Mobile Tracking Based on Support Vector Regressors Ensemble and Game Theory

Abstract

A two-step tracking strategy is proposed to mitigate the adverse effect of non-line-of-sight (NLOS) propagation to the mobile node tracking. This strategy firstly uses support vector regressors ensemble (SVRM) to establish the mapping of node position to radio parameters by supervising learning. Then by modelling the noise as the adversary of position estimator, a game between position estimator and noise is constructed. After that the position estimation from SVRM is smoothed by game theory. Simulations show that the proposed strategy results in the more accurate performance, especially in the harsh environment.

1. Introduction

The location service is foundation of pervasive computing, intelligent transportation, and application of WSN, so the mobile location technologies have drawn a lot of attention [1, 2]. Existing range-based location techniques include ToA, TDoA, and RSSI. For range-based location techniques, if the line-of-sight (LOS) propagation exists between the mobile node and anchor nodes, high location accuracy can usually be achieved using the conventional location algorithms [3]. However, since the direct path from the mobile node to anchor nodes can be blocked by buildings and other obstacles, the transmitted signal could only reach the receiver through reflected, diffracted, or scattered paths called non-line-of-sight (NLOS) paths. The NLOS propagation generally leads to a positive bias in the estimation range and causes a serious error in the mobile location estimation.

Nowadays many methods have been employed to mitigate the adverse effect of NLOS propagation. References [4, 5] have summarized the methods for static position systems. However, these methods are not effective for mobility tracking systems. Recently, the Kalman filtering, unscented Kalman filtering, and particle filter techniques are applied for range measurements smoothing and NLOS error mitigation [6–8]. The EKF-based algorithms are suggested in [9] as a promising alternative to range measurement for smoothing and mitigating NLOS error. A Kalman-based IMM smoother [10] is proposed to estimate the range between the mobile node and the corresponding anchor nodes in the mixed LOS/NLOS conditions. The method in [11] proposed a one-order hidden Markov chain to simultaneously model the transition of the LOS/NLOS condition and the mobile node position.

Although the above algorithms mitigate the influence from the NLOS errors and improve the location accuracy to a certain extent, they always need to use ToA, TDoA, and RSSI to range, so it is hard to overcome the side effect from non-line-of-sight propagation. In order to further mitigate the position tracking error incurred by non-line-of-sight propagation, [12] dealt with mobile position tracking with support vector regression and game theory. To extend the above mobile tracking scheme, this paper uses support vector regressors ensemble (SVRM) to establish the map from radio parameters to node position by supervising learning. Then the position estimation from SVRM model is smoothed by the game theory.

Because the radio parameter is not considered as the distance measure, but as the feature to train the SVRM model, the side effect of non-line-of-sight is efficiently mitigated. Moreover, because of its ability to handle the uncertain and unmodeled noise, game theory can attain theoretically more accuracy of mobile tracking compared to Kalman-filter which can only deal with the Gaussian noise.

The paper is organized as follows. Learning the map from radio parameters to node position to obtain the support vector regressors ensemble is discussed in Section 2. In Section 3, the node tracking is implemented by game theory. The simulation results are given in Section 4, and the paper is concluded in Section 5.

2. Position Estimation with Support Vector Regressors Ensemble

2.1. Position Estimation with Support Vector Regressor

Given training set, $\mathrm{Tr}=\{({\mathbf{m}}_{\mathrm{1}},p_{\mathrm{1}}),({\mathbf{m}}_{\mathrm{2}},p_{\mathrm{2}}),\dots ,({\mathbf{m}}_l,p_l)\}$ , where $m\in {ℝ}^n$ denotes radio parameters from the anchors received by mobile node, for example, ToA or RSSI, and $p\in ℝ$ the coordinates of this mobile node.

