Using the research questions and the attached sources. What are the types of network errors?? How can the network errors be re

Research Article

International Journal of Distributed Sensor Networks 2016, Vol. 12(12) � The Author(s) 2016 DOI: 10.1177/1550147716681793 ijdsn.sagepub.com

A clustering approach for error beacon filtering in underwater wireless sensor networks

Linfeng Liu1,2, Jingli Du2 and Dongyue Guo2

Abstract Underwater wireless sensor networks are the enabling technology for the aquatic environmental monitoring and explor- ing and have attracted much attention recently. Due to the highly hostile and unpredictable underwater environments, some beacon nodes tend to move or be damaged. Therefore, the unknown nodes will be positioned with larger error, which abases the value of data collected by sensor nodes. In order to solve the beacon error problem, this article pro- poses an error beacon filtering algorithm based on K-means clustering. First, the coordinate of each beacon is calculated through an improved trilateration method, and then the beacon with the maximum positioning error is filtered out via the K-means clustering algorithm. The remaining beacons repeat the above processes until the distance error of each beacon does not exceed a preset threshold. The analysis of simulation results indicates that the error beacons can be accurately found and filter out through our proposed error beacon filtering algorithm (based on K-means clustering), and thus the localization accuracy is enhanced. Besides, error beacon filtering algorithm also has a provable low complexity.

Keywords Underwater wireless sensor networks, error beacon filtering, localization algorithm, K-means clustering

Date received: 6 September 2016; accepted: 8 November 2016

Academic Editor: Miguel Ardid

Introduction

Underwater acoustic networks (underwater wireless sensor networks (UWSNs)) consist of abundant low- cost sensor nodes tied to underwater vehicles, and the nodes are deployed to monitor the underwater environ- ment collaboratively over the interest area.

1 In order to

explore the underwater world, UWSNs have attracted wide attention, and many specific applications have emerged, such as environmental monitoring, natural disaster prevention, and distributed tactical surveil- lance, where the node localization is always very signifi- cant.

2 If each sensor node cannot provide its accurate

coordinate, the data collected by sensor nodes may give wrong interpretations for the physical events.

3

However, the underwater environment is more complex than the terrestrial environment, and the underwater

characteristics bring several new challenges as follows. (1) The nodes with limited batteries are more prone to be exhausted, so they should be recharged timely. Unfortunately, it is very hard to access underwater nodes.

4 (2) Radio wave is not feasible underwater,

because it requires a large antenna and a high transmis- sion power, and thus the acoustic communication

1 Laboratory of Computer Network and Information Integration, Ministry

of Education, Southeast University, Nanjing, China 2School of Computer Science & Technology, Nanjing University of Posts

and Telecommunications, Nanjing, China

Corresponding author:

Linfeng Liu, School of Computer Science & Technology, Nanjing

University of Posts and Telecommunications, Nanjing 210003, China.

Email: [email protected]

Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 3.0 License

(http://www.creativecommons.org/licenses/by/3.0/) which permits any use, reproduction and distribution of the work without

further permission provided the original work is attributed as specified on the SAGE and Open Access pages (http://www.uk.sagepub.com/aboutus/

openaccess.htm).

becomes the typical physical layer technology in UWSNs. Nevertheless, the acoustic channel is charac- terized by its limited bandwidth, high bit error rate, path loss, motion-induced Doppler shift, and so on.

5

(3) Underwater sensor nodes are liable to move or be damaged

6,7 due to the water current caused by external

forces such as earthquake, tide, wind velocity, under- water creature touch, or strong electromagnetic inter- ference, which will lead to a dynamic network topology.

8 All above unique characteristics are possible

to result in the damage or inaccurate localization of beacon nodes.

Generally, a typical architecture for three- dimensional (3D) UWSNs is shown in Figure 1, where there are three types of nodes: surface buoys, beacon nodes, and ordinary sensor nodes.

