Decision Fusion Supported by Correlated Auxiliary Data in Wireless Sensor Networks

Abstract

Leakage monitoring is different from sudden incident monitoring because most of the leakage cases involve a slow process that lasts for a long time. During this case monitoring, sensors suffer long exposure to erosion and may lead to errors in the measurement. An approach is proposed to make use of a soft-decision fusion approach according to the Neyman-Pearson criterion to accumulate auxiliary data from multiple sensors. The proposed method optimizes the soft-function and adjusts its range of sensors, which provide auxiliary data to improve the fusion center confidence for making a global decision. The new method encompasses the collection of useful data and weights and combines them according to the corresponding confidence level to make a global decision. In the simulation case of Rayleigh-distributed observations of leakage monitoring, it is proved that the proposed method has a good performance.

1. Introduction

Multiple-sensor distributed decision fusion is now being used in a wide variety of applications [1–5]. The multiple-sensor approach involves the deployment of the sensor nodes in a monitoring area and the collection of environmental information, followed by processing the information, which is then sent to the appropriate users. At the end of the process, the users can decide what they should do according to the real environmental information. During the process, the decision method achieves optimum performance if all the information has been received at the large cost of communication. To improve the accuracy of the results, WSN may be used to address the unreliable nature of the individual nodes. Therefore, redundant data are inevitable. To enhance the capability of identifying and estimating objects, to refine the transmitted information, and to reduce the energy cost during transmission, data collected from sensor nodes distributed at different locations should be constrained with proper fusion rules.

In dense networks, it is difficult to identify each sensor node in a dynamic network. Thus, not all sensor observations are transmitted directly to the fusion center (FC), where all data are fused by using various approaches [3, 4]. In some methods, the preliminary processing of the observations is performed at each node to obtain binary decisions (0 or 1) [2, 5], called local decisions. Next, the decision would be sent to FC, which makes a system decision according to the local decision. The advantages of performing preliminary processing are reduced communication bandwidth requirements and reduced transmission costs. However, the fusion center in binary decision fusion has only binary partial information, which is not sufficient to show all of the details. The result is a loss of performance in this approach. Some previous studies proposed approaches to address this loss of performance. Zhou et al. [6] proposed decision fusion rules based on multibit knowledge of local sensors. Their work indicates that fusion rules based on the commonly used weight are not sufficient to improve the system performance. The decision rules based on statistics and redundancy are introduced. Thuc and Insoo [7] evaluated the credibility of clusters by using fuzzy logic and took them into account when the final decision is made at the fusion center. By mapping observed values to a set {low, medium, high} and providing a fuzzy inference rules set to make a decision, their works have been found to provide good performance when the SNR is near 0. Aziz [8] proposed an approach known as soft-decision, which is based on the Neyman-Pearson criterion. Instead of using a one-bit hard decision, the approach has each local sensor provide the fusion center with a multiple-bit (soft) decision as local decision. This fusion method improves the performance in the cases of Rayleigh-distributed observations. However, Aziz did not consider the case that no sensor found an alarm situation to be satisfied, even though the sum of soft-decision values exceeds the alarm threshold. In the other situation, according to our experience, we found that the sensors in rough conditions may result in some erroneous values at a small rate, which leads to false alarms. When the latter case occurs, the neighbor near the erroneous sensor is still functional. In our work, a novel soft-decision approach for multiple-sensor networks is proposed. Compared with the previous methods, this approach reduces the effect of the sum of soft-decision values and attempts to find sensors with erroneous values.

The remaining parts of this paper are organized as follows. Section 2 shows the soft-function used in this paper. Section 3 presents the fusion rules based on the soft-function, and its performance is discussed in Section 4. Section 5 provides the conclusion of this work.

2. System Model

The model considered involves a network scene of a leakage monitoring system of WSNs in a coal mine. Sensor nodes of a certain density are deployed in the laneway. Once a node sensor detects gas or water seepage on the tunnel walls, the nodes will send the sensing information to the fusion center via a wireless channel. With the efficiency of multipath propagation and fading in a coal mine, the communication environment is a typical nonideal channel. Because the equipment in a coal mine is periodically checked and the batteries of the nodes are changed periodically, we will take the physical error into account but not the energy cost limit. Most systems depend on a global decision that combines the information of all of the nodes in the event area. Thus, the quantization degree plays an important role in the performance of WSNs.

