Decision Fusion for Parallel Sequential Sensors
Decision Fusion for Parallel Sequential Sensors Ji Wang
Pramod Abichandani
Moshe Kam
Department of Electrical and Computer Engineering, Drexel University Philadelphia, PA 19104 {jw899, pva23, kam}@drexel.edu adopt the decision of one of the stopped sensors; (2) all-that-decided rule: once at least one sensor has stopped sampling, we integrate all the decisions of stopped sensors through the 1986 Chair-Varshney decision fusion rule; and (3) all-sensors rule: once at least one sensor has stopped sampling, we combine the available decisions of the stopped sensor and the implied decisions of the remaining sensors. These three rules differ in the information they use once at least one sensor has stopped, namely reached a decision at the end of a stage.
Abstract— Lee and Thomas (1984) have introduced a modified version of Wald’s sequential probability ratio test. The modified version retains most of the features of Wald’s procedure but is easier to analyze and offers efficient truncation procedures. In this study, we use the Lee-Thomas design to analyze the performance of a bank of parallel sequential sensors whose decisions are fused. We evaluate the performance of the sensor bank by two criteria: (1) the probability of error; (2) average sample number (ASN) needed to achieve it. Three rules are studied: (1) first-to-decide rule (Niu and Varshney, 1984): once at least one sensor has stopped sampling, we adopt the decision of one of the stopped sensors; (2) all-that-decided rule: once at least one sensor has stopped sampling, we integrate all the decisions of stopped sensors through the 1986 Chair-Varshney decision fusion rule; and (3) all-sensors rule: once at least one sensor has stopped sampling, we combine the available decisions of the stopped sensor and the implied decisions of the remaining sensors. Performance of the three rules is calculated and gains with respect to the performance of a single sensor are quantified.
The architecture is shown in Fig. 1. Each local sensor collects information about a phenomenon they observe, and make binary decisions based on this information. The decision is to accept the hypothesis H or accept hypothesis K. The decisions follow the MLGDS procedure. They are transmitted to a Fusion Center, where they are integrated to generate the system’s global binary (H or K) decision.
Keyword: sequential detection, fusion rule, multi-sensors
I.
INTRODUCTION
Sequential detection procedures find applications in several areas of research [1-6, 10-11]. Lee and Thomas have proposed [1] a modification to Wald’s Sequential Probability Ratio Test (SPRT) [6]. Wald’s procedure was in turn a significant improvement over previous fixed-sample-size (FSS) detection methods. The Lee-Thomas procedure, titled the memory-less grouped-data sequential (MLGDS) procedure, tests a simple hypothesis against a simple location alternative, based on n independent and identically distributed samples. Specifically, at each stage, the consecutive previous samples are taken, and a test statistic based on them is calculated. A two-threshold test is then made. If the test statistic is above the higher threshold or below the lower threshold, a decision is made. Otherwise the samples are samples are collected for discarded, and the next calculating the next test statistic. Lee-Thomas MLGDS procedure exhibits simplicity in structure and analysis and retains most of the features of Wald’s SPRT.
Fig. 1. The structure of multi-sensors sequential detection fusion system
II.
BACKGROUND
1. Lee and Thomas Modified Sequential Detection Rule for a Single Sensor In [1], Lee and Thomas have studied, using a single sensor, a procedure that tests the hypothesis :
We present three rules for fusing M isolated and identical sensors that use the optimized MLGDS detection procedure. These rules are: (1) first-to-decide rule (Niu and Varshney, 1984): once at least one sensor has stopped sampling, we
,
versus the alternative hypothesis
1
(1a)
:
,
The overall error rate
(1b)
where 0, and are real numbers; are normally and independently distributed each with mean and variance .
of the sequential sensor is therefore
The corresponding average sample number
stage, using the previous MLGDS procedure [1]: At the samples, form a test statistic where ,
(7)
1 is:
(8)
1
,···, 2. Binary Distributed Detection with Multiple Sensors We consider a system that has isolated local Bayesian sensors. Each local sensor performs a decision , which is either 1 for hypothesis or 1 for hypothesis . A false alarm by the ith detector occurs when the decision is 1 but the phenomenon was . A missed detection occurs when the decision is 1 but the phenomenon was . The ; its local sensor’s probability of false alarm is denoted by probability of missed detection is denoted by . The global decision is made so as to minimize the Bayesian cost
and make the following decisions:
(2)
Here R = discard all used samples and proceed to the next stage. 1 A and B are testing thresholds with . They are predetermined so as to achieve the desired test level and power, and to satisfy other conditions, mostly simplifying the procedure.
