Abstract
Student's t-distribution is utilized to derive a novel method for event detection in wireless sensor networks. Numerical analysis is used to show that under the same conditions, the proposed event detection method is comparable to likelihood ratio-based detection method and that it significantly outperforms energy detection method in terms of detection performance. Moreover, the proposed method does not require perfect knowledge of noise variance to set up a decision threshold in terms of a false alarm probability as the likelihood ratio based detection and the energy detection do.
1. Introduction
Recently, wireless sensor networks (WSNs) have been used for a wide variety of applications, including industrial control, home automation, security and military sensing, health monitoring, intelligent agriculture, and environmental sensing [1, 2]. Detection of certain events or targets in the environment is an important application of WSNs [3–6].
There are many detection methods that can be applied to analyze the data of the signals captured from the event of interest, such as the likelihood ratio- (LR-) based detection [7, 8] and energy detection (ED) [9, 10] methods. The LR-based detection method is a generalization of the maximum a posteriori probability, Bayesian, or maximum likelihood decision rules [8], and it compares the fit of two hypotheses (the null hypothesis and the alternative hypothesis) with respect to the likelihood ratio, expressing how many times more likely the observed data fit one hypothesis or the other. LR-based detection is an optimal detection method, but it requires full knowledge of the distributions under the hypotheses of the observations. The principle of the ED method is based on the difference between the energy of the signal emitted by the event and that of the noise. Hence, the detection performance is subject to the uncertainty of noise variance. In addition, when the time-varying nature of the wireless channel (e.g., shadowing, fading) is obvious or the signal-to-noise ratio (SNR) is low, this difference will be small for distinguishing between the signal and the noise. Subsequently, the detection performance of the ED method can be very poor. ED and LR-based detection schemes are local event detection schemes which can be used in a single sensor node. In wireless sensor networks, several nodes can detect the same event in collaborative way for better detection performance. Many existing studies in wireless sensor networks proposed algorithms to detect events by collaboration of sensor nodes [3, 5, 6]. In [5], Zhu et al. proposed a binary decision fusion rule that reaches a global decision on the presence of a target by integrating local decisions made by multiple sensors. In [6], Katenka et al. proposed the local vote decision fusion algorithm in which sensors first correct their decisions using decisions of neighboring sensors and then make a collective decision as a network. Further, in [3], Wei et al. proposed a novel prediction-based data collection protocol to reduce redundant data communications and saving sensor nodes energy. For collaborative detection schemes, the more detection performance we have at each sensor node, the more accurate detection performance we have at the fusion center. Therefore, the authors in the paper mainly focus on local event detection scheme and propose a novel blind event detection scheme. In the proposed scheme, we utilize the test statistic which has Student’s -distribution and further test whether the test statistic is larger than a certain decision threshold or not to detect the presence of an event. Since the distribution under the null hypothesis of the test statistic only depends on the number of samples, the decision threshold can be easily determined by a numerical method with respect to the probability of a false alarm, and the knowledge of the noise variance becomes unnecessary. Furthermore, the proposed detection method can manage the problem of noise uncertainty, and it is also useful for applications where a constant false alarm probability is required. Numerical results reveal that the proposed detection scheme always significantly outperforms ED scheme and has a comparable performance to LR-based detection. To the best of our knowledge, applying a-distribution to event detection in wireless sensor networks is a new approach.
2. Proposed Detection Method
We assume that noise at detector follows a Gaussian distribution with mean zero and variance. Depending on the event status, the signal received at the detector is given as follows: where represents the signal received at the detector;denotes the additive Gaussian noise at the detector; is the amplitude of the signal that is emitted by the event and is received at the detector; is the hypothesis corresponding to the event absence; is the hypothesis corresponding to the event presence. We assume that the power of the signal emitted by the event decays as the distance from the event increases and that the model of power attenuation of the signal is where is the signal power emitted by the event at distance zero; is the distance between the event and the detector; and is the signal decay exponent and takes a value from 2 to 3 [5, 11].
The signal-to-noise ratio (SNR) is Assume that consecutive samples are generated from the received signal by an A/D converter. The mean and variance of these samples are given as follows:
Proposition 1. Let . If the event is absent,has Student’s -distribution with degrees of freedom. Otherwise, if the event is present,has a noncentral Student’s-distribution with degrees of freedom and noncentrality parameter.
The proof is in the Appendix.
The event detection decision can be made by comparing the test statisticto the decision threshold as follows:
3. Performance Analysis
In this section, we present the analytical results of the false alarm probability and the detection probability of the proposed event detection method.
The probability of a false alarm is defined as where denotes the probability.
Let denote the probability distribution function (pdf) ofunder . According to the results obtained in [12] we have where is the Gamma function.
The cumulative distribution function (cdf) ofunder , denoted by , is given by [13] where is the regularized incomplete beta function, and Therefore, the probability of a false alarm is obtained as follows: It is noteworthy that for a given , only depends on the decision threshold, and it does not depend on the noise variance. Hence, the proposed detection method can manage the noise uncertainty, and the decision threshold can be easily determined in terms of the target false alarm probability through numerical method.
The probability of the detection is defined as Let and be the pdf and cdf ofunder , respectively. We then have When , the results in [14] have shown that where is the cdf of the standard normal distribution, , and .
An efficient algorithm to calculate can be found in [14].
4. Simulation Results
Monte Carlo simulations are carried out to evaluate the performance of the proposed method for event detection. The detection performance of the proposed method is compared to those of ED and the LR-based detection methods.
