distributed information fusion with intermittent observations

International Journal of InnovativeComputing, Information and Control ICIC International c⃝2011 ISSN 1349-4198Volume 7, Number 11, November 2011 pp. 6437–6451

DISTRIBUTED INFORMATION FUSION WITH INTERMITTENTOBSERVATIONS FOR LARGE-SCALE SENSOR NETWORKS

Du Yong Kim1, Ju Hong Yoon1, Moongu Jeon1,∗ and Vladimir Shin2,∗

1School of Information and CommunicationsGwangju Institute of Science and Technology

261 Cheomdan, Gwagiro, Gwangju, Republic of Korea{duyong; jhyoon }@gist.ac.kr

2Department of Information and StatisticsGyungsang National University

900 Gajwa-Dong, Jinju 660-701, South Korea∗Corresponding authors: shin [email protected]; [email protected]

Received July 2010; revised November 2010

Abstract. In this paper, we present a robust distributed fusion algorithm to handleintermittent observations via an interacting multiple model (IMM) and a sliding windowstrategy which is applied to large-scale sensor networks. Intermittent observations arefrequently occurred in practice especially when the scale of network becomes larger andsensors are dynamically connected. To solve the problem, we model the communicationchannel as a jump Markov system and a posterior probability distribution of communi-cation channel characteristics is calculated and incorporated into the filter. By doingso, the distributed Kalman filtering can automatically handle the intermittent observa-tion situations. For the implementation of the distributed fusion, a Kalman-Consensusfilter (KCF) is adopted to provide the average consensus based on the estimates of dis-tributed sensors over a large-scale sensor network. In addition, the algorithm is extendedto nonlinear systems so as to be implemented for more general dynamic systems. Theadvantages of proposed algorithm are subsequently verified from target tracking examplesfor a large-scale network with intermittent observations.Keywords: Kalman filtering, Distributed fusion, Intermittent observation, Nonlinearsystems

1. Introduction. In the literature, distributed computing has been a natural break-through for many engineering problems due to its scalability, efficiency and reliability.Distributed computing has been employed in many disciplines, such as data fusion insensor networks [1,3], distributed camera networks [2,13] and mobile robotics [9]. His-torically, the development of distributed signal processing algorithms is not a new topic.It has been investigated over the past decades, and several types of distributed signalprocessing algorithms have been well-known [5,10]. However, further improvements arerequired to satisfy unconstrained real applications, e.g., channel link failure and intermit-tent observations.

Recently, real-world applications of distributed signal processing algorithms have facedpractical issues such as time-varying network topologies and imperfect communicationchannels. As such, network scalability has been discussed in distributed Kalman filteringin an attempt to address issues related to ad-hoc network topologies [4]. To this end, thetopology of a network is understood via algebraic graph theory, with individual networknodes by employing Kalman filter as a micro-filter regarding to the limited communicationbandwidth between neighbouring nodes.

6437

6438 D. Y. KIM, J. H. YOON, M. JEON AND V. SHIN

Another important issue of distributed signal processing implementations is imperfectcommunication channels. Because a huge number of sensors are randomly distributed andcommunicate with each other through wireless channels, communication links occasionallybreak down and become unstable, thereby delaying observations and incurring packetlosses. In previous researches, the communication delay problem has been investigated asan out-of-sequence measurement problem [6].To model an unreliable communication channel, a latent variable for the observation

system is considered. The arrival of the observation is represented by the latent variable;under this formulation, a statistical convergence analysis was performed in [11]. Theintermittent observation is modelled as a conditional probability distribution.

