In view of the traditional increment localization method, only the heteroscedasticity caused by the error accumulation is considered unilaterally a kind of incremental localization algorithm based on multivariate analysis is proposed. The algorithm combines the feasible weighted least squares (FWLS) in the multianalysis with the canonical correlation regression (CCR) and utilizes the FWLS to solve the heteroscedasticity caused by the error accumulation in the process of incremental localization; in the process of estimation, the CCR is used to solve the topology problems between the original beacon nodes and new beacon nodes. The simulation results show that the method can not only effectively restrain the influence caused by the accumulative errors but also can adapt to the different node topological shapes, so as to improve the positioning accuracy of the nodes.
In many application problems related with sensor network, location information of nodes is of great importance to the monitoring activity of the whole network, which plays a critical role in many applications [
Incremental localization is a kind of low energy consumption positioning method which can effectively solve the coverage rate of the monitoring area; through outward extension in turn, each node is localized successively. Due to the successive localization, the previous estimation error will be bound to affect the following estimated accuracy. Such error accumulation inevitably leads to the inconsistency of the variance between the previous error term and the following localized error term; such phenomenon is also known as heteroscedasticity [
Localization in complex environment.
In addition, there is a problem that error of locations through estimation has directivity, for example, in Figure
In pervious incremental algorithms, most incremental localization algorithms are used to adjust heteroscedasticity during localization process and assume that heteroscedasticity is only monotonically increasing but fail to take deployment environment and networking features of sensor network into account. Sensor network is a kind of multi-hope network with relatively worse deployment environment, and incremental pattern of its heteroscedasticity is complicated and diversified for incremental localization algorithm. Furthermore, as same as concurrent localization algorithm, the accuracy of incremental localization is influenced by topology quality of beacon nodes also, and multicollinearity problem [
In a concurrent localization process, distance-coordinates formula is generally transformed into a form of
In the presence of heteroscedasticity, the positions of localization data in location estimation are different; smaller variance of error term of data means higher confidence level of residuals, while bigger variance of error term means lower confidence level of residuals. Therefore, as to the estimation of location in coordinates under the circumstance that heteroscedasticity exists, it usually uses weighted least squares method to discriminate different residuals [
So, variance of error term of distance-coordinates formula exhibits heteroscedasticity which is expressed as
among which
Assume,
Then, the heteroscedasticity of the error term is eliminated, and it is easy to learn that
In order to obtain the optimal solution, we must make
It is assumed that
Obviously, if the row vector of
Through Schwarz inequality [
FWLS is a feasible method which is able to overcome the problem that cannot be implemented by WLS due to the unavailable weight. FWLS uses residuals attained at each computation as weight matrix; therefore, real weight values can be acquired in computation process, and procedure of FWLS algorithm is shown in Algorithm
(1) Firstly, it is essential to estimate the model through OLS method and obtain the estimated value error (2) Utilize the square of the residual term as the (3) Obtain the next-order estimated value and residual value by WLS (4) Go back to Step 2, until the number of iterative times meet the number of times according to the algorithm requirements.
It can be noted that the FWLS algorithm is in marching iteration; the derivation of the optimal estimated value
In concurrent localization algorithm, beacon nodes have a very large influence on final location estimation and possibly cause significant errors when beacon nodes relations are collinear or approximately collinear [
As to incremental localization algorithm, it can use PCA [
Canonical correlation analysis (CCA) is a kind feasible and powerful multivariate analysis method especially appropriate for processing and analysis of two correlated data. At the same time, it is a kind of descending dimension method similar to PCA and is also able to remove some noise information that contains collinear information through recombination of data like PCA. CCA pays more attention to data processing and analysis of correlated data; for this reason, it is more proper for regression algorithm and has higher regression accuracy than PCA.
For the equation
Suppose that there are two groups of data,
in which
The correlation function
To solve this optimization problem of (
Differentiating (
To obtain the optimal solution, let (
Multiply both sides of formula (
Given
Here, the solution of CCA was translated into generalized eigenvalue-eigenvector problem of two matrixes whose scales are
Formula (
The literature [
The existence of multicollinearity usually brings seriously adverse effects on model estimation, testing, and prediction. During localization estimation, multicollinearity not only exists in concurrent localization but also exists in incremental location estimation process. For this reason, we add canonical correlation regression method into FWSL algorithm to acquire optimal prediction direction through correlation analysis of input and output variables and dimension reduction processing then use FWLS method to resolve problems caused by heteroscedasticity. Because the procedure of FWLS-CCR localization algorithm is similar to that of FWLS algorithm, its solution is carried out through iteration, and the algorithm process is shown in Algorithm
Input: Beacon nodes coordinates: Distance from beacon nodes to unknown nodes Output: Estimated coordinates of unknown nodes: (1) Beacon nodes deliver their location information outwards through controllable flooding, if unknown nodes acquire more than 3 beacon nodes, it firstly uses CCR to estimate locations of unknown nodes, and will stop if there is no rest unknown nodes, otherwise, will carry out next step. (2) Uses estimated location to estimate residual vector by the estimation formula: (3) Constructs covariance matrix through FWLS method residual vectors, (4) Uses newly-constructed covariance matrix to rewrite location distance equation, (5) According to new equation, uses CCR method to estimate locations of secondary beacon nodes. (6) If there still are some nodes which locations have not be estimate out in deployment area, skip to Step 2. (7) The algorithm will finish if there is no node to be estimated in deployment area, and it will output coordinates of unknown nodes.