Because the proposed soft model is based on the fusion of local decisions and global decisions, the system includes three layers: a local sensor layer, transmission layer, and fusion center layer. We consider binary hypotheses: $H_0$ denotes no seepage in the event area; $H_1$ denotes the alternative. The system contains N local sensor nodes that operate independently and have a known probability distribution under both hypotheses ( $p(y_i\mid H_0)$ and $p(y_i\mid H_1)$ , $i=1,2,\dots ,N$ ).

2.1. Binary Decision Model

On the local layer, in a traditional study, local node $S_i$ receives the original data vector $Y_i$ in one event, where

$$\begin{array}{cc}& Y_i=\left[y_N,\dots ,y_i,\dots ,y_{\mathrm{0}}\right],\hspace{1em}{y}_i=\left\{\begin{array}{cc}\mathrm{1},\hfill & \text{if}\mathrm{\hspace{0.17em}\hspace{0.17em}}{H}_{\mathrm{1}}\mathrm{\hspace{0.17em}\hspace{0.17em}}\text{is}\mathrm{\hspace{0.17em}\hspace{0.17em}}\text{true},\hfill \\ \mathrm{0},\hfill & \text{if}\mathrm{\hspace{0.17em}\hspace{0.17em}}{H}_{\mathrm{0}}\mathrm{\hspace{0.17em}\hspace{0.17em}}\text{is}\mathrm{\hspace{0.17em}\hspace{0.17em}}\text{true}.\hfill \end{array}\right.\end{array}$$

(1)

Next, the elements of the local decision vector $U_i$ are obtained based on $Y_i$ . Suppose that each sensor's detection probability is $p{d}_i=p[y_i=1\mid H_1]$ , and the false alarm probability is $p{f}_i=p[y_i=0\mid H_0]$ , $i=\mathrm{1,2},\dots ,n$ .

On the transmission layer, the local decision vector $U=[u_N,\dots ,u_i,\dots ,u_0]$ , $u_i\in \{-\mathrm{1,1}\}$ , is transmitted and assumed to be BPSK codes in this study. Due to signal fading and noise in the channel, the results received by the FC are transformed into a new vector R.

On the fusion center layer, the FC makes a final decision $r_f$ based on $R=[r_N,\dots ,r_i,\dots ,r_0]$ from N local sensors. The global detection probability (GDP) will be maximized when the global false alarm probability (GFAP) remains lower than a given value. The binary decision sensor can be described as

$$\begin{array}{cc}& d_i=\left\{\begin{array}{cc}+\mathrm{1},\hfill & \text{if}\mathrm{\hspace{0.17em}\hspace{0.17em}}LR\left(y_i\right)=\frac{P\left(y_i\mid H_1\right)}{P\left(y_i\mid H_0\right)}\ge T_i,\hfill \\ \mathrm{0},\hfill & \text{otherwise},\hfill \end{array}\right.\end{array}$$

(2)

where $T_i$ is the sensor's threshold.

2.2. Soft-Decision Model

Binary decision, that is, 0 and 1, at each local sensor is simple to use but cannot present enough information of the scene. To improve upon binary decision, we attempt to allow some sensors to transmit their local decision with auxiliary information to the fusion center. Therefore, the local decision vector $U=[u_n,\dots ,u_i,\dots ,u_0]$ , $u_i\in (\mathrm{0,1})$ , will be changed. This change can be implemented by a soft membership function [8]. The purpose of the soft membership function is to provide enough information of the observations to the fusion center by providing more binary bits instead of only one bit that represents the local decision by $\{0,1\}$ .