The position estimation of mobile node is to learn the regression function from radio parameters to coordinates by supervising learning. After that, if we get the radio parameter taken by location-unknown node, we can calculate its coordinates by inputting its radio parameter into this function. Usually the regression is nonlinear, so to estimate the nonlinear function between the radio feature and node coordinates, we map the data m into the higher dimensional feature space ℱ where regression is linear between radio feature and node coordinates. Then in feature space linear regression can be formulated as follows:

$$\begin{array}{cc}& p=f\left(\mathit{m}\right)=\varphi {\left(\mathit{m}\right)}^T\mathit{w}+b,\hspace{1em}\text{where}\hspace{0.17em}\hspace{0.17em}\varphi :{ℝ}^n⟶ℱ,\mathit{w}\in ℱ,\end{array}$$

(1) where ϕ is a defined nonlinear mapping, b is a bias term, and w is a coefficient vector. For support vector regression, the objective is to minimize the structure risk by estimating the weight vector w and the objective function can be written as follows:

$$\begin{array}{cc}& \min \sum _{i=\mathrm{1}}^{n}g\left(f\left({\mathit{m}}_i\right)-p_i\right)+\frac{\mathrm{1}}{\mathrm{2}C}{\parallel \mathit{w}\parallel }^{\mathrm{2}},\end{array}$$

(2) where $g(·)$ represents ε-insensitive cost function with the following form:

$$\begin{array}{cc}& g\left(f\left(\mathit{m}\right)-p\right)=\{\begin{array}{cc}\left|f\left(\mathit{m}\right)-p\right|-\epsilon ,\hfill & \text{if}\hspace{0.17em}\left|f\left(\mathit{m}\right)-p\right|\ge \epsilon ,\hfill \\ \mathrm{0},\hfill & \text{otherwise.}\hfill \end{array}\end{array}$$

(3)

By Lagrange multiplier technique, the objective function (2) can be expressed, after using the slack variables, as follows:

$$\begin{array}{l}\underset{\alpha ,{\alpha }^{*}}{\max }\sum _{i=\mathrm{1}}^n\left[{\alpha }_i^{*}\left(p_i-\epsilon \right)-{\alpha }_i^{*}\left(p_i+\epsilon \right)\right]\\ \hspace{1em}-\frac{\mathrm{1}}{\mathrm{2}}\sum _{i=\mathrm{1}}^n\sum _{j=\mathrm{1}}^n\left({\alpha }_i^{*}-{\alpha }_i\right)\left({\alpha }_j^{*}-{\alpha }_j\right)\mathrm{Ker}\left({\mathit{m}}_i,{\mathit{m}}_j\right)\end{array}$$

(4) subject to constraints

$$\begin{array}{cc}& \sum _{i=\mathrm{1}}^n\left({\alpha }_i-{\alpha }_i^{*}\right)=\mathrm{0},\hspace{1em}\mathrm{0}\le {\alpha }_i,\hspace{0.17em}\hspace{0.17em}{\alpha }_i^{*}\le C,\hspace{0.17em}\hspace{0.17em}i=\mathrm{1,2},\dots ,n,\end{array}$$

(5)

$$\begin{array}{cc}& \mathit{w}=\sum _{i=\mathrm{1}}^n\left({\alpha }_i-{\alpha }_i^{*}\right)\varphi \left({\mathit{m}}_i\right),\end{array}$$

(6) where Ker represents the kernel function with $\mathrm{Ker}({\mathit{m}}_i,{\mathit{m}}_j)=\varphi {({\mathit{m}}_i)}^T\varphi ({\mathit{m}}_j)$ and ${\alpha }_i,{\alpha }_i^{*}$ are Lagrange multipliers.

Using (6), (1) can be expressed as

$$\begin{array}{cc}& p=f\left(\mathit{m}\right)=\sum _{i=\mathrm{1}}^n\left({\alpha }_i-{\alpha }_i^{*}\right)\mathrm{Ker}\left({\mathit{m}}_i,\mathit{m}\right)+b.\end{array}$$

(7) The bias b can be calculated using the point ${\mathit{m}}_l$ on the ε-margin since the prediction error for those points is known to be ${\delta }_l=\epsilon \mathrm{sign}({\alpha }_l-{\alpha }_l^{*})$ .

After solving the optimization problem (4) and calculating the b as described above, the output p that denotes the coordinates of the mobile node, corresponding to the new measurement $\mathit{m}$ , can be calculated by (7).

2.2. Constructing Support Vector Regressors Ensemble

To further improve the position estimation, support vector regressors ensemble is constructed by bagging. Bagging [13] is a meta-algorithm to improve classification and regression models in terms of stability and classification accuracy.

