Dependent Randomization in Parallel Binary Decision Fusion

I. Introduction

WE study the parallel decentralized binary detection architecture shown in Fig. 1. The system uses $n$ local detectors (LDs) to observe a binary phenomenon (“target/no target”). The objective is to decide if a target is present (hypothesis $H_1$) or absent (hypothesis $H_0$). We use $P_1$ to denote the a priori probability that a target is present (hypothesis $H_1$) and use $P_0 = 1-P_1$ to denote the a priori probability that a target is absent (hypothesis $H_0$). The local observations collected by the $k{\rm{th}}$ LD are denoted $y_k$. All the local observations are assumed to be statistically independent, conditioned on the hypothesis, therefore, $Pr(y_1,\dots,y_n|H_j) = \prod_k Pr(y_k|H_j),k\in \{1,\dots, n\},j\in\{0,1\}$. Each LD compresses its local observations into a local decision; the local decision of the $k\rm{th}$ LD is $u_k = \gamma_k(y_k)$, $u_k\in\{0,1\}$ and $U = \{u_1,u_2,\ldots,u_n\}$. Here $u_k = 0$ means that the $k\rm{th}$ LD prefers hypothesis $H_0$, and $u_k = 1$ means that the $k\rm{th}$ LD prefers hypothesis $H_1$. A Data Fusion Center (DFC) combines all the local decisions to generate a global decision $u_0 = \gamma_0(U)$, $u_0\in\{0,1\}$, where $u_0 = 0$ indicates preference for hypothesis $H_0$, and $u_0 = 1$ indicates preference for hypothesis $H_1$. The probability of false alarm of the DFC is $P_f = Pr(u_0 = 1|H_0)$. The probability of detection of the DFC is $P_d = Pr(u_0 = 1|H_1)$. The tuple ($P_f,P_d$) is considered the operating point of the detection system. Similarly, the local operating point of the $k\rm{th}$ LD is ($P_{fk},P_{dk}$), where $P_{fk} = Pr(u_k = 1|H_0)$ and $P_{dk} = Pr(u_k = 1|H_1)$. We will sometimes refer to $(P_f^A,P_d^A)$ as “point A”.

Figure  1.  Parallel decentralized detection network.

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The architecture of Fig. 1 was investigated extensively (e.g., [1]–[10]). Its design in this study aims to satisfy a Neyman-Pearson criterion, namely to maximize the global probability of detection $P_d$ while keeping the global probability of false alarm $P_f$ not larger than a specified value $\alpha$ ($0<\alpha <1$).

To achieve this objective, the implementation tasks are to determine the local decision rules (mapping $\gamma_k$ from $y_k$ to $u_k$, $k = 1,\ldots,n$) and the global decision rule (mapping $\gamma_0$ from $U$ to $u_0$). The local decision rule of the $k{\rm{th}}$ LD, $\gamma_k(.)$, determines how the $k{\rm{th}}$ LD compresses its local observations $y_{k}$ into its local decision $u_k$ as $u_k = \gamma_k(y_k)$. The global decision rule, $\gamma_0$, represents how the DFC integrates all the local decisions into the global decision $u_0$ as $u_0 = \gamma_0(U)$. The combination of all the local decision rules is $\gamma_{LD} = \{\gamma_1,\ldots,\gamma_n\}$. The combination of the global decision rule and all the local decision rules is the detection strategy, $\gamma = \{\gamma_0,\gamma_{LD}\} = \{\gamma_0,\ldots,\gamma_n\}$.

Among other approaches, the existing literature provides analysis of three detection strategies for the system in Fig. 1.

1) A Deterministic Strategy: Each LD uses a deterministic local decision rule and the DFC uses a deterministic global decision rule [1].

2) A Strategy With Randomization at the DFC Only: Each LD uses a deterministic local decision rule and the DFC uses a randomized global decision rule [2], [7].

3) Dependent Randomization: all the LDs and the DFC use randomized decision rules. The randomization between the LDs and the DFC is coordinated and synchronized [5].

The need for dependent randomization (or “scheduled test”) was introduced in [4] and [5]. In [4], it was indicated that dependent randomization is certainly feasible but “may perform poorly if synchronization is lost.” In [5] scenarios were studied where dependent randomization can improve performance over a deterministic strategy and independent randomization. In this study, we find conditions under which dependent randomization is beneficial, quantify the effect of synchronization loss, and demonstrate how to recover (partially) from synchronization loss. Specifically, this paper analyzes the following cases.

4) Dependent Randomization (DFC Unsynchronized): All the LDs and the DFC use randomized decision rules. The LDs are coordinated and synchronized with each other. The DFC uses a randomized global decision rule but is not synchronized with the LDs.

5) Dependent Randomization (LDs Unsynchronized): All the LDs and the DFC use randomized decision rules. The randomization between some LDs and the DFC is coordinated and synchronized. Other LDs use randomized local decision rules independently of the other LDs and the DFC.

In Section II we present the three detection strategies (a)–(c) and summarize their characteristics. In Section III we present two examples of parallel decentralized detection system which we use to exemplify different operating conditions and performance. Section IV discusses strategies needed when the DFC loses synchronization with the LDs (d). Section V discusses strategies needed when some LDs lose synchronization with other LDs and the DFC (e).

II. Detection Strategy for Parallel Decentralized Detection Networks

A Deterministic Strategy

A detection strategy is deterministic if each LD uses a single deterministic local decision rule and the DFC uses a single deterministic global decision rule. The operating point of the system $A = (P_f^A,P_d^A)$ is determined by the deterministic strategy $\gamma^A = \{\gamma_0^A,\ldots,\gamma_{n}^A\}$. The corresponding operating point of the $k\rm{th}$ LD in the system, determined by $\gamma_k^A$, is ($P_{fk}^A,P_{dk}^A$).

It is shown in [5] that under the assumption that the local observations $y_1,\ldots,y_n$ are conditionally independent given the hypothesis $H_0$ or $H_1$, for satisfying a Neyman-Pearson criterion, both $\gamma_0$ and $\gamma_k$ are likelihood ratio tests of the form

where

Here, $\eta_0$ is the threshold of the global decision rule and $\eta_k$ is the threshold of the local decision rule of the $k\rm{th}$ LD. These thresholds are often part of the design, and they determine the trade-off between the probability of false alarm and the probability of detection. Under a Neyman-Pearson criterion, $\eta_0,\eta_1,\ldots,\eta_n$ are designed to maximize the global probability of detection while keeping the global probability of false alarm not greater than $\alpha\in(0,1)$. The value of $\alpha$ is determined by the designer, based on the level of tolerance of false alarm in the specific application.

For a binary parallel decentralized detection system with $n$ LDs, the local decision vector $U$ has $2^n$ possible values, which translates into $2^{2^n}$ possible global decision rules. However, not every global decision rules is eligible for consideration as a potentially optimal decision rule. According to (1), the global decision rule is a likelihood ratio test and $u_0$ is a non-decreasing function of $\Lambda(U)$. Thomopoulos et al. [2] showed that the optimal deterministic global decision rule that satisfies the Neyman-Pearson criterion must be a monotonic fusion rule (per Lemma 1 of [2], function $d$). A fusion rule is monotonic if, for every combination of local decisions $U = \{u_1,\ldots,u_n\}$, switching one of the local decision from $0$ to $1$ can only cause the global decision $u_0$ to switch from $u_0 = 0$ to $u_0 = 1$ and not from $u_0 = 1$ to $u_0 = 0$. An algorithm that calculates all the monotonic fusion rules of a system with $n$ LDs is provided in [11]. Since some monotonic fusion rules dominate others (would always result in better performance than others), the eligible optimal deterministic global decision rules would be a subset of all the monotonic fusion rules.

The probability of false alarm and the probability of detection of the architecture shown in Fig. 1 are

The probability of false alarm and the probability of detection at the $k\rm{th}$ LD are

where $\Lambda(y_k)$ is the likelihood ratio of $y_k$.

When all the local operating points $(P_{fk}, P_{dk}), k = 1,\ldots,n$ are known, then (4) can be written as ([3, pp. 567–568])

where $\mathbf{U_{-1}}(.)$ is the unit step function

In (6), the value of the unit step function provides the global decision $u_0$ for a given local decision set $U = \{u_1,\ldots,u_n\}$.

If the local operating points are identical, $(P_{fk},P_{dk}) = (pf, pd), k = 1,\ldots,n$, the global decision rule (1) becomes a “$k$ out of $n$” rule, which means if $k$ or more LDs in the system decide ‘$1$’, then $u_0 = 1$; otherwise, $u_0 = 0$. In this circumstance, the global probability of false alarm and the global probability of detection are

We consider two cases of finding a deterministic strategy, depending on the local observations.

1) Local Observations Contain No Point Masses of Probability: Hoballah and Varshney [1] studied this case, using a Person-by-Person optimization (PBPO) approach to synthesize $\gamma_0$ and $\gamma_k$ in (1) and (2). Acharya et al. [8] proposed a method for solving for the optimal $\gamma_0$ and $\gamma_k$ simultaneously when the LDs are identical.

2) Local Observations Contain Point Masses of Probability: If the local observations are discrete and finite, the probability distribution of the local observations contain point masses of probability. In this case, the number of achievable local operating points $\{(P_{f1}, P_{d1}),\dots,(P_{fn}, P_{dn})\}$, corresponding to the finite set of local decision rules $\{\gamma_1,\ldots,\gamma_n\}$, is finite. For each combination of a monotonic global fusion rule and local operating points, we can calculate the operating point of the system $(P_f,P_d)$ by using (6). We can then calculate all the operating points of the system by running a search on all the combinations of a monotonic global fusion rule and local operating points and find the optimal deterministic strategy satisfying the Neyman-Pearson criterion.