p (vt |γt ) ={

N (0, R) , if γt = 1N (0, σ2I) , if γt = 0

(1)

where vt, γt, R, σ are zero mean white Gaussian observation noise, latent variable, noisecovariance with no loss, and unreliable noise deviation (i.e., σ → ∞ means the absence ofobservation), respectively. According to the conducted research under this formulation,the latent variable γt is assumed to be a Bernoulli process and depending on the state spacemodel there exists a critical arrival probability at which the estimation error covarianceis bounded [11]. They calculate the critical arrival probability to design the networkcontrolled system. Because the algebraic Ricatti equation becomes a stochastic differentialequation, only bound analysis is available. When observation noise is controlled using alatent variable, it is not easy to determine or model the value of the hyper-parameter σthat describes the characteristics of the communication channel.In this paper, rather than designing the hyper-parameter of observation noise, the

characteristics of the communication channel are modelled by using the multiple modeladaptive estimation (MMAE) approach [8]. Thus, instead of designing the critical valueof σ, we propose to directly estimate the state considering a set of observation models(i.e., observation received and loss). By solving MMAE problem, the filter adaptivelyincorporates the current channel status so that accurate estimate is achieved withoutdivergence.To continuously adjust the observation mode switching, we propose two algorithms.

First, an interacting multiple model (IMM) filter is applied to solve the MMAE problem.IMM approach models the candidate observation system models as an ‘on’ and ‘off’ model.Then, mode probability is calculated by online to decide the current mode of observationsystem. Second, by using a sliding window-based approach, the posterior probability oflink failure is calculated and incorporated into the information fusion filtering. The slidingwindow collects the most recent set of observations which are then sequentially processedto calculate the posterior probability of the mode of observation (absence or presence).Finally, we applied two estimators into the distributed Kalman filtering to ensure thatthe intermittent observation situations are effectively handled in a large-sensor network.The novelty of this work is that, to the best of authors’ knowledge, it is the first

attempt to propose a framework that can be implemented in large-scale systems withunreliable channels due to the unconstrained environment. There have been probabilisticdiscussions about the effect of unreliable channel characteristic in [11,16]. However, workin [11] requires more elaborate and offline models and there is no consideration about thedesign of adaptive estimator in [16]. Moreover, the estimator in proposed approach canautomatically adjust the uncertain communication channels of large-scale sensor networks.The remainder of paper is organized as follows. Section 2 explains the problem formu-

lation. Preliminaries including information fusion filtering and MMAE are then given inSection 3. We provide details of the proposed algorithm in Section 4 and the extension to

DISTRIBUTED INFORMATION FUSION 6439

nonlinear systems. The evaluations of advantages of the proposed algorithm are providedin Section 5. Finally we conclude this paper in Section 6.

2. Problem Statement. Consider the discrete-time dynamical linear system:

xt+1 = Atxt + wt

yit = C itxt + vit, t = 0, 1, . . . , i = 1, . . . , N

(2)

where At ∈ ℜn×n is the system matrix, Cit ∈ ℜm×n is the observation matrix for the

‘i’th sensor among N sensors, xt ∈ ℜn is the state vector, yit ∈ ℜm is the output vector(observation) of the ‘i’th sensor in the network, and wt ∈ ℜn and vit ∈ ℜm are Gaussianrandom vectors with zero mean and covariance Q ≥ 0 and Ri > 0, respectively. Inaddition, wt is independent of ws for s < t, and the initial state vector x0 is also assumedto be Gaussian with zero mean and covariance P0.

Here, the main goal is to obtain an accurate estimate given multiple observations inminimum mean square error (MMSE) criterion, i.e., E

(xt

∣∣y1t , . . . , yNt ), under an unreli-

able communication channel. To achieve the goal, we have two issues; fusion problem andintermittent observation problem in large-scale sensor networks.

Fusion problem under multiple sensory systems has been discussed in many ways. Inrelatively small networks, a central fusion scheme is used. The central fusion schemedirectly collects measurements from all the linked sensors and the filter processes thecollected measurements as one matrix. When the scale of networks becomes larger andsensors are dynamically connected, however, the central fusion scheme is not efficient.That is because a large amount of data and complicated communication problems shouldbe solved at once in the fusion centre. Due to the scalability and large amount of data, de-centralized or distributed fusion architectures are proposed and implemented [3,5]. Amongmany approaches we adapt a distributed Kalman filtering approach proposed in [4] to dealwith a large-scale sensor network problem. Here, dynamic connections between sensorsare understood by using an algebraic graph theory.

