Dynamic Coalition Formation for Efficient Sleep Time Allocation in ...
12th International Conference on Information Fusion Seattle, WA, USA, July 6-9, 2009
Dynamic Coalition Formation for Efficient Sleep Time Allocation in Wireless Sensor Networks Using Cooperative Game Theory Omid Namvar Gharehshiran, Vikram Krishnamurthy Department of Electrical and Computer Engineering University of British Columbia Vancouver, BC, Canada V6T 1Z4 {omidn, vikramk}@ece.ubc.ca Abstract – This paper proposes a dynamic coalition formation algorithm for efficient sleep time allocation in a wireless sensor network (WSN) deployed to localize targets based on cooperation among the nodes. The sleep time allocation problem is formulated as a non-convex cooperative game and the concept of the core is exploited to solve this problem. In this formulation, determinant of the Cram´er-Rao lower bound (CRLB) is used to evaluate the accuracy of estimations. Finally, an algorithm is proposed based on a best-reply rule which converges to the core with probability one if the core of the game is non-empty. Keywords: Cooperative game theory, dynamic coalition formation, localization and tracking, wireless sensor network (WSN).
1
Introduction
A WSN is a distributed embedded system consisting of a large number of low-cost, low-power and energyconstrained sensor nodes. These nodes communicate over a wireless channel, performing distributed sensing and collaborative data processing tasks for various vital military and civilian applications. In this paper, we consider the problem of using a WSN for localization of a target based on noisy bearing (angle) measurements at individual nodes. Since estimating the position of a target in two dimensions needs at least two angle measurements (to perform triangularization), it is natural for the sensors to cooperate. In theory, the best estimate is obtained by employing every node in the WSN to collect and share direction-of-arrival (DOA) measurements. However, due to battery life constraints, it is acceptable to permit a slight increase in the position error by letting some nodes go to sleep. Our main goal is to find efficient sleep time allocations so that targets are localized with an acceptable accuracy, at the same time battery resources being used as efficient as possible. As is commonly used in the tracking literature (e.g. [1] and [2]), we use the determinant of CRLB as the measure of estimation accuracy in order to derive
978-0-9824438-0-4 ©2009 ISIF
the characteristic function for the sleep time allocation game. We refer to this measure as stochastic observability. Main Results: We use a dynamic coalition formation algorithm for allocating sleep times to the sensors. Localization is essentially achieved by cooperation among the nodes. It is reasonable to allocate more sleep times to sensors in a coalition which provides more information about the target’s location. To quantify the amount of sleep time that can be allocated to a coalition of nodes, we derive a characteristic function based on the stochastic observability required in localizing each target and put the required constraints on it to fit in the framework of cooperative game theory. Finally, we propose an algorithm which relies on the concept of the core in order to find a solution for the sleep time allocation game. Related Work: Minimizing energy consumption in WSNs has been a central topic in recent years. Effective techniques have been proposed to put sensors into a low-power or sleep mode while maintaining full coverage at a specified degree of redundancy (e.g. [3], [4] and [5]). In [6], a cluster-based sensor network is considered, wherein the sensor nodes communicate directly with the cluster head. It was shown that the nodes which are farther from the cluster head consume more power, and hence it was suggested that they be put to sleep more frequently. There also has been much work on sensor activation using non-cooperative game theory, e.g. [7] and [8]. In the context of these related works, we should emphasize that our attention is primarily focused on efficient sleep time allocation in cooperative target localization without any power control or synchronization mechanism. Our interest is evaluating the most efficient amount of sleep time that each sensor can achieve based on the relative configuration of the nodes and targets and under constraints on localization accuracy. Organization: The rest of this paper is organized as follows. In Sec. 2, the sleep time allocation problem is formulated and relative material from game theory is provided. Sec. 3 incorporates an overview to the solution and the proposed algorithm. In Sec. 4, two numerical ex-
672
amples are given. Finally, concluding remarks are made in Sec. 5.
