Journal of Sensors

Volume 2016, Article ID 8914769, 9 pages

http://dx.doi.org/10.1155/2016/8914769

## A Method for Selecting Optimal Number of Sensors to Improve the Credibility

Liaoning IC Technology Key Laboratory, School of Electronics Science & Technology, Dalian University of Technology, Dalian, Liaoning 116023, China

Received 19 June 2015; Revised 25 September 2015; Accepted 29 September 2015

Academic Editor: Maria Luz Rodríguez-Méndez

Copyright © 2016 Yi Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

With the development of sensors, it is possible to embed many sensors within a certain space, which makes the monitor and alarm system with multisensor possible. There are two important parameters in a monitor and alarm system, namely, the false alarm rate and the missed alarm rate. In this work, a method for selecting optimal number of sensors in the sensor array is presented to improve the credibility. The influence factors of the weights and the false alarm rate and the missed alarm rate of one sensor and total number of sensors are discussed. An experimental setup was developed. The monitoring methods of common strategies and the proposed optimal number of sensors strategy are compared graphically by the receiver operating characteristic curves and the area under receiver operating characteristic curve values. The receiver operating characteristic curves graphically prove that the optimal number of sensors’ method presents the best performance, and it is shown that the optimal number of sensors’ method has the highest area under receiver operating characteristic value (0.9631). This method may aid future users of the monitor and alarm system by providing an optimal number of sensors strategy for high credibility.

#### 1. Introduction

With the development of sensor technologies, sensors have been improved with smaller size, lower power consumption, and better anti-interference ability, which makes it possible to embed many sensors within a certain local space. Multisensor information fusion pursues process redundant or complementary information from the multiple sources provided by the sensors to achieve results that are not feasible from a single sensor [1, 2]. The monitor and alarm system (MAS) based on multisensor information fusion technologies has been widely used in many fields, such as the disease diagnosis [3, 4], the image fusion [1, 5], the environment monitoring [6], and the security surveillance.

In security surveillance of high-risk industries, such as the traffic system [7], the vibration fault diagnosis [8], the recursive track [9], the medical surveillance [10], and the monitoring of hazardous materials [11], the credibility is one of the most important topics which attract extensive attention. A receiver operating characteristic (ROC) graph, which can combine FAR and MAR to one evaluation criteria, is a graphical plot that illustrates the performance of a binary classifier system as its discrimination threshold is varied [12–14]. To compare the discrimination we may want to reduce the ROC performance to a single scalar value representing the expected performance. A common method is to calculate the area under the ROC curve (AUC) [15, 16]. A single performance indicator from the AUC can summarise the ROC curve as the higher the AUC value, the better the performance of the method. The credibility in the region of security surveillance means low false alarm rate (FAR) and low missed alarm rate (MAR) of the MAS [17]. The false alarm (FA) means that the MAS is triggered when it should not be triggered, while the missed alarm (MA) means that the MAS is not triggered when it should be triggered [18]. Smaller FAR or smaller MAR could be easily obtained by a sensor array and the proper tradeoff alarm strategy in an MAS. However, FAR and MAR are associated and it is difficult to simultaneously minimize the FAR and MAR in one MAS. To obtain small FAR and MAR simultaneously is the goal of high credibility MAS; thus the tradeoff between FAR and MAR is a fundamental problem which has attracted extensive interest by many researchers [19–23].

The FAR and MAR of an MAS are determined by the total number of sensors (TNS), the false alarm rate and the missed alarm rate of each sensor, and the alarm strategy. In this work, to make comprehensive estimation for the credibility of an MAS, the equivalent false alarm rate (EFAR), which reflects the FAR and MAR as well as the weights, is defined by the authors. The weights denote the loss of the FAR relative to the MAR. A quantitative method to select the optimal number of sensors (ONS) for minimizing the EFAR is given. The research results show that the proposed ONS strategy could improve the credibility when the TNS of the MAS is constant.

