0732 7449IWKNN 7110 1463 0002 0710 7605

5 Conclusion

This paper presents an Isomap enhanced localizationmethodIWKNN The proposed method combines Isomap andWKNN for Bluetooth positioning Firstly we calculate low-dimensional embeddings for RSSI data by Isomap Then thedistance of different RSSI vectors is measured by Euclidean

distance of these low-dimensional embeddings Finally theunknown testing position is calculated by WKNN Experi-ment indicates the distances of low-dimensional embeddingsare more robust than those of high-dimensional ones Andthe proposed IWKNN is robust and effective

The main contribution of this paper could be concludedas follows (1) introduce Isomap to Bluetooth positioning


TriangulationIoTBayesGMM

WKNN-BayesWKNNIWKNN

2 4 6 8 100Error (m)

0

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ulat

ive p

roba

bilit

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)

Figure 9 The cumulative error distribution of different methods

problem (2) use low-dimensional embeddings to representhigh-dimensional RSSI data (3) adopt distances of low-dimensional embeddings to measure the distance of RSSIvectors (4) combine Isomap with WKNN for Bluetoothpositioning

Disclosure

The method presented in this paper has been applied forpatent

Competing Interests

The authors declare that they have no competing interests

References

[1] A PetrenkoA SizoWQian et al ldquoExploringmobility indoorsan application of sensor-based and GIS systemsrdquo Transactionsin GIS vol 18 no 3 pp 351ndash369 2014

[2] W Wang and R Zhou ldquoResearch on the anti-collision systemof surface coal mine based on the highly accurate GPS locationtechnologyrdquo in Proceedings of the 9th International Conferenceon Electronic Measurement and Instruments (ICEMI rsquo09) pp3196ndash3199 Beijing China August 2009

[3] A Narzullaev and Y Park ldquoNovel calibration algorithm forreceived signal strength based indoor real-time locating sys-temsrdquo AEUmdashInternational Journal of Electronics and Commu-nications vol 67 no 7 pp 637ndash644 2013

[4] K Kaemarungsi and P Krishnamurthy ldquoAnalysis of WLANrsquosreceived signal strength indication for indoor location finger-printingrdquo Pervasive and Mobile Computing vol 8 no 2 pp292ndash316 2012

[5] R Faragher and R Harle ldquoLocation fingerprinting with blue-tooth low energy beaconsrdquo IEEE Journal on Selected Areas inCommunications vol 33 no 11 pp 2418ndash2428 2015

[6] H K Fard Y Chen and K K Son ldquoIndoor positioning ofmobile devices with agile iBeacon deploymentrdquo in Proceed-ings of the 28th IEEE Canadian Conference on Electrical andComputer Engineering (CCECE rsquo15) pp 275ndash279 IEEE HalifaxCanada May 2015

[7] F SubhanHHasbullah A Rozyyev and S T Bakhsh ldquoAnalysisof bluetooth signal parameters for indoor positioning systemsrdquoinProceedings of the IEEE International Conference onComputerInformation Science (ICCIS rsquo12 ) vol 2 pp 784ndash789 KualaLumpur Malaysia June 2012

[8] F Palumbo P Barsocchi S Chessa and J C Augusto ldquoAstigmergic approach to indoor localization using Bluetooth LowEnergy beaconsrdquo in Proceedings of the 12th IEEE InternationalConference on Advanced Video and Signal Based Surveillance(AVSS rsquo15) pp 1ndash6 Karlsruhe Germany August 2015

[9] T Pulkkinen J Verwijnen and P Nurmi ldquoWiFi positioningwith propagation-based calibrationrdquo in Proceedings of the 14thInternational Symposium on Information Processing in SensorNetworks (IPSN rsquo15) pp 366ndash367 Seattle Wash USA April2015

[10] M E Rida F Liu Y Jadi A A A Algawhari and AAskourih ldquoIndoor location position based on bluetooth signalstrengthrdquo in Proceedings of the 2nd International Conference onInformation Science and Control Engineering (ICISCE rsquo15) pp769ndash773 Shanghai China April 2015

[11] S Boonsriwai and A Apavatjrut ldquoIndoor wifi localization onmobile devicesrdquo in Proceedings of the 10th IEEE InternationalConference on Electrical Engineering Electronics ComputerTelecommunications and Information Technology (ECTI-CONrsquo13) pp 1ndash5 Krabi Thailand May 2013

