System-level Calibration for Data Fusion in Wireless ... - CiteSeerX

System-level Calibration for Data Fusion in Wireless Sensor Networks Rui Tan1 ,

Zhaohui Yuan1∗, Guoliang Xing2 , Xue Liu3 , 1 City University of Hong Kong, HKSAR 2 Michigan State University, USA 3 McGill University, Canada

Abstract

calibration becomes intractable when the network scales to tens or hundreds of sensors. Different from the device-level calibration, system-level calibration aims to optimize the overall system performance by tuning all the sensors in a network. System-level calibration often leads to significantly lower overhead by taking advantage of the knowledge about a sensor network, such as how the local information observed by individual sensors is processed in the network. Data fusion [25] is a widely adopted signal processing technique that can improve the system performance by jointly considering the measurements of multiple sensors. In practice, various data fusion schemes have been employed by sensor network systems for target detection [14, 18], localization [17], and classification [7, 8]. In this paper, we study the system-level calibration in fusion-based WSNs. The major challenge of calibration for data fusion is to exploit the complex correlation between system performance and the characteristics of individual sensors. On the one hand, the system performance of fusion-based networks is tightly coupled with the measurements of multiple sensors. On the other hand, system-level calibration must avoid centralized data collection and processing due to the resource constraints of wireless sensors. In this paper, we propose a novel two-tier system-level calibration approach to address the aforementioned challenges. In the first tier, each sensor learns its local sensing model using in-place measurements, and only transmits a few model parameters to the fusion head node. In the second tier, the fusion head calibrates each sensor’s model to a common sensing model. Several challenges must be addressed for realizing such a two-tier approach. First, the measurements of sensors are often corrupted by noise from the physical environment. Model learning from noisy data is challenging for low-power sensors with limited computation and memory resources. At the same time, the local model must accurately preserve the systematic bias of a sensor that is needed for system-level calibration. Moreover, in the second-tier calibration, the correlation between biased local sensing models and the overall system performance must be carefully considered in order to achieve the optimal system performance at run time. We make the following key contributions in this paper.

Systematic biases in sensor measurements undermine the performance of wireless sensor networks in missioncritical applications such as target detection and tracking. Traditional device-level calibration approaches become intractable for moderate to large-scale networks due to limited access of individual sensors after deployment. In this paper, we propose a two-tier system-level calibration approach for a class of sensor networks that employ data fusion to improve the overall system performance. In the first tier, each sensor learns its local sensing model from noisy measurements using an online algorithm and only transmits a few model parameters. In the second tier, sensors’ local sensing models are then calibrated to a common system sensing model. Our approach fairly distributes computation overhead among sensors and significantly reduces the communication overhead of system-level calibration. Based on this approach, we develop an optimal model calibration scheme that maximizes the target detection probability of a sensor network under bounded false alarm rate. The simulations based on synthetic data as well as real data traces collected by 18 sensors show that our system-level calibration scheme can improve the detection performance of a sensor network by up to 50%.

1

Introduction

Deploying wireless sensor networks (WSNs) for missioncritical applications such as target detection [5, 18], localization [17, 23], and security surveillance [14] imposes new challenges for sensor calibration. The measurement inaccuracy caused by hardware drift [4, 22], complex deployment terrain, and random noise [4] must be dealt with properly if sensor data are to be useful. In traditional calibration approaches, the sensor readings are mapped to the true value by a calibration function obtained under controlled environments [22] or from manufacturer’s specification. However, such a device-level calibration scheme fails to account for the post-deployment factors such as non-uniform terrain condition and aging characteristics. Moreover, device-by-device ∗ The

Jianguo Yao3

first two authors are listed in alphabetic order

1

• We propose a novel two-tier calibration approach for data fusion in WSNs. Our approach fairly distributes the computation overhead among individual sensors. Moreover, each sensor only needs to transmit little information in order to achieve the optimal system-level calibration. • We formally formulate the problem of system-level calibration for target detection based on the two-tier approach. A linear regression algorithm is proposed for each sensor to learn the local target signal decay model. The algorithm processes sensor measurements in an online fashion and thus incurs little computation and memory overhead. We then develop a method to calibrate the biased local signal decay models and show that it leads to the maximum target detection probability under bounded false alarm rate. • We evaluate our approach through extensive simulations based on synthetic data as well as the real traces collected by 18 nodes in a vehicle detection experiments [1]. The results show that our approach can achieve satisfactory performance under a range of realistic settings of Signal-to-Noise Ratio (SNR) and sensor density. In particular, our system-level calibration method can improve the detection performance of a sensor network by up to 50%.