First, a simulation is performed under a condition where the number of samples is 30, the SNR at the detector is −4 , and the noise is white Gaussian noise with zero mean and unit variance. Both a Rayleigh fading channel [10] and log-normal shadowing channel with 6 of standard deviation [15] are also considered in this simulation. The ROC curves of the proposed method and of the comparison methods are both shown in Figure 1. Under the same channel conditions, the detection performance of the proposed method is comparable to that of LR-based detection method, and it is superior to that of the ED method.
Second, the detection performance of the proposed method is verified under a condition that the number of samples is 20, the false alarm probability is fixed at 0.1, and the SNR at the detector varies from −15 to 5 . In this case, we first choose the(decision threshold) for a given false alarm probability based on (10) through a numerical calculation and we calculate the test statistic based on where and are given by (4). After that, we compare the test statisticwith decision threshold , for the event decision. As seen in Figure 2, the detection probability of the proposed method is similar to that of the LR-based detection method, and it is higher than that of ED method. When the SNR becomes higher than 3 dB, the detection probabilities of all detectors reach 1.
Finally, the problem of noise uncertainty [16] is considered when , , and the noise uncertainty is 1, 2, and 3 . As shown in Figure 3, both the proposed and the LR-based detection methods have a marginal performance loss due to the noise uncertainty. This result is reasonable since in the proposed method, the change in the noise variance only impacts the detection probability and the false alarm probability is independent of the noise variance. In the case of the LR-based detection method, the noise variance appears in both the numerator and the denominator of the test statistic, so the impact of noise uncertainty is limited. However, in the case of ED method, when the noise variance changes, both the false alarm probability and the detection probability are affected since the distributions under and under of the test statistic depend on the noise variance. Consequently, ED method has a significant performance loss due to noise uncertainty.
Remarks. This paper only considers local event detection. However, the proposed scheme can be applied to any collaborative detection. For example, let us consider collaborative detection for a wireless sensor network that is composed of sensor nodes and one fusion center. sensor nodes adopt the proposed local detection scheme, make a local decision, and send their local decision to the fusion center. The fusion center will merge all local decisions according to the fusion rule when those sensor nodes send their local decision. We can denote the probability of detection and false alarm for the th sensor node as and , which is defined as (6) and (12), respectively. Then, the detectionand false alarm probabilities at the fusion center can be expressed as a function of the probabilities of each sensor node in the following manner [17]: where is the summation of all possible combinations of decisions, is the group of sensor nodes that has decided the event absence, is the group of sensor nodes that has decided the event presence, and is the decision rule.
From sensor nodes, the fusion center will get a local decision vector where represents the decision of the event presence while represents the decision of the event absence at the respectiveth sensor node .
The most general fusion rule that can be used in the fusion center is that of “out of,” and when it is adopted in fusion center, the event decision is carried out in the following manner:
For example, with three sensor nodes (), the probability of detection at the fusion center is given by
If the fusion rule is (the “AND” rule), then we have
Therefore, we have.
If the fusion rule is (the “Majority” rule), then we have
Therefore, we have .
If the fusion rule is (the “OR” rule), then we have
Therefore, we have
Similar expressions can be obtained for the probability of a false alarm at the fusion center from (15).
Figure 4 shows the detection probability and the false alarm probability at the fusion center according to SNR [dB] when three sensor nodes are cooperated to detect the event, that is, . In this case, the false alarm probability of each sensor node () is maintained at 1.0 by choosing the decision thresholdfor local decision based on (10), and all SNR of three sensor nodes are same and are changed from −4 and 3 dB. Log-normal shadowing channel with 6 of standard deviation are considered in the simulation. Since the false alarm probability of each node is fixed at 1.0, the false alarm probability at the fusion center is constant regardless of SNR of each sensor node for given fusion rule. But, the detection probability at the fusion center is improved as the SNR increases. OR rule provides the best detection probability while providing the worst false alarm probability among OR rule, AND rule, and Majority rule. On the other hand, AND rule provides the worst detection probability while providing the best false alarm probability among OR rule, AND rule, and Majority rule. However, it is noteworthy that the proposed scheme provides much better detection probability at the fusion center than ED detection scheme when the same fusion rule is applied at the fusion center while providing same false alarm probability to that of ED detection scheme.
5. Conclusion
Event detection is a fundamental problem in wireless sensor networks. In this paper, we have proposed an event detection method based on Student’s -distribution. The advantage of the proposed detection method is that it does not require any knowledge of the noise variance or SNR when the detection threshold is chosen in terms of a false alarm probability. Simulation results have shown that under same conditions, the proposed detection method always outperforms ED method and matches the performance of LR-based detection method.
Appendix
Proof of Proposition 1. If the event is absent, follows a normal distribution with mean 0 and variance. Hence has a standard normal distribution since the sample meanfollows a normal distribution with mean 0 and variance . Moreover, Cochran’s theorem has shown that has a chi-square distribution with degrees of freedom [18]. It is readily shown that has a Student’s-distribution with degrees of freedom [12]. Note that . Thus,follows Student’s -distribution with degrees of freedom.
If the event is present, follows a normal distribution with meanand variance . Thus, has a standard normal distribution sincefollows a normal distribution with mean and variance . In addition, Cochran’s theorem has shown that has a chi-square distribution with degrees of freedom. It is readily shown that has a noncentral Student’s-distribution with degree of freedom and noncentrality parameter . Note that . As a result,has noncentral Student’s-distribution with degrees of freedom and noncentrality parameter.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by 2014 Research Funds of Hyundai Heavy Industries for University of Ulsan.