Information space of sensor networks: Lagrangian,energy-momentum tensor, and applications

Mo TAO, Shoping WANG, Hong CHEN, Hn PAN, Jin SHI,Yuwei ZHANG

a School of Automation Science and Electrical Engineering, Beihang University, Beijing 100083, China

b Wuhan Second Ship Design and Research Institute, Wuhan 430205, China

c School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai 200240, China

KEYWORDS Energy-momentum tensor;Information science;Information space;Sensor networks;Target tracking

Abstract It is a challenge to investigate the interrelationship between the geometric structure and performance of sensor networks due to the increasingly complex and diverse architecture of them.This paper presents two new formulations for the information space of sensor networks, including Lagrangian and energy–momentum tensor, which are expected to integrate sensor networks target tracking and performance evaluation from a unified perspective.The proposed method presents two geometric objects to represent the dynamic state and manifold structure of the information space of sensor networks.Based on that, the authors conduct the property analysis and target tracking of sensor networks.To the best of our knowledge, it is the first time to investigate and analyze the information energy–momentum tensor of sensor networks and evaluate the performance of sensor networks in the context of target tracking.Simulations and examples confirm the competitive performance of the proposed method.

1.Introduction

With the rapid development and enormous breakthroughs in sensor technology,sensor networks have been widely and successfully applied to many practical applications, such as remote sensing and environmental monitoring1.Utilizing multi-sensor information is essential for target tracking in sensor networks,which has several advantages:(A)fidelity observations;(B)information processing locally;(C)object tracking robustly.However, characterized by the diversity of multisensor information, richness of unknown target and complexity of sensor network,it is difficult to evaluate the tracking performance of sensor networks.In this context, there is a growing need for reliable performance evaluation on target tracking of sensor networks.Meanwhile, the unique network characteristics present many challenges including dynamic and unreliable environment, random deployment, etc.

Commonly, target tracking aims to estimate or predict the target position in sensor networks.It can be classified into different categories.According to the number of tracking targets,it can be divided into single target tracking2–5and multiple target tracking6–9.Concerning the shape of the target, it can be divided into continuous target tracking10–13and discrete target tracking14–17.Recently, extensive researches of target tracking are surveyed18,19.However,the performance evaluation of target tracking within sensor networks is challenging, owing to topology changes and node failures.Specially, the impact of sensor measurement on target detecting and tracking performance is still unclear.Moreover, it is an urgent need to study problems systematically.

The concept of energy momentum tensor is originated from physics.It is a symmetric second-order tensor.Energy momentum tensor can describe the movement of matter in a system,and has been widely studied in many fields such as electromagnetics20–22, relativistic fluid mechanics23–25, general relativity gravitational field26–28, quantum and particle dynamics29–31,etc.Particularly, in the Einstein field equation, the energy momentum tensor Tabappears at the right side of the equation to represent the material term.With the advancement of information theory, modern control theory and various emerging inter-disciplinary theories, for the intelligent sensors and wireless communication technologies,the concept of energy tensor is not limited to a vision,but also a theoretical tool that can be widely utilized in the future.

Information geometry has proven its effectiveness in discovering and exploiting the local structure of multi-sensor information in sensor networks32,33.Because of the capability to analyze the characteristics of sensor networks from a unified perspective,it has attracted increasing attention.The theory of information geometry is recognized as a complex and powerful tool34,35.In this framework, information is regarded as geometric objects.Then, it can deeply explore the structure and characteristics of sensor network systems.

In view of the above considerations, we propose two new formulations for the information space of sensor networks.First, we construct the Lagrangian of information space of sensor networks to model target’s kinetics.Second, energy–momentum tensor is extended to represent the tracking performance of sensor networks.This formulation fills the gap between information manifold and target tracking performance of sensor networks.The proposed scheme is verified and the contributions of this paper are twofold:

(1) Two new formulations are developed for the information space of sensor networks,which include Lagrangian and energy–momentum tensor.These formulations are presented to represent dynamic state and generalize the structure of space–time information manifold in sensor networks, respectively.

(2) The characteristics of the proposed formulations for sensor networks are investigated and analyzed.We demonstrate the superiority of our framework through numerical results and simulations in a target tracking application.

The remainder of this paper is organized as follows.In Section 2,the information space of sensor networks is introduced,and the Hilbert action and Lagrangian on a manifold are investigated.In Section 3,information energy momentum tensor for sensor networks is explored.The numerical experiments are described in Section 4, and the conclusion is drawn in Section 5.

2.Information space of sensor networks

This section demonstrates the theoretical basis of constructing information space (information manifold) through sensor measurement model, the method of potential function on the manifold, and the representation of Lagrangian and Hilbert action on manifolds.