9 The surface buoys

can get the coordinates from their equipped global positioning system (GPS). The beacon nodes are sub- merged underwater, and thus GPS is not feasible for beacon nodes. They should communicate with the sur- face buoys to obtain their X-Y plane coordinates (the Z-axis coordinates can be estimated via the pressure sensors). Besides, the beacon nodes also help ordinary sensor nodes do localizations. However, the localiza- tions by beacons are usually unavailable due to the bea- con damage (such as the hit from water current and the touch of underwater creatures) and signal interference in the underwater environment. Hence, some beacon nodes probably provide the inaccurate coordinate information for the localizations of ordinary nodes, and these beacon nodes are referred to as the error bea- cons. To this end, this work proposes an error beacon filtering algorithm (EBFA), which can effectively improve the localization accuracy through filtering out the error beacons.

Several error beacon nodes which cannot provide accurate references for ordinary nodes should be fil- tered out to avoid the aggravation of localization error. However, it is very difficult to find the error beacon nodes because we usually have no pre-knowledge about

the error ones, and thus the error beacon nodes can be filtered out according to the mutual localization results among all beacon nodes. This work is the extension of our early work,

10 the main differences between the two

papers are as follows: (1) the algorithm is given more illustrations, (2) the theoretical analysis of algorithm has been improved and extended, and (3) more simula- tions have been done and supplied.

Related works

Localization schemes have been extensively investi- gated in wireless sensor networks or UWSNs, and these schemes can be divided into two groups: anchor-based schemes and anchor-free schemes.

11 In the anchor-

based schemes, the beacon nodes get their coordinates in advance through carrying GPS receiver or even they are artificially pre-configured. The beacon nodes broadcast periodically their coordinate information. Subsequently, the ordinary nodes estimate their coordi- nates by calculating the distances or angles to the near- est beacon nodes, especially, some measurement techniques such as received signal strength indicator (RSSI), time of arrival (TOA), and time difference of arrival (TDOA) are usually utilized in the process. Zhang et al.

12 proposed a multi-anchor nodes colla-

borative localization (MANCL) algorithm. First, the well-localized nodes within one hop are prone to become the reference nodes if the ordinary nodes can- not receive four beacon signals, and the selection criter- ion is related to the energy, trust value, and distance. Then, an improved Euclidean distance estimation method is adopted to localize the ordinary nodes. Finally, the remaining un-localized ordinary nodes complete their localizations with the help of two-hop anchor nodes.

The anchor-free schemes 13

determine the ordinary nodes’ coordinates through exploiting the connectivity or distance information among nodes, and thus the assistances of beacon nodes are unnecessary. The anchor-free schemes are especially suitable for the net- works where nodes are hardly deployed, such as the battlefield environment or special warfare environment. Generally, the network protocol without beacon nodes is more complex than that with beacon nodes.

In addition, the anchor-based schemes can further be classified into the static beacon node localization and mobile beacon node localization. In Cheng et al.,

14

an underwater positioning scheme (UPS) is proposed, where ordinary nodes record the receiving time of bea- con messages, and then the time difference is trans- formed into the range distance after receiving four beacon messages. Finally, the ordinary nodes apply the trilateration method to estimate their own coordinates. UPS reduces the communication overhead and does

Figure 1. A typical architecture of UWSNs.

2 International Journal of Distributed Sensor Networks

not require the time synchronization, so the cost of UPS is relatively low. In Rahman et al.,

15 with the help

of a mobile beacon node, Cayley–Menger is used to determine the node coordinates. The distance between nodes is measured through combining the radio and acoustic signals which are free from the phenomenon of multi-path fading. In Zhang and Liang,

16 the dis-

tance between nodes is calculated by a new ranging method named round-trip time of flight (RTOF), and then the ordinary nodes complete localizations using an improved particle swarm optimization (PSO) algo- rithm, which adds a Gaussian decreasing inertia weight and a kind of competition mechanism. This scheme can improve the localization accuracy and localization effi- ciency with less beacon nodes. Nonetheless, the above literatures do not take into account the mobility of underwater nodes, which are only applicable in static underwater networks.