Where CAB = the cost of declaring that the hypothesis is A when B is present. When CHH = CKK = 0, and CHK = CHK = 1, CB becomes the probability of error.
One possible choice for the two thresholds (for the particular case we study) is proposed in [1] – the thresholds are placed symmetrically about /2 . and then become
where
/2
, and
(3a)
/2
,
(3b)
The system makes its global decision , using the local sensors’ decisions and performance probabilities based on the Chair-Varshney rule ([9-pp. 61-63],[8]) sgn
log
1
is a parameter, which is a positive real number.
If we use these symmetric thresholds and P and that . P The probability of error
at the P
The probability of detection
log
log ∆
where, ∆ is the threshold
stage is
∆ .
at the P
, it follows
(9)
1
The Chair-Varshney rule assumes that the local-sensor decisions are statistically independent conditioned on the hypothesis.
stage is .
Here is the a priori probability of hypothesis the a priori probability of hypothesis . The probability that no decision was made at the stage) is (and hence we proceed to the 1
(10)
(4)
(5)
;
Here sgn
is sgn stage
is the algebraic sign function defined by 1, 1, 1 or
1 with probability
is the unit step function defined by 1
1, 0,
(6)
2
0 . 0
1 , 2
0 0 0
If more than one of the sensors stopped at the end of the stage, the Fusion Center randomly chooses one of the stopping sensors’ decision to provide the global decision. For a certain 6-sensor system, Fig. 2 shows the probability that k sensors stopped simultaneously, where 1 … 6. The probability of k >1 is higher than the probability that k = 1. Furthermore, the larger the number of local sensors used by the system, the higher the probability of having multiple sensors stop simutaneously at the end of the stage. This observation suggests the next rule, which we refer to as the All-that-decided fusion rule.
The system’s global false alarm rate and global are calculated as follows [8]: missed detection rate ∆ (11a)
1 1 ∆
0.4
(11b)
1
0.35 0.3 Probability
1 where, 0,1 . The summation is performed over all possible combinations of local decisions.
0.25 0.2 0.15 0.1
When the local sensors are identical, and . The global probability of false alarm becomes [8]: ∑
1
.
0.05 0
(12a)
0 ,
the
decision
(12c)
where
log
log
is the smallest integer that is larger than III.
5
6
7
Here the local decisions are
and · log
3 4 Number of Stopping Sensors
Under this rule, when at least one sensor has stopped, all the sensors that reached a decision simultaneously are taken into account for calculating the global decision. As these sensors that reached a decision either accept or accept , we use the Chair-Varshney binary decision fusion rule to integrate these decisions (see Section II-2).
·
log
2
2. All-that-decided Fusion Rule
(12b)
With 1 and thresholds in (12) are given by
1
Fig. 2. A 6 sensor system is considered. The probability is shown that when at least one sensor stopped at the end of a stage, the number of simultaneously stopping sensors was 1, 2, 3, 4, 5 or 6.
The global missed detection rate becomes: 1
0
1, 1,
(12d)
if the if the
sensor stopped and favored sensor stopped and favored
(13)
The local sensor’s false alarm rate and missed detection rate are: .
|
(14)
|
FUSION RULES
1. First-to-decide Fusion Rule
P
(15)
The global decision can be obtained by equation (9); “1" for K and “-1” for H. The final global decision is
The First-to-decided rule was proposed by Niu and Varshney [11]. MLGDS sensors transmitting their decisions to the Fusion Center. Under this rule, the Fusion Center accepts the first decision that one of these sensors reported as the global decision, provided of course that at least one of them stopped (namely reached a decision). If exactly one sensor stopped, the Fusion Center accepts its decision as global.