B The Potential of Randomization of Decision Rules

Since the vector $U$ is finite-dimensional (and binary), $\Lambda(U)$ in (4) has a finite number of values, each with a corresponding probability of false alarm. The value of $U$ that corresponds to the highest probability of false alarm $\beta$ that satisfies $\beta\leq\alpha$ may have a significant gap $\alpha-\beta$ compared to $\alpha$.

Fig. 2 shows the receiver operating characteristic (ROC) curve [12] of a system with discrete local observations (this curve comes from the system we will later introduce in Section III-A). All possible operating points of the system are shown as blue circles. The probability of false alarm constraint $\alpha$ is shown as the dash line. In this circumstance, the best operating point is $\omega_3$ and $P_f^{\omega_3}<\alpha$.

Figure  2.  Deterministic strategy with isolated operating points.

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Performance of the architecture of Fig. 1 under the circumstance such as the one described in Fig. 2 can benefit from randomization. By randomization we mean that one or more of the decision makers in the system (an LD or the DFC) is selecting its decision rule ($\gamma_k$ or $\gamma_0$) at each time instant by selecting one rule from a finite set of decision rules. The $k\rm{th}$ LD selects a rule from among $\{\gamma_k^1,\dots,\gamma_k^i,\ldots,\gamma_k^N\}$, for some positive integer $N$. The rule $\gamma_k^i$ is selected with probability $p_k^i$ and $\sum^N_{i = 1} p_k^i = 1$. The DFC selects a rule from among $\{\gamma_0^1,\dots, \gamma_0^i,\ldots,\gamma_0^M\}$, where $\gamma_0^i$ is selected with probability $p_0^i$ and $\sum^M_{i = 1} p_0^i = 1$. Randomization will be considered when the specified probability of false alarm constraint $\alpha$ is not achievable by a deterministic strategy (such as in Fig. 2) or when the deterministic strategy achieves the values of false alarm constraint $\alpha$ but the system’s ROC curve is not concave.

It is possible that only the DFC employs randomization (e.g., [2], [4], [7], [9]) or that a subset of the of LDs and the DFC employ randomization (independently or dependently). The term “dependent randomization” is used when both the LDs and the DFC employ randomization, and when, in addition, their switching between decision rules is coordinated and synchronized ([4]–[6]).

One possible design has the system of Fig. 1 operate at one of two operating points, $A$ and $B$. At each time step, one of the two is selected ($A$ with probabilities $p$ and $B$ with probability $1-p$). “Operation at point $A$” means that the DFC selects $\gamma_0^A$ and simultaneously each LD ($k = 1,2,\ldots,n$) selects $\gamma_k^A$. “Operation at point $B$” means that the DFC selects $\gamma_0^B$ and simultaneously each LD ($k = 1,2,\ldots,n$) selects $\gamma_k^B$. By changing the value of $p$, the system can effectively operate anywhere along the line segment that connects $A$ and $B$ (every combinations of $(P_f,P_d)$ along this line segment is realizable). We denote the operating point generated by the randomized strategy as $C = (P_f^C,P_d^C)$, where

In order to satisfy the constraint on the probability of false alarm, we require $P_f^C\leq \alpha$. When $P_f^C = \alpha$, the probability of selecting point A, $p$, is

If $\gamma_{LD}^A = \gamma_{LD}^B$ ($\gamma_{k}^A = \gamma_{k}^B$ for all $k$), the randomization occurs only at the DFC. If $\gamma_{LD}^A\neq\gamma_{LD}^B$ and the selection of operating at point $A$ and point $B$ is coordinated and synchronized between the LDs and the DFC, then the scheme is known as dependent randomization.

Fig. 3 shows how randomization connects the isolated operating points shown in the ROC curve of Fig. 2. For example, randomization allows the system to operate at point $C$, rather than at $\omega_3$, thereby achieving a higher probability of detection while not violating the constraint on probability of false alarm. The red curve is the ROC curve of the system employing randomization, which consists of straight line segments connecting all the previously-isolated operating points (the blue circles). In this example, to achieve the highest probability of detection subject to $P_f\leq \alpha$, we select $A = \omega_3$ and $B = \omega_4$. The operating point $C$ achieved by randomization is shown as the black circle. $P_f^C = \alpha = pP_f^{\omega_3}+ (1-p) P_f^{\omega_4}$ while $p$ is calculated by (11).

Figure  3.  Randomization can improve detection performance by “connecting” the isolated operating points.

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1) Randomization at the DFC Only: The authors of [2], [4], [7], [9], [13] studied strategies requiring that the DFC implement randomization, when the local decision rules are deterministic ($\gamma_k^A = \gamma_k^B = \gamma_k, \forall k$). Each local decision rule is of the form (2). The DFC selects either $\gamma_0^A$ or $\gamma_0^B$ at each time step.

Thomopoulos et al. [2] showed that under a Neyman-Pearson criterion, a desired value of global false alarm $\alpha$, can always be achieved by a strategy with randomization at the DFC [2]. In [7] and [13] examples were presented to show that a strategy with randomization at the DFC is able to achieve higher probability of detection than the one achieved by a deterministic detection strategy.

2) Dependent Randomization: In dependent randomization (or “a scheduled test” as it is called in [4]), both the DFC and the LDs participate in the randomization. At each time step, the system makes a selection between two deterministic strategies, $\gamma^A = \{\gamma_0^A,\ldots,\gamma_{n}^A\}$ and $\gamma^B = \{\gamma_0^B,\ldots,\gamma_{n}^B\}$ ([4]–[6]). The system can operate on the line segment connecting any two operating points realizable by the deterministic strategy. The ROC curve of the system with dependent randomization is the upper boundary of the convex hull of all the operating points achieved by the deterministic strategy. In other words, dependent randomization can make ROC curve of the system concave.

Dependent randomization requires a coordinated action between the DFC and the LDs. The DFC and the LDs would switch simultaneously together, back and forth, between $\gamma_0^A$ (for the DFC) and $\gamma_{LD}^A = \{\gamma_1^A,\ldots,\gamma_n^A\}$ (for the LDs); and $\gamma_0^B$ (for the DFC) and $\gamma_{LD}^B = \{\gamma_1^B,\ldots,\gamma_n^B\}$ (for the LDs). This synchronization challenge is discussed in [5, p. 301], [6], [9]. Among the means to achieve synchronization between the DFC and the LDs is the use of identical pseudo-code generators (or stored sequences of identical pseudo-code) at the DFC and the LDs.

Strategy with randomization at the DFC only (Section II-B-1)) can be considered as a special case of dependent randomization, with $\gamma_{LD}^A = \gamma_{LD}^B$. Randomization at the DFC only does not require synchronization between the DFC and the LDs but it does not necessarily result in a concave team ROC curve.

Table I summarizes the input and output of three different designs of a parallel decentralized detection system of Fig. 1.

Table  I.  Input and Output Of Three Different Designs of a Parallel Decentralized Detection System of Fig. 1 Under a Neyman-Pearson Criterion

Input for the design
1. The number of local detectors, $n$
2. The probability of false alarm constraint, $\alpha$
3. Conditional probability distributions of the local observations, $P(y_k|H_0)$ and $P(y_k|H_1)$, $k = 1,\ldots,n$
Output of a design
Deterministic strategy	1. One global operating point $(P_f,P_d)$ 2. The corresponding local operating points, $(P_{fk},P_{dk}),k = 1,\ldots,n$
Randomization at the DFC only	1. Two global operating points $A = (P_f^A,P_d^A)$ and $B = (P_f^B,P_d^B)$ 2. The corresponding local operating points of $A$ and $B$: $(P_{fk}^A,P_{dk}^A)$ and $(P_{fk}^B,P_{dk}^B), k = 1,\ldots,n$ 3. The probability of selecting A, $p$, calculated by (11) 4. The operating point $C = (P_f^C,P_d^C)$, calculated by (9) and (10)	The local operating points at A and B are identical $(P_{fk}^A,P_{dk}^A) = (P_{fk}^B,P_{dk}^B),k = 1,\ldots,n$
Dependent randomization		The local operating points at A and B are different $(P_{fk}^A,P_{dk}^A)\neq(P_{fk}^B,P_{dk}^B),k = 1,\ldots,n$

 | Show Table

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III. Numerical Examples

In the remaining of the paper, we will use the following two examples to illustrate performance under different strategies.

A Example 1: A 2-LD System With Continuous Local Observations

We consider a system with two LDs ($n = 2$). The local observations are identical logistic random variables (as done in [4]). The conditional probability distribution of the local observations are

The operating point ($P_{fk},P_{dk}$) of the $k\rm{th}$ LD ($k = 1,2$) can be calculated as

where $\tau_k$ is a function of $y_k$

We want to design the system under a Neyman-Pearson criterion with $\alpha = 0.2009$ (this is an arbitrary choice; any value between $0$ and 1 can be used). The nontrivial (monotonic) global decision rules are the $AND$ rule ($u_0 = u_1 \& u_2$) and the $OR$ rule ($u_0 = u_1 | u_2$) [4]. For each global decision rule, the operating points of the two LDs are identical, $(P_{f1},P_{d1}) = (P_{f2},P_{d2})$. The system operating points can be shown to be: $(P_f,P_d) = ((P_{f1})^2,(P_{d1})^2)$ under the $AND$ rule; and $(P_f,P_d) = ((P_{f1})^2+2P_{f1}(1-P_{f1}),(P_{d1})^2+2P_{d1}(1- P_{d1}))$ under the $OR$ rule.