The mobility issue in node localization has also been reviewed. Ojha and Misra

17 used spatially correlated

mobility pattern of UWSNs to estimate the node coor- dinates. In the initial stage of localization, there are only three beacon nodes. If an ordinary node cannot get enough information, it will assume that the node moves according to some specific rules, and the future coordinates can be easily predicted. When the ordinary nodes can communicate with at least three beacon nodes at the original and predicted positions, and then the coordinates of ordinary nodes can be determined. The outstanding advantage of this algorithm is that it is energy efficient as a result of the ‘‘silent localization.’’ In Zhu et al.,

18 a localization scheme based on mobility

prediction for UWSNs is introduced. The localization process is divided into two parts: the beacon nodes uti- lize the modified covariance algorithm to estimate their prediction models to reduce the position error, while the ordinary nodes choose the well-localized reference nodes to get their positions and speed by a node-selection strat- egy. The algorithm increases the localization coverage and decreases the localization error compared with scalable localization scheme with mobility prediction (SLMP) algorithm.

19

But the prediction gives a poor accuracy especially when the underwater environment is hostile. A multi- hop location (MLA) in UWSNs is also proposed in Zhu et al.,

20 where the routing nodes are introduced to

solve the problem of isolated nodes. First, the shortest paths from beacon nodes to ordinary nodes are found through a greedy approach. Subsequently, the shortest paths are fitted into a straight distance using the cosine method. Finally, the trilateration is repeatedly per- formed to localize the ordinary nodes. This algorithm has much higher localization accuracy than determined maximum likelihood (DML) algorithm.

21

Some researchers also take notice of the measure- ment errors in localization process. Liu et al.

22

combined the time synchronization and the node locali- zation, which corrects the bias in the range estimation and improves the propagation delay in estimation when the stratification effect of underwater medium is con- sidered. In addition, in order to further increase the localization accuracy, an advanced tracking algorithm interacting multiple model (IMM) is employed to han- dle the mobile case. Wu and Li

23 proposed an

improved underwater acoustic network localization algorithm, which considers the measurement error caused by the sound velocity distortion and signal refraction. It uses an improved linear difference method to correct the measurement offset, which improves the localization accuracy. Simultaneously, a strategy simi- lar to the greedy algorithm reduces the redundancy of the calculation results.

However, none of these works take the issue of error beacons into consideration. If beacon nodes move, its coordinate information will become obsolete or even wrong. Therefore, the ordinary nodes will be positioned more inaccurately under the assistance of these error beacons. To deal with the error beacon problem, this article proposes the EBFA based on K-means cluster- ing. The coordinate of each beacon is calculated by an improved trilateration, and then the error beacons are filtered out by the K-means clustering algorithm.

24

EBFA based on K-means clustering

Suppose that plenty of sensors nodes are deployed in a 3D underwater space D2IR3. A small part of beacon nodes provides error reference coordinates. Let BN denote the beacon nodes set, where BN = {b1, b2, ., bn}. The number of the beacon nodes is n. Suppose that each node can transmit and receive messages with enough power and obtain the distance between nodes through RSSI.

Algorithm description

The EBFA based on K-means clustering will calculate the coordinate of each beacon by an improved trila- teration, and then the distance differences exceeding a distance threshold are divided into two categories by K-means clustering method. Afterward, the beacon with the maximum positioning error is filtered out. All error beacons will be found until the distance differ- ences are lower than the threshold. The following steps explain EBFA in detail:

Step 1. The first beacon bi is selected randomly and five nearest beacons of bi are found. Four of the nearest beacons will be used to position bi, and this process will be repeated C45 times (C

4 5 denotes the

number of combinations), which produces C45 coor- dinate results of bi. In detail, suppose b1, b2, b3, and