We regard another issue, the intermittent observation problem, as an estimation prob-lem of channel characteristic. Then, the filter is proposed to use the estimated status ofchannel to adaptively adjust measurements. Specifically, in the proposed algorithm, weestimate another state referred to as the communication characteristic state θt to handleintermittent observations. With such a framework, each sensor calculates the posteriorprobability of mode p (θt |yit ), then it is incorporated into the distributed Kalman filteringequation as a current status of the communication channel.

3. Preliminaries. In this preliminary section, an information fusion filter [3,5] is intro-duced as a basic tool for fusing distributed sensors over the network. And the multiplemodel adaptive estimation is discussed for use in the mode probability calculation tomanage intermittent observations.

3.1. Information fusion filter. In the centralized fusion set up, the observation systemin (2) can be reformulated into a composite form

Ct =[(C1

t )T, · · · ,

(CN

t

)T]T, vt =

[(v1t )

T, · · · ,

(vNt

)T]TR = diag

{R1, · · · , RN

}, Yt =

[(y1t )

T, · · · ,

(yNt

)T]T (3)

Then, the information form of Kalman filtering equations (information filter) are givenas follows.


Observation Update:

St = CTt R

−1Ct

zt = CTt R

−1Yt

Mt =(P−1t + St

)−1

xt = xt +Mt [zt − Stxt]

(4)

Time Update:

xt+1 = Atxt

Pt+1 = AtMtATt +Q

(5)

where St and zt represent the contribution terms of the state and information. To derivethe decentralized fusion filter, a mathematically equivalent decentralized form of the in-formation filter can then be obtained from the parallelization of the contribution termsas

St = CTt R

−1Ct =∑N

i=1

(C i

t

)T (Ri

)−1C i

t (6)

zt = CTt R

−1Yt =∑N

i=1

(Ci

t

)T (Ri

)−1yit (7)

Therefore, (4)-(7) define the information fusion filter that will be used in the proposedalgorithm for distributed fusion. Because of mathematical equivalence, optimality of thedistributed fusion algorithm is guaranteed.

3.2. Multiple model adaptive estimation. When a system has parametric uncertain-ties it can be modelled with a set of multiple models. A well-known example of multiplemodels in the state estimation is the tracking problem for manoeuvring targets. Targetmanoeuvres have a set of distinctive models, for example, constant velocity, constant ac-celeration, turning motion, etc. By pre-setting a possible set of models, the system isexpected to operate as one of the models. Existing solutions for the MMAE problem areto use a Lainiotis Kalman filter (LKF) [8] in static mode and interacting multiple modelfilter (IMM filter) [7] in dynamic mode switching, respectively. In the multiple modelsetting, the state space model (2) can be represented by considering the mode state θt,such that

xt+1 = Atxt + wt

yit = C it

(θjt)xt + vit

(θjt)

t = 0, 1, . . . , i = 1, . . . , N, j = 1, 2(8)

where θt represents the time-varying model of observation system as

θjt =

{1, j = 1 (signal present)0, j = 2 (signal absent)

(9)

Under this assumption, the multiple model observation system defined in (8)-(9) ba-sically includes the intermittent observation model in (1). Solutions given in LKF andIMM can then utilize the likelihood probability p

(yit∣∣θjt ) to determine the current mode,

which is subsequently used to obtain an accurate state estimate. Given the initial priorprobability of each model p

(θj0), the recursion for posterior probability using Bayes rule

is given as (10)

p(θjt+1

∣∣yit+1

)=

p(yit+1

∣∣θjt+1

)∑2j=1 p

(yit+1

∣∣θjt+1

)p (θjt+1

∣∣yit ) , j = 1, 2 (10)


The likelihood probability p(yit+1

∣∣θjt+1

)can then be calculated from the normalized

residual as

p(yit+1

∣∣θjt+1

)=

∣∣P it+1

(θjt+1

)∣∣−1/2exp

(−(yit+1

(θjt+1

))T (P it+1

(θjt+1

))−1yit+1

(θjt+1

))P it+1

(θjt+1

)= C i

t+1

(θjt+1

)Mt

(C i

t+1

(θjt+1

))T+Ri

yit+1

(θjt+1

)= yit+1 − Ci

t+1

(θjt+1

)xit

(11)