the calibration parameters. The approach presented in this paper also belongs to the macro-calibration. However, different from these existing macro-calibration approaches, in our two-tier calibration architecture, each sensor estimates a local measurement model and only the model parameters need to be transmitted for system-level calibration, hence much communication overhead is avoided. Existing methods can also be classified into non-blind, blind, and semi-blind calibrations. For the non-blind calibration approaches [9, 22, 26], the parameters of sensors are adjusted using sensor measurements and known ground truth inputs. For instance, the readings of extra high-quality sensors are employed as the ground truth in [9, 22]. In [26], the distance between ranging sensors are known during the calibration phase. In contrast, the blind calibration approaches [3, 4] do not require ground truth inputs. In [4], the discrepancies among co-located sensors are eliminated by exploring the spatial correlation in their measurements, hence the ground truth is unnecessary. Balzano et al. [3] theoretically prove that the sensors can be partially calibrated using blind measurements. However, the blind approaches often require that the deployment is dense enough [3, 4]. As a compromise, the semi-blind calibration approaches [15, 20, 21] require partial ground truth information. In [15, 21], the sensor locations are calibrated using accurate or coarse position information of a subset of nodes. In [20], an uncalibrated sensor calibrates itself when rendezvousing a calibrated sensor. The calibration approach presented in this paper falls into the semi-blind category, in which sensors can calibrate their energy measurements as long as the physical position (instead of the accurate energy profile) of the target is known. Data fusion [25] has been proposed as an effective signal processing technique to improve the system-wide performance in WSNs. Several data fusion schemes have been incorporated into sensor network systems designed for surveillance applications [7, 8]. Our approach calibrates the sensor measurements in data fusion to optimize the system detection performance. To the best of our knowledge, the calibration for data fusion has not been systematically studied.

The rest of this paper is organized as follows. Section 2 reviews related work. Section 3 introduces the motivation and preliminaries. The two-tier calibration approach and the formulation of our problem are presented in Section 4. The local model estimation and system-level model calibration are in Section 5 and 6, respectively. We offer simulation results in Section 7. Section 8 concludes this paper.

2

Related Work

Sensor calibration is a fundamental problem in WSNs. The previous literatures on calibration [3, 4, 9, 15, 20– 22, 26] can be categorized into micro-/macro- and nonblind/blind/semi-blind approaches. The micro-calibration refers to the method that tunes each individual sensor to output accurate readings. In [22], each chemical sensor is carefully calibrated in laboratory to obtain the mapping from its reading to the true value. In contrast, the macro-calibration [4, 9, 26] focuses on optimizing the overall system performance. In [9], the biases of light sensors are estimated by solving a group of equations that integrate all sensors’ measurements, meanwhile the impact of random noise is minimized. Similarly, in [26], the parameters of ranging sensors are estimated by regression based on pair-wise range measurements. In [4], the calibration function, which maps two sensors’ outputs, is adjusted so that the consistency among a group of sensors is maximized. These macro-calibration approaches require the transmission of all or partial raw measurements to a central node for computing

3

Problem Statement and Preliminaries

In Section 3.1, we first introduce the motivation of system-level calibration for data fusion. The technical preliminaries including sensor measurement and data fusion models are described in Section 3.2 and 3.3, respectively.

3.1

Problem Statement

Sensor calibration refers to the process of identifying and correcting systematic errors (bias) of sensors. Various factors attribute to the bias of sensors. First, manufacturing processes inevitably introduce variation in electric characteristics of sensor circuits. Second, the measurements of sensors 2

0.3 0.25

node 41 2/(x/5.2)2 node 48 2/(x/8)1.9 CDF

Energy measurement y

0.4 0.35

0.2 0.15 0.1 0.05 0 20

40 60 80 100 120 140 Distance from target d (m)

Fig. 1. Normalized energy measurement vs. the distance from the vehicle (in the unit of meters).

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 noise CDF 0.1 0 -2 -1 0 1 2 3 4 5 Noise energy (×10−3 )

data fusion literature [5,17,23] often assumes identical sensing characteristics for all sensors and hence cannot be used in uncalibrated networks. This paper aims to calibrate sensors in fusion-based sensor networks such that the system performance is optimized.

3.2

Sensor Measurement Model

The energy of many physical signals (e.g., acoustic and electronmagnetic) attenuates with the distance from the signal source. Suppose sensor i is di meters away from the target that emits a signal of energy S. The attenuated signal energy si at the position of sensor i follows the signal decay model given by S si = , (1) (di /ri )ki

6

Fig. 2. The CDF of noise energy. The dashed line is the CDF of the normal distribution N (0.001, 0.0012 ).

where ri is the reference distance determined by target’s and sensor’s shapes, ki is the decay factor which is typically from 1 to 5. We note that the reference distance and decay factor vary with sensors due to the reasons discussed in Section 3.1. Our objective is to calibrate sensors such that they have an identical signal decay model. For many practical applications, the measurements of sensors are contaminated by additive noise. The energy measurement of sensor i (denoted by yi ) is given by

are largely affected by the terrain of deployment. For instance, in the scenario of vehicle detection [1, 7, 8, 18], complex terrain often causes sensors to yield different sensitivity. In this paper, we do not distinguish the sensor bias due to different causes as doing so in large-scale network deployment is intractable. In addition to systematic bias, random noise (from environment or hardware itself) also causes sensor measurements to be inaccurate. However, random noise often yields predictable distribution (e.g., Gaussian). Our sensor calibration approach exploits this property and is resilient to the influence of random noise. We use an example based on the real data traces from the DARPA SensIT vehicle detection experiments [1, 8] to illustrate the variation in sensors’ characteristics. In the experiments, 75 WINS NG 2.0 nodes are deployed to detect military vehicles driving through the monitored area. Fig. 1 plots the acoustic energy measurements taken by node 41 and 48, versus the distance from a driving vehicle. We can see from Fig. 1 that the energy measurements of both sensors decay exponentially with the distance. However, they follow different decay functions. For instance, when the distance from the vehicle is 40 m, the measurements of node 41 and 48 are about 0.05 and 0.1, respectively. This example illustrates that sensors could yield considerably different sensing characteristics after deployment. Calibration in WSNs is a challenging task as sensors often have limited access after being deployed. Moreover, the manual device-by-device calibration becomes intractable when the network scales to tens of sensors. A conventional calibration method is to correct the readings of sensors to a common benchmark (e.g., readings of high-precision sensors or ground truth) [9,22,26]. However, such an approach often incurs high overhead in large-scale sensor networks. In contrast, we adopt a system-level approach that calibrates the sensing model of each sensor to optimize the system-wide performance of a network. In particular, we focus on the calibration problem in a class of sensor networks that employ data fusion to improve the system performance. Existing