Liu et al. 3

b4 position bi, then the coordinate of bi is calculated as

L b1, b2, b3, b4 bi

=

L b1, b2, b3 bi

d b1, b2, b3 bi

+ L

b1, b2, b4 bi

d b1, b2, b4 bi

+ L

b1, b3, b4 bi

d b1, b3, b4 bi

+ L

b2, b3, b4 bi

d b2, b3, b4 bi

1

d b1, b2, b3 bi

+ 1 d

b1, b2, b4 bi

+ 1 d

b1, b3, b4 bi

+ 1 d

b2, b3, b4 bi

ð1Þ

where L b1, b2, b3 bi

= (x b1, b2, b3, b4 i , y

b1, b2, b3, b4 i , z

b1, b2, b3, b4 i ) rep-

resents the coordinate of bi, and d b1, b2, b3 bi

represents the mean distance from b1, b2, and b3 to bi.

Step 2. The distance difference between the esti- mated coordinate and real one of bi is computed as

D b1, b2, b3, b4 bi

=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x

b1, b2, b3, b4 i � xi

� �2 + y

b1, b2, b3, b4 i � yi

� �2 + z

b1, b2, b3, b4 i � zi

� �2q ð2Þ

Moreover, each beacon should reserve a variable X[bi] to record the number of found error beacons, and X½bi� 0 initially.

Step 3. The distance differences exceeding a thresh- old will be divided into two categories: accurate and inaccurate, by the K-means clustering method (set K = 2).

25

Step 4. The localization results of a beacon are com- pared. If the beacon is considered inaccurate, then X½bi� X bi½ �+ 1. The beacon with the maximum positioning error is filtered out and marked as an error beacon. Step 5. Steps 1–4 are repeated until all distance dif- ferences are lower than the threshold after removing the found error beacons.

The following example describes the EBFA algorithm briefly. Suppose that there are 10 nodes b1, b2, ., b10 and each node requires the localization. As

shown in Figure 2, b2, b3, b4, b5, and b6 position b1, and b5, b6, b7, b8, and b9 position b2. The distance differ- ences exceeding the threshold occur in the localization process of b1 and b2. The categories of beacons are shown in Table 1, where D

(�) bi denotes the distance differ-

ence between the estimated coordinate and real one, and ( � ) indicates the beacon set for the localization of beacon bi. d is a predefined distance threshold, which is set according to the network environments. The value of X[bi] is given in Table 2.

As is shown in Table 2, the value of X[b2] is the max- imum, so b2 is marked as an error beacon. The remain- ing nine beacons repeat the process after removing b2.

Time complexity of EBFA

The time complexity of EBFA is mainly contributed by the Step 1 to Step 4. The time complexity of Step 1 is O(n2); the time complexity of Step 2 is O(n); the sorting time complexity of Step 3 is O(2nt);O(n), where t is the count of iterations; and the time complexity of Step 4 is O(n). The time complexity of EBFA is O(n2), which is acceptable.

Figure 2. Example diagram of node localization.

Table 1. The diagram of beacons classification.

Beacon Neighboring aided beacons D (�) bi

d Accurate/inaccurate category

b1 b2, b3, b4, b5 11.5 0.5 Inaccurate category b2, b3, b4, b6 20.5 0.5 Inaccurate category b2, b3, b5, b6 0.25 0.5 Dbi �d b2, b4, b5, b6 0.15 0.5 Dbi �d b3, b4, b5, b6 0.6 0.5 Accurate category

b2 b5, b6, b7, b8 16.3 0.5 Inaccurate category b5, b6, b7,b9 22.4 0.5 Inaccurate category b5, b6, b8, b9 0.34 0.5 Dbi �d b5, b7, b8, b9 0.1 0.5 Dbi �d b6, b7, b8, b9 0.09 0.5 Dbi �d

4 International Journal of Distributed Sensor Networks

Mathematical analysis

In general, the deployment of sensor nodes tossed from the air to the ground obeys the normal distribution. Let X-coordinate, Y-coordinate, and Z-coordinate of the ordinary node obey the following distribution: X � N(mx, dx), Y � N(my, dy), and Z � N(mz, dz), respec- tively. The real coordinate and the estimated coordi- nate of the beacon node bi are denoted by (xi, yi, zi) and (xi, yi, zi), respectively. The real distance and the esti- mated distance between beacons is di and di, respec- tively. The impact of the error beacons is analyzed as follows.