The ensemble is made of regressors built on a bootstrap sample of the training set [14]. A bootstrap sample is generated by uniformly sampling $L^{\prime }$ instances from the training set with L samples with replacement $(L^{\prime }\le L)$ . T bootstrap samples $S_t\hspace{0.17em}(t=1,\dots ,T)$ are generated and the base support vector regressor is trained and built from each bootstrap. A final regressor is built whose output is weighted average of the output of the base regressors. The algorithm of bagging used in this paper is shown in Algorithm 1. The corresponding resampling subroutine is shown in Algorithm 2.

Algorithm 1 (the bagging algorithm).

 

Input

• (i)

A training set $\mathrm{Tr}=\{({\mathit{m}}_1,p_1),({\mathit{m}}_2,p_2),\dots ,({\mathit{m}}_l,p_l)\}$ , where $m\in {ℝ}^n$ denotes radio parameters from the anchors received by mobile node and $p\in ℝ$ the coordinates of this mobile node;

(ii)

one-against-one support vector regressor;

(iii)

integer T specifying the number of iteration; integer $L^{\prime }$ specifying the number of bootstrap samples.

Trainin Phase

For $t=\mathrm{1},\dots ,T$ ,

• (i)

take a bootstrap sample $S_t$ with sample number $L^{\prime }$ from the training set $\mathrm{Tr}$ using the resampling subroutine;

(ii)

train support vector regressor with $S_t$ and receive the regressor $f_t$ ;

(iii)

add $f_t$ to the ensemble, E.

Output. For a testing set z, $p_f(z)=W_i^t{\sum }_{t=\mathrm{1}}^L{f}_t(z)$ .

Algorithm 2 (resampling subroutine).

 

Input. Weight vector $W_i^t$ and training set $\mathrm{Tr}=\{({\mathit{m}}_1,p_1),({\mathit{m}}_2,p_2),\dots ,({\mathit{m}}_l,p_l)\}$ .

Resampling Process

• (1)

Set the data index set $I_i^t=\varphi$ .

(2)

Normalize $W_i^t=W_i^t/{\sum }_{i=1}^L{W}_i^t$ and compute the cumulative sum vector of $W_i^t$ , ${\mathit{C}}_i$ .

(3)

Generate the uniformly distributed random $R_i$ .

For $i=1,\dots ,L$ , find maximum value max in ${\mathit{C}}_i$ which is less than $R_i$ and its index in ${\mathit{C}}_i$ is j.

If max is empty, $I_i^t=1$ ; else $I_i^t=j+1$ .

Output. ${\mathrm{Tr}}_t={\mathrm{Tr}}_i\mid i=I_i^t$ .

For a given bootstrap sample, an instance in the training set has probability $1-{(1-1/L^{\prime })}^{L^{\prime }}$ of being selected at least once in the $L^{\prime }$ times instances that are randomly selected from the training set. This perturbation causes different regressors to be built, which have different certain diversities.

3. Mobile Tracking Based on Game Theory

To implement mobile node tracking, we need to smooth the node position estimate in Section 2, that is, to filter the position estimation noise. Because noise characteristics are unknown and uncertain, it is unreasonable to utilize the Kalman filter to smooth the position estimate. So we utilize the game theory to address this problem. In this scheme, we assume that the noise is the adversary of the estimator.

Assume that the mobile node measures the radio parameters at each interval $\mathrm{\Delta }T$ that are putted into support vector regression model to attain the mobile node position. Let the mobile node position estimate be ${\mathit{y}}^{\prime }(t)={[y_1^{\prime }(t),y_2^{\prime }(t)]}^T$ at time t and let state vector be ${\mathit{x}}^{\prime }(t)={[x_1^{\prime }(t),x_2^{\prime }(t),x_3^{\prime }(t),x_4^{\prime }(t)]}^T$ , where $x_1^{\prime }(t),x_2^{\prime }(t)$ denote the $x,y$ coordinates. respectively, of mobile location, whereas $x_3^{\prime }(t),x_4^{\prime }(t)$ denote the $x,y$ coordinates, respectively, of velocity vector at time t. Then at $k\mathrm{\Delta }T$ , the state and position estimate equations can be formulated as follows:

$$\begin{array}{cc}& {\mathit{x}}_{k+\mathrm{1}}^{\prime }={\mathit{A}}^{\prime }{\mathit{x}}_k^{\prime }+\mathit{O}{\mathit{u}}_k+\mathit{B}{w}_k+{\delta }_k,\\ {\mathit{y}}_k^{\prime }={\mathit{H}}^{\prime }{\mathit{x}}_k^{\prime }+{\nu }_k,\end{array}$$