In Fig. 4, the ROC curves of this system, using the $AND$ rule and the $OR$ rule, are shown in red and blue, respectively. The ROC curve of the $AND$ rule is given by

Figure  4.  The ROC curves of the 2-LD system when the DFC uses an $AND$ rule (red curve) and when it uses an $OR$ rule (blue curve). The upper boundary of the two curves is the ROC curve of the system with deterministic strategy.

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The ROC curve of the $OR$ rule is given by

Although the two individual ROC curves are both concave, the team ROC curve of the system (which is the upper boundary of the two curves) is not. The point of intersection of the two ROC curves is $O = (P_f^O,P_d^O) = (0.1859,0.8141)$.

Referring to Fig. 4, if the 2-LD system uses a deterministic strategy to achieve the highest possible $P_d$, then the DFC employs an $AND$ rule when the desired $\alpha$ is less than or equal to $P_f^O$ and employs an $OR$ rule when the the desired $\alpha$ is greater than $P_f^O$.

Referring to Fig. 5 (which shows the ROC curve for $P_f\in[0.14,0.28]$), let A be the point where the common tangent of the two ROC curves touches the “$AND$ rule ROC curve.” Let B be the point where the common tangent of the two ROC curves touches the “$OR$ rule ROC curve.” If the maximum allowable value of the probability of false alarm, $\alpha$, satisfies $P_f^A<\alpha<P_f^B$, then dependent randomization would be useful. The ROC curve of the system with dependent randomization as the black curve. In Fig. 5, for the 2-LD system, the points of tangency are $A = (P_f^A,P_d^A) = (0.1581, 0.7870)$ on the “$AND$ rule ROC curve” and $B = (P_f^B,P_d^B) = (0.2437,0.8652)$ on the “$OR$ rule ROC curve.” The system operates at A when both LDs operate at $(P_{f1}^A,P_{d1}^A) = (P_{f2}^A,P_{d2}^A) = (0.3976,0.8871)$ and simultaneously the DFC uses the $AND$ rule. The system operates at B when both LDs operate at $(P_{f1}^B,P_{d1}^B) = (P_{f2}^B,P_{d2}^B) = (0.1304,0.6328)$ and simultaneously the DFC uses the $OR$ rule. To make the team ROC curve concave, we apply dependent randomization whenever the desired probability of false alarm $\alpha$ satisfies $0.1581 = P_f^A < \alpha < P_f^B = 0.2437$. In this range, at each time step the system operates at $A$ with probability $p$ and at $B$ with probability $1-p$. The equivalent operating point $C$ on the line segment AB is provided by (9) and (10). Otherwise, if $\alpha<P_f^A$ we use the $AND$ rule, and if $P_f^B<\alpha$ we use the $OR$ rule.

Figure  5.  The operating points of the 2-LD system employing different detection strategies under the Neyman-Pearson criterion with $\alpha = 0.2009$: (a) deterministic strategy (and randomization at the DFC) (blue circle); (b) dependent randomization (black circle).

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Fig. 5 shows the operating points of the 2-LD system employing different detection strategies. The blue circle $G = (P_f^G,P_d^G) = (0.2009,0.8217)$ is the operating point of the system employing deterministic strategy. The system operates at $G$ when both LDs operate at $(P_{f1}^G,P_{d1}^G) = (P_{f2}^G,P_{d2}^G) = (0.4482, 0.9065)$ and the DFC uses the $AND$ fusion rule.

In this example the operating point achieved by the system employing Randomization at the DFC only is also $G$.

The value of $\alpha$ ($\alpha = 0.2009$) in our case satisfies $0.1581 = P_f^A < \alpha < P_f^B = 0.2437$. Therefore, dependent randomization can improve the probability of detection of the system at that value of $\alpha$. The black circle $C = (P_f^C,P_d^C) = (0.2009,0.8261)$ is the operating point of the system employing dependent randomization. $C$ is generated by operating at $A = (P_f^A,P_d^A) = (0.1581,0.7870)$ with probability $p = 0.5$ and at $B = (P_f^B,P_d^B) = (0.2437,0.8652)$ with probability $1-p = 0.5$. $p = \frac{0.2437-0.2009}{0.2437-0.1581}$, calculated by (11).

Under the Neyman-Pearson criterion with $\alpha = 0.2009$, the input and output of three different designs of the 2-LD system (corresponding to Table I) are shown in Table II. As shown in Table II, dependent randomization is able to increase the probability of detection from $P_d = 0.8217$ to $P_d = 0.8261$ with the same probability of false alarm $P_f = 0.2009$.

Table  II.  Input and Output of Three Different Designs of a 2-LD System

Input for the design
1. The number of local detectors, $n=2$
2. The probability of false alarm constraint, $\alpha=0.2009$
3. Conditional probability distributions of the local observations, $P(y_k|H_0)$ and $P(y_k|H_1)$, $k=1,2$, shown in (12)
Output of a design
Deterministic strategy	1. System operating point ${G} = ({P}_{{f}}^{{G}},{P}_{{d}}^{{G}}) = (\bf{0.2009},\bf{0.8217})$
	2. The system operates at $G$ when both LDs operate at $(P_{f1}^G,P_{d1}^G)=(P_{f2}^G,P_{d2}^G)=(0.4482,0.9065)$ and the DFC uses the $AND$ fusion rule
Randomization at the DFC	Same as deterministic strategy (Randomization at the DFC does not improve the system performance since the local observations are continuous)
Dependent randomization	1. Two operating points $A=(P_f^A,P_d^A)=(0.1581,0.7870)$ and $B=(P_f^B,P_d^B)=(0.2437,0.8652)$
	2. The system operates at A when both LDs operate at $(P_{f1}^A,P_{d1}^A)=(P_{f2}^A,P_{d2}^A)=(0.3976,0.8871)$ and the DFC uses the $AND$ fusion rule The system operates at B when both LDs operate at $(P_{f1}^B,P_{d1}^B)=(P_{f2}^B,P_{d2}^B)=(0.1304,0.6328)$ and the DFC uses the $OR$ fusion rule
	3. The probability of selecting A is $p=0.5$
	4. The resulting operating point is ${{C}} = ({{P}}_{{f}}^{{C}},{{P}}_{{d}}^{{C}}) = ({\bf{0}}.{\bf{2009}},{\bf{0}}.{\bf{8261}})$

 | Show Table

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B Example 2: A 3-LD System With Discrete Local Observations

We consider the 3-LD implementation of the network shown in Fig. 1 ($n = 3$). The local observations of the three LDs in the system have identical discrete probability distributions, as shown in Fig. 6, where the conditional probabilities $P(y_k|H_i)$ are given for $k = 1,2,3$, and $i = 0,1$.

Figure  6.  The conditional probability distributions of the local observations.

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We assume that the local observations are statistically independent, conditioned on the hypothesis. From Fig. 6, each LD has 4 distinct local decision rules, corresponding to 4 distinct local observation thresholds, $\tau_1$ (anywhere in the range $\tau_1<0$), $\tau_2\ (0<\tau_2<1)$, $\tau_3\ (1<\tau_3<2),\tau_4\ (\tau_4>2)$. At the $k{\rm{th}}$ LD, $k = 1,2,3$, if $\tau_1$ is used, $P_{fk} = 1,P_{dk} = 1$; if $\tau_2$ is used, $P_{fk} = 0.2,P_{dk} = 0.7$; if $\tau_3$ is used, $P_{fk} = 0.1,P_{dk} = 0.6$; if $\tau_4$ is used, $P_{fk} = 0,P_{dk} = 0$. Each LD therefore has 4 possible local operating points ($P_{fk},P_{dk}$), which are $(0,0)$, $(0.1,0.6)$, $(0.2,0.7)$, and $(1,1)$.

Under a Neyman-Pearson criterion with the probability of false alarm constraint $\alpha = 0.1708$, the input and output of three designs of this 3-LD system are given in Table III. The table shows that in order to satisfy the probability of false alarm constraint $P_f\leq\alpha$, the deterministic strategy provides a design with $P_d = 0.7960$; randomization at the DFC improves the probability of detection to $P_d = 0.8208$; dependent randomization improves the probability of detection further to $P_d = 0.8448$.