2
Problem Formulation
The problem of prolonging the lifetime of a WSN can be formulated as a maximization problem for the sleep time of each node under constraints on stochastic observability. Formally, the problem can be stated as maximize τi ∈Di
subject to
τi
∀i ∈ N
¯ det(J(τ )) ≥ O(k) ∀k ∈ K
(1)
where • N = {1, 2, . . . , N } is the set of sensor indices. • K = {1, 2, . . . , K} is the set of target indices. • τ = (τ1 , τ2 , . . . , τN ) is the vector of sleep times. • Di is the set of allocations possible for sensor i. ¯ • O(k) is the required accuracy in localizing target k. In this section, notations and measurement model are described. Some relevant subjects from game theory are introduced. Furthermore, DOA-based localization in two-dimensional space is formulated. Finally, the characteristic function is derived based on the expression given for stochastic observability in two-dimensional space.
2.1
Notation and Terminology
Let N = {1, 2, . . . , N } denote the set of sensors. Any subset S ⊂ N is called a coalition. Those subsets which only contain one sensor are called singleton coalitions, i.e. {i}. The set of all coalitions is denoted by C and is called the coalition structure and the set of all possible coalition structures is denoted by C. Finally, the set of all nonempty coalitions, i.e. 2N \{∅}, is denoted by P.
2.2
Measurement Model
We start with an abstract measurement model. Consider a network comprising of N nodes. Let the L dimensional vector p denote the unknown parameter that the network aims to estimate. Each node i records a measurement zi = hi (p) + vi
(2)
Here, zi ∈ RD is the measurement vector, hi is an arbitrary vector-valued but differentiable function of p, and vi ’s are mutually independent Gaussian random vectors with zero mean and covariance matrix Ri . In the following formulation, the set of measurements (zi ’s) collected by a coalition of sensors is denoted by Z. If p is regarded as a non-random parameter, the CRLB ˆ , the establishes a lower bound on the covariance of p ˆ , there exists estimate of p. If C denotes covariance of p J given by J(τ ) = Ep(Z|p) {[p ln(p(Z|p))]T [p ln(p(Z|p))]} (3) such that C − J−1 is positive semi-definite.
In the equation given in (3), J is the Fisher Information Matrix (FIM) of the estimated parameter p. In general, direct inversion of the FIM is computationally expensive and gives no insight to the essence of localization. Therefore, stochastic observability (the determinant of this matrix) is used as a measure for estimation accuracy which can be attributed to how accurate an estimate is by noting that it determines the volume of the 1 − σ confidence ellipsoid around the estimate [9]. Proposition 1. In a coalition comprising of N sensors, based on the measurement model given in (2), J can be expressed as J(τ ) =
N
[p hi ]T R−1 i [p hi ].
(4)
i=1
Target Localization in Two-Dimensional Space: In two dimensions, we have hi = θi , L = 2 and D = 1. We assume that each sensor records a number of measurements proportional to its active time. Therefore, the estimated DOA θˆi is related to the true DOA θ˜i via θˆi = θ˜i + ηi
(5)
where ηi ∼ N (0, σi2 ) denotes the error in the estimation of the bearing for sensor i in the interval of interest. In order to take the sleep time of each sensor into consideration, we assume that σi2 ∝
1 f (T − τi )
(6)