[12] H N Thi and K N T Vu ldquoA positioning algorithm in theinternet of thingsrdquo in Proceedings of the 8th InternationalConference on Ubiquitous and Future Networks (ICUFN rsquo16) pp594ndash598 Vienna Austria July 2016

[13] F Lemic A Behboodi V Handziski and A Wolisz ldquoExper-imental decomposition of the performance of fingerprinting-based localization algorithmsrdquo in Proceedings of the 5th IEEEInternational Conference on Indoor Positioning and Indoor Nav-igation (IPIN rsquo14) pp 355ndash364 Busan South Korea October2014

[14] T Jaffre P-M Grigis S Papanastasiou and E Peytchev ldquoOnthe efficacy of WiFi indoor positioning in a practical settingrdquoin Proceedings of the 18th IEEE Symposium on Computers andCommunications (ISCC rsquo13) pp 699ndash704 July 2013

[15] G Caso L De Nardis and M-G Di Benedetto ldquoFrequentistinference for WiFi fingerprinting 3D indoor positioningrdquo inProceedings of the IEEE International Conference on Communi-cation Workshop (ICCW rsquo15) pp 809ndash814 IEEE London UKJune 2015

[16] B Shin J H Lee T Lee and H S Kim ldquoEnhanced weightedknearest neighbor algorithm for indoor wi-fi positioning sys-temsrdquo in Proceedings of the 8th International Conference onComputing Technology and Information Management (ICCMrsquo12) vol 2 pp 574ndash577 IEEE Berlin Germany 2012

[17] J Cheng Y Cai Q Zhang J Cheng and C Yan ldquoA new three-dimensional indoor positioning mechanism based on wirelessLANrdquoMathematical Problems in Engineering vol 2014 ArticleID 862347 7 pages 2014


[18] Z Mu X Yubin and M Lin ldquoANN indoor position deter-mination based on area correlation in WLAN environmentrdquoin Proceedings of the International Conference on WirelessCommunications and Signal Processing (WCSP rsquo09) pp 1ndash4IEEE Nanjing China November 2009

[19] C Zhe ldquoThe research and implementation of indoor locationpositioning system based on low-power bluetooth and locationfingerprintrdquo 2014

[20] M Fras K Wasko and T Wierzowiecki Personal Wi-Fi BasedIndoor Localization of Mobile Devices in Active EnvironmentSpringer New York NY USA 2016

[21] M Alfakih M Keche and H Benoudnine ldquoGaussian mixturemodeling for indoor positioning wifi systemsrdquo in Proceedingsof the 3rd International Conference on Control Engineering andInformation Technology (CEIT rsquo15) Tlemcen Algeria May 2015

[22] J Kim M Ji J il Jeon S Park and Y Cho ldquoK-NN basedpositioning performance estimation for fingerprinting local-izationrdquo in Proceedings of the 8th International Conference onUbiquitous and Future Networks (ICUFN rsquo16) pp 468ndash470Vienna Austria July 2016

[23] N T Thuong H T Phong D-T Do P V Hieu and DT Loc ldquoAndroid application for WiFi based indoor positionsystem design and performance analysisrdquo in Proceedings of theInternational Conference on Information Networking (ICOINrsquo16) pp 416ndash419 Kota Kinabalu Malaysia January 2016

[24] L Zhang X Liu J Song C Gurrin and Z Zhu ldquoA compre-hensive study of bluetooth fingerprinting-based algorithms forlocalizationrdquo in Proceedings of the 27th International Conferenceon Advanced Information Networking and Applications Work-shops (WAINA rsquo13) pp 300ndash305 Barcelona Spain March 2013

[25] W Zhan G Dai and H Liu ldquoResearch overview of manifoldlearning algorithmrdquo Advances in Information Sciences andService Sciences vol 5 no 1 pp 58ndash67 2013

[26] I Borg and P J Groenen Modern Multidimensional ScalingTheory and Applications Springer Science amp Business MediaBerlin Germany 2005

[27] S T Roweis and L K Saul ldquoNonlinear dimensionality reduc-tion by locally linear embeddingrdquo Science vol 290 no 5500pp 2323ndash2326 2000

[28] J B Tenenbaum V De Silva and J C Langford ldquoA globalgeometric framework for nonlinear dimensionality reductionrdquoScience vol 290 no 5500 pp 2319ndash2323 2000

[29] G Welch and G Bishop An Introduction to the Kalman FilterUniversity of North Carolina at Chapel Hill Chapel Hill NCUSA 1995