2.1.Information space and potential function

This subsection presents the information space of sensor networks.As pointed out in Ref.36,the multivariate normal distribution measured by the sensor networks belongs to the exponential family of distributions, which mainly includes Gaussian distribution, Gamma distribution and Poisson distribution.

where ⊕is the direct sum operator, Span() is the span operator,and ?prepresents a lateral vector field.Eq.(4)ensures that the differential structure is immersed completely.The representation of a multivariable Gaussian manifold obtained from the sensor networks can be achieved by affine immersion in Euclidean space Rn+1.

Suppose that S has the Fisher information metric g, and(θ,Ξ) indicates the natural standard coordinate system.The manifold M can be represented in Euclidean space Rn+1by potential function graph.Generally, manifold S can be expressed by affine immersion {f,? }38,39:

2.2.Hilbert action and Lagrangian on manifold

The Hilbert action or the Einstein-Hilbert action proposed in 1915 by David Hilbert is the action that can derive the Einstein’s gravitational field equation in the general theory of relativity (by taking the variation to obtain the motion equation of the space-time metric).There are many advantages for the Einstein gravitational field equation which is derived from the Hilbert action.First,it is simple to unify the general theory of relativity and other classical field theories that are also expressed in the form of actions, such as Maxwell’s theory.Second, by looking for the symmetry contained in this action,the conserved quantity can be easily judged according to Noether theorem.In general relativity, the action is generally regarded as a functional of the metric (and the matter field),and its connection is a Levi-Civita connection.

The Lagrangian of a dynamic system, also known as the Lagrangian function, is a function that describes the dynamic state of the entire physical system.If the Lagrangian of a system is known, the Lagrangian can be directly substituted into the Lagrangian equation, and the equation of motion of this system can be obtained by calculation.

The Lagrangian is extremely important because it can help us understand the physical behavior of a physical system.The physical quantities involved in Lagrangian are kinetic energy and potential energy.The values of these physical quantities will not change with the choice of generalized coordinates.Not only the Lagrangian can be widely used in classical mechanics, but also its expression can be easily extended to other academic fields, such as circuit science, quantum mechanics, particle physics, etc.The Lagrangian also has an excellent property, which means that the law of conservation can be easily found from its expression.

In this paper, the Lagrangian is introduced into the information field to analyze the information system.Some new discoveries and explanations will be obtained.

The complete action form of a field in theory includes the Einstein-Hilbert action and the Lagrangian L that can describe any information field, and then

3.Lagrangian and energy-momentum tensor for information space of sensor networks

For the energy-momentum tensor of sensor networks,it is first necessary to derive the Hilbert action through the potential function and the Lagrangian in the corresponding information space.Then,by variation and derivation of the Hilbert action,the energy momentum tensor of the information space can be carried out.

The following section studies an angle sensor network,and demonstrates how to establish the corresponding information space through the sensor measurement model.Then, we deduce the research process of Lagrangian, Hilbert action and energy-momentum tensor.The derivation results and some properties of the reflected sensor network information space are also analyzed.

3.1.Information space of sensor networks

Fig.1 Spatial position of angle sensor networks and target.

where gαβis the contravariant index of gαβ.For every point in Riemann manifold, its scalar curvature is a specific real number,and the number is decided by the intrinsic geometric structure of manifold that surrounds the point.

Fig.2 Circle formed by information distance in information space.

By calculation, it is found that in the studied information space:

Zero Ricci curvature denotes that the manifold is locally flat everywhere.This kind of manifolds is called Ricci flat manifolds.And the Ricci curvature tensor provides the rate of change of information along a given direction on the manifold, and zero Ricci curvature reflects the rate of isotropic change.

3.2.Potential function of sensor networks

For the sensor network in this study,the measurement error is assumed to be the characteristics of multivariate Gaussian distribution:

Take η1=-10,ξ1=0,η2=10,ξ2=0,σ1=1,σ2=1, and the three-dimensional images of the potential function are obtained.

As shown in Fig.3,the potential function of the two-angle sensor network is symmetric, and it is worth noting that the locations of the sensors are two singularities in the information space.As in the general theory of relativity,if the gravitational source material is relatively concentrated, the energy–momentum tensor of the matter in the so-called vacuum outside the gravitational source is zero, the spacetime at the gravitational source is curved, and the Riemann curvature tensor is impossible to be a zero tensor,but the Ricci tensor,curvature scalar and Einstein tensor here are zero tensors37.And in the threedimensional information space of multi-source information,the‘‘gravitation-generating gravitational source material”is the sensor; there is only one sensor network (a point in the information space)in the infinite information space,which will cause the space there to bend,here the Riemann curvature tensor cannot be a zero tensor,but its Ricci tensor,curvature scalar and Einstein tensor are zero tensors36.