First, the real coordinate of the localized node is cal- culated from the following equation set

(x � xa)2 + (y � ya)2 + (z � za)2 = da 2

(x � xb)2 + (y � yb)2 + (z � zb)2 = db 2

(x � xc)2 + (y � yc)2 + (z � zc)2 = dc 2

8>< >: ð3Þ

where a, b, and c are the aided beacons. (xa, ya, za) denotes the real coordinate of the beacon a, and da is the real distance from a to the coordinate (x, y, z) (the coordinate of the localized node). Thus, the real coordi- nate of the localized nodes is expressed as

x

y

z

0 @

1 A= �A�1 �B + �Dð Þ ð4Þ

where A = 2(xa � xc) 2(ya � yc) 2(za � zc) 2(xb � xc) 2(yb � yc) 2(zb � zc) 2(xa � xb) 2(ya � yb) 2(za � zb)

0 @

1 A,

B = xa

2 � xc2 + ya2 � yc 2 + za2 � zc2 xb

2 � xc2 + yb2 � yc 2 + zb2 � zc2 xc

2 � xb2 + ya2 � yb2 + za2 � zb2

0 @

1 A, and

D = dc

2 � da 2

dc 2 � db

2

db 2 � da

2

0 [email protected]

1 CA. D indicates that the localization

results are related with the distance. A and B show the localization results are also related with the coordinates of aided beacons. To simplify the formulations, let

A = a1 b1 c1 a2 b2 c2 a3 b3 c3

0 @

1 A, then we obtain that

�A�1 = �A �

A �� �� = 1A�� ��

b2c3 � c2b3 c1b3 � b1c3 b1c2 � c1b2 c2a3 � a2c3 a1c3 � c1a3 a2c1 � a1c2 a2b3 � b2a3 b1a3 � a1b3 a1b2 � a2b1

0 @

1 A

where j�Aj= a1(b2c3 � c2b3)� a2(b1c3 � c1b3) + a3(b1c2 �c1b2).

The error from trilateration algorithm executions

should be taken into account. Set di = di + jid (i = a, b, c), hence the measured coordinate of the localized

nodes is calculated as �x �y �z

0 @

1 A= �A�1(�B + D). Therefore,

the localization error of trilateration algorithm is

D�x D�y D�z

0 @

1 A= �x�y

�z

0 @

1 A� xy

z

0 @

1 A= �A�1 �B + Dð Þ� �A�1 �B + �Dð Þ

= �A�1 D � �Dð Þ ð5Þ

Moreover, the localization error is expressed by the scalar ERf

ERf = D�x 2 + D�y2 + D�z2 ð6Þ

Formula (6) transforms the localization error into a scalar, and then the error can be analyzed from each axis. Let xi = xi + jix, yi = yi + jiy, and zi = zi + jiz (i = a, b, c). Thus, the measured coordinate of loca-

lized nodes is expressed as x0

y0

z0

0 @

1 A= A�1 B + Dð Þ.

Therefore, the localization error is written as

Dx0

Dy0

Dz0

0 @

1 A= x

0

y0

z0

0 @

1 A� xy

z

0 @

1 A= A�1 B + Dð Þ� �A�1 �B + �Dð Þ

ð7Þ

Then the localization error expressed by the scalar ERs is rewritten as ERs = Dx

02 + Dy02 + Dz02.The sign of ERs � ERf are discussed from Case I and Case II.