(8) where ${\mathit{u}}_k$ denotes the two-dimensional acceleration component, $w_k$ is white noise sequence, and ${\delta }_k$ is the noise introduced by an adversary:

$$\begin{array}{cc}& {\mathbf{A}}^{\prime }=\left[\begin{array}{cccc}\mathrm{1}& \mathrm{0}& \mathrm{\Delta }T& \mathrm{0}\\ \mathrm{0}& \mathrm{1}& \mathrm{0}& \mathrm{\Delta }T\\ \mathrm{0}& \mathrm{0}& \mathrm{1}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{1}\end{array}\right],\hspace{1em}\hspace{1em}\mathbf{O}=\left[\begin{array}{cc}\mathrm{0}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}\\ \mathrm{\Delta }T& \mathrm{0}\\ \mathrm{0}& \mathrm{\Delta }T\end{array}\right],\\ {\mathbf{H}}^{\prime }={\left[\begin{array}{cc}\mathrm{1}& \mathrm{0}\\ \mathrm{0}& \mathrm{1}\\ \mathrm{0}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}\end{array}\right]}^T.\end{array}$$

(9)

Let ${\mathit{x}}_k=[{\mathit{x}}_k^{\prime },{\mathit{u}}_k{]}^T$ , ${\mathit{y}}_k={\mathit{y}}_k^{\prime }$ , then the equation system of node state and position estimate can be rewritten as follows:

$$\begin{array}{cc}& {\mathit{x}}_{k+\mathrm{1}}=\mathit{A}{\mathit{x}}_k+\mathit{B}{w}_k+{\delta }_k\\ {\mathit{y}}_k=\mathit{H}{\mathit{x}}_k+{\nu }_k,\end{array}$$

(10) where $\mathbf{A}=\left[\begin{array}{cc}{\mathbf{A}}^{\prime }& \mathbf{O}\end{array}\right]$ , $\mathbf{H}=\left[\begin{array}{cc}{\mathbf{H}}^{\prime }& {\mathbf{0}}_{\mathrm{2}\times \mathrm{2}}\end{array}\right]$ , and ${\nu }_k$ , $w_k$ are mutually uncorrelated unity-variance white noise sequence.

Then the mobile tracking problem is to find an estimate ${\widehat{\mathit{x}}}_{k+1}$ of ${\mathit{x}}_{k+1}$ given the $({\mathit{y}}_0,{\mathit{y}}_1,\dots ,{\mathit{y}}_k)$ . We assume that the estimate is unbiased and has the following structure:

$$\begin{array}{cc}& {\widehat{\mathit{x}}}_0=0,\\ {\widehat{\mathit{x}}}_{k+\mathrm{1}}=\mathit{A}{\widehat{\mathit{x}}}_k+{\mathit{K}}_k\left({\mathit{y}}_k-{\mathit{H}}_k{\widehat{\mathit{x}}}_k\right).\end{array}$$

(11)

Suppose ${\delta }_k$ as the noise introduced by an adversary that has the goal of maximizing the estimation error. Assume ${\delta }_k={\mathit{L}}_k({\mathit{G}}_k({\mathit{x}}_k-{\widehat{\mathit{x}}}_k)+n_k)$ , where ${\mathit{L}}_k$ denotes the gain to be determined, ${\mathit{G}}_k$ is the given matrix, and $n_k$ is the noise sequence.

The estimation error is defined as follows:

$$\begin{array}{cc}& {\mathit{e}}_k={\mathit{x}}_k-{\widehat{\mathit{x}}}_k.\end{array}$$

(12)

It can be shown from the preceding equations that the dynamic system describing the evolution of the estimation error is given as follows:

$$\begin{array}{cc}& {\mathit{e}}_{\mathrm{0}}={\mathit{x}}_{\mathrm{0}},\\ {\mathit{e}}_{k+\mathrm{1}}=\left(\mathit{A}-{\mathit{K}}_k\mathit{H}+{\mathit{L}}_k{\mathit{G}}_k\right){\mathit{e}}_k+\mathit{B}{w}_k+{\mathit{L}}_k{n}_k-{\mathit{K}}_k{\nu }_k.\end{array}$$