3.3.Lagrangian of sensor networks’information space

The Lagrangian of sensor networks’information space is named after the mathematician and astronomer Joseph Lagrange.The Lagrangian of a dynamic system, also known as the Lagrangian function, is a function that describes the dynamic state of the entire physical system.For a general classical physical system, it is usually defined as kinetic energy minus potential energy, and the equation is expressed as.

where L is the Lagrangian, K is the kinetic energy, and V is the potential energy.

Usually, the parameters of kinetic energy are the generalized velocity ˙q1, ˙q2,..., ˙qN, where the above dot indicates the total derivative with respect to time t, and the parameters of potential energy are the generalized coordinates q1,q2,...,qN;t.Therefore, the parameters of the Lagrangian are q1,q2,...,qN; ˙q1, ˙q2,..., ˙qN;t.To calculate the Lagrangian, a suitable generalized coordinate should be selected first.Assuming that these parameters(generalized coordinates and generalized velocity) are independent of each other, the Lagrangian equation can be used to find the motion equation of the system.

Assuming that the Lagrangian of a physical system is L,the motion of this physical system can be expressed as Lagrangian equation:

where t is the time, qiis the generalized coordinate, and ˙qiis the generalized speed.

Through the study of the above potential function,it can be found that the potential function can transform any point(except the singularity) on the information space into a value, so the following definition is given.

Definition 1.In the information space of angle sensors (M,g),one of the scalar fields fMis taken as Eq.(26).

Fig.3 Potential function of angle sensor network.

Q.E.D.

As for the sensor network composed of two angle sensors shown in Fig.1, its Lagrangian is given by

It is demonstrated in Fig.4 that the Lagrangian of the studied sensor network is symmetric, and the location of the two angle sensors are the singularities in the information space.

3.4.Energy-momentum tensor of sensor networks’information space

In this subsection,a new formulation for information space of sensor networks, i.e., energy–momentum tensor, is proposed.First, we present some fact from Lagrangian of sensor networks’information space.These provide some preparations for energy–momentum tensor.This plays an important role in some geometric arguments.Second, the details of the energy–momentum tensor of the information space of sensor networks are presented.

Fig.4 Lagrangian of angle sensor network’s information space.

The general formula of the energy–momentum tensor divergence of the information space is given in this subsection.

The information Lagrangian derived in this paper is a function that describes the overall state of an information space(or information manifold), which can help researchers to deeply understand the features and behavior of the information space.The information energy–momentum tensor derived from the information Lagrangian is a physical quantity that describes the information distribution in the information space,and represents an intrinsic property of the information space.

Based on the above analysis, this paper takes the information energy–momentum tensor as the expression of the characteristics of the sensor network.The target tracking experiment will be designed through this tensor,and the following conjecture is proposed.Conjecture 1.If the divergence of the energy–momentum tensor in the information space of the sensor network is always zero, there is a conservation quantity or conservation law in the information space.

Fig.5 Energy-momentum tensor T11.

Fig.6 Energy-momentum tensor T22.

Fig.7 Energy-momentum tensors T12 and T21.

4.Experiments

Fig.8 Determinant of energy–momentum tensor.

For target tracking in sensor networks, the challenge is the observability of the target.It can be obtained by the maneuvering of the sensor44,and it is essential to use the relative movement between the target and the sensor to obtain the maximum target observation information.Generally, the movement and state change of the target can be described by the Markov process, and the sensor trajectory scheduling problem could be regarded as a Partially Observable Markov Decision Process(POMDP).The trajectory optimization of the sensor can be realized based on a certain cost function or optimization criterion, and these criteria are usually related to the measured Fisher information or mutual information.The above optimization problem can be transformed into a smooth variational problem.The solution to the resulting problem is given by the corresponding Euler-Lagrange equation.The energy–momentum tensor of the information space of sensor networks provides an alternative way to solve the above problem.The optimal sensor trajectory scheduling research in this section is based on the energy–momentum tensor in the information space.

In order to in depth explain the role of the energy–momentum tensor in the research of information fusion and target tracking, an application scenario is set, as shown in Fig.9.

Fig.9 Angle sensor network and target.

Fig.9 shows a schematic diagram of the single-step optimization problem of trajectory planning for target tracking.The angle sensor at the origin O observes the target at the point P.In order to maximize the acquisition of measurement information, the sensor needs to be maneuvered.Assuming that the single-step maneuvering radius of the sensor is r, the next moment can appear at any point(such as A)on the circle with O as the origin and r as the radius to obtain the measurement of the target,so as to use the two observations before and after to realize the estimation of the target state.Obviously,the different maneuvering angle φ makes the different accuracy of the target state estimation.To obtain the largest amount of information and achieve the highest estimation accuracy, it is necessary to find the optimal maneuvering angle φopt.

