6. Performance analysis

where q represents the fingerprint dataset and p, which is the dataset of the test points, represents the APs that the mobile device received. Because φ is a strictly convex function and SD pð Þ ; q equals zero if and only if p = q, this family of distortions is termed JSD. The geometric interpretation is represented in Figure 3, where divergence represents

In general, for a positive definite matrix, the Jensen-Bregman divergence contains the general-

<sup>¼</sup> <sup>2</sup>h i Qp; <sup>p</sup> <sup>þ</sup> <sup>2</sup>h i Qq; <sup>q</sup> � <sup>2</sup>h i Q pð Þ <sup>þ</sup> <sup>q</sup> ; <sup>p</sup> <sup>þ</sup> <sup>q</sup> 4

ð Þ h i Qp; p þ h i Qq; q � 2h i Qp; q

<sup>2</sup> � <sup>φ</sup> <sup>p</sup> <sup>þ</sup> <sup>q</sup>

2 

and the midpoint of the segment ½ð Þ <sup>p</sup>;φð Þ<sup>p</sup> ;

(28)

2 ;φ <sup>p</sup>þ<sup>q</sup> 2

ized quadratic distance, which is known as the Mahalanobis distance:

¼ 1 4

¼ 1 4

¼ 1 4 k k p � q 2 Q

To improve accuracy, we present Algorithm 2:

SD pð Þ¼ ; <sup>q</sup> <sup>φ</sup>ð Þþ <sup>p</sup> <sup>φ</sup>ð Þ<sup>q</sup>

h i Q pð Þ � q ; p � q

the vertical distance between <sup>p</sup>þ<sup>q</sup>

Figure 3. Interpreting the Jensen-Bregman divergence.

150 Machine Learning - Advanced Techniques and Emerging Applications

ð Þ� q; φð Þq .

The proposed algorithm evaluations will be presented in the subsequent subsections; the algorithms were implemented on the first floor of the CEAS at WMU. To collect the data sample, a Samsung S5 smartphone with operating system 4.4.2 was used. The proposed algorithms were implemented on an HP Pavilion using Java software with an Eclipse framework. Cisco Linksys E2500 Advanced Simultaneous Dual-Band Wireless-N Routers were used in the area of interest. Most of this work discounted the variation of the RSS from the APs.

To evaluate the performance of the different fingerprinting techniques, the localization error was computed as the Euclidean distance between the actual reported coordinates of the test points and the coordinates of the mobile user during the online phase. The number of RSS of the APs and the number of nearest neighbors were noted, as they can affect the accuracy of the algorithms. The number of APs can play an important role in the accuracy of the distance error, which can distinguish near RPs from those further away.

To evaluate the performance of the different fingerprinting techniques, the localization error was computed as the Euclidean distance between the actual reported coordinates of the test points and the coordinates of the mobile user during the online phase. The number of RSS of the APs and the number of nearest neighbors were noted, as they can affect the accuracy of the algorithms. The number of APs can play an important role in the accuracy of the distance error, which can distinguish near RPs from those further away.

In order to measure the impact of the APs on the accuracy, we used a specific number of nearest neighbors with a variety of APs. However, that resulted in a longer RSS scanning interval, which slowed the process down. As a result, the online phase comprised five time samples, which took 1 s for Wi-Fi scanning on the device. To investigate the accuracy of our proposed algorithm, different algorithms were used, such as PNN and KNN, and compared with our proposed algorithm. Different numbers of nearest neighbors were used to estimate the location of the object and to evaluate the performance of our system framework.

Figure 4 shows the impact of different APs when five nearest neighbors were used. The lowest localization error was obtained when 22 APs were used: 0.98 m for kJBD, 1.12 m for kJSD, 1.16 m for KLMvG, 1.34 m for PNN, and 1.38 m for kNN. Greater accuracy was obtained when more nearest neighbors were used, as illustrated in Figure 5.

The lowest localization accuracy was also obtained when 22 APs were used: 0.92 m for kJBD, 1.01 m for kJSD, 1.02 m for KLMvG, 1.097 m for PNN, and 1.19 m for kNN. More improvements in system accuracy were noticed when 80 nearest neighbors were used: 0.865 m for kJBD, 0.96 m for kJSD, 0.99 m for KLMvG, 0.995 m for PNN, and 1.12 m for kNN, as shown in Figure 6.