Case I. If jix = jiy = jiz = 0, one gets ERs � ERf = 0 easily; Case II. If jix 6¼ 0, jiy 6¼ 0, and jiz = 0, then

A = �A + 2(jcx � jax) 2(jcy � jay) 0 2(jcx � jbx) 2(jcy � jby) 0 2(jbx � jax) 2(jby � jay) 0

0 @

1 A= m1 e1 0m2 e2 0

m3 e3 0

0 @

1 A,

D = �D + G = �D + jcd

2 � 2dcjcd + 2dajad � jad 2 jcd

2 � 2dcjcd + 2dbjbd � jbd 2 jbd

2 � 2dbjbd + 2dajad � jad 2

0 @

1 A,

and

Table 2. The diagram of value of X[bi].

X[b1] X[b2] X[b3] X[b4] X[b5] X[b6] X[b7] X[b8] X[b9] X[b10]

2 4 2 2 3 3 2 1 1 0

Liu et al. 5

B = B + F = B +

jax 2 � 2xajax + 2xcjcx � jcx2 + 2yajay + 2ycjcy � jcy2

jbx 2 � 2xbjbx + 2xcjcx � jcx2 + 2ybjby + 2ycjcy � jcy2

jax 2 � 2xajax + 2xbjbx � jbx2 + 2ybjby + 2ycjcy � jcy2

0 [email protected]

1 CA:

B is transformed into the sum of the real beacon value �B and the beacon error F. D is transformed into the

sum of the real distance value �D and the distance error

G. In order to facilitate the analysis, B + �D and F + G

are jointly analyzed. Define B + �D = n1 n2 n3

0 @

1 A and

F + G = f1 + g1 f2 + g2 f3 + g3

0 @

1 A, so Formula (7) can be rewritten

as

Dx0

Dy0

Dz0

0 @

1 A= A�1 B + Dð Þ� �A�1 �B + �Dð Þ= A�1

�B + �D + F + Gð Þ� �A�1 �B + �Dð Þ ð8Þ

Furthermore, ERs � ERf can be expressed as (Dx02 � D�x2) + (Dy02 � D�y2) + (Dz02 � D�z2). To observe the sign of ERs � ERf , (Dx0)2 � (D�x)2, (Dy0)2 � (D�y)2, and (Dz0)2 � (D�z)2 are verified respectively in the Appendix 1. Without loss of generality, the error from Y-axis and Z-axis is ignored temporarily, and thus there are g2 = g3 = 0, n2 = n3 = 0, and f2 = f3 = 0, then there is Dx02 � D�x2.0, Dy02 � D�y2.0, and Dz02 � D�z2.0, the derivation and proof of which are also given in the Appendix 1.

Therefore, ERs � ERf � 0. That is, the localization error with error beacons is higher than that without error beacons which have been filtered out.

Simulations

EBFA is evaluated by observing the performance varia- tion when adopting different model parameters (such as the number of beacon nodes and the number of error beacons) and by comparing EBFA with other algo- rithms. Table 3 shows the values of the parameters.

The accurate discovery of the error beacons is extremely critical to the localization accuracies of ordinary nodes. This simulation measures the number of found error beacons with different number of beacon nodes. As shown in Figure 3, three plots (the proportion is assigned as 20%, 30%, and 40%, respectively) are observed. The plot with a larger rate is higher than the other plots because there are more error beacon nodes to be found. Besides, the number of found error beacons also grows with the increase in the total number of beacon nodes and the proportion of error beacon nodes. When the number of real error beacons is fixed, more beacons will bring

more accurate judgments for the error beacons, and thus more error beacons can be found. However, when there are excessive error beacons, the number of found error beacons is much different from the number of real error beacons. When the proportion is set 20% and the number of beacons reaches 60, the number of found error beacons (about 13) is approximately equal to the number of real error beacons (60 3 20% = 12), which indicates that EBFA can effectively detect almost all error beacons espe- cially when the number of beacon nodes and the propor- tion of error beacons are not very large.

In Figure 4, the variance metric denotes the stability of the number of found error beacons. The variance is expressed as

Pcount i = 1 (a½i�� (

Pcount j = 1 a½j�=count))

2=count, where count is the number of execution times, and a[i] is the number of found error beacons at the ith execution. The plot with a larger rate has a sharper fluctuation than the others, which is attributed to the fact that the variable number of the found error beacons becomes larger when there are more error beacons. Moreover, more error beacons give rise to a larger localization error of ordinary nodes as well. Note that some of the error beacons are still not being detected, especially

Table 3. Simulation parameters.