In this paper, the main purpose of the work is to apply MMAE solution in the dis-tributed sensor network to efficiently handle intermittent observations. To adaptively es-timate the state vector with intermittent observation circumstances, an IMM-based KCFand a sliding window-based KCF are proposed, respectively. The IMM-based approachfor intermittent observations can be more effective because IMM is particularly advanta-geous when the observation mode is arbitrarily switched without knowing the switchingtimes. But this approach requires both prior knowledge of the probability of switchingbetween modes and additional computations. In the second approach, we propose com-putationally more efficient approach that the prior knowledge about the transition andadditional computations are avoided by using a sliding window-type algorithm instead ofsacrificing a little accuracy. Details of the proposed algorithms are given in the followingsection.

4. Proposed Algorithm.

4.1. Basic framework. By incorporating IMM and the sliding window approach intothe filtering algorithm to adaptively adjust intermittent observations in the distributedsensor network we propose two algorithms. In the IMM-based approach, intermittentobservations are handled by calculating the mode probability of the observation system(8)-(9) with a special mixing process of the mode probability, estimates, and covariance.

The second approach, the sliding window approach – which considers a recent observa-tion set from the ith sensor yit−∆+1:t =

{yit−∆+1, y

it−∆+2, · · · , yit

}– is proposed to calculate

the mode probability p(θjt

∣∣yit−∆+1:t

), where ∆ is the window length. This approach can

be seen as a modified version of the LKF algorithm.

Let us assume that the mode probability µit

∆= p (θt |yit ) is available then it can be

incorporated into the information fusion filtering equation to handle the intermittentcommunication channels as follows,

Observation Update:

St =∑N

i=1µit

(Ci

t

)T (Ri

)−1Ci

t︸︷︷︸U it

zt =∑N

i=1µit

(C i

t

)T (Ri

)−1yit︸︷︷︸

uit

Mt =(P−1t + St

)−1

xt = xt +Mt [zt − Stxt] (12)

Time Update:

xt+1 = Atxt

Pt+1 = AtMtATt +Q

(13)


Figure 1. A large-scale sensor network with an ad-hoc topology (100 nodes)

Under the modified information filtering framework of (12)-(13), we propose two algo-rithms for distributed Kalman filtering that have intermittent observations by calculatingthe mod probability using IMM and sliding window-based LKFs, respectively.

4.2. Distributed information fusion filtering with intermittent observation viaIMM approach. The information fusion filter has often been used for decentralizedfusion algorithms in sensor networks but can only be used in local sensor nodes in alarge-scale network. The scalability and topology of large-scale networks are not typi-cally considered even though they are crucial factors in real situations. To satisfy theserequirements, the distributed Kalman filtering algorithm was recently proposed that usesa consensus algorithm, referred to as a Kalman consensus filter (KCF) [4].Unlike other data fusion algorithms, there is no fusion centre; insteads individual sensor

nodes calculate their own state estimates and communicate messages (contribution termsand local estimate of each node) to make a global agreement to converge to a certainvalue. As mentioned in the introduction, the scalability and topology of this network isunderstood using algebraic graph theory so that the time-varying topology is consideredin the individual estimator [4].Suppose there is a large scale network with an ad-hoc topology described by the undi-

rected graph G = (V,E) and N nodes. Vertices V = {1, 2, · · · , N} denote the sensornodes and the edges E ⊂ V × V , refer to the communication links between the sensornodes. An example of a large-scale sensor network of randomly distributed sensors is dis-played in Figure 1. Here, the KCF serves as a micro filter of a network that only sharesmessages with its neighbors Li: Ji = Li ∪ {i}.The IMM filter is a well-known method for MMAE problems requiring dynamic mode

switching. Therefore, it is a reasonable solution that the micro-filter in distributed infor-mation filtering is modified via IMM method to handle the intermittent observations. InAlgorithm 1, we provide the proposed IMM-based information fusion filter.As shown in Algorithm 1, every node in the IMM-based AKCF requires all the above

calculations for each mode. In addition, mixing for weight and estimate is needed inorder to adjust the arbitrary switching of modes. Usually in practice, the observationmode switching does not frequently occur, because the loss of packets can be thoughtof as a rare event. When the frequently switching is detected in the linked node, we


