A Monte Carlo localization method based on differential evolution optimization applied into economic forecasting in mobile wireless sensor networks

In the era of big data, economic forecasting is crucial. The nodes’ localization in wireless sensor networks (WSNs) refers to the process of obtaining their own or monitoring the geographic position of the object in a certain way [1]. It is important to obtain the sensor node’s location, and the monitoring data will be meaningless without location information. For instance, in precision agriculture, the sensor nodes can gather data of light intensity, humidity, and temperature, which must be accompanied by the coordinates of the collectors [2]. Without such positional information, the observer cannot match the data with the region and make an appropriate decision. Meanwhile, the exact location information of sensor nodes are of great help in improving the efficiency of network routing [3]. Since the wireless sensor network consists of a large number of sensors and the topology often change especially for the environment with mobile nodes, each sensor nodes need to be equipped with a positioning system, such as GPS. Due to the high cost of GPS, it is not suitable for low-power and low-cost requirements of sensor nodes [4]. In addition, for some special application scenarios (such as shopping malls), the positioning performance of GPS will be affected [5]. Generally, there are two types of nodes in wireless sensor networks, which are called as anchor node and blind node. Anchor nodes, which are usually configured manually or equipped with a GPS receiver to obtain their location information, can obtain position coordinates by themselves. However, the proportion of anchor nodes in all sensor nodes is relatively small. Comparatively, blind node can only acquire its position information by using the localization algorithm.

The rest of the paper is structured as follows. The motivation for this work is discussed in Section 2. In Section 3, we derive Monte Carlo localization methods based on differential evolution optimization (MCL-DE) for valid samples in mobile wireless sensor networks. In Section 4, a comparative performance evaluation is carried out. Finally, concluding remarks and future work are given in Section 5.

So far, the research of node’s localization algorithm in wireless sensor networks has been widely carried out. The main purpose of sensor localization is to determine the location of sensors in WSNs via noisy measurements, and most of the methods for localization can be classified into geometrical techniques, multidimensional scaling, stochastic proximity embedding, convex and nonconvex optimization, and hybrid. In range-based measurement localization, the major task is to find the accurate position in non-line-of-sight (NLOS) paths. These range-based measurements may include time-of-arrival (TOA) [6], time-difference-of-arrival (TDOA) [7], angle-of-arrival (AOA) [8], and received signal strength (RSS). After evaluating the distance between the nodes, the position of the blind node can be obtained based on three edge-measuring or maximum likelihood methods [9].

Range-based localization requires additional hardware and power consumption, so nodes can achieve accurate positioning resolution. However, the demand to reduce hardware dependency and energy cost has been the focus of academia and industry, and some researchers also proposed a range-free localization algorithm [10]. Usually, range-free location algorithms demonstrate poor performance in the aspect of positioning accuracy than the range-based localization algorithm, but it does not need additional hardware support and can meet many requirements in the scenarios with rough localization effect. In [10], an indoor localization strategy for mini-UAV in the presence of obstacles is proposed, in which the signal propagation state is identified according to the prior probability and statistics of TDOA and RSS measurements. In [11], a NLOS identification and weaken algorithm with machine learning is proposed to identify and weaken the NLOS error by means of support vector machine (SVM), which can employ a large number of data samples to train the SVM classifier. A voting matrix is constructed to weaken the error of non-line-of-sight and obtain the candidate position in accordance with the error characteristics of LOS measurements and NLOS measurements. Then, the residual weighted method is used to obtain the final positioning results. Base on the range distance in each sampling period, Cui et al. [12] use multidimensional scaling localization algorithm to evaluate the location of the target and fit the result of the estimation by polynomial. The estimation results of current position can be corrected effectively, and the method is proven to achieve high positioning accuracy in indoor environment.

In recent years, interacting multiple model (IMM) combined with filtering technology has become a hot research topic. Chen et al. [13] combine IMM and extended Kalman filtering to achieve accurate positioning in NLOS environment. Zhang et al. [14] propose a Kalman filter model based on interacted multiple objectives to filter the measured distance under the LOS/NLOS mixed environment, in which the IMM algorithm is applied to filter the distance, and then the extended Kalman filtering algorithm is used to realize the positioning. Under the IMM framework, Ru et al. [15] employ hidden Markov random field to solve the nonlinear Bayesian estimation problem and improve the positioning accuracy. Nevertheless, the above methods are put forward in the premise of accurate NLOS error parameters. But in the actual environment, the parameters of NLOS error are usually unknown. In [16], an Advanced DV-Hop localization algorithm is proposed to reduce the localization error without requiring additional hardware and computational costs. The hop-size of the anchor node is obtained base on the distance measurement of unknown nodes, and the weighted least square algorithm is introduced to decrease the inherent error in the estimated distance between the anchor and an unknown node. In [17], a mixed localization algorithm for wireless sensor networks based on APIT is proposed to deal with the problem of low localization accuracy with dense distribution of beacon nodes and low coverage ratio in the sparse case.