Parameter Description Value

Dj j Deployment space 100 m3100 m3100 m N Number of nodes 300 n Number of beacon nodes 60 K Number of categories 2 rate Proportion of error beacon

nodes in beacon nodes 20%

RCmax Maximum communication range

10 m

d Distance threshold 5 m Count Number of execution times 100

Figure 3. Number of found error beacons versus rates of error beacons.

6 International Journal of Distributed Sensor Networks

when the number of beacon nodes becomes larger, which is attributed to the random deployment of bea- cons and the original localization error.

In EBFA, the error beacons are filtered out accord- ing to a distance threshold, that is, the beacons with larger localization error are excluded. Therefore, the value of threshold has a significant influence on the per- formance of EBFA. Both Figures 5 and 6 illustrate the impacts of the threshold when rate is set 20%. In Figure 5, it can be found that when the threshold is set 10, the number of found error beacons is approximately equal to the number of real error beacons because the proper setting of threshold helps EBFA to find the error beacon nodes accurately. Nevertheless, when the threshold is too small, some accurate beacons are also mistakenly labeled as the error beacons. Moreover, when the threshold is too large, most of the error bea- cons are not found because EBFA cannot differentiate the accurate beacons and error beacons by a large threshold. Consequently, the proper setting of the threshold is important to the EBFA performance.

As depicted in Figure 6, the threshold has an obvious impact on the variance number of found error nodes as well. In general, with the increase in the number of bea- con nodes, the plots of variance continue to rise up. When the threshold is 5, the variance number is larger than the others, this is because the number of found error beacons is larger, and thus the deployment of error beacons becomes more random accordingly.

Figure 7 compares the number of found beacons of EBFA, Centroid, and Trilateration. Apparently, EBFA overcomes Centroid and Trilateration absolutely. In Centroid, almost all nodes are labeled invalid. This is because Centroid has a stricter requirement for the node distribution and it assumes that the nodes obey the uniform distribution, which does not tally with the random deployment in our simulations. Therefore, it is hard to find error beacons exactly. In Trilateration, all nodes involved in localization will be marked as error beacons provided that this localization result is wrong. Hence, some of the error beacons cannot be found. In EBFA, most of the error beacons can be found in an iterative way, and probabilities of falsely marking the beacon nodes are very small, but the found error bea- cons are usually a bit more than the real ones. The rea- son is that some accurate beacons served for the localizations of the neighboring error beacons are also prone to be regarded as the error beacons.

As shown in Figure 8, the localization error denotes the coordinate derivations of ordinary nodes from the beacon localizations. The results indicate that EBFA can achieve the lowest localization error after effectively filtering most of the error beacons. Moreover, the scal- ability of EBFA is also better than those of the other two algorithms, this is because the proportion of the number of found error beacons stays the same approxi- mately, as shown in Figure 3.

In summary, when the proper distance threshold is set, EBFA can accurately filter out most of the error

Figure 4. Variance versus number of beacons.

Figure 5. Found error beacons versus distance threshold.

Figure 6. Variance versus different thresholds.

Liu et al. 7

</

Collepals.com Plagiarism Free Papers

Are you looking for custom essay writing service or even dissertation writing services? Just request for our write my paper service, and we'll match you with the best essay writer in your subject! With an exceptional team of professional academic experts in a wide range of subjects, we can guarantee you an unrivaled quality of custom-written papers.

Get ZERO PLAGIARISM, HUMAN WRITTEN ESSAYS

Why Hire Collepals.com writers to do your paper?

Quality- We are experienced and have access to ample research materials.

We write plagiarism Free Content

Confidential- We never share or sell your personal information to third parties.

Support-Chat with us today! We are always waiting to answer all your questions.