Table  III.  Input and Output of Three Different Designs of a 3-LD System

Input for the design
1. The number of local detectors, $n = 3$
2. The probability of false alarm constraint, $\alpha = 0.1708$
3. Conditional probability distributions of the local observations, $P(y_k|H_0)$ and $P(y_k|H_1)$, $k = 1,2,3$, shown in Fig. 6
Output of a design
Deterministic strategy	1. System operating point ${{G}} = ({{P}}_{{f}}^{{G}},{{P}}_{{d}}^{{G}}) = ({\bf{0}}.{\bf{1360}},{\bf{0}}.{\bf{7960}})$
	2. One way to achieve $G$ is that the LDs operate at $\{(0.1,0.6), (0.2,0.7),(0.2,0.7)\}$ while the DFC uses the fusion rule $u_0=u_1|(u_2\And u_3)$
Randomization at the DFC	1. Two operating points $A = (P_f^A,P_d^A) = (0.1180,0.7680)$ and $B = (P_f^B,P_d^B) = (0.1900,0.8400)$
	2. When the LDs operate at $(0.2,0.7),(0.2,0.7),(0.1,0.6)$, the DFC can use the fusion rule $u_0=u_2|(u_1\And u_3)$ to achieve point $A = (0.1180,0.7680)$ and use the fusion rule $u_0=u_1|u_2$ to achieve point $B = (0.1900,0.8400)$
	3. The probability of selecting A is $p = 0.2667$
	4. The resulting operating point is ${{E}} = ({{P}}_{{f}}^{{E}},{{P}}_{{d}}^{{E}}) = ({\bf{0}}.{\bf{1708}},{\bf{0}}.{\bf{8208}})$
Dependent randomization	1. Two operating points $A = (P_f^A,P_d^A) = (0.104,0.784)$ and $B = (P_f^B,P_d^B) = (0.271,0.936)$
	2. A is achieved when $(P_{fk},P_{dk}) = (0.2,0.7),k = 1,2,3$, and the DFC uses a “2 out of 3 rule”, B is achieved when $(P_{fk},P_{dk}) = (0.1,0.6),k = 1,2,3$, and the DFC uses a “1 out of 3 rule”
	3. The probability of selecting A is $p = 0.6$
	4. The resulting operating point is ${{C}} = ({{P}}_{{f}}^{{C}},{{P}}_{{d}}^{{C}}) = ({\bf{0}}.{\bf{1708}},{\bf{0}}.{\bf{8448}})$

 | Show Table

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Fig. 7 shows the ROC curves of the 3-LD system with three different designs for $0.075<P_f<0.3$. The operating points of the 3-LD system employing (a) deterministic strategy is $G = (0.1360,0.7960)$, shown by the blue circle; (b) randomization at the DFC is $E = (0.1708,0.8208)$, shown by the red circle; and (c) deterministic strategy is $C = (0.1708, 0.8448)$, shown by the black circle.

Figure  7.  The operating points of the 3-LD system employing different detection strategies under the Neyman-Pearson criterion with probability of false alarm constraint $\alpha = 0.1708$: (a) deterministic strategy (blue circle); (b) randomization at the DFC (red circle); (c) dependent randomization (black circle). The ROC curves of the 3-LD system employing different detection strategies: (a) deterministic strategy (blue); (b) randomization at the DFC (red); (c) dependent randomization (black).

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More details of the numerical examples are available in https://github.com/moshekam/Dependent-Randomization.

IV. Loss of Synchronization Between the DFC and The Group of LDs

Dependent randomization assumes synchronization between all the LDs and the DFC. When the synchronization is lost, unless a corrective action is taken, the system may exceed the allowed probability of false alarm $\alpha$ for which it was designed under a Neyman-Pearson criterion.

We will demonstrate the approach to corrective action on the 2-LD example in Section III-A (also see [9]). The ROC curves for the $AND$ rule and the $OR$ rule are shown in Figs. 8 and 9 (also see Fig. 4). Recall that A and B are, respectively, the points of tangency of the original $AND$ rule and $OR$ rule ROC curves. We assume that the 2 LDs are still synchronized with each other but the synchronization between the DFC and the group of LDs was lost. Under these circumstances, the DFC selects $\gamma_{0}^A$ with probability $p$ and $\gamma_{0}^B$ with probability $1-p$. The group of LDs select $\gamma_{1}^A,\ldots,\gamma_{n}^A$ with probability $p$ and $\gamma_{1}^B,\ldots,\gamma_{n}^B$ with probability $1-p$. However the DFC selection is not coordinated with the selection of the group of LDs. There are four possible detection strategies: $(\gamma_{0}^A, \gamma_{LD}^A)$, $(\gamma_{0}^B, \gamma_{LD}^B)$, $(\gamma_{0}^B, \gamma_{LD}^A)$, and $(\gamma_{0}^A, \gamma_{LD}^B)$. They correspond, respectively, to the four operating points $A$, $B$, $M1$, and $M2$, shown by green circles in Fig. 8. Operating point $A$ is selected with probability $p^2$, operating point $B$ is selected with probability $(1-p)^2$, operating point $M1$ is selected with probability $(1-p)p$, operating point $M2$ is selected with probability $p(1-p)$.

Figure  8.  $A$, $B$, $M1$, and $M2$, shown by green circles, are the possible operating points of the 2-LD system (Section III-A) when the synchronization between the LDs and the DFC is lost. The black circle, $C$, shows the operating point of the synchronized system. The cyan circle, $W^*$, shows the equivalent operating point of the system when it lost synchronization. $C^*$ is the equivalent operating point after the corrective action is taken.

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Figure  9.  Zooming in on the ROC curve of the 2-LD system employing dependent randomization.

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Let $W^*$ represent an equivalent operating point which results from this combination. It is a weighted average of the four points $A$, $B$, $M1$, and $M2$. For this operating point $W^*$, the probability of false alarm $P_f^{W^*}$ and the probability of detection $P_d^{W^*}$ are

In Fig. 8, the operating point $W^*$ is shown as a cyan circle. It is possible that $P_{f}^{W^*}>\alpha$, where $\alpha$ was our upper bound for the system’s probability of false alarm (we wanted $P_f\leq \alpha$).

A special case occurs when the global fusion rules at point $A$ and point $B$ are the same, $\gamma_0^A = \gamma_0^B$. In this case, the DFC does not participate in the randomization. Points $A$ and $M1$ would be the same; points $B$ and $M2$ would be the same. The operating point $W^* = (P_f^{W^*},P_d^{W^*})$, calculated by (17) and (18), would be exactly the operating point of the system employing dependent randomization (without losing synchronization), $C = (P_f^C,P_d^C)$, calculated by (9) and (10). No corrective action is needed in this situation. In all other cases, a corrective action by the DFC may help.

A Corrective Action After the Group of LDs Lost Synchronization With the DFC

If the DFC realizes that the synchronization with the group of LDs was lost, it may have the opportunity to take a corrective action to try to satisfy the probability of false alarm constraint $P_f\leq\alpha$. The DFC can do so by changing the probability of selecting point $A$ from $p$ to a certain $q,\,0\leq q\leq 1$. Let the DFC choose $\gamma_0^A$ with probability $q$ (which may be different from $p$ that was used for (17) and (18)) and $\gamma_0^B$ with probability $1-q$. A new operating point $C^* = (P_f^{C^*},P_d^{C^*}$) is created with

The role of $p$ in (19) and (20) is due to the continued use of the probability $p$ (calculated before the loss of synchronization) to select the local decision rules at the operating point $A$ by the LDs. The role of $q$ is due to the selection of the global decision rule at the operating point $A$ by the DFC with probability $q$ (calculated after the loss of synchronization).

Let $Q_0$ be the operating point when $q = 0$ and $Q_1$ be the operating point when $q = 1$

$Q_0$ is located on the line segment connecting $B$ and $M1$; $Q_1$ is located on the line segment connecting $A$ and $M2$. Both $P_f^{C^*}$ and $P_d^{C^*}$ are affine functions of $q$ (see (19) and (20)). By changing the value of $q$ from 0 to 1, the system operating point will move from $Q_0$ to $Q_1$. Both $W^*$ and $C^*$ are located on the line segment connecting $Q_0$ and $Q_1$. By selecting an appropriate value of $q$ to determine $C^*$, the probability of false alarm constraint $P_f\leq\alpha$ may still be satisfied by point $C^*$ (shown by cyan square in Figs. 8 and 9), but with a lower probability of detection ($P_d^{C^*}$) compared to $P_d^C$ (that was calculated for the synchronized system, see (9) and (10)).

The value of $q$, the new probability that determines how the DFC hops between $\gamma_0^A$ and $\gamma_0^B$, can be derived from (19) and (20)

It is usable only if $0\leq q\leq 1$. Otherwise, a corrective action is not possible.

Recall that before we lost synchronization, dependent randomization would be useful if the probability of false alarm constraint $\alpha \in ( P_f^{A} ,P_f^{B})$ (assuming $P_f^{B}>P_f^{A}$). Satisfying the probability of false alarm constraint after losing synchronization (when the DFC selects $\gamma_0^A$ with probability $q$) requires that $\alpha \geq P_f^{Q_0}$ if $P_f^{Q_0}<P_f^{Q_1}$; or $\alpha\geq P_f^{Q_1}$ if $P_f^{Q_1}\leq P_f^{Q_0}$.

Using (25), the conditions on the new probability of randomization at the DFC $q$ which satisfies $P_f^{C^*}<\alpha$ after the loss of synchronization are summarized in Table IV.

Table  IV.  Conditions on $q$ (Probability That the DFC Selects $\gamma_0^a$) to Satisfy the Neyman-Pearson Constraint After the DFC Loses Synchronization With the LDs Group

	Value of $\alpha$	Existence of $q$	Value of $P_f^{C^*}$
$P_f^{Q_0}<P_f^{Q_1}$	$\alpha\in(-\infty ,P_f^{Q_0})$	$q$ does not exist	–
	$\alpha\in[P_f^{Q_0},P_f^{Q_1}]$	$q\in[0,1]$	$P_f^{C^*} = \alpha$
	$\alpha\in(P_f^{Q_1},\infty)$	$q = 1$	$P_f^{C^*} = P_f^{Q_1}$
$P_f^{Q_0}\geq P_f^{Q_1}$	$\alpha\in(-\infty ,P_f^{Q_1})$	$q$ does not exist	–
	$\alpha\in[P_f^{Q_1},P_f^{Q_0}]$	$q\in[0,1]$	$P_f^{C^*} = \alpha$
	$\alpha\in(P_f^{Q_0},\infty)$	$q = 0$	$P_f^{C^*} = P_f^{Q_0}$

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So far we have referred to the 2-LD example in Section III-A. In the general case, points $A$ and $B$ reside on two different ROC curves (each one corresponds to a different decision rule of the DFC). $A$ represents the deterministic operating point of the system with one of the possible deterministic strategy $\gamma^A$ (the one that corresponds to $(P_f^A,P_d^A)$). $B$ represents the deterministic operating point of the system with one of the possible deterministic strategy $\gamma^B$ (the one that corresponds to $(P_f^B,P_d^B)$). $A$ and $B$ reside on the joint tangent to the two ROC curves, and represent the end points of a line segment on which an operating point resides that satisfies a probability of false alarm constraint $\alpha$ ($P_f^A<\alpha<P_f^B$) while maximizing the probability of the detection using dependent randomization. When the synchronization between the DFC and the LDs is lost, we can satisfy the probability of false alarm constraint $\alpha$ by assigning a new probability of randomization for the DFC, $q$ (calculated by (25)).