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Distributed Sensor Networks


Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014


ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014


Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications


Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia


Biomedical Imaging


ArtificialNeural Systems

Advances in


RoboticsJournal of



Computational Intelligence and Neuroscience

Industrial EngineeringJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in



Figure 1 A picture of beacon

Cheng et al [17] introduce SVM (Support VectorMachine) to classify the RSSI vectors Mu et al [18] classifythe RSSI data by ANN (Artificial Neural Network) Zhe [19]develops a positioning method based on Bayes method Heuses more flexible models to estimate distributions of beaconsignals Fras et al [20] proposed a newmethod that combinesBayes and WKNN Alfakih et al [21] develop a positioningalgorithm based on GMM (Gaussian Mixture Model) Theycalculate the probability that the testing RSSI belongs to eachreference RSSI dataThe final location is calculated by sum ofweighted coordinates of reference locations Different fromBayes method GMM is to calculate the probability based onGaussian Mixture function

Kim et al [22] and Thuong et al [23] estimate thepositioning performance for KNN algorithm Jaffre et al [14]compare KNN and Bayes positioning method with severaldifferent distance measurements and value 119870 Experimentalresult states that KNN and Euclidean distance can get thesmallest mean positioning error when 119870 = 4 Zhang etal [24] use beacons as emitter and compare the positioningperformance of ANN SVM and improved KNN The paperreports that the improved KNN algorithm has the highestpositioning accuracy

Because of the instability of Bluetooth received RSSIvectors have a large variety This means that RSSI vectors offaraway positions may have smaller Euclidean distances thanthat of neighborhood It leads to choosing a faraway referenceposition as candidate location sometimes This enlarges thelocalization error of Euclidean distance based methods suchas KNN and WKNN So just using Euclidean distance tomeasure similarity of RSSI vectors is not accurate Bluetoothpositioning requires a more robust distance measurementmethod

Manifold learning [25] finds the low-dimensional embed-ding for high-dimensional data Researchers have pro-posed different algorithms for manifold learning such asMDS (Multidimensional Scaling) [26] LLE (Locally LinearEmbedding) [27] and Isomap [28] The geodesic distance isa useful distance metric of Isomap in manifold learning Itcan be used to measure the similarity of high-dimensional

vectors Figure 3 illustrates the comparison of Euclidian andgeodesic distances

We consider Bluetooth indoor positioning as a high-dimensional data matching problem Calculate low-dimen-sional embeddings for the training and testing RSSI dataUse the Euclidian distance of low-dimensional embeddingsto measure the approaching extent of two given RSSI vectorsThe final location is calculated by WKNN Figure 4 demon-strates the flow chart of proposed method

Figure 5 illustrates the comparison of positioning resultsof WKNN and proposed IWKNN Figure 5(a) gives a local-ization illustration by WKNN based on Euclidian distancewhile Figure 5(b) shows a positioning process by proposedIWKNN In Figure 5 the red rectangles are real testingpositions The pink dots are selected 119870 nearest trainingpositions (119870 = 4) The green hollow dots are obtainedfinal position Figure 5 shows that the proposed method canselect more proper nearest positions than WKNN achievinga better localization

This paper is organized as follows Section 2 introducesthe mathematical model of indoor positioning problem Sec-tion 3 illustrates the proposed localization algorithmbased onIsomap andWKNN Section 4 shows the experimental resultand analysis Section 5 concludes the whole paper

2 The Mathematical Model ofBluetooth Positioning

This section presents the mathematical model of Bluetoothpositioning problem

Here let us consider a general Bluetooth positioningproblem We establish a rectangular coordinate system in aregion 119898 beacons are distributed evenly in this area Thena cell phone in this region would receive and measure RSSIsfrom all beacons They can be grouped as a RSSI vector TheBluetooth positioning problem is to determine the coordinateof cell phone in the rectangular coordinate system

Suppose there are total 119899 reference positions Administra-tor should record Bluetooth RSSI at each reference positionfor a period of time When a user is standing at some place inthis area he can measure the Bluetooth RSSI by cell phone orother equipmentThen the coordinate of person can be calcu-lated based on the newly measured and recorded RSSI data