Algorithm 1. IMM-based AKCF of node i

Given ,i j

tP,i j

tx , parameter ε , mode probability ( )1 1

j i

t tp yθ − −, and the transition matrix kjπ , between

( )1 1

k i

t tp yθ − −and ( )1 1

j i

t tp yθ − − where 1,2, 1,2.j k= =

1. IMM procedure

A. Predicted mode probability

( ) ( )2

1 1 11

j i j i

t t kj t tkp y p yθ π θ− − −=

=∑

B. Mixing weight

( ) ( ) ( )1 1 1 1 1, /k j i k i j i

t t t kj t t t tp y p y p yθ θ π θ θ− − − − −=

C. Mixing estimate

( )2, ,

1 1 1 1 1 11, ,i j i j i i k k j i

t t t t t t t tkx E x y x p yθ θ θ− − − − − −=

= = ∑

D. Mixing covariance

( )( ) ( )2, , , , , , , ,

1 1 1 1 1 1 1 11ˆ ˆ( ) ,

Ti j i k i j i k i j i k i k i j i

t t t t t t t t tkP P x x x x p yθ θ− − − − − − − −=

= + − −∑

2. KCF procedure

A. Obtain measurement ( ) , 1,..., .i i j i

t t t t ty C x v i Nθ= + =

B. Calculate mode likelihood ( )i j

t tp y θ using (11).

C. Update the mode probability as

( ) ( )( ) ( ) ( )12

11

i j

t tj i j i

t t t ti j j i

t t t tj

p yp y p y

p y p y

θθ θ

θ θ −

−=

=∑

D. Compute the contribution term of information state and matrix such that

( ) ( ) ( ) 1,

Ti j j i i i i

t t t t tu p y C R yθ−

=

( )( ) ( ) 1, .

Ti j j i i i i

t t t t tU p y C R Cθ−

=

E. Broadcast the message ( ), , , ,, , i j i j i j i j

t t t tm u U x= to neighbors in iL .

F. Collect the messages ( ), , , ,, ,r j r j r j r j

t t t tm u U x= from neighbors.

G. Aggregate the information states and matrices of neighbors including node i : { }.i iJ L i= ∪

, , , ,, i i

i j r j i j r j

t t t tr J r Jz u S U

∈ ∈= =∑ ∑

H. Compute the Kalman-Consensus estimate

( )( )( )( )

-11, ,

,

-11

,

, 1,

, 2,

i j i j

t ti j

t

i j

t

P S j

M

P j

−

−

+ == =

,

( ) ( ),

, , , , , , ,

1 ,ˆ

1 i

i j

i j i j i j i i j i j r j i jt

t t t t t t t tr Ji j

t

Mx x M z S x x x

Mε+ ∈

= + − + −+ ∑

I. Update stage

, ,

1

, ,

1 1

,

ˆ

i j i j T

t t t t

i j i j

t t t

P AM A Q

x A x

+

+ +

← +

←


can exclude the node and consider it as a permanent failure. Under this assumption,we subsequently propose the second algorithm to reduce the computational cost and stillshow reasonable performance based on the use of LKF with sliding a window approach.