B Numerical examples

1) 2-LD System: The design input and output of the 2-LD system employing dependent randomization with $\alpha = 0.2009$ was shown in Table II. When the DFC lost synchronization with the LDs group in the 2-LD system employing dependent randomization, the input and output of the redesigned algorithm are shown in Table V and Fig. 8. Before the loss of synchronization the system operated at $C = (0.2009,0.8261)$ (Table II). After the loss of synchronization the system operates at $C^* = (0.2009,0.7005)$.

Table  V.  The Output of the 2-LD System Employing Dependent Randomization When the DFC Lost Synchronization With the LDs Group Before and After a Corrective Action is Taken

Output of the non-synchronized 2-LD system before the corrective action is taken ($\alpha = 0.2009$)
1. The probability of false alarm constraint, $\alpha = 0.2009$
2. The probability of selecting A, $p = 0.5$, calculated by (11)
3. Four possible operating points when the DFC lost synchronization with the LDs group, $A = (0.1581,0.7870)$, $B = (0.2437,0.8652)$, $M1 = (0.6371,0.9873)$, and $M2 = (0.0170,0.4004)$
4. The operating point of the non-synchronized system, $W^* = (0.2640,0.7600)$, calculated by (17) and (18)
Output of the non-synchronized 2-LD system after the corrective action is taken ($\alpha = 0.2009$)
1. The new probability for the DFC selecting $\gamma_0^A$, $q = 0.6787$, calculated by (25)
2. The fulfillment of the prerequisite of the correction action, $0<q<1$
3. The operating point of the non-synchronized system after the corrective action is taken, ${{{C}}^*} = ({\bf{0}}.{\bf{2009}},{\bf{0}}.{\bf{7005}})$, calculated by (19) and (20)

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2) 3-LD System: The design input and output of the 3-LD system employing dependent randomization with $\alpha = 0.1708$ was shown in Table III. When the DFC lost synchronization with the LDs group in the 3-LD system employing dependent randomization, the input and output of the redesigned algorithm are shown in Table VI and Fig. 10. Before the loss of synchronization the system operated at $C = (0.1708, 0.8448)$ (Table II). After the loss of synchronization the system operates at $C^* = (0.1708,0.7974)$.

Table  VI.  THe Output of the 3-LD System Employing Dependent Randomization When the DFC Lost Synchronization With the LDs Group Before and After a Corrective Action is Taken

Output of the non-synchronized 3-LD system before the corrective action is taken ($\alpha = 0.1708$)
1. The probability of false alarm constraint, $\alpha = 0.1708$
2. The probability of selecting A, $p = 0.6$, calculated by (11)
3. Four possible operating points when the DFC lost synchronization with the LDs group, $A = (0.1040,0.7840)$, $B = (0.2710,0.9360)$, $M1 = (0.4880,0.9730)$, and $M2 = (0.0280,0.6480)$
4. The operating point of the non-synchronized system, $W^* = (0.2046,0.8210)$, calculated by (17) and (18)
Output of the non-synchronized 3-LD system after the corrective action is taken ($\alpha = 0.1708$)
1. The new probability for the DFC selecting $\gamma_0^A$, $q = 0.7033$, calculated by (25)
2. The fulfillment of the prerequisite of the correction action, $0<q<1$
3. The operating point of the non-synchronized system after the corrective action is taken, ${{{C}}^*} = ({\bf{0}}.{\bf{1708}},{\bf{0}}.{\bf{7974}})$, calculated by (19) and (20)

 | Show Table

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Figure  10.  $A$, $B$, $M1$, and $M2$, shown by green circles, are the possible operating points of the 3-LD system (Section III-A) when the synchronization between the LDs and the DFC is lost. The black circle, $C$, shows the operating point of the synchronized system. The cyan circle, $W^*$, shows the equivalent operating point of the system when it lost synchronization. $C^*$ is the equivalent operating point after the corrective action is taken.

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More details of the numerical examples are available in https://github.com/moshekam/Dependent-Randomization.

V. Partial Loss of Synchronization Among the Local Detectors

We assume that a decision fusion architecture was designed per Section II-B-2) (Dependent randomization) to maximize the probability of detection under a probability of false alarm constraint. This approach means that the LDs and the DFC are designed to operate at any given time at one of two operating points (say A and B). They operate at operating point A (corresponding to deterministic strategy $\gamma^A = \{\gamma_0^A,\gamma_1^A,\ldots,\gamma_n^A\}$) with probability $p$ and at operating point B (corresponding to deterministic strategy $\gamma^B = \{\gamma_0^B,\gamma_1^B,\ldots,\gamma_n^B\}$) with probability $1-p$. A and B are on the upper boundary of the convex hull of all the operating points which are achievable by deterministic strategies of the system. If a value of $p$, $0<p<1$, exists that would keep the probability of false alarm of the system at the maximal allowable level, $\alpha$, the resulting operating point of the system $C = (P_f^C = \alpha,P_d^C)$ would be optimal.

In the previous section (IV: Loss of Synchronization Between the DFC and the LDs Group), we assumed that in spite of loss of synchronization between the DFC and the LDs group, all LDs were still synchronized with each other. Under this circumstance, the DFC can in some cases change the probability $p$ of hopping between $A$ and $B$ to satisfy the probability of false alarm constraint, but generally at the expense of reaching a lower probability of detection than $P_d^C$.

In this section, we assume a synchronization failure of the following characteristics:

a) Only $m$ ($1\leq m\leq n-1$) LDs are synchronized with the DFC and with each other. We call them group $Y$.

b) The remaining $n-m$ LDs are not synchronized with the DFC, nor are they synchronized with each other. We call them group $\overline{Y}$.

c) The DFC is aware of a) and b) and of the identity of members in $Y$ and $\overline{Y}$. The LDs are not.

Each LD of the system (say LD $k$) flips a coin, and, based on the outcome, it follows the local decision rule $\gamma_k^A$ or the other local decision rule, $\gamma_k^B$ ($\gamma_k^A$ is used with probability $p$ and $\gamma_k^B$ is used with probability $1-p$). If LD $k$ belongs to $Y$ it uses the “joint coin” flipped simultaneously and synchronously by all the LDs in $Y$ and the DFC. If LD $k$ belongs to $\overline{Y}$ then it flips its own coin, which is not synchronized with either the “joint coin” used by the LDs in $Y$ and the DFC, or the “separate” $n-m-1$ coins of the other members of $\overline{Y}$.

Since $n-m$ LDs are now unsynchronized, the resulting operating point of the system, $W' = (P_f^{W'},P_d^{W'})$, is a combinations of $2^{n-m+1}$ possible operating points. If no correction is made, $P_f^{W'}$ is highly likely to exceed the level $\alpha$ which was satisfied (per the constraint $P_f\leq \alpha$) before synchronization was lost. In this section, we redesign the global fusion rules at the DFC to satisfy $P_f\leq\alpha$, and show the resulting performance cost (the reduction in the probability of detection).

The input of the redesigned algorithm is shown in Table VII.

Table  VII.  Input of the Redesigned Algorithm When Each LD in $\overline{Y}$ Lost Synchronization the DFC and Each Other

Input of the redesigned algorithm
The design input of dependent randomization (Table I)	1. The number of local detectors, $n$
	2. The probability of false alarm constraint, $\alpha$
The design output of dependent randomization (Table I)	3. The local operating points of $A$ and $B$: $(P_{fk}^A,P_{dk}^A)$ and $(P_{fk}^B,P_{dk}^B), k = 1,\ldots,n$
	4. The probability of selecting A, $p$, calculated by (11)
Information of synchronized LDs	5. Numbers of synchronized LDs, $m$ (we assume $Y = \{LD1,\ldots,LDm\}$ are synchronized)
	6. Identity of all synchronized LDs, $Y = \{LD1,\ldots,LDm\}$

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A Calculating the Local Operating Points After the LDs in $\overline{Y}$ Lost Synchronization

The local operating points of the $m$ LDs in $Y$ are $\{(P_{f1}^i,P_{d1}^i),\dots,(P_{fm}^i,P_{dm}^i)\}$, $i\in\{A,B\}$. For the $j{\rm{th}}$ LD in $\overline{Y}$ ($j = m+1,\ldots,n$), the expected value of the probability of false alarm is $pP_{fj}^A+(1-p)P_{fj}^B$ and the expected value of the probability of detection is $pP_{dj}^A+(1-p)P_{dj}^B$. The local operating points of the $n-m$ LDs in $\overline{Y}$ are $\{(pP_{fm+1}^A+(1-p)$ $P_{fm+1}^B, pP_{dm+1}^A+(1-p)P_{dm+1}^B),\dots,(pP_{fn}^A+(1-p)P_{fn}^B, pP_{dn}^A+(1-p)$ $P_{dn}^B)\}$. The equivalent local operating points of the system are $\Phi^i = \{(P_{f1}^i,P_{d1}^i),\dots,(P_{fm}^i,P_{dm}^i),(pP_{fm+1}^A+(1-p)P_{fm+1}^B, pP_{dm+1}^A+ (1-p)P_{dm+1}^B),\dots,(pP_{fn}^A+(1-p)P_{fn}^B, pP_{dn}^A+(1-p)P_{dn}^B) \}$. Therefore, at each time step, we use system A, shown in Fig. 11, with probability $p$, and system B, shown in Fig. 12, with probability $1-p$. The local operating points of $\{LD\ m+1 ,\ldots, LD\ n\}$ in system A and system B are the same.