The use of the membership function [8] has a good performance for sending more information to the FC, whereas the soft-decision approach depends on the choice of the function. We do not use the function in [8]; instead, we select the function shown in Figure 1 because the region of $y_i$ mapping to $v_i\in [0.5,1]$ is larger. In general, for a given sensor, the soft membership function value depends on how the observation is obscure. As shown in Figure 1, in the two confidence regions, $[0,T_{\min }]$ or $[T_{\max },1]$ , it is the determination of whether $v_i$ is 1 or 0 is clear. However, when an observation falls into $[T_{\min },T_{\max }]$ , a sensor may not make the correct decision according to threshold T. As a result, if sensor i obtains a sensor observation $y_i$ , a soft-decision $v_i$ will be transmitted to FC. As the threshold and the difference between the sensor's observation increase, the corresponding confidence level increases more rapidly.

Figure 1

Two soft membership functions.

The membership function shown in Figure 1 can be represented as in formula (3) according to the local sensor observation:

$$\begin{array}{cc}& f_i\left(y_i\right)=\left\{\begin{array}{cc}\mathrm{0},\hfill & y_i\in \left[\mathrm{0},T_{\min }\right],\hfill \\ \mathrm{1}-{\left[\frac{\left(y_i-T_{\min }\right)}{\left(T_{\max }-T_{\min }\right)}\right]}^{\mathrm{2}},\hfill & y_i\in \left[T_{\min },T_{\max }\right],\hfill \\ \mathrm{1},\hfill & y_i\in \left[T_{\max },\mathrm{1}\right].\mathrm{\hspace{0.17em}\hspace{0.17em}}\hfill \end{array}\right.\end{array}$$

(3)

The actual values of $T_{\min }$ and $T_{\max }$ depend on the real environment and statistics of the signal range of $H_0$ and $H_1$ . We chose the function of formula (3) to be able to obtain more information in the ambiguous region of the observations. According to (3), the confidence region on deciding $H_0$ and $H_1$ is

$$\begin{array}{cc}& {cfd}_i=\left\{\begin{array}{cc}\mathrm{1},\hfill & \text{if\hspace{0.17em}\hspace{0.17em}}{y}_i\in \left[\mathrm{0},T_{\min }\right],\text{\hspace{0.17em}\hspace{0.17em}decide\hspace{0.17em}\hspace{0.17em}}{H}_0,\hfill \\ v_i,\hfill & \text{if\hspace{0.17em}\hspace{0.17em}}{y}_i\in \left[T_{\min },T_{\max }\right],\hfill \\ \hfill & \text{decide\hspace{0.17em}\hspace{0.17em}}{H}_0\text{\hspace{0.17em}\hspace{0.17em}with\hspace{0.17em}\hspace{0.17em}a\hspace{0.17em}\hspace{0.17em}confidence\hspace{0.17em}\hspace{0.17em}}{v}_i,\hfill \\ 1-v_i,\hfill & \text{if\hspace{0.17em}\hspace{0.17em}}{y}_i\in \left[T_{\min },T_{\max }\right],\hfill \\ \hfill & \text{decide\hspace{0.17em}\hspace{0.17em}}{H}_1\text{\hspace{0.17em}\hspace{0.17em}with\hspace{0.17em}\hspace{0.17em}a\hspace{0.17em}\hspace{0.17em}confidence\hspace{0.17em}\hspace{0.17em}}1-v_i,\hfill \\ \mathrm{1},\hfill & \text{if\hspace{0.17em}\hspace{0.17em}}{y}_i\in \left[T_{\max },\mathrm{1}\right],\text{\hspace{0.17em}\hspace{0.17em}decide\hspace{0.17em}\hspace{0.17em}}{H}_1.\hfill \end{array}\right.\end{array}$$

(4)

Because the sensor will send more information on the transmission layer, the quantization degree of confidence region is not simple binary bits. Suppose that the sensor i local decision with confidence is presented by l sensor bits, $v_i=\left\{y_{i,}^l\right\}$ . FC will receive a vector $V^l$ and then make a final decision $v_f$ . We denote the vector by V in order to distinguish soft-decision from R mentioned above.

3. Fusion Rules of the Soft-Decision

Several assumptions should be accepted before we discuss the decision fusion rules. First, all local sensor nodes work independently and have the same capability parameters, $pd$ , $pf$ . Second, all local sensor nodes use the same local decision rules, and the FC trusts all of the local sensor nodes indiscriminatingly.