4.3. Distributed information fusion filtering with intermittent observation viaa sliding window-based LKF. LKF is basically the solution of the MMAE problemfor the static mode case. Therefore, to implement LKF in the dynamic mode switchingcase, the mode probability should be initialized. In the proposed method, we set thesize of window fixed and the window is receding and a set of observation in windowbeing processed to calculate the mode probability at current time. In brief, details of thealgorithm are as follows.Given the initial prior probability, using (10) the mode probability is calculated until it

converges. The initial length of window ∆ is then set as the current time of convergence,and the sliding window that contains the most recent set of observations is created andstarts to calculate mode probability p

(θjt

∣∣yit−∆:t

). Here, it is easy to decide the initial size

of window ∆ by several experiments and we can interpret ∆ as the minimum permissibleswitching duration. For every sliding window, we set the initial probability as,

p(θjt−∆

∣∣yit−∆

)=

{α, j = 1

1− α, j = 2(14)

where α is the prior probability of arrival of observation. Figure 2 illustrates the slidingwindow scheme for calculating the mode probability. In the proposed approach, themode probability asymptotically converges either at the signal presence or signal absencecompared with certain threshold within the window.

Figure 2. Sliding window approach to determine mode probability calcu-lation at individual sensors

It should be noted that the sliding window-based AKCF does not inherently take intoaccount the mixing procedure as in the IMM method so that it is less adaptive whenthe mode is quickly switching. However, it has significant advantages in terms of com-putational time because the filter bank is not deployed. Therefore, the sliding window


Algorithm 2. Sliding window-based AKCF of node i

approach enables moderate observation mode switching, as it monitors the temporal his-tory of the mode probability.

4.4. Computational complexity. The additional mode probability calculation processincreases the computational complexity of the proposed algorithm compared to the clas-sical KCF algorithm. Because the Kalman filtering has o (n3) complexity, the total com-putational complexity of the KCF algorithm is o (n3)×N , where n is the dimension of thestate vector and N is the number of sensors in the network. On the other hand, the com-plexity of the IMM-based AKCF is o (2n3) ×N because two multiple modes (i.e., signalpresent and absent) are separately implemented in parallel for each node. The alternativemethod, the sliding window-based AKCF has almost same complexity compared to theKCF but it is slightly increased due to the computation of the mode probability. It has rel-atively smaller computational burden compared to the IMM-based AKCF because thereis no two parallel filter implementations. Therefore, if we assume that the communica-tion link status is not seriously unstable (i.e., extremely fast observation mode switching)then, we expect the sliding window-based AKCF can be the reasonable alternative for theIMM-based AKCF.

5. Extension to Nonlinear Systems. In many situations, the state dynamic of theobject is not often described by linear systems. Recently, data fusion algorithms based onnonlinear filters are investigated using particle filter [14], and sigma-point filter [15] that


can be classified as nonparametric and parametric, respectively . To extend our frameworkfor nonlinear systems, there are two requirements for nonlinear filters. First, because wehave Gaussian assumption, it should be parametric filter (Kalman filtering). Second, if thefirst is satisfied, it must have the information Kalman form. Unfortunately, sigma-pointKalman filter (known as unscented Kalman filter) does not have the information form; weadopt the sigma-point information Kalman filter (SPIF) introduced in [15]. Because SPIFinherently has the information Kalman form, it can be easily applied to the proposed KCFframework. Thus, we propose to use the SPIF as a micro-filter of the KCF and a newextension is called consensus sigma-point information fusion filter (CSPIF).

Figure 3. Simulated link status of selected sensor nodes in the example(unknown for estimators)

In SPIF, by applying the statistical linearization technique, it successfully obtains thelinearized state space in order to overcome the performance degradation from nonlinearity.Basic sigma-point filters use a set of sigma-points to approximate the nonlinear transfor-mation (nonlinear dynamic system) of Gaussian density. This approximation method iscalled the unscented transformation (UT) and it is used for the main tool for derivation ofsigma-point filters. To apply the sigma-point based nonlinear filter for the information fu-sion, system and measurement model are statistically linearized by using UT. UT is usedto compute the linearization matrices and linearization offset based on the sigma-pointapproach.Let u = g (∗) be a nonlinear function to be linearized as u = g (∗) ≈ Ax+ c. To imple-

ment SPIF, we need coefficient matrix A and vector c for the approximation of nonlineardynamic system and measurement system model. For the detail of the procedure, see thereference [15].


