RI-MDS: Multidimensional Scaling Iterative Localization Algorithm Using RSSI in Wireless Sensor Networks

To improve the feasibility and the convenience of localization methods for wireless sensor networks, a localization algorithm RI-MDS (RSSI-based iterative-multidimensional scaling) is proposed. The RI-MDS method is centralized and mainly focuses on improving localization accuracy. It collects RSSI vectors as ranging basis and combines the metric MDS method and the nonmetric MDS method to accomplish the relative localization. Then it uses the maximum likelihood method in affine transformation to transform the relative coordinates to absolute ones. Our method has no need for additional equipment on the WSN nodes. Simulation and field experiments show that the average localization error and the localization error ratio of the RI-MDS method are relatively lower and thus they are more feasible.


Introduction
Wireless sensor networks (WSNs) are very important for the IOT (Internet of Things). They are ad hoc networks via wireless communication and consist of a large number of smart nodes [1]. WSNs are widely used in applications such as medical, military, and environment surveillance. We deploy WSNs for information perception, collection, and management and send information to the observer. Usually, the sensor nodes are deployed randomly and the location of the nodes cannot be known. But in many applications the positions of the sensor nodes must be estimated first. As a result, the sensor localization is one of the important research tasks.
Due to the high cost and high energy consumption, it is impossible to use a GPS (Global Positioning System) on every sensor node. In general, we get the positions of sensor nodes by indirect estimation. Localization techniques for sensor nodes can be classified into range-based techniques and range-free techniques [2]. Range-based techniques estimate location by the received signal strength indicator (RSSI) [3], the time of arrival (TOA) [4], the angle of arrival (AOA) [5], and so forth. They perform localization via geometric calculation. The range-based method using RSSI is the most feasible and suitable due to the hardware overhead, the energy saving, and the technical stability. But rangebased localization methods have physical limitation and measurement error. Range-free localization methods do not use additional devices. So, range-free localization methods estimate the positions according to the network connectivity, and the localization output is an interval with considerable error. According to the difference of localization reference, the localization algorithm can be classified into absolute positioning scheme and relative positioning scheme. The former produces global standard coordinate, and the latter produces local relative coordinate. Each method has its own pros and cons. The multidimensional scaling (MDS) localization method can fulfill both schemes according to the configuration of network.
The MDS algorithms also can be classified into metric MDS and nonmetric MDS (NMDS) according to whether it needs ranging or not [6]. The diversity matrix of metric MDS method comes from ranging, and the data are accurate comparatively. So it can reflect the similarity and the diversity of the entity quantitatively. While the nonmetric MDS has no special demand on the distance between nodes. Usually, the nonmetric MDS can gain a good performance only if the similarity or diversity of the node distance satisfies 2 International Journal of Distributed Sensor Networks monotone increasing or monotone decreasing. It is found that the MDS localization methods always have significant error. The nonmetric MDS can get the approximate optimal solution via iterations, so it has higher tolerance to error. But the localization accuracy of nonmetric MDS greatly depends on initial configuration.
In this paper, we proposed a new type of MDS localization method-RSSI-based iterative-MDS (RI-MDS) method. It constructs the dissimilarity matrix RSSI values and combines the metric MDS scheme with the nonmetric MDS scheme. The initial position coordinates are calculated via metric MDS method and construct the relative position relationship via iterative nonmetric MDS. Then we transform the relative localization coordinates to the absolute localization coordinates using mathematical method. With the rational cooperation of these two kinds of MDS methods, the localization accuracy was improved. In addition, our RI-MDS needs no special device equipped. Extensive simulation experiments and field experiments show that the performance and the practicability of RI-MDS method are good. The main contributions can be summarized as follows: (1) improving the localization accuracy by combining the metric MDS method and the nonmetric MDS method and (2) improving the robustness by introducing the maximum likelihood estimation (MLE) method to the affine transformation process.
The rest of this paper is organized as follows. A brief introduction to the state of the art of the MDS localization methods is presented in Section 2. In Section 3, the details of our RI-MDS method are given. Performance analysis of our RI-MDS algorithm is provided in Section 4, and conclusions are drawn in Section 5.