where f (·) is an increasing function and T denotes the length of the decision epoch. This means that the variance of estimation error for each node is inversely related to the active time of each sensor through an increasing function and can be verified by the law of large numbers. If the target is stationary, θˆi is a function of the position of the ith node relative to the target. Given that the true position of the target and the position of the ith node are [xt , yt ] and [xi , yi ], respectively, the true DOA is given by yt − yi ) (7) θˆi = arctan( xt − xi In this case, J(τ ) can be expressed as N − sin(θi ) cos(θi ) sin2 (θi ) αf (T − τi ) J= ri2 σi2 cos2 (θi ) − sin(θi ) cos(θi ) i=1 (8) where ri = (xt − xi )2 + (yt − yi )2 is the relative distance of the ith sensor from the target and α is the proportionality constant in (6). Proposition 2. In two-dimensional space, using the measurement model given in (5), determinant of J(τ )
673
for a coalition comprising of N nodes can be expressed as N N gij (9) det(J(τ )) = i=1 j=1
2
α gij = 2
2.3
f (T − τi ) ri2 σi2
f (T − τj ) rj2 σj2
sin2 (θi − θj ) (10)
Relevant Concepts from Cooperative Game Theory
In this subsection, we formulate the sleep time allocation problem as a cooperative game and state some basic concepts and definitions from cooperative game theory. The motivation behind this formulation is to allow a set of nodes to group into one body in order to consume the energy resources in the most efficient way. As a means to solve this problem, we use the concept of the core. A cooperative game with transferable utility (TU game) is defined by the set of sensor nodes N and a real-valued characteristic function v : P −→ R. In our problem, the characteristic function gives the total sleep time that can be gained by a coalition as a result of sensors cooperation and the payoff for each sensor τi is the share that it demands from the coalition it belongs to. A vector of payoffs, denoted by τ = (τ1 , . . . , τn ), is called an allocation. Suppose δ is the smallest time unit that each sensor can practically be in the sleep mode. Then, each sensor’s demand is restricted to the integer multiples of δ in the closed interval [v({i}), T − δ]. This set is denoted by Di . n R is called feasible if we have An allocation τ ∈ i∈N τi ≤ maxC∈C S∈C v(S) and is called efficient if equality holds. Furthermore, assume that an allocation τ has been proposed by the sensors. If a group of sensors can form a coalition which provides its members a larger sleep time, this coalition will block the proposal. In other words, a coalition S will block an allocation if we have i∈S τi < v(S). Definition 1. An allocation τ is called a core allocation if it is feasible and is not blocked by any S ∈ P. Formally, τ ∈ Rn is a core allocation if it satisfies the following conditions: i∈N
τi ≤ max C⊂C
S∈C
v(S)
Dynamic Coalition Formation for Efficient Sleep Time Allocation in Wireless Sensor Networks Using Cooperative Game Theory Omid Namvar Gharehshiran, Vikram Krishnamurthy Department of Electrical and Computer Engineering University of British Columbia Vancouver, BC, Canada V6T 1Z4 {omidn, vikramk}@ece.ubc.ca Abstract – This paper proposes a dynamic coalition formation algorithm for efficient sleep time allocation in a wireless sensor network (WSN) deployed to localize targets based on cooperation among the nodes. The sleep time allocation problem is formulated as a non-convex cooperative game and the concept of the core is exploited to solve this problem. In this formulation, determinant of the Cram´er-Rao lower bound (CRLB) is used to evaluate the accuracy of estimations. Finally, an algorithm is proposed based on a best-reply rule which converges to the core with probability one if the core of the game is non-empty. Keywords: Cooperative game theory, dynamic coalition formation, localization and tracking, wireless sensor network (WSN).