Let nonlinear state dynamic and measurement systems are given as

xt+1 = a (xt) + wt, zt+1 = h (xt) + vt (15)

Then, we obtain the statistically linearized models from UT approximation,

xt+1 = Axt + cxt + wt, zt+1 = Cxt + czt + vt (16)

where A, C, cx and cz are coefficient matrices of statistically linearized dynamic system

and measurement system and offset vector, respectively. w ∼ N(0, Q

)and v ∼ N

(0, R

)are the process noise and measurement noise with linearization noise. All the modelparameters are obtained from the statistical linearization technique. Then, by insertingnew matrices A, C instead of A, C and add offset vectors of cx and cz it is straightforwardto extend our algorithm to nonlinear systems.

Figure 4. Comparison of estimated trajectories (left: ground truth withKCF, right: ground truth with sliding window-based AKCF, blue line:ground truth, dotted red line: estimates, green arrow: starting point)

6. Experiment Results. To validate the advantages of the proposed algorithms andthen compare them, target tracking examples are considered.

6.1. Experiment 1. First, linear dynamic of circular object motion is tested and non-linear system example is followed.

Given the target dynamics of a circular movement

xt+1 = Axt +Bwt

where A0 = 2

[0 −11 0

], B0 = 52I2, A = I2 + εA0 + ε2

2A2

0 + ε2

6A3

0, and B = εB0.

In addition, I2 is a 2 × 2 identity matrix which is a discretized model with a step-sizeε = 0.015, and the initial position and uncertainty are x0 = (15,−10)T , and P0 = 10I2,respectively. A moving target having a circular motion can then be observed via the large-scale sensor network in Figure 1 of 100 sensor nodes. Here, the sensor nodes measure thetarget position with intermittent sensor observations linked to the node, i.e.,

yit = Cit

(θjt)xt + vit, t = 0, 1, . . . , i = 1, . . . , 100, j = 1, 2


where either C it

(θjt)=

{ [1 0

], j = 1[

0 0], j = 2

or C it

(θjt)=

{ [0 1

], j = 1[

0 0], j = 2

.

In the observation model, individual sensor measures either x-position or y-position.For each sensor node we model the channel status variable θjt that describes the signalpresent or absent. To simulate the true observation mode for each node, the data lossin the communication channel link is modeled, i.e., P

(θj=2t

)≡ P (u (0, 1) < 0.01), where

u (0, 1) is a uniform random distribution. We also model the evolution of channel statusvariable as the first Markov chain. The observation noise for each sensor is white Gaussiannoise with vit ∼ N

(0, 302

√i). For the sliding window-based AKCF, the window length ∆

is set to 3. In this example, the target is not fully observable by individual sensors andintermittent observations occur randomly. Figure 3 illustrates the time history of truestatus of links in communication channel of selected node pairs during the experimentationtime. Note that it is not known for designed estimator.

Figure 5. Monte Carlo MSE comparison

From a practical point of view, the arbitrary switching model is reasonable for describingintermittent observations because we really do not know exactly when there are packetlosses in channels. Therefore, in every single experiment, the time history of link statusis randomly generated so that the performance verification is reasonable.In intermittent observation situations, estimated trajectories of the sliding window-

based AKCF and KCF are compared in Figure 4. In this figure, we compare the slidingwindow-based AKCF with KCF because to clearly see that the adaptive estimation of linkstatus in the proposed method is effective. We do not compare the proposed method withexisting works of [11] or [16] because the results in [11] do not consider the design of specificestimator for large-scale network. And [16] requires more parameter and inherently notadaptive estimator. We believe that it is sufficient to show the comparison with the KCFto verify the adaptive channel status consideration.The experimental results clearly show that performance of KCF is seriously degraded

due to the effect of intermittent observations. In contrast to the KCF, the proposed


AKCF adaptively adjusts for intermittent observations, thereby allowing it to accuratelyestimate the object position.