Figure  11.  System A is used with probability $p$.

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Figure  12.  System B is used with probability $1-p$.

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B Calculating the ROC Curves of System A and System B

In order to satisfy the Neyman-Pearson criterion, under the new condition we can try to redesign the global fusion rules at the DFC. At each time step, the DFC will now use $\gamma_0^{A'}$ (was $\gamma_0^{A}$) with probability $p$ and $\gamma_0^{B'}$ (was $\gamma_0^{B}$) with probability $1-p$. Namely, $\gamma_0^{A'}$ will be used by system A and $\gamma_0^{B'}$ will be used by system B. Unlike $\gamma_0^{A}$ and $\gamma_0^{B}$, which are deterministic fusion rules, $\gamma_0^{A'}$ or $\gamma_0^{B'}$ could be a randomized fusion rule. We will select $\gamma_0^{A'}$ and $\gamma_0^{B'}$ from among the monotonic fusion rules [2] for both systems. For each one of $\gamma_0^{A'}$ and $\gamma_0^{B'}$, we need in general two monotonic fusion rules and a randomization scheme to hop between them.

We use (6) to calculate the operating points, corresponding to all the monotonic fusion rules, for both system A and system B. The calculated operating points are isolated in the $P_f-P_d$ plane. We denote ROC curve $A$ (ROC curve $B$) as the ROC curve of system A (system B). The ROC curve of a system with isolated operating points in the $P_f-P_d$ plane is the upper boundary of the convex hull of those isolated operating points, which is a concave piecewise-linear curve. Therefore both ROC curve $A$ and ROC curve $B$ are concave piecewise-linear curves. Finding the fusion rule $\gamma_0^{A'}$ for system A is equivalent to finding an operating point of system A on ROC curve A. Similarly, finding a fusion rule $\gamma_0^{B'}$ for system B is equivalent to finding an operating point of system B on ROC curve B.

ROC curve A can be drawn as a sequence of straight line segments $\omega_1^A\omega_2^A, \omega_2^A\omega_3^A, \ldots, \omega_{mA-1}^A\omega_{mA}^A$, where $\Omega^A = \{\omega_1^A = (0,0),\omega_2^A,\ldots,\omega_{mA-1}^A,\omega_{mA}^A = (1,1)\}$ are points in the $P_f-P_d$ plane. Similarly, we can find $\Omega^B = \{\omega_1^B = (0,0),\omega_2^B,\ldots,\omega_{mB-1}^B, \omega_{mB}^B = (1,1)\}$ for the ROC curve B. Each of points in $\Omega^A$ and $\Omega^B$ is realizable by a deterministic (monotonic) fusion rule. Meanwhile, each one of the other operating points on ROC curve $A$ (ROC curve $B$), those that are not in $\Omega^A(\Omega^B)$, can be realized by hopping between two points in $\Omega^A$ ($\Omega^B$) by using randomization at the DFC.

In Fig. 13, the ROC curve of the 2-LD system (Section III-A) with dependent randomization is shown as the black curve. Recall (Figs. 4 and 5) that to create this curve, we have integrated two different ROC curves (one corresponding to an $AND$ global decision rule and one corresponding to an $OR$ global decision rule). We calculated the points of tangency $A$ and $B$ of the ROC curves for the $AND$ rule and the $OR$ rule respectively, and connected them with a straight line. If $\alpha\in(P_f^A,P_f^B)$ then the highest achievable probability of detection, corresponding to $\alpha$, was on the straight-line segment that joins points $A$ and $B$. For $\alpha$ shown in Fig. 5, the highest probability of detection is denoted $P_d^C$, achieved at point $C$, which is the the midpoint of line segment connecting $A$ and $B$ (in this example $p = 0.5$).

Figure  13.  The ROC curve of the system with dependent randomization is shown by the black curve. For $\alpha\in(P_f^{A},P_f^{B})$, $C$ is the desired operating point of the system (black circle). $A$ and $B$ (green circles) are the operating points used to generate $C$ through a randomization procedure. When the second LD loses synchronization ($Y = \{LD 1\}$, $\overline{Y} = \{LD 2\}$), if $A$ is selected the ROC curve $A$ is effective (shown in red); if $B$ is selected the ROC curve $B$ is effective (shown in blue).

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Suppose that $LD1$ and the DFC continue to be synchronized with each other ($Y = \{LD1\}$) but $LD2$ lost synchronization with $LD1$ and with the DFC ($\overline{Y} = \{LD2\}$). In Fig. 13 we show two ROC curves for this new situation. ROC curve $A$ (red) is obtained when the members of $Y$ (just $LD1$ in our case) select $\gamma_1^A$ (when this happen $LD2$, oblivious to $LD1$ and the DFC, selects $\gamma_2^A$ with probability $p$ and $\gamma_2^B$ with probability $1-p$). ROC curve $B$ (blue) is obtained when the members of $Y$ ($LD1$) select $\gamma_1^B$ ($LD2$ still selects $\gamma_2^A$ with probability $p$ and $\gamma_2^B$ with probability $1-p$). In Fig. 13, $\Omega^A$ are shown by the red circles and $\Omega^B$ are shown by the blue circles. Our next task is to select the points $A'$ (on ROC curve A) and $B'$ (on ROC curve B) such that we can hop between them and meet the following objectives: a) satisfy the probability of false alarm constraint $P_f\leq\alpha$; b) maximize the probability of detection $P_d$.

C Satisfying the Probability of False Alarm Constraint and Maximizing the Probability of Detection

Let $a = (P_f^a,P_d^a)$ be an operating point of system A on ROC curve A and $b = (P_f^b,P_d^b)$ be an operating point of system B on ROC curve B. Points $a$ and $b$ are selected such that upper limit of the probability of false alarm of the system, $\alpha$, satisfies $P_f^a\leq\alpha\leq P_f^b$ or $P_f^b\leq\alpha\leq P_f^a$. In order to meet the probability of false alarm constraint ($P_f = \alpha$), points $a$ and $b$ need to be found such that

This design would yield the probability of detection

We now proceed to locate the specific point on ROC curve A, denoted $A'$ (so $a = A'$), and the specific point on ROC curve B, denoted as $B'$ (so $b = B'$) that allow the system to maximize the probability of detection while satisfying the probability of false alarm constraint. The optimal resulting system operating point $C' = (P_f^{C'},P_d^{C'})$ is on the line segment connecting $A'$ and $B'$, where

It can be shown that for $P_d^{C'}$ in (29) to be the maximum probability of detection either i) $A'\in\Omega^A$ or ii) $B'\in\Omega^B$ or both ($A'\in\Omega^A$ and $B'\in\Omega^B$). The proof can be found in https://github.com/moshekam/Dependent-Randomization. Here we provide an intuitive explanation: the resulting operating point of the system would be on the line segment connecting a point on ROC curve A and a point on ROC curve B. We can “lift” the line segment to improve the probability of detection of the resulting operating point. During the process of “lifting”, the length of the line segment is adjustable to keep its endpoints on different ROC curves. The resulting probability of detection would stop growing when the line segment is about to “leave” the ROC curves. The line segment can only “leave” the ROC curve A at a point in $\Omega^A$ and the ROC curve B at a point in $\Omega^B$.

D Finding $A'$ From $\Omega^A$ if $A'\in\Omega^A$ and $B'$ From $\Omega^B$ if $B'\in\Omega^B$ (at Least One of $A'$ and $B'$ Can be Found)

We can examine every point in $\Omega^A$ to find $A'$ and every point in $\Omega^B$ to find $B'$ by using the following steps:

1) For each operating point $a = (P_f^a,P_d^a)\in\Omega^A$, we calculate first the probability of the false alarm of the corresponding operating point $b = (P_f^b,P_d^b)$ on ROC curve B by using (26) ($P_f^b = \frac{\alpha-pP_f^a}{1-p}$). Since each probability of detection is paired with exactly one probability of false alarm on ROC curve B, $P_f^b$ can be used to locate $b$ on ROC curve B and define $P_d^b$. The resulting probability of detection of the system can be calculated by using (27). We store the resulting probability of detection and the corresponding $a$ and $b$ for each $a\in\Omega^A$.

2) For each operating point $b = (P_f^b,P_d^b)\in\Omega^B$, we calculate first the probability of the false alarm of the corresponding operating point on ROC curve A ($P_f^a = \frac{\alpha-(1-p)P_f^b}{p}$) by using (26). $P_f^a$ can be used to locate $a$ on ROC curve A and define $P_d^a$. The resulting probability of detection of the system can be calculated by using (27). We store the resulting probability of detection and the corresponding $a$ and $b$ for each $b\in\Omega^B$.