yi = si + ni ,

(2)

where ni is the energy of noise experienced by sensor i. In practice, a sensor’s measurement is computed by the average over a number of (≥ 30) samples [7,8,23]. From the Central Limit Theorem, the noise energy ni follows the normal distribution [23], formally, ni ∼ N (µi , σi2 ), where µi and σi2 are the mean and variance of the noise ni . The above signal decay and noise models have been widely assumed in the literature of signal detection [23, 25] and also have been empirically verified [12,17]. In Fig. 1, the curves are the curve fittings of the data points to the signal decay model in (1). We can see that the model parameters, i.e., ri and ki , vary with sensors. For instance, the reference distances for node 41 and 48 (i.e., r41 and r48 ) are 5.2 m and 8 m, respectively. Fig. 2 plots the Cumulative Distribution Function (CDF) of background noise1 measured by a sensor in the SensIT experiments [1, 8]. We can see that the CDF of measured noise well matches the CDF of the normal distribution N (0.001, 0.0012 ). Table 1 summarizes the notation used in this paper.

3.3

Multi-sensor Data Fusion Model

Data fusion [25] has been proposed as an effective signal processing technique to improve the system performance of sensor networks. A sensor network that employs data fusion 1 The

measurements are regarded to be noise when the vehicle is far enough away from the sensor (e.g., 400 m).

3

Table 1. Summary of Notation∗ Symbol

Definition

S µi , σi2 ri , ki di si

original energy emitted by the target the mean and variance of noise the reference distance and decay factor distance from controlled target / surveillance spot S attenuated signal energy, si = ki (di /ri )

ni Gaussian noise, ni ∼ N (µi , σi2 ) yi energy measurement, yi = si + ni bi the z-intercept of linear fitting of measurements N the number of sensors in a detection cluster α the upper bound of false alarm rate H0 / H1 the hypothesis that the target is absent / present ∗ The symbols with subscript i are respect to sensor i.

Fig. 3. Two system-level calibration schemes.

is often organized into clusters. The cluster head is responsible for making a decision regarding the presence of target by fusing the information gathered by member sensors. As sensors can only carry out limited processing due to resource constraints, we adopt a simple data fusion scheme as follows [5]. Each cluster head makes the detection decision by comparing the sum of measurements reported by member sensors against a detection threshold T . Suppose there are N member sensors in a cluster, ∑N the sum of measurements (denoted by Y ) is Y = i=1 yi . If Y ≥ T , the cluster head decides 1; otherwise, it decides 0. The detection of a target is inherently stochastic due to the noise in sensor measurements. The system detection performance is characterized by two metrics, namely, the false alarm rate (denoted by PF ) and the detection probability (denoted by PD ). PF is the probability of making a positive decision when no target is present, and PD is the probability that a present target is correctly detected. Under the aforementioned data fusion model, PF and PD are given by PF = P(Y ≥ T |H0 ),

note that our approach does not require the knowledge of target signal energy as long as the location information is available. Sensors and the target may obtain their positions through GPS or a localization service [13] in the network. There are two possible schemes to such a calibration problem. The straightforward scheme works in a centralized fashion. For example, in Fig. 3, each member sensor of cluster A sends the raw measurements to the cluster head A1, which computes the calibration parameters for each individual sensor. However, a large number of measurements are often required to accurately characterize the sensing model of a single sensor such as the signal decay model in (1). Hence, such a scheme will introduce high communication overhead. The second scheme is based on a two-tier architecture. In Fig. 3, each sensor in cluster B learns its sensing model which is characterized by a few model parameters based on the raw measurements, and only sends the model parameters to the cluster head B1. The cluster head then calibrates each sensor’s sensing model such that the expected system performance of detecting targets at run time is maximized. Such a scheme not only distributes the computation load to each sensor, but also avoids costly transmission of raw data. In this paper, we will adopt this scheme and the details are discussed in the following. In the local calibration (i.e., the first tier), each sensor i estimates its noise and signal decay models using in-place measurements of controlled targets. The sensors periodically (e.g., every 5 seconds) measure the energy of signals emitted by a controlled target driving through the deployment region. To reduce the impact of noise, each sensor takes a number of measurements when the target is at a certain position. Only the estimated sensing model parameters of each sensor are then transmitted to the cluster head which only incurs low overhead. Several parameter estimation methods such as Maximum Likelihood Estimation can be used to solve such a problem. We will adopt the online Least Squares method because of its low computation and memory overhead. In the system-level calibration (i.e., the second tier), the cluster head computes the calibration parameters for each

PD = P(Y ≥ T |H1 ),

where H0 and H1 represent the cases where the target is absent and present, respectively.

4

Overview of Approach

In this section, we first present our two-tier system-level calibration approach that aims to maximize system detection performance, then formally formulate the problem.

