3D Many-objective DV-hop Localization Model With NSGA3

: Wireless sensor location is a challenging task issue in the Internet of Things (IoT). Distance vector - hop (DV - hop) algorithm provides a range - free positioning scheme, but its position prediction method based on least square method brings a large positioning error. To overcome this issue, this paper constructs a three - dimensional (3D) many - objective positioning model. Specifically, we consider many factors, such as the error characteristics of the estimated distance, the distribution characteristics of nodes and the computational cost. Based on these factors, we propose a many - objective 3D - DV - hop positioning model, and propose a data preprocessing strategy and outlier removal strategy. Finally, a fashionable many - objective optimization algorithm is employed to solve the model. The experimental results show that the model proposed in this paper has great advantages in accuracy and robustness, and is superior to the current single and multi - objective positioning model.

population is generally randomly generated, which leads to a slow convergence speed. 3) Outlier interference. Not all estimated distances between anchor nodes and unknown nodes will improve the positioning performance. On the contrary, some estimated distances will bring noise, and corresponding anchor nodes are outliers. Such as Fig. 1, denotes the base station node, denotes the corresponding estimated distance between a common node and . As shown in Figure 1, the nodes detected by 1 . are red, and the nodes detected by other nodes are blue. In contrast, the distribution of blue points is more concentrated, and the red nodes are more divergent. This means that red nodes are more likely to be outliers.

Fig. 1. Outlier detection in WSNs.
To solve these problems, we propose a many-objective positioning model and employ NSGA3 [33] to solve this model. The main innovations and contributions of this paper are as follows:  To address the issue that the existing positioning model does not fully consider the factors affecting the positioning accuracy, we propose a many-objective positioning model. The model includes the estimated distance of anchor nodes, weight design, and the error distribution characteristics of estimated distance.
 To reduce computational cost, a data preprocessing is proposed in this paper. It accelerates the population convergence by improving the quality of the initial value of the population.
 To reduce the noise interference caused by outliers, an outlier removal strategy is designed in this paper. It makes the model insensitive to outliers and increases the robustness of the model.

3D DV-hop positioning model with optimization algorithms
The DV-hop (3D) positioning algorithm consists of three processes, including broadcast, distance measurement and positioning. However, the 3 rd stage of the DV-hop algorithm is usually replaced by the optimization algorithm, and the process is as follows.
1 st stage: broadcast. Each BSN broadcasts packets with location and hop count to the network, and other nodes forward the packets and record the minimum hop count (which refers to the minimum hop count between nodes). It should be noted that the hop count information increases as the number of packets forwarding increases. 3 rd stage: positioning. In this stage, because the least squares method has large positioning errors, scholars use various optimization algorithms to replace it. And the positioning model is as follows: where, 1k F denotes the traditional objective function, n denotes the number of BSNs, ( , , ) Apparently, the positioning precision of the model is determined by the estimated distance ( ik d ), and since the terrain of the node deployment has a large impact on _ i Per dis , the estimated distance ( ik d ) is unstable. In order to reduce the sensitivity to ik d , Cai [15] proposed the second objective function, as follows: where, ik dis is calculated by Eq. (5) and (6).  Fig. 2 shows the distribution of nodes with peak topography and its sectional view. In Fig.2 (a), the sensor nodes (the number is 130) are following uniformly distribution in this area, which contain the base station nodes (BSNs, the number is 30) and common nodes (CNs). And we assume that Fig. 2   According to the node distribution of Fig. 2 (a), Table 1 can be obtained.  [27.44, 36.74] Table 1 shows the results of the average distance ( _ Per avgdis ), estimated distance and estimated errors. And we noticed that the value of lower bound of confidence error interval is larger than CR (25m). Emphatically, in literature [14], Wang find that in normal circumstances, the maximum estimated error value is CR. Therefore, we can know that for node ( 2 BSN ), node ( 1 BSN ) is an outlier. More generally, when the node is placed on the opposite side of the peak, the probability that the BSN is an outlier will increase. Unfortunately, the existing method is helpless for solving this problem. To effectively deal with this problem, this paper presents a many-objective positioning model and outlier removal strategy to optimize CNs.