, ,
if if
1 1
(16)
While the All-that-decided rule makes use of all available decisions of the local sensors, there is still unused information in the system, namely the sampled data of all the sensors that have not yet reached a decision. In the next section, we will 3
introduce an All-sensors fusion rule that also takes into account this sampling data by implying a decision for the unstopped sensors.
IV.
We evaluate the performance of the three fusion rules in terms of the average sampling number (ASN) and the global error rate.
3. All-sensors Fusion Rule We assume that the system is at the first stage when one or more sensors have stopped. In addition to collecting and integrating the decisions of all the sensors that stopped, we now force all the sensors that have not stopped yet to give us their implied decisions. All local sensors are redesigned to have an additional threshold : ·
2
1. Global ASN Performance For all three rules, the entire system stops sampling once at least one sensor stopped and made a decision. Therefore, the global ASNs of all three rules are the same. We compare this global ASN, which we denote ASNg, to the ASN of a single sensor ASN1 (see eq. (8)). The probability at least one sensor stopped at the stage is:
(17)
,
The following rule is used for the sensors that have not stopped:
where, is a positive integer. samples sampled by each of the parallel system stopped.
(18)
With this third threshold decision to the Fusion Center.
2
1
3
(19)
is the number of sensors before the
2
1
all sensors provide a binary
1 (24a)
Since
For the sensors that stopped |
,
|
,
1, it follows that
(20)
P
1
(21)
If |
1 1 1
and
For the sensors that that did not stop but provided an implied decision
|
(24)
The global average sample number in this case is
The threshold is selected so that the undecided sequential sensors minimize the Bayesian risk [Section II-2]. When we minimize the probability of error, the threshold becomes ·
PERFORMANCE ANALYSIS OF THE THREE FUSION RULES
∞, lim
(25)
(26)
1
(27)
(22)
P
For all three fusion rules we thus get a shorter sampling time (a lower ASN) ASNg
Pramod Abichandani
Moshe Kam
Department of Electrical and Computer Engineering, Drexel University Philadelphia, PA 19104 {jw899, pva23, kam}@drexel.edu adopt the decision of one of the stopped sensors; (2) all-that-decided rule: once at least one sensor has stopped sampling, we integrate all the decisions of stopped sensors through the 1986 Chair-Varshney decision fusion rule; and (3) all-sensors rule: once at least one sensor has stopped sampling, we combine the available decisions of the stopped sensor and the implied decisions of the remaining sensors. These three rules differ in the information they use once at least one sensor has stopped, namely reached a decision at the end of a stage.
Abstract— Lee and Thomas (1984) have introduced a modified version of Wald’s sequential probability ratio test. The modified version retains most of the features of Wald’s procedure but is easier to analyze and offers efficient truncation procedures. In this study, we use the Lee-Thomas design to analyze the performance of a bank of parallel sequential sensors whose decisions are fused. We evaluate the performance of the sensor bank by two criteria: (1) the probability of error; (2) average sample number (ASN) needed to achieve it. Three rules are studied: (1) first-to-decide rule (Niu and Varshney, 1984): once at least one sensor has stopped sampling, we adopt the decision of one of the stopped sensors; (2) all-that-decided rule: once at least one sensor has stopped sampling, we integrate all the decisions of stopped sensors through the 1986 Chair-Varshney decision fusion rule; and (3) all-sensors rule: once at least one sensor has stopped sampling, we combine the available decisions of the stopped sensor and the implied decisions of the remaining sensors. Performance of the three rules is calculated and gains with respect to the performance of a single sensor are quantified.
The architecture is shown in Fig. 1. Each local sensor collects information about a phenomenon they observe, and make binary decisions based on this information. The decision is to accept the hypothesis H or accept hypothesis K. The decisions follow the MLGDS procedure. They are transmitted to a Fusion Center, where they are integrated to generate the system’s global binary (H or K) decision.