4.1

System Architecture

We use the scenario of military vehicle surveillance to illustrate our proposed calibration approach. In order to calibrate the sensors that are deployed to monitor military vehicles as shown in Fig. 3, we can drive a vehicle which acts the controlled target through the monitored region. The network then calibrates the sensors based on their in-place measurements of the controlled target such that the system’s performance of detecting targets at run time is maximized. We 4

sensor such that the overall system performance is maximized. Due to the resource constraints of sensors, we adopt a simple linear calibration scheme as follows. The calibrated measurement of sensor i (denoted by yˆi ) is given by

With the model parameters obtained in each sensor’s local calibration, the system-level calibration aims to calibrate each sensor’s measurement model to maximize the system detection performance at each surveillance spot. We focus on a single surveillance spot in the rest of this paper. When the target is present, the calibrated measurement of sensor i is given by yˆi |H1 = γi · yi |H1 = γi · si + γi · ni , where si follows the signal decay model in (1) and ni is the Gaussian noise. We denote the calibrated signal energy sˆi = γi · si . After the calibration, the calibrated signal energy of each sensor should follow a common signal decay model which is given by S0 sˆi = , (3) (di /r)k

yˆi = γi · yi , where γi is the calibration coefficient of sensor i. The objective is to determine the calibration coefficients of all sensors involved in the data fusion, such that the system detection performance is maximized. We are only concerned with the detection performances at a number of physical locations which are referred to as surveillance spots. Surveillance spots can be chosen before network deployment according to application requirements or identified by the network autonomously at run time. For instance, in Fig. 3, the surveillance spots can be chosen along the roads in the monitored area. For each surveillance spot, the cluster head computes a calibration coefficient γi for each sensor i such that the detection performance at the surveillance spot is maximized. The calibrated network detects target as follows. To detect whether a target is present at a particular surveillance spot, each sensor i multiplies its measurement yi by the calibration coefficient γi to output the calibrated measurement yˆi . Then, the cluster head fuses all calibrated measurements as described in Section 3.3 to make a detection decision regarding whether a target is present at the surveillance spot.

4.2

where S 0 , r, and k are the common source energy, reference distance, and decay factor for all calibrated sensors, respectively. Note that di is the distance between sensor i and the surveillance spot. We note that there exists a trade-off between the false alarm rate PF and detection probability PD . Specifically, a higher PD is always achieved at the price of higher PF [25]. Therefore, a common requirement of many systems is to maximize the PD while the PF is bounded. Our objective is to find the calibration coefficients to maximize the system detection performance, i.e., to maximize the PD under an upper bound of PF , α ∈ (0, 1), specified by application. The system-level calibration problem can be formally formulated as follows.

Problem Formulation

Problem 2. Suppose there are N sensors in a detection cluster. Given each sensor’s sensing model {µi , σi2 , ri , ki |1 ≤ i ≤ N } and the distances from the surveillance spot {di |1 ≤ i ≤ N }, find a common signal decay model (S 0 , r, k) and a list of calibration coefficients {γi |1 ≤ i ≤ N }, such that the detection probability PD is maximized subject to that the false alarm rate PF is upper-bounded by α (i.e., PF ≤ α, α ∈ (0, 1)).

The objective of local calibration at each sensor is to learn the Gaussian noise model and the signal decay model in (1), which are characterized by µi , σi2 , ri , and ri . For a single sensor i, the inputs are a collection of data pairs (di , yi ), i.e., the measurement yi when the target is di meters away. The major challenge is how to learn these models using stochastically noisy measurements. To cope with the noise, sensor i samples a number of energy measurements when the controlled target is at a certain position. These measurements can be used to compute a statistic such as average so that the impact of noise is mitigated. Our local calibration problem is formally formulated as follows.

This is a constrained optimization problem. We study the optimal solution of this problem in Section 6.

5

Problem 1. Suppose at time step t, sensor i is di (t) meters away from the controlled target and takes M measurements Yi (t) = {yi [1], yi [2], . . . , yi [M ]}, where yi [·] is given by (2). How to estimate the noise and signal decay models (µi , σi2 , ri , ki ) of sensor i using its noisy measurements {di (t), Yi (t)|t = 1, 2, . . .}?

Online Local Calibration

In this section, we present our approach of local calibration. We propose a solution based on linear regression technique in Section 5.1 and its online variant in Section 5.2.

5.1

Measurement Model Estimation

We first present how to estimate the noise model. When no target is present, sensor i only measures noise, i.e., yi |H0 = ni . Accordingly, each sensor can estimate the noise model using a number of measurements in absence of target. In practice, the measurements can be treated as noises when the target is far enough away from the sensor. The mean and variance of the noise can be estimated by the

Basically, this is an over-provision parameter estimation problem [2], as a large number of measurements are usually available to train the models. However, we note that the parameter estimation method must incur low computation and memory overhead due to the resource constraints of sensors. This problem will be studied in Section 5. 5