Data preprocessing strategy
Data preprocessing is frequently used in industrial manufacture, especially in data mining and image processing. And in the positioning of WSNs, in order to reduce the time consumption of the algorithm, scholars usually choose the algorithm with fast convergence to solve the model. However, this approach breaks the balance between convergence speed and ability. Therefore, to reduce the time cost and ensure the convergence ability of the algorithm, a data preprocessing strategy is proposed in this paper.
In 2019, Wang [14] found that in the 2D region, the error between the estimated and actual distance is following Gaussian distribution. According to this finding, an assumption can be obtained, which is, if the error of the 3D region is also following the Gaussian distribution, this conclusion (where, the conclusion represents the error follows Gaussian distribution) can improve the initialization of the population. Therefore, we have statistical statistics on the error characteristics of the 3D region, and the results are shown in Fig. 3 and Table 2. Where, the number of BSN is 30, total nodes number is 130, CR is 25 and runtime is 500.  Fig. 3 shows the test sets and the error characteristics (where, the test set is one of 500 tests.). From the Fig. 3 (a) and (b), we can know that the error is consistent with expectations and is following Gaussian distribution. Table   2 shows the fitting function and relevant parameters. Obviously, in different networks, the values of  are tending to 0, and the  tending to 1 3 CR . On this basis, the DV-hop algorithm uses the least squares method to solve the position of the node, and the error characteristics of DV-hop follow this discipline.   Fig. 4 (a), the population of Fig. 4 (b) has a better initial value at initialization, which significantly increases the convergence rate of the population.  Table 1, we can acquire that the probability that the BSN is an outlier is increased as the number of hops between it and the CN increases, which can lead to an increase in error. To reduce the interference of outliers on positioning, this paper develops an outlier removal strategy. And then, to determine the number of outliers, we performed a simulation error test on BSNs, where, the parameters are the same as in subsection 4.1, and the result is show in Fig. 5. Apparently, the ratio within the standard deviation interval is tending to 2 3 . Therefore, this paper regards nodes outside the interval as outliers, and its number accounts for 1 3 of the total. And the removal process is shown in Algorithm 1. The removal of outliers is achieved by indexing the estimated distances, weights, and BSNs that satisfy the condition.

Many-objective Localization model
In 2019, Cai [15] developed the multi-objective DV-hop model, she took the average hop distance as the second optimization objective in the positioning model. It is worth noting that the error characteristics and weight are also important factors affecting positioning accuracy. In this paper, we present the other two objectives based on the error characteristics and weight.
According to section 3.1, we know that the error of estimated distance is roughly following the Gaussian distribution, 1 3 (0, ) error CR : . Therefore, A Gaussian error correction is performed as follows.
where, _ ik Dis GE represents the estimated distance with the Gaussian error correction, The third objective function is defined as follows: Generally, there are different effects on nodes when hop counts between BSNs and CNs are not consistent. It means that an appropriate weight design will bring the gain of positioning performance. Therefore, this paper considers the weight model as one of the objectives in the construction of a many-objective model. The weight model is as follows.
where, i  denotes the weight value and it is calculated as follows, which means that the weight decreases with the hop count increase.
Consequently, the many-objective positioning model is constructed as follows.

Many-objective DV-hop with NSGA3
This section describes the algorithm for solving the model, NSGA3, which is developed for handling more than three objectives optimization problems in practical. In this paper, it is applied to solve the many-objective DV-hop positioning model, and the flow chart is shown in Fig. 6. where, t represents the current iteration, P(t) denotes the parent population in the iteration t, Q(t) denotes the child population in the iteration t, R(t) represents the population before non-dominant sort. And it is continued in this paper that the reference-point-based non-dominated sorting strategy in the literature.

Parameters and evaluation indicator
In this section, the parameter values of NSGA3 and 3DDV-hop are shown in Table 3. Particularly, in stand NSGA3, the value of population is 156 and maximum iteration is 500 when the objective function is 4. However, it is obvious obtained from the table 3, the number of populations and maximum iteration are much smaller than the standard NSGA3 because the data preprocessing strategy is adopted. And it can reduce computational cost distinctly. In addition, the test set of this paper is the same as Fig. 3, including random and multimodal network. And the evaluation indicator is as follows. (13) where,  To test the performance of the algorithm (NSGA3-3DDV-hop), abundant simulation is executed in this section, and compare with other algorithms. Table 4 shows the variation of errors in different CRs. It is not difficult to find that the error value is ∞ when the number of nodes is 25 in the random network. The reason for this phenomenon is that there are some sensor nodes in the deployed nodes that cannot be detected by other nodes. It makes the position of some nodes impossible to calculate, so that the error tends to ∞. In Fig. 7, with the CR increases, the APE of different algorithms generally shows a downward trend. And the NSGA3-3DDV-hop not only reveals prominent precision advantages, but also has more excellent robustness than other algorithms.     Table 5 and Fig. 8 shows that the APE in different nodes. NSGA3-3DDV-hop emerges huge advantage in positioning accuracy. From Fig. 8, it is obvious that the APE of NSGA3-3DDV-hop is distinctly better than single-objective model (GA-3DDV-hop and OCS-3DDV-hop) and 3DDV-hop in random network. And it has outstanding performance than other algorithms when the nodes larger than 95 in multimodal network.   Table 6 and Fig. 9 shows that the APE with different BSNs. And we can obtain that the APE of NSGA3-3DDV-hop is always superior to other algorithms in different number of BSNs and network. That is because the outlier removal strategy (ORS) is adopted to reduce the interference of the outliers (which denotes bad BSNS) during the positioning process.