3) Let $P_d^{C'}$ be the highest probability of detection found for all the pairs examined in steps 1) and 2). $A'$ and $B'$ are the pair of values of $a$ and $b$ which corresponded to the highest $P_d^{C'}$ for the final design.

Computational complexity: in steps 1) and 2) we examine $m_A+m_B$ points ($m_A$ points in $\Omega^A$ and $m_B$ points in $\Omega^B$) in order to find $A'$ if $A'\in\Omega^A$ and $B'$ if $B'\in\Omega^B$. An algorithm which requires the examination of at most ${\rm{log}}_2(m_A+m_B)$ points in $\Omega^A$ and $\Omega^B$ is available (https://github.com/moshekam/Dependent-Randomization).

E Finding $A'$ if $A'\notin\Omega^A$ and $B'$ if $B'\notin\Omega^B$ (Applying Randomization at the DFC)

In most cases, when $P_d^{C'}$ is maximized, one of the points $A'$ and $B'$ is realized by randomization at the DFC and the other is realized by a deterministic strategy (since it is an element of $\Omega^A$ or $\Omega^B$). We first discuss the situation that $B'$ is a randomized operating point, $A'\in\Omega^A$, and $B'\notin\Omega^B$. In this circumstance $P_f^{B'} = \frac{\alpha-pP_f^{A'}}{1-p}$ (from (26)). $B'$ is on a line segment connecting two operating points in $\Omega^B$, denoted as $\omega_a^B$ and $\omega_b^B$.

Let the probabilities needed to redesign $B'$ by hopping between using $\omega_a^B$ and $\omega_b^B$ be $q'$ and $1-q'$, respectively. Then $P_f^{B'}$ can be expressed as

Therefore, $P_f^{C'}$ is a weighted sum of $P_f^{A'}$, $P_f^{\omega_a^B}$, and $P_f^{\omega_b^B}$

Since $P_f^{C'} = \alpha$, $q'$ can be calculated as

The probability of detection $P_d^{C'}$ is

Similarly, when $A'$ is a randomized operating point ($A'\notin\Omega^A$ and $B'\in\Omega^B$), $P_d^{C'}$ becomes

$\omega_a^A$ and $\omega_b^A$ are the two end points of the line segment on the ROC curve $A$ which $A'$ locates on. Let $q''$ and $1-q''$ be the probabilities of using $\omega_a^A$ and $\omega_b^A$ respectively. $q''$ satisfies

$q''$ is

In the case $A'\in\Omega^A$ and $B'\in\Omega^B$, both $A'$ and $B'$ are realized by deterministic strategies. The probability of detection can be calculated by (29).

F Numerical Examples

1) 2-LD System: In Fig. 14, the black curve is the ROC curve of the 2-LD example in Section III-A when using dependent randomization. The design input and output of the 2-LD system employing dependent randomization with $\alpha = 0.2009$ was shown in Table II. The operating point of the system is $C = (0.2009,0.8261)$, shown by the black circle. The operating point $C$ is generated by operating at $A = (0.1581,0.7870)$ with probability $p = 0.5$ and at $B = (0.2437,0.8652)$ with probability $1-p$.

Figure  14.  $C$ is the desired operating point of the system with dependent randomization (black circle). When $Y = \{LD 1\}$ and $\overline{Y} = \{LD 2\}$, $A$, B, $M1'$, and $M2'$ are the four possible operating points (green circles). $W'$ is the equivalent operating point (purple circle). The ROC curve $A$ is shown as the red curve. The ROC curve $B$ is shown as the blue curve. $C'$ is the operating point with maximized probability of detection given $\alpha = 0.2009$, shown by the purple square.

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If $Y = \{LD 1\}$, $\overline{Y} = \{LD 2\}$, there are four possible operating points: $A$, $B$, $M1' = (0.0518,0.5614)$, and $M2' = (0.4761, 0.9586)$, determined by $(\gamma_{0}^A, \gamma_{1}^A,\gamma_{2}^A)$, $(\gamma_{0}^B, \gamma_{1}^B,\gamma_{2}^B)$, $(\gamma_{0}^A, \gamma_{1}^A,\gamma_{2}^B)$, and $(\gamma_{0}^B, \gamma_{1}^B,\gamma_{2}^A)$. $A$, $B$, $M1'$, and $M2'$ are shown by the green circles. The resulting operating point $W' = (0.2324,0.7930)$, shown by the purple circle is calculated as

In this case, $\Phi^A = \{(P_{f1}^A,P_{d1}^A),(pP_{f2}^A+(1-p)P_{f2}^B, pP_{d2}^A+ (1-p)P_{d2}^B)\}$. The ROC curve $A$ is shown as the red curve; $\Phi^B = \{(P_{f1}^B,P_{d1}^B),(pP_{f2}^A+(1-p)P_{f2}^B, pP_{d2}^A+(1-p)P_{d2}^B)\}$. The ROC curve $B$ is shown as the blue curve. $\Omega^A$ and $\Omega^B$ then can be found, shown in Table VIII. By using the proposed algorithm, we find that when $A' = (0.1049,0.6742)$ and $B' = (0.2968,0.8410)$, the probability of detection is maximized. $A'\in\Omega^A$ and $B'\notin\Omega^B$. $A'$ is shown as the cyan circle. It is achieved when the DFC uses the AND fusion rule ($\gamma _0^{A'}(u1,u2) = {u_1}\& {u_2}$). $P_f^{B'}$ can be calculated by (26) and $B$ is shown by the purple triangle. $B'$ is generated by the randomized fusion rule $\gamma_0^{B'}$, which requires the system hopping between $\omega_a^B = (0.1304,0.6328)$ and $\omega_b^B = (0.3599, 0.9199)$. $\omega_a^B$ (achieved by the fusion rule such that $u_0 = u_1$) is used with probability $q'$ and $\omega_b^B$ (achieved by the OR fusion rule) is used with probability $1-q'$, where $q' = 0.2748$, calculated by (32). $\omega_a^B$ and $\omega_b^B$ are shown as the cyan triangles.

Table  VIII.  the Output the 2-LD System Employing Dependent Randomization Before and After a Corrective Action is Taken When $Y = \{LD 1\}$ and $\overline{Y} = \{LD 2\}$

Input of the redesigned algorithm for the 2-LD system when $LD 2$ lost synchronization
The design input of dependent randomization (Table II)	1. The number of local detectors, $n = 2$
	2. The probability of false alarm constraint, $\alpha = 0.2009$
The design output of dependent randomization (Table II)	3. The local operating points of $A$: $(P_{f1}^A,P_{d1}^A) = (P_{f2}^A,P_{d2}^A) = (0.3976,0.8871)$, 　The local operating points of $B$: $(P_{f1}^B,P_{d1}^B) = (P_{f2}^B,P_{d2}^B) = (0.1304,0.6328)$
	4. The probability of selecting $A$, $p = 0.5$
Information at the synchronized LDs	5. Numbers of synchronized LDs, $m = 1$
	6. Identity of all synchronized LDs, $Y = \{LD1\}$
Output of the redesigned algorithm for the 2-LD system when $LD 2$ lost synchronization
1. $\Omega^A = \{(0,0), (0.1049,0.6742), (0.3976,0.8871), (0.5566,0.9729), (1,1)\}$, $\Omega^B = \{(0,0), (0.0344,0.4809), (0.1304,0.6328) ,(0.3599,0.9119) ,(1,1) \}$
2. Two operating points $A' = (0.1049,0.6742)\in\Omega^A$ and $B' = (0.2968,0.8410)\notin\Omega^B$, which allow $P_f^{C'}$ from (28) satisfying the probability of false alarm constraint $\alpha$ and achieving the highest probability of detection
3. The deterministic fusion rule $\gamma_0^{A'}$(AND fusion rule) used to achieve $A'$
4. The randomized fusion rule $\gamma_0^{B'}$ used to achieve $B'$, which requires the system operating at $\omega_a^B = (0.1304,0.6328)$(achieved by the fusion rule such that $u_0 = u_1$) with probability $q'$ and $\omega_b^B = (0.3599,0.9199)$(achieved by the OR fusion rule) with probability $1-q'$, where $q' = 0.2748$, calculated by (32)
5. The operating point of the non-synchronized system after the corrective action is taken, ${{C'}} = ({\bf{0}}.{\bf{2009}},{\bf{0}}.{\bf{7547}})$, calculated by (31) and (33), which is achieved by operating at $A'$ with probability $p$ and $B'$ with probability $1-p$

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In this example, when the DFC realizes that the $LD 2$ loses synchronization, if $\gamma_1^A$ is selected at the $LD 1$ ($p = 0.5$), the system operates at point $A'$; if $\gamma_1^B$ is selected at the $LD 1$ ($1-p = 0.5$), the system operates at point $\omega_a^B$ with probability $q' = 0.2748$ and operates at point $\omega_b^B$ with probability $1-q' = 0.7252$. The maximized probability of detection is $P_d^{C'} = 0.7547$, calculated from (29). $C' = (0.2009,0.7547)$ is shown by the purple square. Due to the loss of synchronization we drop from $P_d^C = 0.8261$ to $P_d^{C'} = 0.7547$.

Table VIII provides a summary of the input and output of the redesigned algorithm for the 2-LD system employing dependent randomization when $LD2$ lost synchronization.

Fig. 15 and Table IX compare the operating points of the 2-LD system under the Neyman-Pearson criterion with $\alpha = 0.2009$ for the following detection strategies:

Figure  15.  The operating points of the 2-LD system employing different detection strategies under the Neyman-Pearson criterion with $\alpha = 0.2009$.