1
Introduction
A WSN is a distributed embedded system consisting of a large number of low-cost, low-power and energyconstrained sensor nodes. These nodes communicate over a wireless channel, performing distributed sensing and collaborative data processing tasks for various vital military and civilian applications. In this paper, we consider the problem of using a WSN for localization of a target based on noisy bearing (angle) measurements at individual nodes. Since estimating the position of a target in two dimensions needs at least two angle measurements (to perform triangularization), it is natural for the sensors to cooperate. In theory, the best estimate is obtained by employing every node in the WSN to collect and share direction-of-arrival (DOA) measurements. However, due to battery life constraints, it is acceptable to permit a slight increase in the position error by letting some nodes go to sleep. Our main goal is to find efficient sleep time allocations so that targets are localized with an acceptable accuracy, at the same time battery resources being used as efficient as possible. As is commonly used in the tracking literature (e.g. [1] and [2]), we use the determinant of CRLB as the measure of estimation accuracy in order to derive
978-0-9824438-0-4 ©2009 ISIF
the characteristic function for the sleep time allocation game. We refer to this measure as stochastic observability. Main Results: We use a dynamic coalition formation algorithm for allocating sleep times to the sensors. Localization is essentially achieved by cooperation among the nodes. It is reasonable to allocate more sleep times to sensors in a coalition which provides more information about the target’s location. To quantify the amount of sleep time that can be allocated to a coalition of nodes, we derive a characteristic function based on the stochastic observability required in localizing each target and put the required constraints on it to fit in the framework of cooperative game theory. Finally, we propose an algorithm which relies on the concept of the core in order to find a solution for the sleep time allocation game. Related Work: Minimizing energy consumption in WSNs has been a central topic in recent years. Effective techniques have been proposed to put sensors into a low-power or sleep mode while maintaining full coverage at a specified degree of redundancy (e.g. [3], [4] and [5]). In [6], a cluster-based sensor network is considered, wherein the sensor nodes communicate directly with the cluster head. It was shown that the nodes which are farther from the cluster head consume more power, and hence it was suggested that they be put to sleep more frequently. There also has been much work on sensor activation using non-cooperative game theory, e.g. [7] and [8]. In the context of these related works, we should emphasize that our attention is primarily focused on efficient sleep time allocation in cooperative target localization without any power control or synchronization mechanism. Our interest is evaluating the most efficient amount of sleep time that each sensor can achieve based on the relative configuration of the nodes and targets and under constraints on localization accuracy. Organization: The rest of this paper is organized as follows. In Sec. 2, the sleep time allocation problem is formulated and relative material from game theory is provided. Sec. 3 incorporates an overview to the solution and the proposed algorithm. In Sec. 4, two numerical ex-
672
amples are given. Finally, concluding remarks are made in Sec. 5.
2
Problem Formulation
The problem of prolonging the lifetime of a WSN can be formulated as a maximization problem for the sleep time of each node under constraints on stochastic observability. Formally, the problem can be stated as maximize τi ∈Di
subject to
τi
∀i ∈ N
¯ det(J(τ )) ≥ O(k) ∀k ∈ K
(1)
where • N = {1, 2, . . . , N } is the set of sensor indices. • K = {1, 2, . . . , K} is the set of target indices. • τ = (τ1 , τ2 , . . . , τN ) is the vector of sleep times. • Di is the set of allocations possible for sensor i. ¯ • O(k) is the required accuracy in localizing target k. In this section, notations and measurement model are described. Some relevant subjects from game theory are introduced. Furthermore, DOA-based localization in two-dimensional space is formulated. Finally, the characteristic function is derived based on the expression given for stochastic observability in two-dimensional space.
2.1
Notation and Terminology
Let N = {1, 2, . . . , N } denote the set of sensors. Any subset S ⊂ N is called a coalition. Those subsets which only contain one sensor are called singleton coalitions, i.e. {i}. The set of all coalitions is denoted by C and is called the coalition structure and the set of all possible coalition structures is denoted by C. Finally, the set of all nonempty coalitions, i.e. 2N \{∅}, is denoted by P.
2.2
Measurement Model
We start with an abstract measurement model. Consider a network comprising of N nodes. Let the L dimensional vector p denote the unknown parameter that the network aims to estimate. Each node i records a measurement zi = hi (p) + vi
(2)
Here, zi ∈ RD is the measurement vector, hi is an arbitrary vector-valued but differentiable function of p, and vi ’s are mutually independent Gaussian random vectors with zero mean and covariance matrix Ri . In the following formulation, the set of measurements (zi ’s) collected by a coalition of sensors is denoted by Z. If p is regarded as a non-random parameter, the CRLB ˆ , the establishes a lower bound on the covariance of p ˆ , there exists estimate of p. If C denotes covariance of p J given by J(τ ) = Ep(Z|p) {[p ln(p(Z|p))]T [p ln(p(Z|p))]} (3) such that C − J−1 is positive semi-definite.