We derive the determinant of the energy–momentum tensor to eliminate the item of (η2,ξ2), i.e.

Take η1=0,ξ1=0,x=20,y=0,σ1=1,σ2=1, and the determinant of the energy–momentum tensor is given by

Fig.10 shows the development of the energy–momentum tensor of the information space when the maneuver radius r=5 m is fixed.As the figure demonstrates, the image of the energy–momentum tensor of the information space is symmetrical.At the same time,it is found that the maneuvering angle is not simply the larger the better, or the smaller the better.Then the energy–momentum tensor of the information space changes with the sensor’s maneuvering distance-angle (r,φ ),as indicated in Fig.11.

As the five given maneuvering distances r show in Fig.11,the maximum energy–momentum tensor corresponding to the maximum maneuvering distance r=30 m is not the maximum; the maximum energy–momentum tensor appears in the case of maneuvering distance r=25 m.When the maneuvering distance is fixed as r=5 m, the energy–momentum tensor reaches its maximum value at the maneuvering angle φ=±1.09360861 rad,and the corresponding energy–momentum tensor value is 1.60113563×10-6.

Fig.13 displays the distribution of the determinant of the energy–momentum tensor in the studied information space.It can be seen that the determinant of the energy–momentum tensor at the location of the tracking target is significantly higher than those at other locations.The distribution of the determinant of the energy–momentum tensor in the information space reflects the distribution of the sensors in the information space, or we can say that the distribution of the sensors determines the distribution of the energy–momentum tensor in the corresponding information space.

Table 1 shows how the value of the determinant of the energy–momentum tensor varies with distance-direction(r,φ ).Fig.14 shows the corresponding motion geometry of the sensor under three given radii.According to the energy–momentum tensor measured by the sensor network composed of two angle measurements given by Eq.(43), the energy–momentum tensor of the sensor maneuvering at a given radius and different angles can be calculated.

Fig.10 Energy-momentum tensor of information space(r=5 m).

Fig.11 Change of energy–momentum tensor with sensor’s maneuvering distance-angle (r,φ ).

Fig.12 Change of energy–momentum tensor with maneuvering distance r.

Fig.13 Distribution of the determinant of energy–momentum tensor.

The polar coordinate diagram shown in Fig.15 demonstrates the information gain obtained by maneuvering the sensor to different angles with the given radius.We assume that the sensor is located at the origin of polar coordinates,the target is located in the 0°direction,and the distance from the sensor is d1=20.

Table 1 Max determinant of energy–momentum tensor with r and φ.

Fig.14 Sensor movement geometry under different maneuvering radii.

Fig.14 indicates the information gain obtained by maneuvering the sensor in the direction of 0°–360°when the radius is 5 m,25 m and 40 m respectively.It can be seen from the results in the figure that for each given maneuvering radius,the sensor has two best maneuvering directions (marked by dots in the Fig.14).The sensor maneuvering to these two angles will obtain the maximum energy–momentum tensor.At the same time, due to the symmetry of the sensor’s maneuvering direction, the two best maneuvering angles are symmetrical about the 0°direction,which can be expressed as±φopt.In addition,it can be seen that among the three given radii,the energy–momentum tensor corresponding to the maneuvering radius r of 25 m is the largest.

Remark 2.Thus, we present analytic properties by energy–momentum tensor in the context of information space of sensor networks, i.e.

(1) The energy–momentum tensor first becomes larger and then becomes smaller.

(2) The change of the energy–momentum tensor presents a very strong nonlinearity.

(3) As the maneuvering radius increases, the maneuvering angle passes: large-small-large.

Fig.15 Energy-momentum tensor distribution of information space.

(4) Relatively speaking, the closer the distance to the target is, the larger the energy–momentum tensor is, but the trend is not linear.When r = 25 m and φ = ±0.355951122 rad, the energy–momentum tensor reaches the maximum.

(5) The area of the disc represents the energy–momentum tensor determinant.

(6) When the maneuvering radius r=40 m,the maneuvering angle is close to 60°.

5.Conclusions

In this paper, two new formulations based on the theory of information geometry, i.e., Lagrangian and energy–momentum tensor,are proposed to investigate the manifold structure of the information space of sensor networks.These two formulations allow the proposed method to represent dynamic state and generalize the structure of space–time information manifold,respectively.Their properties are analyzed with complete proofs.The proposed method realizes integrated sensor networks’target tracking and performance evaluation from a unified perspective of information geometry, and is validated in application of target tracking in sensor networks and performance analysis.Numerical results from the target tracking experiments of sensor networks show the effectiveness of the proposed method.This study could be applied to optimize the trajectory of mobile sensors such as robots, or for fixed sensor networks to maximize the information obtained during the task.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgement

This work was supported by the National Natural Science Foundation of China (No.51875014).