Related Work
The MDS technology is widely used in psychometrics and psychophysics. It itself is a data analysis method on a group of data structures which represent distances using geometric figure. It is first applied in localization of wireless sensor networks by Shang et al. [7]. They illustrated how the MDS can solve the localization problem by taking a pile of suspended small ball as an example; that is, we called their method classical MDS algorithm (MDS-MAP). Later, Shang and Ruml improved the MDS-MAP in [8]. The MDS localization method has strong robustness and higher accuracy even in sparse network. Furthermore, the MDS method can fulfill both the absolute localization and the relative localization. Lots of MDS-like WSNs localization methods have been proposed in [9][10][11]. The MDS method itself belongs to the centralized computational algorithm. It runs high strength computations on the sink nodes or on the backend server. There are two ways to improve it: (1) to develop new MDS methods based on distributed computation and (2) to improve the localization accuracy of the MDS method. In view of the fact that the MDS method has a high computational time complexity, [12,13] use the distributed MDS methods. They divided the whole network into several clusters. In the cluster, the head node fulfills the MDS method to work out the node local positions and converts the local position coordinates to the global position coordinates by using the coordinates of some communal nodes. The distributed MDS method increases the network overhead and causes accumulative errors, although it can improve the performance of localization algorithm. In this paper, we focus on how to improve the localization accuracy of centralized MDS method. In [14], Amar et al. divide the sensor nodes into two groups according to the connectivity. Then they adopted MDS method in the group with the higher connectivity and extended the localization process to another group by the least square method. The error accumulation makes the localization accuracy that should be improved. Ma et al. [15] showed that the distance of nodes can be estimated by the hop counts which can be represented by a real number. However, the improvement of localization accuracy is trivial. Other researchers applied the range-based MDS algorithm using the mobile anchor nodes [16]. Although the redundancy of dissimilarity matrix is increased, the anchor node with mobility is necessary, and the design of mobile path is thus complex. A linear WLS-based weighted MDS estimating method is proposed in [17], which can get an accuracy close to Cramér-Rao lower bound (CRLB) for sufficiently small noise conditions; it is modified to operate in partially connected WSN scenarios. The literature [18] estimates the distance of indirect communication nodes using the shortest possible theoretical distance instead of Dijkstra algorithm. Although it improves the dissimilarity matrix accuracy to some extent, the improvement of localization accuracy is limited because of the lack of the subsequent error eliminate process. The most important research questions relating to centralized MDS method are how to generate the relative coordinates which can really reflect the monotonous relationship of nodes and how to eliminate the error produced in the previous step.

The Basic Idea of RI-MDS.
The range-free MDS methods take advantage of the relationship in which the distances and the diversities of nodes are monotonous. It improves localization accuracy via iterative computation. Range-free methods have the great tolerance to the measurement error via iterative calculations. But the localization accuracy of the range-free MDS methods tends to be influenced largely by initial configuration. The value of RSSI suffers the impact of environment, but we can acquire the relatively actual initial location information of nodes by simplicity treatment. The impact of environment can be eliminated in some degree via iterative calculation of NMDS. The localization error also can be further eliminated in the coordinate translation process by applying maximum likelihood estimation (MLE).
According to the analysis mentioned above, we proposed the RI-MDS localization method which adopted the RSSI values to construct the dissimilarity matrix and utilized the NMDS method to calculate the relative coordinates. Sensor nodes collect the RSSI values times to construct the distance vectors and transmit the distance vectors to the sink node. Sink node constructs the dissimilarity matrix using the distance vectors. By combining metric MDS method with the nonmetric MDS method, sink node acquires the relative coordinates of sensor nodes, and then the relative coordinates are transformed to the absolute coordinates. The algorithm flow of RI-MDS is depicted in Figure 1.

Algorithm Design and Description. The description of RI-MDS algorithm follows.
Algorithm Input. It is RSSI distance vectors of nodes.
Algorithm Output. It is the absolute location coordinates of nodes.
The steps of RI-MDS algorithm are as follows.
Step 1. Sensor nodes collect the pairwise RSSI values for times to form the distance vector and transmit these distance vectors to sink node.
Step 2. Sink node calculates the average RSSI values and acquires the ranging values according to (1). The dissimilarity matrix [ ] was constructed based on the ranging values: where denotes distance and and are constants which are related to the antenna gain and environment. The ranging value is influenced by the environment factor greatly, but it can be degraded by the subsequent NMDS iterations.
Step 3. According to the known distance information, the average distance of single hop ave can be acquired. Distances of nodes which cannot communicate directly are estimated by the shortest path algorithm; then, the integrity of [ ] is provided by adding these distances. Calculating the relative coordinate 0 by using the classical MDS algorithm on the dissimilarity matrix [ ] is acquired in Step 2. We can get the distance matrix [ ] based on 0 .
Step 4. Calculate the relative position coordinate of each node via nonmetric MDS algorithm.
Step 5. Select several anchor nodes (more than 3 noncollinear nodes in two-dimensional space and more than 4 noncoplanar nodes in three-dimensional space). Convert the relative coordinate of each node to the absolute one by utilizing the affine transform based on the absolute coordinates of these anchor nodes.
Step 4 is a key step; it adopted nonmetric MDS method to compute the coordinates of nodes iteratively such that the STRESS is satisfied. The detailed description follows.
Input of Step 4. The threshold value of the stress coefficient and the initial position coordinate 0 are acquired in Step 3.
Output of Step 4. It is the relative position coordinates of each node.
Initialization. Initialize the iterations as 0.
The process of Step 4 follows.
Step 4.  Step 4.2. The iterations counter = + 1; calculate the new coordinate of each node by utilizing the steepest descent method: where stands for the number of nodes and represents the iteration increment of the steepest descent method. Properly speaking, the value of should be recalculated in every iteration. But what the value of can influence is not the result of algorithm but is the convergence. So we adopted = 0.2 [20]. 4 International Journal of Distributed Sensor Networks The last step of the RI-MDS is coordinate transformation.