Keyword: sequential detection, fusion rule, multi-sensors
I.
INTRODUCTION
Sequential detection procedures find applications in several areas of research [1-6, 10-11]. Lee and Thomas have proposed [1] a modification to Wald’s Sequential Probability Ratio Test (SPRT) [6]. Wald’s procedure was in turn a significant improvement over previous fixed-sample-size (FSS) detection methods. The Lee-Thomas procedure, titled the memory-less grouped-data sequential (MLGDS) procedure, tests a simple hypothesis against a simple location alternative, based on n independent and identically distributed samples. Specifically, at each stage, the consecutive previous samples are taken, and a test statistic based on them is calculated. A two-threshold test is then made. If the test statistic is above the higher threshold or below the lower threshold, a decision is made. Otherwise the samples are samples are collected for discarded, and the next calculating the next test statistic. Lee-Thomas MLGDS procedure exhibits simplicity in structure and analysis and retains most of the features of Wald’s SPRT.
Fig. 1. The structure of multi-sensors sequential detection fusion system
II.
BACKGROUND
1. Lee and Thomas Modified Sequential Detection Rule for a Single Sensor In [1], Lee and Thomas have studied, using a single sensor, a procedure that tests the hypothesis :
We present three rules for fusing M isolated and identical sensors that use the optimized MLGDS detection procedure. These rules are: (1) first-to-decide rule (Niu and Varshney, 1984): once at least one sensor has stopped sampling, we
,
versus the alternative hypothesis
1
(1a)
:
,
The overall error rate
(1b)
where 0, and are real numbers; are normally and independently distributed each with mean and variance .
of the sequential sensor is therefore
The corresponding average sample number
stage, using the previous MLGDS procedure [1]: At the samples, form a test statistic where ,
(7)
1 is:
(8)
1
,···, 2. Binary Distributed Detection with Multiple Sensors We consider a system that has isolated local Bayesian sensors. Each local sensor performs a decision , which is either 1 for hypothesis or 1 for hypothesis . A false alarm by the ith detector occurs when the decision is 1 but the phenomenon was . A missed detection occurs when the decision is 1 but the phenomenon was . The ; its local sensor’s probability of false alarm is denoted by probability of missed detection is denoted by . The global decision is made so as to minimize the Bayesian cost
and make the following decisions:
(2)
Here R = discard all used samples and proceed to the next stage. 1 A and B are testing thresholds with . They are predetermined so as to achieve the desired test level and power, and to satisfy other conditions, mostly simplifying the procedure.
Where CAB = the cost of declaring that the hypothesis is A when B is present. When CHH = CKK = 0, and CHK = CHK = 1, CB becomes the probability of error.
One possible choice for the two thresholds (for the particular case we study) is proposed in [1] – the thresholds are placed symmetrically about /2 . and then become
where
/2
, and
(3a)
/2
,
(3b)
The system makes its global decision , using the local sensors’ decisions and performance probabilities based on the Chair-Varshney rule ([9-pp. 61-63],[8]) sgn
log
1
is a parameter, which is a positive real number.
If we use these symmetric thresholds and P and that . P The probability of error
at the P
The probability of detection
log
log ∆
where, ∆ is the threshold
stage is
∆ .
at the P
, it follows
(9)
1
The Chair-Varshney rule assumes that the local-sensor decisions are statistically independent conditioned on the hypothesis.
stage is .
Here is the a priori probability of hypothesis the a priori probability of hypothesis . The probability that no decision was made at the stage) is (and hence we proceed to the 1
(10)
(4)
(5)
;
Here sgn
is sgn stage
is the algebraic sign function defined by 1, 1, 1 or
1 with probability
is the unit step function defined by 1
1, 0,
(6)
2
0 . 0
1 , 2
0 0 0
If more than one of the sensors stopped at the end of the stage, the Fusion Center randomly chooses one of the stopping sensors’ decision to provide the global decision. For a certain 6-sensor system, Fig. 2 shows the probability that k sensors stopped simultaneously, where 1 … 6. The probability of k >1 is higher than the probability that k = 1. Furthermore, the larger the number of local sensors used by the system, the higher the probability of having multiple sensors stop simutaneously at the end of the stage. This observation suggests the next rule, which we refer to as the All-that-decided fusion rule.