3.1. Quantizing the Local Sensor Decision with Confidence

When the local sensor decides $H_0$ or $H_1$ with full confidence, the decision can be denoted by 0 or 1. Thus, we pay more attention to the soft-decision space to map the input values, observations $y_i$ , into $v_i$ . To distinguish the different confidence levels, we introduce the concept of quantization level.

Definition 1 (quantization level (QL)).

Suppose local sensor i makes the soft-decision $v_i$ (b bits) based on observation $y_i$ represented by B binary bits, such that $b\le B$ . Let $\Omega \hspace{0.17em}\hspace{0.17em}(0,2^b)$ be the local soft-decision space that divides Ω into q (≤ $2^b$ ) sets, such that ${\Omega }_i\cap {\Omega }_j=\text{Ø}$ ( $i,j=1,\dots ,q$ ), to present different confidence. Each value in one set ${\Omega }_q$ has the same confidence $v_q$ . ${\Omega }_q$ is the QL.

Quantization level is to evaluate confidence level in range $\{T_{\min },T_{\max }\}$ . To compute the QL, we use the Lloyd quantizing method [9], such that the mean-squared quantization error is minimized. If its confidence value in set ${\Omega }_q$ , sensor i's detection probability is $p{d}_{iq}$ , the same as $p{f}_{iq}$ . Assuming that the probability density function (PDF) of the soft membership function (3) is denoted by $P(f)$ , the representative points $v_j$ of set ${\Omega }_j$ are obtained as

$$\begin{array}{cc}& v_j=\frac{{\int }_{{\Omega }_j}fP\left(f\right)df}{{\int }_{{\Omega }_j}P\left(f\right)df},\hspace{1em}j=1,\dots ,q.\end{array}$$

(5)

The partition of Ω is

$$\begin{array}{cc}& {\Omega }_j=\left\{\begin{array}{cc}0\hfill & j=0,\hfill \\ \frac{\left(v_j+v_{j+1}\right)}{2},\hfill & j\in \left[1,q-1\right],\hfill \\ 1\hfill & j=q.\hfill \end{array}\right.\end{array}$$

(6)

Including the ambiguous observations, the confidence is affected by the neighbors of the sensor. Sensor i's neighbors belong to subset of nodes monitoring the same object with i, and in the geographical location they are very close. According to that, in order to calculate conveniently, we appointed some nodes near a sensor as its neighbors in our simulation. Besides this method, neighbor set can be got in another method based on the spatial and temporal data [10]. If a sensor i knows that its neighbors agree with its submitted value, then one may treat its observation as being valuable.

Definition 2 (auxiliary confidence).

Sensor i and its neighbors undertake the task of identifying the presence or absence of a phenomenon of interest (PoI) and make their local soft-decision based on their own observation. If sensor i's neighbors hold the same opinion (i.e., the PoI is absent or not), sensor i would improve its soft-decision QL. The soft-decision of sensor i is its auxiliary confidence.

Suppose that the number of neighbors of sensor i is m; we denote the confidence coming from each neighbor as

$$\begin{array}{cc}& {acfd}_i=\left\{\begin{array}{cc}\frac{x}{m+1}{cfd}_i,\hfill & y_i\in \left(T_{\min },T_{\max }\right),\hfill \\ 1,\hfill & \text{otherwise},\hfill \end{array}\right.\end{array}$$

(7)

where x is the number of neighbors that hold the same opinions with sensor i; that is, i's neighbors soft-decision value is greater than that of i's. If sensor i has a definite decision, it would have a full confidence of 1. Otherwise, if there is no other sensor that agrees with sensor i, then sensor i will send its observation with a confidence value of $1/(m+1)$ . Through the coefficient of auxiliary confidence, the sensor i and its neighbors are working together with each other.