(13)

However, this is an inappropriate term for a minimax problem because the adversary can arbitrarily increase ${\mathit{e}}_k$ by arbitrarily increasing ${\mathit{L}}_k$ . To prevent this, we decompose ${\mathit{e}}_k$ as follows:

$$\begin{array}{cc}& {\mathit{e}}_k={\mathit{e}}_{\mathrm{1},k}+{\mathit{e}}_{\mathrm{2},k},\end{array}$$

(14) where ${\mathit{e}}_{1,k}$ , ${\mathit{e}}_{2,k}$ evolve as follows:

$$\begin{array}{cc}& {\mathit{e}}_{\mathrm{1,0}}={\mathit{x}}_{\mathrm{0}},\\ {\mathit{e}}_{\mathrm{1},k+\mathrm{1}}=\left(\mathit{A}-{\mathit{K}}_k\mathit{H}+{\mathit{L}}_k{\mathit{G}}_k\right){\mathit{e}}_{\mathrm{1},k}+\mathit{B}{w}_k-{\mathit{K}}_k{\nu }_k,\\ {\mathit{e}}_{\mathrm{2,0}}={\mathit{x}}_{\mathrm{0}},\\ {\mathit{e}}_{\mathrm{2},k+\mathrm{1}}=\left(\mathit{A}-{\mathit{K}}_k\mathit{H}+{\mathit{L}}_k{\mathit{G}}_k\right)e_{\mathrm{2},k}+{\mathit{L}}_k{n}_k.\end{array}$$

(15)

From the above analysis, we can see that the estimator minimizes the objective function by searching ${\mathit{K}}_k$ , but the noise (adversary) maximizes the objective function by finding ${\mathit{L}}_k$ . Thus the estimator and noise constitute two side of a game.

For this game, we define the objective functions as

$$\begin{array}{cc}& J\left({\mathit{K}}_k,{\mathit{L}}_k\right)=\text{trace}\sum _{k=\mathrm{0}}^N{\eta }_{k}E\left({\mathit{e}}_{\mathrm{1},k}{\mathit{e}}_{\mathrm{1},k}^T-{\mathit{e}}_{\mathrm{2},k}{\mathit{e}}_{\mathrm{2},k}^T\right),\end{array}$$

(16) where ${\eta }_k$ is any positive definite weighting matrix.

According the difference game theory, the optimal solution to the objective function (16) is the saddle point $({\mathit{K}}_k^{*},{\mathit{L}}_k^{*})$ of difference game and satisfies the following equation:

$$\begin{array}{cc}& J\left({\mathit{K}}_k^{*},{\mathit{L}}_k\right)\le J\left({\mathit{K}}_k^{*},{\mathit{L}}_k^{*}\right)\le J\left({\mathit{K}}_k,{\mathit{L}}_k^{*}\right),\hspace{1em}\forall \left({\mathit{K}}_k^{},{\mathit{L}}_k^{}\right).\end{array}$$

(17)

For brevity, we rewrite the objective function and define the matrix ${\mathit{F}}_k$ as follows:

$$\begin{array}{cc}& {\mathit{F}}_k=\mathit{A}-{\mathit{K}}_k\mathit{H}+{\mathit{L}}_k{\mathit{G}}_k.\end{array}$$

(18)

Define the following matrix difference equation:

$$\begin{array}{cc}& {\mathit{Q}}_{\mathrm{0}}=E\left({\mathit{x}}_{\mathrm{0}}{\mathit{x}}_{\mathrm{0}}^T\right),\\ {\mathit{Q}}_{k+\mathrm{1}}={\mathit{F}}_k{\mathit{Q}}_k{\mathit{F}}_k^T+\mathit{B}{\mathit{B}}^T+{\mathit{K}}_k{\mathit{K}}_k^T-{\mathit{L}}_k{\mathit{L}}_k^T.\end{array}$$

(19)

Then the objective function is reformulated as follows:

$$\begin{array}{cc}& J\left({\mathit{K}}_k,{\mathit{L}}_k\right)=\text{trace}\sum _{k=\mathrm{0}}^N{\mathit{w}}_k{\mathit{Q}}_k.\end{array}$$