For the quantitative comparison, mean square error (MSE) results obtained from 1000Monte Carlo simulations are illustrated in Figure 5. The same experiment was performedfor three methods: IMM-based AKCF, sliding window-based AKCF, and KCF. Note thatin every Monte Carlo run, distributed sensor networks for different topologies are realizedand communication channel conditions are randomly selected. Figure 5 shows us the MSEbetween the ground truth object trajectory and estimated trajectory for each method. Ascan be seen in Figures 4 and 5, KCF quickly starts to diverge due to the effect from theintermittent observation. The performance degradation of KCF is evident because thereis no consideration of link status of neighboring nodes. On the other hand, two proposedadaptive method successfully handle the intermittent observations from neighboring nodesdue to the estimated channel status online.

Whereas tracking accuracies were easily compared between KCF and sliding windowbased AKCF in Figure 4, the two proposed AKCF algorithms show us almost the sameperformances. This result experimentally proves that even if sliding window has lowcomplexity, there is no significant degradation of performance.

Figure 6. MSE of CSPIF and SPIF from four set of nodes

In summary, conducted MSE comparison confirms that the two proposed algorithms(IMM-based AKCF and sliding window-based AKCF) are robust and accurate againstintermittent observations in a distributed sensor network. In addition, under mild condi-tions (intermittent observation rarely occur), the sliding window-based AKCF efficientlyhandles uncertainty in communication channels with fewer computations required com-pared to the IMM-based approach, and without no loss of advantages. Note that two


proposed algorithms can be used in complementary manner that when switching is fre-quent: IMM-based, and rare: sliding window-based algorithm, respectively. And if theobservation mode switching is too frequent, we immediately regard the link is permanentlyin failure and block.

6.2. Experiment 2. For a nonlinear example, we have a model of moving object withsinusoidal stochastic vector composed of its x- and y-axis position given in (17).

pxk+1 = cos (pxkpyk) + 2 sin (pyk) + wx

k , pyk+1 = pyk sin (pxk) + cos (pxk) sin (p

yk) + wy

k (17)

The position of object is measured by the sensor network of nodes with measurementsystem

zik+1 =

√(pxk − sxi )

2 + (pyk − syi )2 + vik, i = 1, . . . , 100 (18)

where sxi and syi are the ith sensor node location in the network. The initial position[px0 , p

y0] and sensor noise vik are white Gaussian and its variance is proportional to the

distance between object and sensor node location.This example is fully nonlinear where standard Kalman filter is not useful. The perfor-

mance of the proposed algorithm is compared with the estimate of individual SPIF nodewithout communication to show the improvement due to data fusion without degradationfrom nonlinearity.The simulation result of selected nodes are compared by pairs and displayed in Figure 6,

and every sensor node approaches the average consensus and its accuracy is significantlyimproved without degradation not affecting the distributed fusion. The result proves thatthe proposed algorithm handles scalability of the network in the estimation of nonlineardynamic system efficiently even without any global fusion center. Compared to the resultusing particle filter [14], with light computational cost nonlinearity of dynamic systemstate is reasonably handled.

7. Conclusion. In this paper, the state estimation problem for a large-scale sensor net-work with intermittent observations was discussed. Two adaptive algorithms were sub-sequently suggested to alleviate inaccuracies caused by imperfect communication links inthis type of sensor network. Unlike other works, the proposed approach automaticallymanages data loss in the channel without requiring additional indicators. Additionally,the proposed algorithm is extended to nonlinear dynamic systems by designing the micro-filters based on sigma-point information filter (SPIF). From target tracking examples,under the reasonable assumption significant improvements are confirmed and the com-putational complexity is reduced by using an alternative method without degrading theperceived improvements and also effective even for nonlinear systems. To now, there isno rigorous analysis for critical value of switching rate to use the sliding window-basedAKCF. For the future work, we have plans to analyze the theoretical bound and criticalvalue how to intelligently apply the approach without heuristics.

Acknowledgment. This work was partly supported by National Research Foundation ofKorea Grant funded by the Korea Government (No. 2010-001-2932) and from UnmannedTechnology Research Center at KAIST, originally funded by DAPA, ADD.

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