Node localization process based on FWLS-CCR is shown in Figure
Example and phases of LE FWLS-CCR.
Obviously, the coordinates of node A can be calculated and estimated according to the beacon nodes of
In incremental localization approach, localization of nodes is implemented in batches. Owing to distance measurement error, there is a certain difference between the estimated value and the practical value of first level beacon node. As for second level beacon nodes, their estimated values are as well influenced by measurement error and the intrinsic error of first level beacon nodes. In addition, it is shown in Figure
In this Section, we compare the time complexities for localization as required by LE-FWLS-CCR and other popular incremental location estimation algorithms, namely, WLS-based location estimation (LE-WLS) proposed in the literature [
LE-FWLS-CCR: basically, the complexity of our algorithm is dominated by two parts: canonical correlation regression and feasible weighted least squares. The complexity of computing CCR is mainly determined by the core algorithm CCA, whose complexity is
LE-WLS: WLS-based location estimation method using the WLS method as the core algorithm, and the complexity of WLS is
LE-IILA: the complexity of SQP method is the main reason in LE-IILA algorithm. If the limited number of iterations, the computational complexity of SQP is
The literature [
This section will analyze and evaluate LE-FWLS-CCR localization algorithm on Matlab platform. In simulation experiment, it is supposed that nodes are deployed in a two-dimensional monitoring area and adopt transformation of RSSI signals to distance for matrix of distance among nodes. In order to compare impartiality of experimental results, this section adopts signal model proposed in the literature [
among which
Due to higher coverage of incremental algorithm, the experiment in this section mainly examines the accuracy of localization of nodes with ALE as evaluation basis, and the definition of ALE is as follows:
In the formula,
This experiment also compares the method proposed in this paper, the localization algorithm based on FWLS-CCR (LE-FWLS-CCR), with WLS-based location estimation (LE-WLS) proposed in the literature [
The experiments based on distance-measuring model have set four experimental scenes: random deployment nodes in square area, regular deployment nodes in square area, random deployment nodes in C-shape area and regular deployment nodes in C-shape area, in which C-shape area, is formed because of a bigger barrier, mainly used to evaluate localization performance with lager barrier, that is, in case of non-line-of-sight. In order to decrease the effect of single one experiment, each group of experiments will be repeated for 50 times in each scene, finally the average indicators of the 50 experiments will be reported. The experiments will examine accuracy of final localization results of unknown nodes with incremental quantity of beacon nodes. In these experiments, the valid communication radius of nodes is supposed to be 60 m.
Regular deployment of nodes in monitoring area principally aims to explore effects of collineation of beacon nodes on localization accuracy, while regular deployment in C-shape area is used to observe effects of non-line-of-sight caused by barriers in monitoring area on localization accuracy.
In this group of experiments, regular deployment of nodes is within a
Localization results of regular deployment in square area.
The circumstance that there is a barrier in regular deployment area was described in Figure
Localization results of regular deployment in C-shape area.
It can be seen from Figures
Because incremental localization method is used to locate nodes through gradually increment, as shown in Figure
Figure
Average localization error of regular deployment.
Random deployment is more close to actual situation. The experiments in this scene are used mainly is to discuss whether this algorithm is proper for various actual situations or not. In the same way, experiments about random deployment are classified into two groups, in C-shape area with barrier and in square area without barrier. In this group of experiments, there are 200 nodes randomly deployed in a
Figure
Localization results of random deployment in square area.
Figure
Localization results of random deployment in C-shape area.
Figure
Average localization error of random deployment in square area.
This paper uses actually measured data set provided by Neal Patwari of Utah State University. The experiment was arranged in a standard office area that is a
Comparisons of average localization errors based on actual RSSI measurement data.
Wireless communication radius (m) | LE-WLS ALE | LE-IILA ALE | LE-FWLS-CCR ALE |
---|---|---|---|
6.5 | 99.1% | 46.6% | 20.8% |
7 | 81.8% | 39.6% | 19.6% |
7.5 | 70.23% | 41.3% | 18.4% |
8 | 83.51% | 49.12% | 15.51% |
Figure
Localization results of actually measured data.
This paper combines FWLS method and CCR method in a localization process that uses FLWS method to solve problems in location estimation caused by heteroscedasticity and uses dimensionality reduction algorithm CCR in multivariate analysis to deal with topology between original and newly added beacon nodes and error accumulation problems. The results of many groups of experiments indicate that the method proposed in this paper can effectively resolve heteroscedasticity, accumulative error, and multicollinearity problems, and its localization results are stable and have higher accuracy than previous incremental localization methods.
The paper is sponsored by Natural Science Foundation of China (61005008); Provincial University Natural Science Research Foundation of Jiangsu Education Department (11KJD510002, 12KJD510006); Natural Science Foundation of Jiangsu (BK2012082).