The system’s global false alarm rate and global are calculated as follows [8]: missed detection rate ∆ (11a)
1 1 ∆
0.4
(11b)
1
0.35 0.3 Probability
1 where, 0,1 . The summation is performed over all possible combinations of local decisions.
0.25 0.2 0.15 0.1
When the local sensors are identical, and . The global probability of false alarm becomes [8]: ∑
1
.
0.05 0
(12a)
0 ,
the
decision
(12c)
where
log
log
is the smallest integer that is larger than III.
5
6
7
Here the local decisions are
and · log
3 4 Number of Stopping Sensors
Under this rule, when at least one sensor has stopped, all the sensors that reached a decision simultaneously are taken into account for calculating the global decision. As these sensors that reached a decision either accept or accept , we use the Chair-Varshney binary decision fusion rule to integrate these decisions (see Section II-2).
·
log
2
2. All-that-decided Fusion Rule
(12b)
With 1 and thresholds in (12) are given by
1
Fig. 2. A 6 sensor system is considered. The probability is shown that when at least one sensor stopped at the end of a stage, the number of simultaneously stopping sensors was 1, 2, 3, 4, 5 or 6.
The global missed detection rate becomes: 1
0
1, 1,
(12d)
if the if the
sensor stopped and favored sensor stopped and favored
(13)
The local sensor’s false alarm rate and missed detection rate are: .
|
(14)
|
FUSION RULES
1. First-to-decide Fusion Rule
P
(15)
The global decision can be obtained by equation (9); “1" for K and “-1” for H. The final global decision is
The First-to-decided rule was proposed by Niu and Varshney [11]. MLGDS sensors transmitting their decisions to the Fusion Center. Under this rule, the Fusion Center accepts the first decision that one of these sensors reported as the global decision, provided of course that at least one of them stopped (namely reached a decision). If exactly one sensor stopped, the Fusion Center accepts its decision as global.
, ,
if if
1 1
(16)
While the All-that-decided rule makes use of all available decisions of the local sensors, there is still unused information in the system, namely the sampled data of all the sensors that have not yet reached a decision. In the next section, we will 3
introduce an All-sensors fusion rule that also takes into account this sampling data by implying a decision for the unstopped sensors.
IV.
We evaluate the performance of the three fusion rules in terms of the average sampling number (ASN) and the global error rate.
3. All-sensors Fusion Rule We assume that the system is at the first stage when one or more sensors have stopped. In addition to collecting and integrating the decisions of all the sensors that stopped, we now force all the sensors that have not stopped yet to give us their implied decisions. All local sensors are redesigned to have an additional threshold : ·
2
1. Global ASN Performance For all three rules, the entire system stops sampling once at least one sensor stopped and made a decision. Therefore, the global ASNs of all three rules are the same. We compare this global ASN, which we denote ASNg, to the ASN of a single sensor ASN1 (see eq. (8)). The probability at least one sensor stopped at the stage is:
(17)
,
The following rule is used for the sensors that have not stopped:
where, is a positive integer. samples sampled by each of the parallel system stopped.
(18)
With this third threshold decision to the Fusion Center.
2
1
3
(19)
is the number of sensors before the
2
1
all sensors provide a binary
1 (24a)
Since
For the sensors that stopped |
,
|
,
1, it follows that
(20)
P
1
(21)
If |
1 1 1
and
For the sensors that that did not stop but provided an implied decision
|
(24)
The global average sample number in this case is
The threshold is selected so that the undecided sequential sensors minimize the Bayesian risk [Section II-2]. When we minimize the probability of error, the threshold becomes ·
PERFORMANCE ANALYSIS OF THE THREE FUSION RULES
∞, lim
(25)
(26)
1
(27)
(22)
P
For all three fusion rules we thus get a shorter sampling time (a lower ASN) ASNg