Therefore, the transformed data points {zi (t), xi (t)|t = 1, 2, . . .} should lie on the straight line with slope of ki and z-intercept of bi . The Least Square linear fitting of the data points are given by [19]

sample respectively. ∑M mean and variance, ∑M Specifically, µi = 1 1 2 2 y (j)|H and σ = i 0 i j=1 j=1 (yi (j)|H0 − µi ) , M M −1 where M is the number of samples that are used to estimate the noise model. Now we discuss how to estimate the signal decay model in (1). As the measurements are noisy, we cannot directly use them to estimate the signal decay model. A common approach to reduce the impact of noise is to average multiple measurements [27]. However, such a method requires a large number of samples when the sensor experiences heavy noise. We propose a detection-based approach as follows, which explicitly considers the impact of stochastic noise on detection performance. As described in Section 4.2, at time step t, sensor i takes M measurements (i.e., Yi (t)) when the target is di (t) meters away from it. We ignore the movement of the target during these M measurements. For instance, if the sensor samples 10 energy measurements in 0.5 seconds (i.e., 0.05 seconds per measurement) [1, 8], the movement of the vehicle is about 2.5 m as the average vehicle speed in the SensIT experiments [1] is 5 m/s. Therefore, the movement of the vehicle during these 10 measurements can be ignored as the distance from the vehicle is usually tens to hundreds of meters [1]. For each measurement yi ∈ Yi (t), sensor i makes a detection by comparing yi against a threshold η 2 . The detection probability at time step t (denoted by PDi (t)) can be estimated by the ratio of the number of measurements that exceed η to M . Hence, sensor i has a statistic PDi (t) for each time step t. Sensor i then estimates the signal decay model by Least Square Estimation techniques using the statistics {PDi (t), di (t)|t = 1, 2, . . .} according to the relationship between the statistic PDi (t) and the signal decay model, which is derived as follows. When the target is present, the measurement of sensor i follows the normal distribution, i.e., yi |H1 = si + ni ∼ N (si + µi , σi2 ). Therefore, under the aforementioned detection rule, the detection probability of sensor i is ( ) η − si − µ PDi = P(yi ≥ η|H1 ) = 1 − Φ , σ

ki =

bi = z¯i − ki · x ¯i ,

(6)

where z¯i and x ¯i are the sample means, cov(xi , zi ) and σx2i are the covariance and variance (with( respect ) to xi ) of the data points, respectively. As bi = ln Sriki , the reference distance ri can be computed by the estimates of ki and bi which are given by (6), ( ri =

ebi S

) k1

i

.

(7)

Note that the target’s original energy S is usually unknown, therefore we cannot compute the exact value of ri from the linear fitting. However, our analysis in Section 6 shows that the system detection performance only depends the zintercept bi and the exact value of ri is unnecessary. Therefore, the measurement model of sensor i can be represented by a 4-tuple (µi , σi2 , ki , bi ). Sensor i then transmits such 4tuple to the cluster head after its local calibration.

5.2

Online Model Estimation

In Section 5.1, we adopt the linear regression technique to estimate the slope ki and z-intercept bi in (5). According to (6), sensor i has to store all previous data points to compute the estimates of ki and bi . Such an approach is not suitable for memory-constrained sensors. For instance, if a sensor generates a data pair (PDi , di ) every 0.5 seconds and the duration of the training process is 5 minutes, the sensor needs to store 600 data pairs, which represents significant overhead for low-cost sensors with only a few Kbytes of memory [6]. In this section, we will introduce an online variant of the linear regression to estimate ki and bi in (5). The online algorithm works in a real-time fashion, i.e., the estimates of ki and bi are updated for each time when sensor i obtains a data point. The estimates converge after the sensor obtains sufficient data points. We adopt the widely used Recursive Least Square Estimation (RLSE) approach [2, 16, 24]. After reformulating (5) with vectors as zi (t) = φTi (t) · θi where φi (t) = [xi (t), 1]T and θi = [ki , bi ]T , the recursive estimator of θi is given by ( ) θi (t) = θi (t − 1) + Li (t) zi (t) − φTi (t)θi (t − 1) , (8) ( )−1 , L(t) = P (t − 1)φ(t) 1 + φT (t)P (t − 1)φ(t)

where η is the detection threshold and Φ(·) is the CDF of the standard normal distribution, formally, Φ(x) = ∫ x −t2 /2 √1 e dt. By replacing si with (1) and taking log2π −∞ arithmic transformation, we have ( ) ( ) ln η−µi −σi Φ−1 (1−PDi ) = −ki ln di +ln Sriki , (4) where Φ−1 (·) is the ( inverse function of Φ(·). For ) time step t, let zi (t) = ln η −(µi − σ)i Φ−1 (1 − PDi (t)) , xi (t) = − ln di (t), and bi = ln Sriki . From (4), we have zi (t) = ki · xi (t) + bi .

cov(xi , zi ) , σx2i

P (t) = (I − L(k)φT (t))P (t − 1), (5)

where I is the identity matrix. Eq. (8) updates the estimates at each time step based on the error between the model output and the predicted output. The initial estimates θi (0) can

threshold η can be set to be the mean or median of Yi (t) to avoid the saturation of PDi (t), i.e., the case where PDi (t) equals 0 or 1. 2 The

6

offline online

1 2 3 4 5 6 7 8 9 10 Time step t

(a) Estimate of k

Estimate of b

Estimate of k

14 12 10 8 6 4 2 0 -2

70 60 50 40 30 20 10 0 -10

T ∗ can be derived as

offline online

T∗ =

N ∑

v uN u∑ −1 γi µi + Φ (1 − α) · t γi2 σi2 .