#### 2. The Method for Selecting Optimal Number of Sensors

The parameters and are used to represent the false alarm rate and the missed alarm rate of a single sensor in the sensor array. The parameters and are equivalent to the* false positive rate* and one minus* true positive rate,* respectively, in ROC space. In a practical application, the values of and can be evaluated by the average value, where is the number of the negatives incorrectly classified divided by the total negatives and is the number of the positives incorrectly classified divided by the total positives. We propose an EFAR model with the following assumption.

The sum of and is less than 1. The expression (i.e.,* true positive rate* in ROC space) represents the correct alarm decision rate. Therefore, the assumption demands that each sensor in the array produces useful information for detection. In ROC space, this assumption means that the (*false positive rate* and* true positive rate*) pairs should be in the upper triangular region, because the pairs in the diagonal represent the strategy of random guessing and in lower right triangle will perform even worse.

We may choose the alarm stratagem as follows: the result of the MAS is positive if sensors in the system give positive results. Here is the selected number of sensors (SNS). Assuming that the TNS in MAS is , then is in the range from 1 to . Obviously, corresponds to the lowest MAR but the highest FAR; and possesses the lowest FAR but the highest MAR. To study the optimized (ONS) of the system, the FAR and MAR associated with are derived from Bernoulli trials as follows:where is the combination of selecting items from a set . The FAR exists only when the output of the MAS is in alarm state; therefore, the expression is the sum of the false alarm rate with more than sensors being in alarm state. In contrast, the MAR exists only when the output of the MAS is in nonalarm state, and consequently the expression denotes the sum of the missed alarm rate with less than sensors being in nonalarm state.

The objective function and constraints are as follows:where the coefficients and denote the weight of FAR and MAR, respectively. The EFAR is the normalized rate of the FAR and MAR and is equal to . Bigger shows that the FAR is more important than MAR in the monitoring strategy in the MAS.

The objective function and constraints are nonlinear programming with integer variables. In order to get the solution of the problem, a variable is introduced. If the index is the ONS of the MAS, then the range is from 1 to in the forward difference and from 2 to in the backward difference. A method based on recurrence relations of forward difference and backward difference is proposed for selecting the ONS. Suppose that EFAR() denotes the solution of (2), the expressions are as follows:

According to the first formula in (3), the expression can be changed into

As , (4) can be changed into

Further, (5) can be changed into

Taking natural logarithms on both sides, (6) can be changed into

According to the assumption, the sum of and is less than 1. The expression and .

Therefore, the result of forward difference is as follows:

According to the second formula in (3), the expression can be changed into

As , (9) can be changed into

Further, (10) can be changed into

Taking the natural logarithms on both sides, (11) can be changed into

According to the assumption, the sum of and is less than 1. The expression could be expressed as and . Therefore, the result of the forward difference is as follows:

The results of (3) are as follows:where the notation denotes the ONS, which is given as follows:

Then the solutions of (2) can be divided into two situations: namely, is an integer or not an integer:

The EFAR denotes the comprehensive effect of FAR and MAR. In what follows, a case study was presented, namely, the relationship between FAR, MAR, and EFAR. We assume that the TNS is 10, is 0.5, and false alarm rate and the missed alarm rate are 0.3 and 0.4, respectively. Figure 1 shows the FAR, MAR, and EFAR as a function of SNS. With the increase of SNS, the FAR decreases (Figure 1(a)) and MAR increases (Figure 1(b)). For a fixed TNS, the feasible range of SNS should be an integer which is greater than or equal to 1 and less than or equal to TNS . The more the number of sensors chosen is, the smaller the FAR and the bigger the MAR of the MAS would be. The reason why MAR increases is that the probability of more sensors in alarm state is smaller than less sensors in alarm state. If bigger SNS is chosen, the probability of MA would increase. The tradeoffs of FAR and MAR are substituted by EFAR which shows concave upward in the feasible range (Figure 1(c)). The minimum value of EFAR is 0.16 (Figure 1(c)), at SNS = 5, which is much less than the EFAR at SNS = 1 or SNS = 10.