As a fundamental requirement in Internet of Things (IoT) and other wireless sensor applications, localization awareness is an important part of network design. Due to the complexity of related algorithms, researchers in this area have acquired significant progress but still exists quite a few deficiencies. Most of the solutions are static localization algorithms for node’s positioning, but the performance of those algorithms in mobile wireless sensor network is unsatisfactory. In the real scene, such as target tracking, the sensor nodes are usually mobile and the node localization mechanism should be suitable for mobile wireless sensor networks. Considering that the mobile sensors change their locations frequently over time, Monte Carlo localization algorithm utilizes the moving characteristics of nodes and employs the probability distribution function (PDF) in the previous time slot to estimate the current location by using a weighted particle filter. However, it also has the problem of insufficient number of valid samples, which further affects the node’s localization accuracy. Therefore, it is necessary to increase the number of valid samples to improve the localization accuracy under low anchor node density.

In this section, we will conduct the experiments to compare our algorithm with the traditional method, for example, RMCL [18] and DLS [19] in terms of the localization precision, sample size, maximum velocity, and the density of anchor nodes. In order to verify the indicators objectively, the parameters of the experiments are set to be identical in different scenarios. The specific parameters are set in details as follows: the number of sensor nodes is 320, the deployment area is 500 × 500 m²; the transmission radius of sensor node is r = 50 m; the number of valid samples is N = 50. The motion process of the node employs RWP model [20], and the maximum moving speed v_max = 0.2r. Besides, the main parameters of differential evolution are set as: η=0.8, ρ₀ = 0.4, and ρ* = 0.9.

To evaluate the effects on the localization algorithm by setting different parameters, the average localization error and the number of candidate samples are regarded as key indicators. Among them, the number of candidate samples reflects the number of times as the sampling process is executed for obtaining the valid samples. Usually, the lesser the number of candidate samples is, the higher the success rate of sampling can obtain.

Figure 1 shows the localization performance of the samples of different generations with differential evolution.

Once the sampling box and a certain number of initial samples are acquired, the proposed differential evolution algorithm will be executed, which uses the sample weight as objective function. Since the observation results from the normal nodes only involve anchor nodes, the sample weight is equal to 0 or 1. Next, the derived sample weight is equal to 1 and it can be remained in the next generation. Otherwise, the parents’ sample can only reserve in the next generation. After several generations being produced, plenty of samples can be satisfied with the filtering conditions, most of which is close to the actual position of the node. As can be seen from Fig. 1, compared with the initial samples, the samples of subsequent generation with differential evolution are closer to the actual location of the node.

Figure 2 shows the comparison of convergence of the algorithm. As can be seen from the result, the localization accuracy of all algorithms demonstrates absolutely low-quality in the initial stage. With the time goes, the number of valid samples is getting more and more, and the accuracy is corrected by iteration. Therefore, the localization error decreases and fluctuates in a relatively stable range. During the whole convergence phase, we can observe that the average error in MCL-DE shows approximately 21.23% lower than RMCL, about 35.42% lower than DLS. Generally, the speed and direction of the node are unknown except for the maximum velocity. The maximum velocity and previous position can be utilized to predict the constrained sampling area in current time.

When the maximum moving speed is slow, the size of the sampling box based on the moving velocity is relative small. Apparently, the actual position of the nodes will be of high probability to fall outside of the sampling box and the localization results will be very poor. With the increase of the maximum velocity, the node will be able to establish a better sampling box, which can contain the actual position with high probability, to reduce the localization error. However, excessive value of the maximum velocity of node can constraint the size of sampling box and result in the increase of the average node’s localization error and it can be seen from Fig. 3. In the process of the whole trend, the MCL-DE algorithm always keeps the advantage over other methods in localization accuracy.

The number of the candidate samples depends on the size of sampling box, and Fig. 4 demonstrates the relationship between the maximal moving velocity and the number of candidate samples. For DLS, since the effect of sampling is entirely determined by the maximum velocity of the node, the increase of the maximum velocity of movement will extend the range of the sample box as well as increase the number of final candidate samples. However, the sampling constraint box of MCL-DE is mainly composed of one-hop and two-hop neighbor anchor nodes, and the size of the sample box will be primarily restricted by the anchor nodes. That is, when the maximum velocity of motion is increased, the number of candidate samples will not change obviously due to the stable observation results from the anchor nodes.

Next, the experiment is conducted to verify the effect on localization error in different density of anchor nodes. As can be seen from the Fig. 5, the samples being selected optimally can approximate to the actual position, and the results with weighted average calculation make the average localization error as small as possible. From the results, the trend of average localization error in all algorithms shows very clearly. Comparatively, the average location error in different densities of anchor nodes in MCL-DE is less than RMCL and DLS. Especially, when the density of anchor nodes is set to 1, the average localization error of MCL-DE can be less than RMCL about 16.59% and DLS about 33.43%.

Moreover, the increase of the anchor node’s density will bring about more constraint conditions for filtering process. As a result, the filtration rate of samples will increase. As shown in Fig. 6, with the increase of anchor node’s density, the number of candidate samples in RMCL and RMCL grows remarkably. In contrast, the growth of DLS algorithm is greater because it does not make use of the anchor node’s information at the sampling stage. Due to fixed samples for initial operation, the number of candidate samples remains constantly.

After obtaining a certain number of initial samples, we select the weight of the sample as the objective function of the optimization and present the differential evolution algorithm to obtain valid samples rather than perform initial sample filtering and resampling. Finally, the node position is estimated by making the sample close to the actual location of the node instead of being filtered out. The simulation result demonstrates that the proposed algorithm provides a better position estimation with less localization error. In the future, we will study and validate the signal strength indicator to improve the performance of our method in aspect of computational and communication costs. And we are also planning to discuss the challenges and open research issues related to the parameters and focus on the localization accuracy of range-free schemes.