i=1

(9)

i=1

When the target is present, the calibrated measurement of sensor i follows the normal distribution, i.e., yˆi |H1 = sˆi + γi ni ∼ N (ˆ si + γi µi , γi2 σi2 ). Therefore, the sum of calibrated measurements ∑Nalso follows the( normal distribution, i.e., Yˆ |H1 )= ˆi |H1 ∼ i=1 y ∑N ∑N ∑N 2 2 N ˆi + i=1 γi µi , i=1 γi σi . Hence, the sysi=1 s tem detection probability is given by   ∑N ∑N T − s ˆ − γ µ i i i i=1 i=1 . √ PD = P(Yˆ ≥ T |H1 ) = 1−Φ  ∑N 2 2 i=1 γi σi

1 2 3 4 5 6 7 8 9 10 Time step t

(b) Estimate of b

Fig. 4. The convergence of RLSE.

be set to be the best guesses of ki and bi so that the estimator can quickly converge. More detailed derivation and convergence proof can be found in [16, 24]. For each time step, only total 22 float multiplications and 17 float additions are needed to update the estimator state. Moreover, only the estimator state, i.e., θ2×1 , L2×1 , and P2×2 are needed to be stored. Such computation and memory overhead is affordable for low-power wireless sensors such as MICA motes [6]. Fig. 4 plots the numerical results that demonstrate the convergence of the RLSE algorithm. From the figure, we can see that the results of RLSE converge to the offline results in ten time steps. Therefore, it is safe to adopt the RLSE in local calibration, as a large number of ( 10) training data points are usually available for each sensor in practice.

By replacing T with the optimal detection threshold T ∗ (given by (9)) to bound the system false alarm rate, we have   ∑N s ˆ i  . (10) PD = 1 − Φ Φ−1 (1 − α) − √∑ i=1 N 2 2 i=1 γi σi

6.2

Optimal Calibration

In this section, we first derive the system detection performance of a calibrated network under the data fusion model described in Section 3.3, and then discuss how to find the optimal system-level calibration coefficients.

From the derivation in Section 6.1, if the detection threshold at the cluster head is set to be T ∗ given by (9), the system false alarm rate is α and the detection probability of the calibrated network is given by (10). Therefore, Problem 2 formulated in Section 4.2 reduces to find the common signal decay model (S 0 , r, k) and calibration coefficients {γi |1 ≤ i ≤ N } such that the detection probability PD given by (10) is maximized. The optimal solution is given by the following theorem.

6.1

Theorem 1. The optimal common signal decay model and calibration coefficients are given by

6

Optimal System-level Model Calibration

Calibrated System Detection Performance

∗

sˆi = d−k , i

When no target is present, the calibrated measurement of sensor i follows the normal distribution, i.e., yˆi |H0 = γi ni ∼ N (γi µi , γi2 σi2 ). Therefore, the sum of calibrated ˆ measurements follows distribution, (∑the normal∑ ) i.e., Y |H0 = ∑N N N 2 2 ˆi |H0 ∼ N i=1 y i=1 γi µi , i=1 γi σi , where N is the number of sensors in the detection cluster. Hence, the system false alarm rate is given by   ∑N T − γ µ i i , PF = P(Yˆ ≥ T |H0 ) = 1 − Φ  √∑ i=1 N 2 2 i=1 γi σi

∗

γi = dki i −k e−bi ,

(11)

∗

respectively, where k maximizes the following function ∑N −k i=1 di Λ(k) = √∑ . (12) N 2 −2bi d2ki −2k i i=1 σi e Proof. From (1) and (3), the calibration coefficient can be calculated as γi =

S 0 rk sˆi = · · dki −k = S 0 rk dki i −k e−bi , si S riki i

(13)

where ri is replaced with its estimate (7). By replacing sˆi and γi in (10) with (3) and (13), respectively, we have   ∑N −k d i=1 i  (14) PD = 1−ΦΦ−1(1−α)− √∑ N 2 e−2bi d2ki−2k σ i i=1 i

where T is the detection threshold of the data fusion model. As PD is a non-decreasing function of PF [25], it is maximized when PF is set to be the upper bound α. Such a scheme is referred to as the Constant False Alarm Rate detector [25]. Let the PF = α, the optimal detection threshold

= 1 − Φ(Φ−1 (1 − α) − Λ(k)). 7

1400 1300

recorded by 18 nodes at a frequency of 4960 Hz, when an Assault Amphibian Vehicle (AAV) drives through a road. The ground truth data include the positions of sensors and the trajectory of the AAV recorded by a GPS device. The data traces used in our simulations include the acoustic time series recorded for 9 AAV runs (AAV3-11). Fig. 6 plots the layout of the 18 nodes used in our simulations and the trajectory of AAV3. We first evaluate the convergence of the online model estimation algorithm proposed in Section 5. We use the data traces of AAV11 to train the sensors. In the training phase, a sensor generates a data point (PDi , di ) using the time series recorded during 0.75 seconds for every 0.75 × 15 = 11.25 seconds. The signal decay model of the sensor is updated for each training data point by the RLSE algorithm discussed in Section 5.2. To evaluate the detection performance of the calibrated network, we measure the average detection probability of AAV11 as follows. The network optimizes the calibration coefficients as discussed in Section 6 and detects AAV11 for every 0.75 seconds when AAV11 drives through the monitored region. The average detection probability is computed as the fraction of successful detections. Fig. 7 plots the average detection probability versus the number of data points used in the training phase at each sensor. We can see from the figure that our online calibration approach converges in ten steps, which is consistent with the performance evaluation of RLSE in Fig. 4. This is because the impact of noise is largely reduced by the detection-based approach used in local calibration, and the linear fitting of (5) becomes accurate after several data points are obtained. We compare our approach with an non-calibration algorithm, in which the measurements of sensors are fused to detect the target without calibration. We can see from Fig. 7 that our approach outperforms the non-calibration algorithm in terms of average detection probability. Note that as the detection threshold T (given by (9) with γi = 1) in the non-calibration algorithm only depends on the noise model, the detection performance of the network without calibration is independent from the training data volume. We then use the data traces of various AAV runs to evaluate the effectiveness of our approach. We train the network using AAV3 and measure the average detection probability of various runs. Fig. 8 plots the average detection probability of 9 AAV runs. We can see from the figure that in each run, our approach outperforms the non-calibration approach. The performance gain is up to 50%. We note that the low detection probabilities in some runs are due to the fact that the target was far away from the sensors. In our approach, the calibration coefficient γi can also be considered as a weight for the measurement of sensor i. According to (11), the optimized weight jointly accounts for sensor’s sensing model and the distance from the surveillance spot. We observe from the simulations that the sensors close to the target are allocated with high coefficients