Figure 7 illustrates the corresponding cumulative probability distributions of the localization error for the three methods. In particular, the median errors for kJBD were 0.89 m, 0.98 m for kJSD, and 1.02 m for KLMvG. Furthermore, an accuracy of 90% was achieved at 2.13 m for KLMvG and 1.93 m for kJSD, with the best accuracy obtained at 2.13 m for kJSD.

To validate our work, a comparison was made between the proposed algorithms with other algorithms from prior works, such as kNN [14], compressive sensing [28], and the kernelbased method [33], as illustrated in Table 1.

Figure 5. Error distance estimation with respect to APs with 20 nearest neighbors.

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Figure 6. Error distance estimation with respect to APs with 80 nearest neighbors.

Figure 4. Error distance estimation with respect to APs with five nearest neighbors.

Figure 5. Error distance estimation with respect to APs with 20 nearest neighbors.

Figure 4 shows the impact of different APs when five nearest neighbors were used. The lowest localization error was obtained when 22 APs were used: 0.98 m for kJBD, 1.12 m for kJSD, 1.16 m for KLMvG, 1.34 m for PNN, and 1.38 m for kNN. Greater accuracy was obtained when

The lowest localization accuracy was also obtained when 22 APs were used: 0.92 m for kJBD, 1.01 m for kJSD, 1.02 m for KLMvG, 1.097 m for PNN, and 1.19 m for kNN. More improvements in system accuracy were noticed when 80 nearest neighbors were used: 0.865 m for kJBD, 0.96 m for kJSD, 0.99 m for KLMvG, 0.995 m for PNN, and 1.12 m for kNN, as shown in

Figure 7 illustrates the corresponding cumulative probability distributions of the localization error for the three methods. In particular, the median errors for kJBD were 0.89 m, 0.98 m for kJSD, and 1.02 m for KLMvG. Furthermore, an accuracy of 90% was achieved at 2.13 m for

To validate our work, a comparison was made between the proposed algorithms with other algorithms from prior works, such as kNN [14], compressive sensing [28], and the kernel-

KLMvG and 1.93 m for kJSD, with the best accuracy obtained at 2.13 m for kJSD.

Figure 4. Error distance estimation with respect to APs with five nearest neighbors.

more nearest neighbors were used, as illustrated in Figure 5.

152 Machine Learning - Advanced Techniques and Emerging Applications

based method [33], as illustrated in Table 1.

Figure 6.

Figure 6. Error distance estimation with respect to APs with 80 nearest neighbors.

nearest neighbors with 22 APs. We are currently in the process of investigating position prediction error distributions and quantifying the localization variation of Wi-Fi signal distri-

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Electrical and Computer Engineering Department, Almamoon University College, Kalamazoo,

[1] Markets & Markets. Indoor Localization Market by Positioning Systems, Map and Navigation, Location Based Analysis, Monitoring and Emergency Services-Worldwide Market

[2] Torres-Sospedra J, Montoliu R, Trilles S, Belmonte O, Huerta J. Comprehensive analysis of distance and similarity measures for Wi-Fi fingerprinting indoor positioning systems.

[3] Jiang P, Zhang Y, Fu W, Liu H, Su X. Indoor mobile localization based on Wi-Fi fingerprint's important access point. International Journal of Distributed Sensor Networks.

[4] Shchekotov M. Indoor localization methods based on Wi-Fi lateration and signal strength data collection. In: 2015 17th Conference of. Open Innovations Association (FRUCT);

[5] Swangmuang N, Prashant K. An effective location fingerprint model for wireless indoor

[6] Abdullah O, Abdel-Qader I, Bazuin B. A probability neural network-Jensen-Shannon divergence for a fingerprint based localization. In: 2016 Annual Conference on Informa-

[7] Abdullah O, Abdel-Qader I. A PNN-Jensen-Bregman divergence symmetrization for a WLAN indoor positioning system. In: 2016 IEEE International Conference on Electro

[8] Abdullah O, Abdel-Qader I, Bazuin B, Fingerprint-based technique for indoor positioning system via machine learning and convex optimization. In: 2016 IEEE 7th Annual Ubiquitous

bution in space

Author details

Michigan USA

References

2015;11(4):429104

Yaroslavl; 2015

Osamah Ali Abdullah\* and Ikhlas Abdel-Qader

\*Address all correspondence to: osamah.abdullah@wmich.edu

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Figure 7. Experiment results: the CDF of localization error when using 80 nearest neighbors.


Table 1. Position error statistic.