3.2. Decision Rules in the FC

According to the Neyman-Pearson principle, the FC makes the optimal decision using all of the sensor soft-decisions. The likelihood ratio test makes the system Bayes risk the lowest when a single sensor makes its local decision. When more information is transmitted to the FC, more binary bits are used. According to [6, 11], multibits fusion has been taken into account. The fusion rules of multiple-bit decision are also reduced to

$$\begin{array}{cc}& \Lambda ={\displaystyle \sum }_{i=1}^N{c}_i\underset{H_0}{\stackrel{H_1}{\gtrless }}\eta ,\end{array}$$

(8)

where

$$\begin{array}{cc}& c_i=\left\{\begin{array}{cc}\log \left(\frac{{pd}_i}{{pf}_i}\right),\hfill & v_i=\mathrm{1},\hfill \\ \log \left(\frac{\left(\mathrm{1}-{pd}_i\right)}{\left(\mathrm{1}-{pf}_i\right)}\right),\hfill & v_i=\mathrm{0}.\hfill \end{array}\right.\end{array}$$

(9)

η is the threshold of the fusion center, which meets a desired GFAP and is set up as a fixed value in many cases. The coefficients $c_i$ are determined in terms of the false alarm probability ( ${pf}_i$ ) and the detection probability ( ${pd}_i$ ) of each sensor i.

Similarly, the FC combines the soft-decisions of all of the individual sensors in terms of the likelihood ratio test. Because sensor's local decision is supported by its neighbors, FC can decide whether the global decision is trusted based on confidence value.

Theorem 3.

Supposing that each sensor in a network has an average of m neighbors nodes, an auxiliary combiner based on local soft-decision in the fusion center is

$$\begin{array}{cc}& {\Lambda }_{sd}={\displaystyle \sum }_{i=1}^N{c}_i^{\prime }\underset{H_0}{\stackrel{H_1}{\gtrless }}\eta ,\end{array}$$

(10)

where N is the number of sensor nodes and $x_q$ is the average value of sensors at the same QL:

$$\begin{array}{cc}& c_i^{\prime }=\left\{\begin{array}{cc}\ln \left(\frac{\left(\mathrm{1}-{pd}_{i\mathrm{0}}\right)}{\left(\mathrm{1}-{pf}_{i\mathrm{0}}\right)}\right),\hfill & v_{ij}=\mathrm{0},\hfill \\ \frac{v_{ij}{\widehat{x}}_j^{*}}{n_j{\left(m+\mathrm{1}\right)}^{*}}\ln \left(\frac{{pd}_{i\mathrm{1}}}{{pf}_{i\mathrm{1}}}\right),\hfill & v_{ij}\in {\Omega }_j,\hfill \\ \ln \left(\frac{{pd}_{i\mathrm{1}}}{{pf}_{i\mathrm{1}}}\right),\hfill & v_{ij}=\mathrm{1}.\hfill \end{array}\right.\end{array}$$

(11)

Proof.

According to [11], the likelihood ratio test of the multiple-bit decision based on local decision is

$$\begin{array}{cc}& LR\left(v_i\right)=\frac{P\left(V\mid H_1\right)}{P\left(V\mid H_0\right)}={\displaystyle \prod }_{i=1}^N\frac{P\left(v_i\mid H_1\right)}{P\left(v_i\mid H_0\right)},\end{array}$$

(12)

where V is the vector of the soft-decision and N is the total number of sensors. We assume that there are N sensor nodes in total, in which $n_0$ is the number of sensors deciding $H_0$ , $n_1$ is the number of sensors deciding $v_{i1}$ of QL ${\Omega }_1$ , $n_{q-1}$ is the number of sensors deciding $v_{iq-\mathrm{1}}$ of QL ${\Omega }_{q-\mathrm{1}}$ , and $n_q$ is the number of sensors deciding 1 of QL ${\Omega }_q$ . Then we have

$$\begin{array}{cc}& {\displaystyle \sum }_{q=0}^n{n}_q=N.\end{array}$$

(13)

For ${\Omega }_j$ , there may be more than one sensor considered to decide $v_{ij}$ . The auxiliary confidence coefficient of every sensor, $x/(m+1)$ , may be different. Because m is the average number of the sensor's neighbors, x is a predominant influence. To obtain the exact value of x of each sensor in the same QL is difficult. Therefore, let

$$\begin{array}{cc}& {\widehat{x}}_j={\displaystyle \prod }_{x_i\in {\Omega }_j}{x}_{ij}.\end{array}$$