S = 1.5 S = 2.0 S = 2.5

1200 Λ(k)

1100 1000 900 800 700 600 500

0

1

2

3 4 5 6 7 8 Common decay factor k

9

10

Fig. 5. Λ(k) vs. common decay factor k .

From (14), we observe that the system detection probability does not depend on the common source energy S 0 and reference distance r. To simplify the computation on sensors, we set S 0 = 1 and r = 1 m. As Φ(·) is an increasing function, the maximum of Λ(k) maximizes PD . Therefore, the optimal decaying factor k ∗ maximizes Λ(k). By setting S 0 = 1, r = 1, and k = k ∗ , the optimal common signal decay model and calibration coefficients are given by (11). Maximizing Λ(k) in (12) is an unconstrained numerical optimization problem. There are various algorithms for solving such a problem such as the Newton’s method. However, the naive search of k ∗ also suffices as k usually has a narrow reasonable value domain, e.g., (0, 10]. Moreover, the computation overhead can be controlled by the search granularity. The communication overhead is low as only the calibration coefficient γi is needed to be transmitted back to each sensor. We now illustrate the optimization through a numerical example. Only two nodes are involved in this example, and their signal decay models are given in Fig. 1. Their distances from the surveillance spot are 8 m and 18 m, respectively. Fig. 5 plots the objective function Λ(k) versus the common decay factor k given various source energies of the controlled target. The numerical results show that k ∗ is 1.68 m, which does not depend on the source energy of the controlled target. When S = 2, the calibration coefficients of these two sensors are γ41 = 0.036 and γ48 = 0.018 respectively.

7

Simulations

In this section, we conduct simulations based on synthetic data as well as real data traces collected in the DARPA SensIT vehicle detection experiments [1, 8].

7.1

Trace-driven Simulations

We conduct extensive simulations using the real data traces collected in the DARPA SensIT vehicle detection experiments [1,8]. In the experiments, 75 WINS NG 2.0 nodes are deployed to detect military vehicles driving through several intersecting roads. We refer to [8] for detailed setup of the experiments. The dataset used in our simulations includes the ground truth data and the acoustic time series 8

Average PD of AAV11

South-North (meters)

500 400 300 200 100 0

18 WINS nodes AAV3 trajectory 0

100

200 300 West-East (meters)

400

500

Fig. 6. Sensor layout in the SensIT experiments [1].

1

1.2

0.8

1 Average PD

600

0.6 0.4 0.2 0

2-tier calibration non-calibration 1

7.2

0.8 0.6 0.4 0.2

2 3 4 5 6 7 8 9 Training data volume of each sensor

0

10

Fig. 7. The Average PD vs. training data volume.

and therefore the high-quality measurements from these sensors mostly contribute to the detection. In contrast, the non-calibration approach treats all sensors equally and hence yields low performance.

2-tier calibration non-calibration

3

4

5

6 7 8 AAV runs

9

10 11

Fig. 8. The Average PD in different AAV runs.

Table 2. False alarm rate (%) vs. number of sensors N 2-tier calibration non-calibration averaging

Numerical Settings

In addition to trace-driven simulations, we also conduct simulations based on synthetic data. These simulations allow us to test our approach in a wide range of settings on SNR and network density. A number of sensors are uniformly deployed in a circular region with diameter of 200 meters, as illustrated in Fig. 9. Both the reference distance ri and decay factor ki for each sensor i are drawn from the normal distribution N (2, 1). The mean and variance of the Gaussian noise are set to be µ = 1 and σ 2 = 1 for all sensors. The upper bound of false alarm rate is set to be α = 5%. When no target is present, each sensor samples 100 energy measurements and estimates its noise model (i.e., µi , σi2 ) by the method presented in Section 5.1. The controlled target drives through the region at a constant speed of 1 m/s. Each sensor i continuously samples signal energy at a frequency of 100 Hz and generates a statistics PDi every one second as described in Section 5.1. Note that the local detection threshold η is set to be the average of the energy samples that are used to compute a PDi to avoid the saturation of PDi . After the training phase, each sensor obtains its own sensing model characterized by the 4-tuple (µi , σi2 , ki , bi ). The network detects targets as follows. A target randomly appears in the circular surveillance region. For each target, the network optimizes the calibration coefficient γi for each sensor as stated in Section 6.2. To measure the system detection performance, the target appears for a large number of times (1000 in our simulations). The detection probability of the network is calculated as the fraction of successful detections. In addition to the non-calibration algorithm, we also employ another baseline algorithm. In the averaging algorithm, the common reference distance r and decay factor k are set to be the average of each sensor’s ri and ki respectively, i.e.,

5 3.7 1.4 2.6

10 3.9 1.9 2.3

15 4.6 0.8 2.8

20 4.4 1.1 2.7

25 3.5 1.5 1.7

30 4.3 1.2 2.9

35 4.5 1.7 2.4

40 3.7 1.2 2.2

r = r¯i and k = k¯i . Then the calibration coefficient γi is calculated by (13). We note that the reference distance ri is usually unknown in practice as discussed in Section 5.1.