(20)

Now define ${\tilde{\mathit{Q}}}_k$ , ${\Sigma }_k$ as the nonsingular solution as the following set of equations:

$$\begin{array}{cc}& {\tilde{\mathit{Q}}}_0=E\left({\mathit{x}}_0{\mathit{x}}_0^T\right),\\ {\tilde{\mathit{Q}}}_k\left(\mathit{I}-{\mathit{H}}^T\mathit{H}{\Sigma }_k\right)=\left(\mathit{I}-{\tilde{\mathit{Q}}}_k{\mathit{G}}_k^T{\mathit{G}}_k\right){\Sigma }_k,\\ {\tilde{\mathit{Q}}}_{k+\mathrm{1}}=\mathit{A}{\Sigma }_k{\mathit{A}}^T+\mathit{B}{\mathit{B}}^T.\end{array}$$

(21)

And if solutions to these equations do exist, then ${\Sigma }_k$ can be computed as follows:

$$\begin{array}{cc}& {\Sigma }_k={\left({\tilde{\mathit{Q}}}_k{\mathit{H}}^T\mathit{H}-{\tilde{\mathit{Q}}}_k{\mathit{G}}_k^T{\mathit{G}}_k+\mathit{I}\right)}^{-\mathrm{1}}{\tilde{\mathit{Q}}}_k.\end{array}$$

(22)

Theorem 3 (see [15]).

If $(\mathbf{H}{\tilde{\mathbf{Q}}}_k{\mathbf{H}}^T+\mathbf{I})\ge \mathrm{0}$ and $(\mathbf{I}-{\mathbf{G}}_k{\tilde{\mathbf{Q}}}_k{\mathbf{G}}_k^T)\ge \mathrm{0}$ , then ${\mathbf{L}}_k^{*}=\mathbf{A}{\Sigma }_k{\mathbf{G}}_k^T$ , ${\mathbf{K}}_k^{*}=\mathbf{A}{\Sigma }_k{\mathbf{H}}^T$ satisfy the saddle point equilibrium (17).

In brief, the implementation of mobile tracking is shown as follows:

•  (1) 

equation system

$$\begin{array}{cc}& {\mathit{x}}_{k+\mathrm{1}}=\mathit{A}{\mathit{x}}_k+\mathit{B}{w}_k+{\delta }_k,\\ {\mathit{y}}_k=\mathit{H}{\mathit{x}}_k+{\nu }_k,\end{array}$$

(23)

 (2) 

initialization

$$\begin{array}{cc}& {\widehat{\mathit{x}}}_0=0,\\ {\tilde{\mathit{Q}}}_{\mathrm{0}}=E\left({\mathit{x}}_{\mathrm{0}}{\mathit{x}}_{\mathrm{0}}^T\right),\end{array}$$

(24)

 (3) 

at each step $k=\mathrm{0,1},\dots$ , do the following.

•  (a) 

Choose the parameter ${\mathit{G}}_k$ .

 (b) 

Calculate the following equations:

$$\begin{array}{cc}& {\Sigma }_k={\left({\tilde{\mathit{Q}}}_k{\mathit{H}}^T\mathit{H}-{\tilde{\mathit{Q}}}_k{\mathit{G}}_k^T{\mathit{G}}_k+\mathit{I}\right)}^{-\mathrm{1}}{\tilde{\mathit{Q}}}_k,\\ {\tilde{\mathit{Q}}}_{k+\mathrm{1}}=\mathit{A}{\Sigma }_k{\mathit{A}}^T+\mathit{B}{\mathit{B}}^T,\\ {\mathit{K}}_k=\mathit{A}{\Sigma }_k{\mathit{H}}^T,\\ {\widehat{\mathit{x}}}_{k+\mathrm{1}}=\mathit{A}{\widehat{\mathit{x}}}_k+{\mathit{K}}_k\left({\mathit{y}}_k-{\mathit{H}}_k{\widehat{\mathit{x}}}_k\right).\end{array}$$

(25)

 (c) 

If $(\mathit{I}-{\mathit{G}}_k{\tilde{\mathit{Q}}}_k{\mathit{G}}_k^T)\ge 0$ , then the algorithm terminates; otherwise, decrease the parameter ${\mathit{G}}_k$ and go to step (a).