Let 119909119894 represent the coordinate of 119894th reference location119909119894 = [119886119894 119887119894] 119894 = 1 119899 rssi119905119894119895 is the RSSI received fromthe 119895th beacon at the 119894th reference position at time 119905 120572119905119894 isa vector composed of RSSI on the 119894th reference location fromdifferent beacons at time 119905 120572119905119894 = [rssi1199051198941 rssi1199051198942 rssi119905119894119898]119879119894 = 1 119899 Here 119898 is the number of beacons 119899 is thenumber of reference locations

If the RSSI data are separately received at time 119905 =1199051 1199052 119905119897 then the matrix of RSSI data for 119894th referencelocations could be written as

[1205721199051119894 1205721199052119894 120572119905119897119894 ] = (rssi11990511198941 rssi11990521198941 sdot sdot sdot rssi1199051198971198941rssi11990511198942 rssi11990521198942 sdot sdot sdot rssi1199051198971198942 d

rssi1199051119894119898 rssi1199052119894119898 sdot sdot sdot rssi119905119897119894119898

) (1)





Beacon1

Beaconm

[a b]

x1

x2

xn

120572t1

120572t2

120572tn


0 2 4 6 8 10 12 14 16

Geodesic

Euclidean minus100

minus80

minus60

minus40

65

43

21

0



119879 (2)





3 Proposed Method







xi 120572ti


[a b]

Training phase

Testing phase




2

4

62 4 6 8 10 12 14 16

50

100

150

200


2 4 6 8 10 12 14 16

6080

100120140160180200220

2

3

4

5

6









7m

18m


7m

18m











where 119908119895 = 1119889119901119895 + 119888 (11)

























001020304

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

0

minus004minus002

002004

Prob

abili

ty (

)

2 4 60Error (m)

001020304

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

0

minus004minus002

002004

Prob

abili

ty (

)

2 4 60Error (m)

001020304

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04Pr

obab

ility

()

2 4 60Error (m)

0

minus004minus002

002004

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

minus004minus002

0002004

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

0

minus004minus002

002004

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

minus004minus002

0002004

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

0

minus004minus002

002004

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

0

minus004minus002

002004

Prob

abili

ty (

)

2 4 60Error (m)











0

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y (

)

1 2 3 40Error (m)

WKNN (K = 3)

IWKNN (K = 3)


1 2 3 40Error (m)

0

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y (

)

WKNN (K = 4)

IWKNN (K = 4)


1 2 3 40Error (m)

0

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y (

)

WKNN (K = 5)

IWKNN (K = 5)


1 2 3 40Error (m)

0

02

04

06

08

1Cu

mul

ativ

e pro

babi

lity

()

WKNN (K = 6)

IWKNN (K = 6)


1 2 3 40Error (m)

0

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y (

)

WKNN (K = 7)

IWKNN (K = 7)


1 2 3 40Error (m)

0

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y (

)

WKNN (K = 8)

IWKNN (K = 8)


Figure 8 Continued


1 2 3 40Error (m)

0

02

04

06

08

1Cu

mul

ativ

e pro

babi

lity

()

WKNN (K = 9)

IWKNN (K = 9)


1 2 3 40Error (m)

0

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y (

)

EWKNNIEWKNN







5 Conclusion






WKNN-BayesWKNNIWKNN

2 4 6 8 100Error (m)

0

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y (

)



Disclosure


Competing Interests


References






































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in







Beacon1

Beaconm

[a b]

x1

x2

xn

120572t1

120572t2

120572tn


0 2 4 6 8 10 12 14 16

Geodesic

Euclidean minus100

minus80

minus60

minus40

65

43

21

0



119879 (2)





3 Proposed Method







xi 120572ti


[a b]

Training phase

Testing phase




2

4

62 4 6 8 10 12 14 16

50

100

150

200


2 4 6 8 10 12 14 16

6080

100120140160180200220

2

3

4

5

6









7m

18m


7m

18m











where 119908119895 = 1119889119901119895 + 119888 (11)

























001020304

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

0

minus004minus002

002004

Prob

abili

ty (

)

2 4 60Error (m)

001020304

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

0

minus004minus002

002004

Prob

abili

ty (

)

2 4 60Error (m)

001020304

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04Pr

obab

ility

()

2 4 60Error (m)

0

minus004minus002

002004

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

minus004minus002

0002004

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

0

minus004minus002

002004

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

minus004minus002

0002004

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

0

minus004minus002

002004

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

0

minus004minus002

002004

Prob

abili

ty (

)

2 4 60Error (m)











0

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y (

)

1 2 3 40Error (m)

WKNN (K = 3)

IWKNN (K = 3)


1 2 3 40Error (m)