7.3

Simulation Results of Synthetic Data

We first validate that the system false alarm rates under different algorithms are bounded by α (5%). Table 2 lists the measured false alarm rates with various number of deployed sensors. Although the false alarm rates of our approach are slightly higher than those of the two baseline approaches, they fall below the required bound of 5%. We then evaluate the effectiveness of our calibration approach. Fig. 10 plots the error bar of measured detection probability versus the SNR when total 10 sensors are deployed. The SNR is defined as the ratio of source energy to the standard deviation of noise, i.e., SNR = S/σ. Note that the SNRs in the vehicle detection experiments based on MICA2 motes [10, 11] are greater than 50. We can see that the detection probability increases with the SNR. This conforms to the intuition that a louder target can be detected more easily. Moreover, we can see that our approach can outperform other two baseline algorithms by up to 50%, when the SNR is low. Fig. 11 plots the error bar of measured detection probability versus the number of deployed sensors when the SNR is 60. We can see from the figure that the detection probability increases with the number of sensors. This is consistent with the intuition that the system detection performance increases with the network density. Moreover, our calibration approach gains 1-fold detection performance than the other two baseline algorithms. 9

0.8 0.6 0.4

8

2-tier calibration averaging non-calibration

0.2 0

Fig. 9. The surveillance region.

1 Detection probability PD

Detection probability PD

1

0

100

200

300 400 SNR S/σ

500

Fig. 10. PD vs. SNR

Conclusion

600

700

0.8 0.6 0.4 2-tier calibration averaging non-calibration

0.2 0

0

5

10

15 20 25 30 Sensor number N

35

40 45

Fig. 11. PD vs. sensor number

[13] T. He, C. Huang, B. Blum, J. Stankovic, and T. Abdelzaher. Range-free localization schemes for large scale sensor networks. In MobiCom, 2003. [14] T. He, S. Krishnamurthy, J. A. Stankovic, T. Abdelzaher, L. Luo, R. Stoleru, T. Yan, L. Gu, J. Hui, and B. Krogh. Energy-efficient surveillance system using wireless sensor networks. In MobiSys, 2004. [15] A. Ihler, J. Fisher III, R. Moses, A. Willsky, and C. MIT. Nonparametric belief propagation for sensor self-calibration. In IEEE ICASSP, 2004. [16] C. Johnson. Lectures & Adaptive Parameter Estimation. Prentice-Hall, 1988. [17] D. Li and Y.-H. Hu. Energy based collaborative source localization using acoustic micro-sensor array. EUROSIP Journal on Applied Signal Processing, (4), 2003. [18] D. Li, K. Wong, Y.-H. Hu, and A. Sayeed. Detection, classification and tracking of targets in distributed sensor networks. IEEE Signal Process. Mag., 19(2), 2002. [19] R. Lupton. Statistics in Theory and Practice. Princeton University Press, 1993. [20] E. Miluzzo, N. Lane, A. Campbell, and R. Olfati-Saber. CaliBree: A Self-calibration System for Mobile Sensor Networks. In DCOSS, 2008. [21] R. Moses and R. Patterson. Self-calibration of sensor networks. In SPIE, volume 4743, 2002. [22] N. Ramanathan, L. Balzano, M. Burt, D. Estrin, T. Harmon, C. Harvey, J. Jay, E. Kohler, S. Rothenberg, and M. Srivastava. Rapid deployment with confidence: Calibration and fault detection in environmental sensor networks. Technical Report CENS-TR-62, Center for Embedded Networked Sensing, 2006. [23] X. Sheng and Y.-H. Hu. Maximum likelihood multiplesource localization using acoustic energy measurements with wireless sensor networks. IEEE Trans. Signal Processing, 53(1):44–53, January 2005. [24] A. Vahidi, A. Stefanopoulou, and H. Peng. Recursive least squares with forgetting for online estimation of vehicle mass and road grade: theory and experiments. Vehicle System Dynamics, 43(1):31–55, 2005. [25] P. Varshney. Distributed Detection and Data Fusion. Spinger, 1996. [26] K. Whitehouse and D. Culler. Calibration as parameter estimation in sensor networks. In WSNA, 2002. [27] Y. Zhuang, L. Chen, X. Wang, and J. Lian. A weighted moving average-based approach for cleaning sensor data. In ICDCS, 2007.

Data fusion is an effective technique for improving system sensing performance by enabling efficient collaboration among sensors. In this paper, we propose a two-tier system-level calibration approach for fusion-based WSNs. Our approach introduces low computation and communication overhead and therefore is suitable for low-power wireless sensors. The simulations based on synthetic data as well as real data traces show that our approach can significantly boost the system detection performance.

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