(14)

The global decision statistic will be based on each soft-decision space QL. Next, the global decision statistic is weighted by the sum of the local sensor decision statistic. We can rewrite (12) as

$$\begin{array}{cc}& LR\left(v_i\right)={\displaystyle \prod }_{j=0}^q{\displaystyle \prod }_{v_j\in {\Omega }_j}\frac{P\left(v_j\mid H_1\right)}{P\left(v_j\mid H_0\right)}.\end{array}$$

(15)

According to formula (7), formula (15) can be rewritten as

$$\begin{array}{l}LR\left(v_i\right)={\displaystyle \prod }_{i=\mathrm{1}}^{n_{\mathrm{1}}}\frac{P\left(v_i=\mathrm{0}\mid H_{\mathrm{1}}\right)}{P\left(v_i=\mathrm{0}\mid H_{\mathrm{0}}\right)}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\cdots {\displaystyle \prod }_{i=\mathrm{1}}^{n_j}{\displaystyle \prod }_{k=\mathrm{1}}^{n_j}{acfd}_k\frac{P\left(v_i=v_{ij}\mid H_{\mathrm{1}}\right)}{P\left(v_i=v_{ij}\mid H_{\mathrm{0}}\right)}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\cdots {\displaystyle \prod }_{i=\mathrm{1}}^{n_q}\frac{P\left(v_i=\mathrm{1}\mid H_{\mathrm{1}}\right)}{P\left(v_i=\mathrm{1}\mid H_{\mathrm{0}}\right)}.\end{array}$$

(16)

Then, substituting (13) and (14) into (16)

$$\begin{array}{l}LR\left(v_i\right)={\displaystyle \prod }_{i=\mathrm{1}}^{n_{\mathrm{1}}}\frac{P(v_i=\mathrm{0}\mid H_{\mathrm{1}})}{P(v_i=\mathrm{0}\mid H_{\mathrm{0}})}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\cdots {\displaystyle \prod }_{i=\mathrm{1}}^{n_j}\left(\frac{{\widehat{x}}_k}{{\left(\mathrm{1}+m\right)}^{n_j}}\right)\left(v_{ij}\right)\frac{P(v_i=v_{ij}\mid H_{\mathrm{1}})}{P(v_i=v_{ij}\mid H_{\mathrm{0}})}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\cdots {\displaystyle \prod }_{i=\mathrm{1}}^{n_q}\frac{P\left(v_i=\mathrm{1}\mid H_{\mathrm{1}}\right)}{P\left(v_i=\mathrm{1}\mid H_{\mathrm{0}}\right)}.\end{array}$$

(17)

Taking the logarithms in (17), the $LR$ -test is given by

$$\begin{array}{l}\Lambda \left(y_i\right)=\ln \left(LR\left(v_i\right)\right)=\ln \left(\frac{P\left(v\mid H_1\right)}{P\left(v\mid H_0\right)}\right)\\ \hspace{1em}\hspace{1em}\hspace{0.17em}={\displaystyle \sum }_{i=1}^{n_1}\ln \left(\frac{P(v_i=0\mid H_1)}{P(v_i=0\mid H_0)}\right)\\ \hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{1em}+\cdots +{\displaystyle \sum }_{i=1}^{n_j}\frac{{\widehat{x}}_j^{*}}{{\left(1+m\right)}^{*}{n}_j}\ln \left(\frac{P\left(v_i=v_{ij}\mid H_1\right)}{P\left(v_i=v_{ij}\mid H_0\right)}\right)\\ \hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{1em}+\cdots +{\displaystyle \sum }_{i=1}^{n_q}\ln \left(\frac{P\left(v_i=1\mid H_1\right)}{P\left(v_i=1\mid H_0\right)}\right),\end{array}$$

(18)

where ${\widehat{x}}_j^{*}$ is $\log ({\widehat{x}}_j)$ in which ${\widehat{x}}_j$ is defined in (11) and ${\left(m+1\right)}^{*}$ is $\ln (m+1)$ . Simplifying formula (18) of the fusion rule of the data fusion center, we obtain

$$\begin{array}{cc}& {\Lambda }_{sd}={\displaystyle \sum }_{i=1}^N{c}_i^{\prime }\underset{H_0}{\stackrel{H_1}{\gtrless }}\eta ,\end{array}$$