= cos
± sin + , Acquire the estimated values of arguments sin , cos , , and by applying the MLE method on (5). These values are in the sense of least mean square error. And we get the linear transformation equation: = cos ± sin + , Acquire the absolute coordinates of the other -nodes via applying (6) successively.

Algorithm Time Complexity
Analysis. RI-MDS method has seven main steps as mentioned above, covering the Dijkstra shortest path algorithm, classical MDS method, nonmetric MDS method, affine transformation and coordinate transformation, and so forth. Due to the RSSI information can be acquired easily. The RI-MDS method can be fulfilled conveniently. The time complexity analysis of these steps follows.
(1) The time complexity of Dijkstra shortest path algorithm is ( 2 ).
(3) The time complexity of the nonmetric MDS is ( 4 ). This step is the most time-consuming step, but in line with the variation of the stress coefficient the number of iterations changed too. So, the RI-MDS has great flexibility.

Simulation Environment.
To verify the effectiveness and the performance of RI-MDS, simulation experiments are conducted. Due to so many matrix calculations in MDS algorithm, the MATLAB simulator is suitable. For the performance comparison under different nodes deployments, we adopt two kinds of nodes deployments; the random deployment and grid deployment are tested, respectively. In the grid deployment, the nodes are not set at the cross of the gird line preciously but with small deviation to the grid cross.

Wireless Propagation Model.
The classical wireless propagation model with random interference is employed to simulate the RSSI variation. The model mentioned above is described as follows: where stands for the distance between a couple of nodes, 0 is the reference distance, and denotes the path-loss exponent, which can be from 2 to 4 generally. ( 0 ) represents the received signal strength where the distance of sender and receiver is 0 ; ( ) stands for the same meaning when the nodes distance is . is the Gaussian white noise, which can be generated by Gaussian random function where the average value is 0 and the variance is . By utilizing above model, we can simulate the RSSI values where the nodes distance is given.

Simulation Experiment Design and Evaluation Indicators.
The most commonly used indicators are the average localization error (ALE) and the localization error ratio (LER) and these two indicators are given in formulas (8) and (9), respectively: where denotes the number of nodes, represents the measuring coordinate of node , denotes the real coordinate of node , and stands for the communication radius. There are four parts in our experiments. the network connectivity is varied from 8 to 12 with step size 2, and the amount of anchor nodes is 4. The ranging error varies from 5% to 10% with step size 5%.

Simulation Result and Analysis.
Simulation results ( Figure 2) suggest that, along with the network average connectivity increase, the LER of the RI-MDS algorithm declines under both of these two deployments. And the LER of the RI-MDS under grid deployment is better than under random deployment, especially in the low connectivity. By analyzing the principle of RI-MDS algorithm, we know that there are regular distribution characteristics in the grid deployment. And the diversities of nodes are mainly reflected in connectivity. The connectivity of nodes is stable and sufficient; hence the diversity of nodes is similar in redundancy, and the localization accuracy is relative even. So we draw the conclusion mentioned above.
Furthermore, the localization accuracy of civil GPS system is about 10 meters. While most of wireless sensor nodes have a communication radius about from 30 m to 50 m, the localization algorithm can be regarded as being practicable only when the LER is less than 25%. The LER gets the practicability when the network average connectivity is 8 and 14, respectively, under the two deployments as we can see in Figure 2.
The results of second simulation experiment are shown in Figures 3 and 4, respectively. We can draw a conclusion from Figures 3 and 4 that the LER of the RI-MDS method is 5% to 10%, which is lower than the IMDS algorithm regardless of what deployment is used. We can also note that the LER decline trend of the IMDS is slowed down when the network The average localization error rate (%) The average network connectivity Random deployment Grid deployment The average localization error rate (%) The average network connectivity RI-MDS random deployment IMDS random deployment average connectivity reaches more than 16. However, the LER of RI-MDS method is declining along with the connectivity increasing in the same situation. The feature mentioned above became more obvious under the grid deployment. We can deduce, based on the algorithm principle analysis, that since the IMDS method belongs to metric algorithm, and merely takes the ranging distance as the diversity for all nodes. So the data redundancy is increased along with the network connectivity increased under the situation that more location relationships are introduced. And result in localization accuracy increased, whereas there are lots of data collision in the diversity relationships because the diversity information has measure error introduced by simulation. The more the network connectivity is increasing, the more the redundancy collision is increased. The data collision offset The average localization error rate (%) The average network connectivity RI-MDS grid deployment IMDS grid deployment the localization accuracy improvement by introducing more redundancy at a certain extent. Thus the localization accuracy cannot be improved significantly. But the RI-MDS method uses the distance value as the input; thus it reduces the system error in transforming the RSSI to distance. In addition it applies the nonmetric MDS iterative algorithm to eliminate the error introduced by the data redundancy increasing. Thus, the localization accuracy of RI-MDS can still improve obviously under the same situation. Figure 5 indicates that the localization accuracy increase of RI-MDS is small whatever the network connectivity is, under the condition that the amount of anchor nodes is more than 5. Meanwhile, the localization performance is satisfactory when the network connectivity is higher than 10, whatever the amount of anchor nodes is. Therefore, we can draw a conclusion that our RI-MDS scheme can work well with anchor sparse network as long as the network connectivity is not significantly low.
As shown in Figure 6, the localization accuracy of RI-MDS declines with the ranging error increase. However, the performance of RI-MDS is good enough in the case that the network connectivity is higher than 10. We draw the same conclusion mentioned above. Average localization error rate (%)