The PE and confidence error intervals
This section shows the confidence error interval and the error (PE) of the node. Fig.10 provides the error distribution boxplot of different algorithms; apparently, NSGA3-3DDV-hop has the lowest error interval in different network. To reveal its performance more clearly, this paper performs the confidence bounds analysis on simulated data. And Table 7 demonstrates the standard deviation and confidence bounds (where, the parameter settings are the same as in Table 3). In random network, the confidence bounds of NSGA3-3DDV-hop is similar to OCS-3DDV-hop, and in multimodal network, it has preeminent confidence bounds.

The convergence analysis
In order to test the validity of the data preprocessing strategy (DPS), this paper analyzes the convergence of the algorithm with and without DPS, and Fig. 11 provides the convergence curve. Obviously, we find that the NSGA3-3DDV-hop with DPS has a prominent advantage compare to the NSGA3-3DDV-hop without DPS. In Fig.  11 (a), the highlight denotes the global search area, which caused by unconscious initialization, like the Fig. 4 (a). The large search area causes a phenomenon that the initialization error is large, and then local search makes the error tend to be stable. Conversely, in Fig. 11 (b), the highlight denotes the unstable search area, which caused by purposeful initialization (DPS), like the Fig. 4 (b). And the error of DV-Hop is following Gaussian distribution; which determined that the initial population node with DPS has a higher probability of approaching the actual position. Therefore, from Fig. 11(d), when the value of iteration is 1, the NSGA3-3DDV-hop with DPS has lower error; subsequently, the error slightly increased and stabilized. Additionally, from (a)-(f), it is obvious that when the population is 30, the convergence error of NSGA3-3DDV-hop without DPS is higher than NSGA3-3DDV-hop with DPS. And when the population increases, although the positioning error is reduced, it is still inferior to NSGA3-3DDV-hop with DPS. And it which indicates that DPS plays an important role in reducing the time complexity and improving the positioning accuracy.

The time complexity
This section compares the time complexity between different algorithms (where, OB denotes the number of objectives). From Eq. (11), the algorithm optimizes the nodes not as an entirety; therefore, it can find the location of a specific node. And as an added result, it also inevitably brings high time complexity compare with the other three contrasting algorithms. However, on the one hand, due to the adoption of preprocessing strategy, the population and the maximum number of iterations are reduced, resulting in lower time complexity than NSGA-III. On the other hand, compare with existing models, NSGA3-3DDV-hop has an excellent superiority in positioning precision. Therefore, this increase in computational complexity is acceptable in practical applications.

Conclusion
Generally, there are many elements that have a momentous impact on positioning accuracy, which contains estimated distance, computational cost, localization model and potential outliers. These influencing elements did not receive the attention of scholars during the positioning process, resulting in an inaccurate positioning accuracy. To handle these problems, this paper develops a many-objective positioning model, which considers the commonness ( _ ) and individuality ( _ ) of nodes, error characteristics and weight. Moreover, the data preprocessing and outlier removal strategies are adopted to speed up the convergence of the population and improve the positioning accuracy. Finally, the model is embedded in a fashionable NSGA3 many-objective optimization algorithm, i.e. NSGA3-3DDV-hop. The simulation results show that the NSGA3-3DDV-hop has prominent advantages compared with the single and multi-objective models.
In conclusion, the many-objective model has brilliant performance overall in this paper. However, there are also some shortcomings, such as Table 5. And we will analyze the reasons and optimize it with efficient strategies.

Conflict of Interest
Penghong Wang declares that he has no conflict of interest. Hangjuan Li declares that he has no conflict of interest. Xingjuan Cai declares that she has no conflict of interest.

Ethical approval
This article does not contain any studies with human participants performed by any of the authors.

Informed consent
Informed consent was obtained from all individual participants included in the study.