(19)

where ${\eta }^{\prime }$ is the fusion center threshold and

$$\begin{array}{cc}& c_i^{\prime }=\left\{\begin{array}{cc}\ln \left(\frac{\left(\mathrm{1}-p{d}_{i\mathrm{0}}\right)}{\left(\mathrm{1}-p{f}_{i\mathrm{0}}\right)}\right),\hfill & v_{ij}=\mathrm{0},\hfill \\ \frac{v_{ij}{\widehat{x}}_j^{*}}{n_j{\left(m+\mathrm{1}\right)}^{*}}\ln \left(\frac{p{d}_{i\mathrm{1}}}{p{f}_{i\mathrm{1}}}\right),\hfill & v_{ij}\in {\Omega }_j,\hfill \\ \ln \left(\frac{p{d}_{i\mathrm{1}}}{p{f}_{i\mathrm{1}}}\right),\hfill & v_{ij}=\mathrm{1}.\hfill \end{array}\right.\end{array}$$

(20)

This completes the proof of Theorem 3.

4. Detection Performances

The detection performance of the system can be reflected by the probability of the detection $pd$ and the probability of a false alarm $pf$ . We will evaluate the performance as follows. Let $A=\{(v_1,\dots ,v_n);{\sum }_{i=1}^n{c}_i^{\prime }\ge t\}$ be the decision region of the fusion center that belongs to hypothesis $H_1$ . If GFAP is known, then we can obtain the sensors thresholds and $c_i^{\prime }$ according to formula (11). Next, GDP is determined by

$$\begin{array}{cc}& \text{GDP}={\displaystyle \sum }_{v\in A}{\displaystyle \prod }_{i=\mathrm{1}}^{N}P\left(v_i\mid H_{\mathrm{1}}\right).\end{array}$$

(21)

As an example for the performance study, we assume the case of n sensors with Rayleigh-distributed observations, and all of the nodes have the same local decision rules:

$$\begin{array}{cc}& P\left(y\mid H_0\right)=\exp \left(-y_i\right)U\left(y_i\right),\\ P\left(y\mid H_1\right)=\frac{\exp \left(y_i/\left(1+\text{snr}\right)\right)U\left(y_i\right)}{\left(1+\text{snr}\right)},\end{array}$$

(22)

where $\text{snr}$ is the signal-noise ratio at sensor i. In terms of formula (2), y is the signal value, and the corresponding false alarm and detection probabilities at local sensors are as follows [8]:

$$\begin{array}{cc}& p{d}_i=p{f}_i^{1/(1+\text{snr})}.\end{array}$$

(23)

According to [8, 12], for a given global false alarm probability at the FC, the corresponding global decision probability is given by

$$\begin{array}{cc}& \text{GDP}=\phi \left(\frac{\left({\phi }^{-\mathrm{1}}\left(\text{GFAP}\right)-\text{sqrt}{\left(n\right)}^{*}\left(\mathrm{1}+\text{snr}\right)\right)}{\left(\mathrm{1}+\text{snr}\right)}\right),\end{array}$$

(24)

where $\phi (x)={\int }_x^{\infty }(1/\sqrt{2\pi })e^{-y^2/2}dy$ and $t_i$ is the local threshold of the ith sensor. After the soft-function is introduced into the decision, the ambiguous interval should be taken as a percentage of the sensor's threshold; that is, it is a weight of threshold. To determine the coefficients in formula (21), we should compute the value of $v_{ij}$ . According to [13], we should obtain the root of soft-function (3). They are determined as

$$\begin{array}{cc}& {\text{root}}_{\mathrm{1,2}}=T_{\max }\pm \sqrt{{\left(T_{\max }-T_{\min }\right)}^{\mathrm{2}}(1-f)}.\end{array}$$

(25)

Next, we can determine the probability density function of (3) when y is in the ambiguous region [13]:

$$\begin{array}{cc}& P\left(f\right)=\frac{P\left({\text{root}}_{\mathrm{1}}∣H_1\right)}{\left|f^{\prime }\left({\text{root}}_{\mathrm{1}}\right)\right|}+\frac{P\left({\text{root}}_{\mathrm{2}}∣H_1\right)}{\left|f^{\prime }\left({\text{root}}_{\mathrm{2}}\right)\right|}.\end{array}$$

(26)

Therefore, we can obtain the value of $v_{ij}$ by (26) according to [8, 12, 14]. Next, we substitute the value into (11).