0

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y (

)

WKNN (K = 4)

IWKNN (K = 4)


1 2 3 40Error (m)

0

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y (

)

WKNN (K = 5)

IWKNN (K = 5)


1 2 3 40Error (m)

0

02

04

06

08

1Cu

mul

ativ

e pro

babi

lity

()

WKNN (K = 6)

IWKNN (K = 6)


1 2 3 40Error (m)

0

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y (

)

WKNN (K = 7)

IWKNN (K = 7)


1 2 3 40Error (m)

0

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y (

)

WKNN (K = 8)

IWKNN (K = 8)


Figure 8 Continued


1 2 3 40Error (m)

0

02

04

06

08

1Cu

mul

ativ

e pro

babi

lity

()

WKNN (K = 9)

IWKNN (K = 9)


1 2 3 40Error (m)

0

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y (

)

EWKNNIEWKNN







5 Conclusion






WKNN-BayesWKNNIWKNN

2 4 6 8 100Error (m)

0

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y (

)



Disclosure


Competing Interests


References






































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in








xi 120572ti


[a b]

Training phase

Testing phase




2

4

62 4 6 8 10 12 14 16

50

100

150

200


2 4 6 8 10 12 14 16

6080

100120140160180200220

2

3

4

5

6









7m

18m


7m

18m











where 119908119895 = 1119889119901119895 + 119888 (11)

























001020304

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

0

minus004minus002

002004

Prob

abili

ty (

)

2 4 60Error (m)

001020304

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

0

minus004minus002

002004

Prob

abili

ty (

)

2 4 60Error (m)

001020304

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04Pr

obab

ility

()

2 4 60Error (m)

0

minus004minus002

002004

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

minus004minus002

0002004

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

0

minus004minus002

002004

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

minus004minus002

0002004

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

0

minus004minus002

002004

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

0

minus004minus002

002004

Prob

abili

ty (

)

2 4 60Error (m)











0

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y (

)

1 2 3 40Error (m)

WKNN (K = 3)

IWKNN (K = 3)


1 2 3 40Error (m)

0

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y (

)

WKNN (K = 4)

IWKNN (K = 4)


1 2 3 40Error (m)

0

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y (

)

WKNN (K = 5)

IWKNN (K = 5)


1 2 3 40Error (m)

0

02

04

06

08

1Cu

mul

ativ

e pro

babi

lity

()

WKNN (K = 6)

IWKNN (K = 6)


1 2 3 40Error (m)

0

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y (

)

WKNN (K = 7)

IWKNN (K = 7)


1 2 3 40Error (m)

0

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y (

)

WKNN (K = 8)

IWKNN (K = 8)


Figure 8 Continued


1 2 3 40Error (m)

0

02

04

06

08

1Cu

mul

ativ

e pro

babi

lity

()

WKNN (K = 9)

IWKNN (K = 9)


1 2 3 40Error (m)

0

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y (

)

EWKNNIEWKNN







5 Conclusion






WKNN-BayesWKNNIWKNN

2 4 6 8 100Error (m)

0

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y (

)



Disclosure


Competing Interests


References






































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




7m

18m


7m

18m











where 119908119895 = 1119889119901119895 + 119888 (11)

























001020304

Prob

abili

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2 4 60Error (m)

0

02

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Prob

abili

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)

2 4 60Error (m)

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minus004minus002

002004

Prob

abili

ty (

)

2 4 60Error (m)

001020304

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

0

minus004minus002

002004

Prob

abili

ty (

)

2 4 60Error (m)

001020304

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04Pr

obab

ility

()

2 4 60Error (m)

0

minus004minus002

002004

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

minus004minus002

0002004

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

0

minus004minus002

002004

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

minus004minus002

0002004

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

0

minus004minus002

002004

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

0

minus004minus002

002004

Prob

abili

ty (

)

2 4 60Error (m)











0

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y (

)

1 2 3 40Error (m)

WKNN (K = 3)

IWKNN (K = 3)


1 2 3 40Error (m)

0

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y (

)

WKNN (K = 4)

IWKNN (K = 4)


1 2 3 40Error (m)

0

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y (

)

WKNN (K = 5)

IWKNN (K = 5)


1 2 3 40Error (m)

0

02

04

06

08

1Cu

mul

ativ

e pro

babi

lity

()

WKNN (K = 6)

IWKNN (K = 6)


1 2 3 40Error (m)

0

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y (

)

WKNN (K = 7)