In the equation given in (3), J is the Fisher Information Matrix (FIM) of the estimated parameter p. In general, direct inversion of the FIM is computationally expensive and gives no insight to the essence of localization. Therefore, stochastic observability (the determinant of this matrix) is used as a measure for estimation accuracy which can be attributed to how accurate an estimate is by noting that it determines the volume of the 1 − σ confidence ellipsoid around the estimate [9]. Proposition 1. In a coalition comprising of N sensors, based on the measurement model given in (2), J can be expressed as J(τ ) =
N
[p hi ]T R−1 i [p hi ].
(4)
i=1
Target Localization in Two-Dimensional Space: In two dimensions, we have hi = θi , L = 2 and D = 1. We assume that each sensor records a number of measurements proportional to its active time. Therefore, the estimated DOA θˆi is related to the true DOA θ˜i via θˆi = θ˜i + ηi
(5)
where ηi ∼ N (0, σi2 ) denotes the error in the estimation of the bearing for sensor i in the interval of interest. In order to take the sleep time of each sensor into consideration, we assume that σi2 ∝
1 f (T − τi )
(6)
where f (·) is an increasing function and T denotes the length of the decision epoch. This means that the variance of estimation error for each node is inversely related to the active time of each sensor through an increasing function and can be verified by the law of large numbers. If the target is stationary, θˆi is a function of the position of the ith node relative to the target. Given that the true position of the target and the position of the ith node are [xt , yt ] and [xi , yi ], respectively, the true DOA is given by yt − yi ) (7) θˆi = arctan( xt − xi In this case, J(τ ) can be expressed as N − sin(θi ) cos(θi ) sin2 (θi ) αf (T − τi ) J= ri2 σi2 cos2 (θi ) − sin(θi ) cos(θi ) i=1 (8) where ri = (xt − xi )2 + (yt − yi )2 is the relative distance of the ith sensor from the target and α is the proportionality constant in (6). Proposition 2. In two-dimensional space, using the measurement model given in (5), determinant of J(τ )
673
for a coalition comprising of N nodes can be expressed as N N gij (9) det(J(τ )) = i=1 j=1
2
α gij = 2
2.3
f (T − τi ) ri2 σi2
f (T − τj ) rj2 σj2
sin2 (θi − θj ) (10)
Relevant Concepts from Cooperative Game Theory
In this subsection, we formulate the sleep time allocation problem as a cooperative game and state some basic concepts and definitions from cooperative game theory. The motivation behind this formulation is to allow a set of nodes to group into one body in order to consume the energy resources in the most efficient way. As a means to solve this problem, we use the concept of the core. A cooperative game with transferable utility (TU game) is defined by the set of sensor nodes N and a real-valued characteristic function v : P −→ R. In our problem, the characteristic function gives the total sleep time that can be gained by a coalition as a result of sensors cooperation and the payoff for each sensor τi is the share that it demands from the coalition it belongs to. A vector of payoffs, denoted by τ = (τ1 , . . . , τn ), is called an allocation. Suppose δ is the smallest time unit that each sensor can practically be in the sleep mode. Then, each sensor’s demand is restricted to the integer multiples of δ in the closed interval [v({i}), T − δ]. This set is denoted by Di . n R is called feasible if we have An allocation τ ∈ i∈N τi ≤ maxC∈C S∈C v(S) and is called efficient if equality holds. Furthermore, assume that an allocation τ has been proposed by the sensors. If a group of sensors can form a coalition which provides its members a larger sleep time, this coalition will block the proposal. In other words, a coalition S will block an allocation if we have i∈S τi < v(S). Definition 1. An allocation τ is called a core allocation if it is feasible and is not blocked by any S ∈ P. Formally, τ ∈ Rn is a core allocation if it satisfies the following conditions: i∈N
τi ≤ max C⊂C
S∈C
v(S)