Figure 2 compares the receiver operating characteristics of different fusion methods in the normal environment in which no sensor makes a mistake regarding the local decision. There are sixteen sensors ( $n=16$ ) that send their local decision in the case of signal-to-noise ratio $(\text{SNR})=-\mathrm{3}$  dB and the number of neighbors ( $m=5$ , $x=3$ ). We find that the new fusion method (N fusion) is no better than Aziz's fusion method (A fusion). However, when we assume that these sensors are monitoring leakage, we find that the new method works better. Figure 3 compares the performance of Aziz's fusion method, the optimum binary fusion method, and the performance of the proposed approach for 2-bit quantization degree in the case that the sensors find leakage that is not serious or one sensor makes an error in the local decision $H_1$ with an error probability of 0.05. From Figure 3, it is clear that the new method using neighbor's local decision as auxiliary data outperforms the other decision fusion method. The reason for the improved performance is that the observation values would be above the threshold to lead to the local decision $H_1$ when a sudden event occurs. However, the leaking event often lasts for a very long time. At the beginning of the leaking event, it may be not serious enough to cause the fusion center to give a global decision $H_1$ . However, the sum of observation values may be higher than the threshold, resulting in a false alarm.

Figure 2

Comparison of different fusion rules in the case of 12 identical sensors in the Rayleigh-distributed observation.

Figure 3

Comparison of different fusion rules in the cases that sensors find leakage is not serious or when one sensor makes an error in the local decision $H_1$ .

Figure 4 shows the performance of the proposed approach compared with the other approach for different SNR values. This case has the same parameters as those in Figure 3, except for GFAP and SNR. From Figure 4, it is clear that the new method using 2 bits per local sensor decision has a better performance than the method using 3 bits. Therefore, more quantization bits cannot improve the global detection probability. The result is consistent with the conclusion of [8]. Figure 5 shows that the proposed method has a significant improvement when the number of neighboring sensors is small; however, it does not possess a superior increase when the number of neighboring sensors is sufficient ( $m>7$ ). The proposed method operates as the binary methods when the number of neighboring sensors holding the same local decision is small. Figure 5 also shows that the performance of the proposed method with random numbers of neighboring sensors holding the same local decision is close to the case that the number of neighbors holding agreeable decision is half.

Figure 4

Comparison of the different fusion rules with different quantization bits.

Figure 5

Performances of rules with different numbers of sensor nodes.

5. Conclusion

Fusion rules based on a soft-function can improve the performance of the fusion center. However, the improvement is not assured under some nonideal cases. We discuss such nonideal cases and propose novel fusion rules based on a soft-function and the auxiliary data of a sensor node's neighbors, which supports the global decision. We found that the sensor's neighbors in leakage monitoring can obtain information near the event sensor that indicates an alarm and provides these data to the FC to support the local decision of the event sensor. We investigated this process and made use of a soft-decision fusion approach to accumulate these useful data. The research results indicated the following.

• (1)

  Among all of the sensors, the proposed method based on soft-decision utilizes ambiguous data that is not sufficient to denote an exact local decision to support the global decision. This method was adapted to event monitoring that considered the case that the monitoring event is not explosive and lasts for a long time.

• (2)

  This method based on auxiliary data can enhance the system capability when the numbers of neighbors providing the supporting data are adequate. However, the performance would not increase significantly if the auxiliary data are sufficient.

Our work is based on the assumption that the data obeys a given distribution. This assumption may not be the same as the conditions of a practical application. Further research will further improve its quantization degree [6, 15], which can represent QL by using more bits.