Number of anchor nodes
Network connectivity is 8 Network connectivity is 10 Network connectivity is 12 Network connectivity is 14 Average localization error rate (%) Network connectivity 5% ranging error 10% ranging error 15% ranging error 20% ranging error gridding with 60 cm edge length. From N1 to N9 denote the position of 9 sensor nodes, respectively. All of the nodes have the same configuration and run the same program prewritten by Flash Programmer. The field experiment process is described as following.
Step 1. Input the real position coordinates of all nodes by PC data processing program. Assign the beacon nodes which used to transform the relative coordinates to global coordinates.
Step 2. The sink node broadcasts network construction packets, and then all of the nodes join the network, respectively, according to the distribution picture.  Step 3. Sink node schedules the nodes from N1 to N9 and sends the ranging packets in turns. In each round, only one node sends the ranging packet. The other nodes record the RSSI value when receiving the ranging packet.
Step 4. All of the nodes submit the data collected in Step 2 after every node has sent the ranging packet. Then, the sink node transmits these data to the data process program on PC terminal.
Step 5. The data process program calls the RI-MDS algorithm to calculate the localization coordinates and display these coordinates via GUI.
The field experiment repeats 10 times and the number of beacon nodes is 4. We adopt the absolute localization error of every time to make comparison. The LER, maximum localization error, and the minimum localization error are shown in Figure 8.
The field experiment of [21] is based on the same hardware platform and the same scene; the typical value of LER in [21] ranged from 2 m to 6 m, while the LER ranged from 1.5 m to 4 m when using the RI-MDS method. By comparison, it is shown that the RI-MDS method has a better performance and higher localization accuracy.
By comparing the experimental data, if we convert the average localization accuracy of field experiment to the LER value, the value is better than the LER in simulation. Because the number of beacon nodes is nearly one half of the total node numbers and the network size is so small that all nodes can communicate with others, the dissimilarity matrix is full of diversity data. The localization method should not have to supplement the diversity data by the shortest path algorithm, and therefore the introduced error is reduced. So, the localization accuracy of field experiment is better than that in simulation.
So, it is feasible to improve the localization accuracy by some computational cost sacrifice, due to the computational performance raised along with the hardware price declining. The RI-MDS algorithm would be of more economic advantage when the network scale and the localization area are extended continuously. However, the localization information collection process needs the sink node scheduling, and the time complexity is raised with the increase of network dimension. So the node energy consumption and the algorithm feasibility have to meet the challenge with the network scale increasing. We must adopt some particular strategies to solve this problem in the large-scale network mentioned above.

Conclusion
We proposed the RI-MDS localization method based on RSSI ranging. The RI-MDS method takes full advantage of both metric MDS algorithm and nonmetric MDS algorithm. The algorithm design is presented in detail, and the theoretical analysis of RI-MDS method is proposed. We verified the algorithm performance via simulation and performed the field experiments in real WSNs to illustrate the actual performance of RI-MDS. All the experiments prove that the RI-MDS method is feasible and practicable and its performance is better than IMDS algorithm. The RI-MDS method can be applied to the WSNs in the centralized computation applications. In the future, we are going to develop the clusterbased RI-MDS method to meet the demand of high accuracy in anisotropic network.