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Table  IX.  The Operating Points of the 2-LD System Employing Different Detection Strategies Under the Neyman-Pearson Criterion With $\alpha = 0.2009$ (corresponding to Fig. 15)

	Detection strategy	$\alpha = 0.2009$
1	Deterministic strategy	$G = (0.2009,0.8217)$
2	Randomization at the DFC	$G = (0.2009,0.8217)$
3	Dependent randomization (synchronized)	$C = (0.2009,0.8261)$
4	Dependent randomization (the DFC is unsynchronized with the LDs group)	$C^* = (0.2009,0.7005)$
5	Dependent randomization (the $2{{\rm{nd}}}$ LD is unsynchronized with the DFC and other LDs)	$C' = (0.2009,0.7547)$
6	Dependent randomization (all LDs and the DFC are unsynchronized)	$C'' = (0.2009,0.7008)$

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a) Deterministic strategy and randomization at the DFC ($G$, blue circle);

b) Dependent randomization ($C$, black circle);

c) Dependent randomization when the LDs group lost synchronization with the DFC before the redesigned algorithm is applied ($W^* = (0.2640,0.7600)$, cyan circle) and after the redesigned algorithm is applied ($C^*$, cyan square);

d) Dependent randomization when the $LD2$ lost synchronization with the $LD1$ and the DFC before the redesigned algorithm is applied ($W' = (0.2324,0.7930)$, purple circle) and after the redesigned algorithm is applied ($C'$, purple square); and

e) Dependent randomization when both two LDs lost synchronization with each other and the DFC before the redesigned algorithm is applied ($W'' = (0.2044,0.6401)$, yellow circle) and after the redesigned algorithm is applied ($C''$, yellow square).

2) 3-LD System: Returning to the 3-LD system (Section III-B), we show in Fig. 16 what happen when $Y = \{LD 1, LD 2\}$ and $\overline{Y} = \{LD 3\}$ and when the probability of false alarm constraint is $P_f\leq\alpha = 0.1708$. The notation is the same as in Fig. 14. Table X summarizes the input and output of the redesigned algorithm for the 3-LD system employing dependent randomization when $LD3$ lost synchronization.

Figure  16.  $C$ is the desired operating point of the system with dependent randomization (black circle). When $Y = \{LD 1, LD 2\}$ and $\overline{Y} = \{LD 3\}$, $A$, $B$, $M1'$, and $M2'$ are the four possible operating points (green circles). $W'$ is the equivalent operating point (purple circle). ROC curve $A$ and ROC curve $B$ are shown as the red curve and the blue curve, respectively. $C'$ is the operating point maximizing the probability of detection given $\alpha = 0.1708$, shown by the purple square.

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Table  X.  The Output the 3-LD System Employing Dependent Randomization Before and After a Corrective Action is Taken When $Y = \{LD 1,\ LD2\}$ and $\overline{Y} = \{LD 3\}$

Input of the redesigned algorithm for the 3-LD system when $LD 3$ lost synchronization
The design input of dependent randomization (Table III)	1. The number of local detectors, $n = 3$
	2. The probability of false alarm constraint, $\alpha = 0.1708$
The design output of dependent randomization (Table III)	3. The local operating points of $A$: $(P_{fk},P_{dk}) = (0.2,0.7),k = 1,2,3$, 　The local operating points of $B$: $(P_{fk},P_{dk}) = (0.1,0.6),k = 1,2,3$
	4. The probability of selecting $A$, $p = 0.6$
Information of synchronized LDs	5. Numbers of synchronized LDs, $m = 2$
	6. Identity of all synchronized LDs, $Y = \{LD1,LD2\}$
Output of the redesigned algorithm for the 3-LD system when $LD 3$ lost synchronization
1. $\Omega^A = \{(0,0),(0.0064,0.3234),(0.0576,0.6006),(0.0912,0.7672),(0.1936,0.8266),(0.4624,0.9694),(1,1)\}$, $\Omega^B = \{(0,0),(0.0016,0.2376),(0.0100,0.3600),(0.0388,0.6768),(0.1900,0.8400),(0.3196,0.9456),(1,1)\}$
2. Two operating points $A' = (0.0912,0.7672)\in\Omega^A$ and $B' = (0.2902,0.9216)\notin\Omega^B$, which allow $P_f^{C'}$ from (28) satisfying the probability of false alarm constraint $\alpha$ and achieving the highest probability of detection
3. The deterministic fusion rule $\gamma_0^{A'}$(2 out of 3 rule) used to achieve $A'$
4. The randomized fusion rule $\gamma_0^{B'}$ used to achieve $B'$, which requires the system operating at $\omega_a^B = (0.1900,0.8400)$(achieved by the fusion rule such that $u_0 = u_1|u_2$) with probability $q'$ and $\omega_b^B = (0.3196,0.9456)$(achieved by the 1 out of 3 rule) with probability $1-q'$, where $q' = 0.2748$, calculated by (32)
5. The operating point of the non-synchronized system after the corrective action is taken, ${{C'}} = ({\bf{0}}.{\bf{1708}},{\bf{0}}.{\bf{8290}})$, calculated by (31) and (33), which is achieved by operating at $A'$ with probability $p$ and $B'$ with probability $1-p$

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Fig. 17 compares the operating points of the 3-LD system under the Neyman-Pearson criterion with $\alpha = 0.1708$ for the following detection strategies:

Figure  17.  The operating points of the 3-LD system employing different detection strategies under the Neyman-Pearson criterion with $\alpha = 0.1708$.

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a) Deterministic strategy ($G$, blue circle);

b) Randomization at the DFC ($E$, red circle);

c) Dependent randomization ($C$, black circle);

d) Dependent randomization when the LDs group lost synchronization with the DFC before the redesigned algorithm is applied ($W^* = (0.2046,0.8210)$, cyan circle) and after the redesigned algorithm is applied ($C^*$, cyan square);

e) Dependent randomization when the $LD2$ lost synchronization with the $LD1$ and the DFC before the redesigned algorithm is applied ($W' = (0.1826,0.8386)$, purple circle) and after the redesigned algorithm is applied ($C'$, purple square); and

f) Dependent randomization when both two LDs lost synchronization with each other and the DFC before the redesigned algorithm is applied ($W'' = (0.2223,0.8380)$, yellow circle) and after the redesigned algorithm is applied ($C''$, yellow square).

Table XI shows the operating points of the 3-LD system employing different detection strategies under the Neyman-Pearson criterion with a) $\alpha = 0.1708$ and b) $\alpha = 0.05$. Lines 3 and 4 at column “$\alpha = 0.05$” are identical. The reason is that when $\alpha = 0.05$, dependent randomization requires the system hopping between two operating points with the same global fusion rule. In this circumstance, dependent randomization is “randomization at the LDs only” and the DFC does not participate in the randomization. Therefore, the loss of synchronization between the LDs group and the DFC has no influence to the system performance.

Table  XI.  The Operating Points of the 3-LD System Employing Different Detection Strategies Under the Neyman-Pearson Criterion With (a) $\alpha = 0.1708$ (Corresponding to Fig. 17) and (b) $\alpha = 0.05$

	Detection strategy	$\alpha = 0.1708$	$\alpha = 0.05$
1	Deterministic strategy	$G = (0.1360,0.7960)$	$(0.0460,0.6960)$
2	Randomization at the DFC	$E = (0.1708,0.8208)$	$(0.0500,0.7000)$
3	Dependent randomization (synchronized)	$C = (0.1708,0.8448)$	$(0.0500,0.7031)$
4	Dependent randomization (the DFC is unsynchronized with the LDs group)	$C^* = (0.1708,0.7974)$	$(0.0500,0.7031)$
5	Dependent randomization (the $3{{\rm{rd}}}$ LD is unsynchronized with the DFC and other LDs)	$C' = (0.1708,0.8290)$	$(0.0500, 0.5704)$
6	Dependent randomization (all LDs and the DFC are unsynchronized)	$C'' = (0.1708,0.8009)$	$(0.0500,0.5534)$

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An example of a larger system (with seven LDs) is available in https://github.com/moshekam/Dependent-Randomization.

VI. Conclusion

In parallel binary decentralized detection networks, dependent randomization can sometimes make the team’s Receiver Operating Characteristic curve concave (if it was non-concave under other detection schemes). This effect improves the system’s performance under a Neyman-Pearson criterion by realizing a higher probability of detection for the same upper bound on the probability of false alarm. Dependent randomization requires that the DFC and the LDs be synchronized, guided by a coordinated randomization scheme. The DFC and the LDs switch simultaneously together, back and forth, between $\gamma_0^A$ (for the DFC) and $\gamma_{LD}^A = \{\gamma_1^A,\ldots,\gamma_n^A\}$ (for the LDs); and $\gamma_0^B$ (for the DFC) and $\gamma_{LD}^B = \{\gamma_1^B,\ldots,\gamma_n^B\}$ (for the LDs). However, if the synchronization is lost, the system may exceed the permitted probability of false alarm. We study the consequences of synchronization loss in the following two sets of circumstances: a) the DFC is not synchronized with the LDs group; and b) some LDs are not synchronized with other LDs and with the DFC. Corrective action is devised in order to restore the detection system to compliance with the probability of false alarm constraint, at a cost of reduced probability of detection. The implication of these results is that operation of systems in “dependent randomization” mode is actually feasible. It can improve performance of parallel binary decision fusion systems, and its failure mode (loss of synchronization) can be quantified, as well as mitigated. Future work includes applications to pertinent networks, such as those studied in [14]–[18].