4. Simulation Results

In the simulations, support vector regression is implemented by using MATLAB Support Vector Machine Toolbox. A 600 m $\times$ 600 m relevant area with three base stations at locations (187.5, 150), (487.5, 150), and (337.5, 409.8) is considered. The mobile node is moving from location (50, 305) to (575, 230) on the solid track shown in Figure 1 with a constant speed of 10 m/s. TOA measurements are taken at a rate of 5 samples/s.

Figure 1

True track and the SVR and SVRM estimate.

In experiment, we use elliptical scatter model [16] to produce the ToA data. For the elliptical scatter model, elliptical scatters are uniformly distributed inside the ellipse with foci at the base station and mobile. The ToA probability density function is given by the following function:

$$\begin{array}{cc}& p\left(x\right)=\{\begin{array}{cc}\frac{c\left(\mathrm{2}{c}^{\mathrm{2}}{x}^{\mathrm{2}}-D^{\mathrm{2}}\right)}{\mathrm{4}{a}_m{b}_m\sqrt{c^{\mathrm{2}}{x}^{\mathrm{2}}-D^{\mathrm{2}}}},\hfill & \frac{D}{c}\le x\le t_m,\hfill \\ \mathrm{0},\hfill & \text{else},\hfill \end{array}\end{array}$$

(26) where $t_m$ is the maximum delay associated with scatterers within the ellipse, c is the speed of light, and D is the true distance between anchor and mobile node. That is, only multipath components that arrive within $t_m$ seconds are considered. The parameters $a_m$ and $b_m$ are the semimajor axis and semiminor axis values which are given by $a_m=c{t}_m/2,b_m=(1/2)\sqrt{c^2{t}_m^2-D^2}$ .

This TOA distribution is used in the simulation by equating $t_m$ to a multiple of the true TOA between the location of mobile node and the anchor; that is, $t_m=\alpha D/c$ , so that closer locations have higher probability of having smaller NLOS errors, because the probability that signal from mobile node to anchor encounters the scatters decreases with α, that is, the distance between mobile node to anchor.

For the SVR and SVRM model, the radial basis function is used as the kernel and the ε-insensitive cost function with $\epsilon =1$ is employed.

In the first set of simulations, the mobile is tracked in the solid line shown in Figure 1. The location estimates obtained by the SVR and SVRM are also shown in Figure 1 with $C=200$ in (3), $\alpha =1.5$ , and $\mathrm{\Delta }=30$  m, where $\mathrm{\Delta }$ is the distance between the consecutive training locations.

The smoothing results from the Kalman filter and game theory, which take the SVR and SVRM estimates as inputs, are presented in Figure 2. From these two figures, the effect of Kalman filter and game theory is seen clearly. The average error between the true locations and the estimated locations is 40.77 m for the SVR estimates, which improves to 37.32 m after Kalman filtering and further improves to 28.08 m after smoothing by game theory [12]. But the average error between the true locations and the estimated locations is 30.58 m for the SVRM estimates and further improves to 23.08 m after smoothing by game theory. This shows that the position estimate from SVRM is better than that from SVR.

Figure 2

Position estimate after smoothing by Kalman filter and game.

In Figure 3, the average errors after smoothing by Kalman and game theory are plotted for different values of α (from 1.5 to 2.5) with $C=200$ and $\mathrm{\Delta }=\mathrm{30}$  m. Also the estimates from the conventional least squares algorithm are plotted. As expected, the average error increases with α. However, the average error after game theory increases less fast with α than those of other methods. From Figure 3, it is also shown that smoothing SVRM output by game theory can attain more lower position tracking error than smoothing SVR output by game theory.

Figure 3

Average tracking error versus α.

5. Conclusion

In this paper, we utilize SVRM to learning the map from radio parameter to node position and game theory to smooth the SVRM output. Because the radio parameter is not considered as the distance measure, but as the feature to train the SVRM model, the side effect of non-line-of-sight is mitigated. And by modeling the noise as the adversary of position estimator, because of its ability to handle the uncertain and unmodeled noise, game theory can attain more accuracy of mobile tracking compared to Kalman filter. The above method can yield very accurate position estimates even in NLOS environments and attain better performance than the method in [12]. From the simulations, it can be seen that SVRM can reduce the effect of NLOS.