IWKNN (K = 7)


1 2 3 40Error (m)

0

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y (

)

WKNN (K = 8)

IWKNN (K = 8)


Figure 8 Continued


1 2 3 40Error (m)

0

02

04

06

08

1Cu

mul

ativ

e pro

babi

lity

()

WKNN (K = 9)

IWKNN (K = 9)


1 2 3 40Error (m)

0

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y (

)

EWKNNIEWKNN







5 Conclusion






WKNN-BayesWKNNIWKNN

2 4 6 8 100Error (m)

0

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y (

)



Disclosure


Competing Interests


References






































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in



















001020304

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

0

minus004minus002

002004

Prob

abili

ty (

)

2 4 60Error (m)

001020304

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

0

minus004minus002

002004

Prob

abili

ty (

)

2 4 60Error (m)

001020304

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04Pr

obab

ility

()

2 4 60Error (m)

0

minus004minus002

002004

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

0

02

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Prob

abili

ty (

)

2 4 60Error (m)

minus004minus002

0002004

Prob

abili

ty (

)

2 4 60Error (m)

0

02

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Prob

abili

ty (

)

2 4 60Error (m)

0

02

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Prob

abili

ty (

)

2 4 60Error (m)

0

minus004minus002

002004

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

minus004minus002

0002004

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

0

minus004minus002

002004

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

0

02

04

Prob

abili

ty (

)

2 4 60Error (m)

0

minus004minus002

002004

Prob

abili

ty (

)

2 4 60Error (m)











0

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y (

)

1 2 3 40Error (m)

WKNN (K = 3)

IWKNN (K = 3)


1 2 3 40Error (m)

0

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y (

)

WKNN (K = 4)

IWKNN (K = 4)


1 2 3 40Error (m)

0

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y (

)

WKNN (K = 5)

IWKNN (K = 5)


1 2 3 40Error (m)

0

02

04

06

08

1Cu

mul

ativ

e pro

babi

lity

()

WKNN (K = 6)

IWKNN (K = 6)


1 2 3 40Error (m)

0

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y (

)

WKNN (K = 7)

IWKNN (K = 7)


1 2 3 40Error (m)

0

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y (

)

WKNN (K = 8)

IWKNN (K = 8)


Figure 8 Continued


1 2 3 40Error (m)

0

02

04

06

08

1Cu

mul

ativ

e pro

babi

lity

()

WKNN (K = 9)

IWKNN (K = 9)


1 2 3 40Error (m)

0

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y (

)

EWKNNIEWKNN







5 Conclusion






WKNN-BayesWKNNIWKNN

2 4 6 8 100Error (m)

0

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y (

)



Disclosure


Competing Interests


References






































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




0

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y (

)

1 2 3 40Error (m)

WKNN (K = 3)

IWKNN (K = 3)


1 2 3 40Error (m)

0

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y (

)

WKNN (K = 4)

IWKNN (K = 4)


1 2 3 40Error (m)

0

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y (

)

WKNN (K = 5)

IWKNN (K = 5)


1 2 3 40Error (m)

0

02

04

06

08

1Cu

mul

ativ

e pro

babi

lity

()

WKNN (K = 6)

IWKNN (K = 6)


1 2 3 40Error (m)

0

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y (

)

WKNN (K = 7)

IWKNN (K = 7)


1 2 3 40Error (m)

0

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y (

)

WKNN (K = 8)

IWKNN (K = 8)


Figure 8 Continued


1 2 3 40Error (m)

0

02

04

06

08

1Cu

mul

ativ

e pro

babi

lity

()

WKNN (K = 9)

IWKNN (K = 9)


1 2 3 40Error (m)

0

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y (

)

EWKNNIEWKNN







5 Conclusion






WKNN-BayesWKNNIWKNN

2 4 6 8 100Error (m)

0

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y (

)



Disclosure


Competing Interests


References






































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




1 2 3 40Error (m)

0

02

04

06

08

1Cu

mul

ativ

e pro

babi

lity

()

WKNN (K = 9)

IWKNN (K = 9)


1 2 3 40Error (m)

0

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y (

)

EWKNNIEWKNN







5 Conclusion






WKNN-BayesWKNNIWKNN

2 4 6 8 100Error (m)

0

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y (

)



Disclosure


Competing Interests


References






































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





WKNN-BayesWKNNIWKNN

2 4 6 8 100Error (m)

0

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y (

)



Disclosure


Competing Interests


References






































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in



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