Embed Size (px)
Citation preview
IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 23, NO. 8, AUGUST 2012 1279
Spatial Gaussian Process RegressionWith Mobile Sensor Networks
Dongbing Gu, Senior Member, IEEE, and Huosheng Hu, Senior Member, IEEE
Abstract— This paper presents a method of using Gaussianprocess regression to model spatial functions for mobile wirelesssensor networks. A distributed Gaussian process regression(DGPR) approach is developed by using a sparse Gaussianprocess regression method and a compactly supported covariancefunction. The resultant formulation of the DGPR approach onlyrequires neighbor-to-neighbor communication, which enableseach sensor node within a network to produce the regressionresult independently. The collective motion control is imple-mented by using a locational optimization algorithm, whichutilizes the information entropy from the DGPR result. Thecollective mobility of sensor networks plus the online learningcapability of the DGPR approach also enables the mobile sensornetwork to adapt to spatiotemporal functions. Simulation resultsare provided to show the performance of the proposed approachin modeling stationary spatial functions and spatiotemporalfunctions.
Index Terms— Coverage control, Gaussian process regression(GPR), mobile sensor networks, spatiotemporal modeling.
I. INTRODUCTION
EENVIRONMENTAL surveillance in many areas, such asmeteorology, climatology, ecology, oceanography, etc.,
requires the capability of modeling spatial functions, or evenspatiotemporal functions. The distributed nature and the col-lective mobility of mobile sensor networks offer a feasiblesolution to environmental surveillance applications by meansof collectively sampling data and cooperatively modelingspatial functions. Several research projects have targeted to thisresearch area, such as monitoring forest fires using unmannedaerial vehicles (UAVs) in [1], monitoring air quality usingUAVs in [2], monitoring ocean ecology conditions usingunmanned underwater vehicles in [3].
A mobile sensor network is able to make sensory obser-vations of environmental physical phenomena with on-boardsensors, exchange information via on-board wireless commu-nication devices, and explore over the area of interest with themobility. As a result, it is able to produce a predictive modelfor the sampled physical phenomenon and allocate itself in apattern which can generate a more accurate predictive model.
Manuscript received January 17, 2011; revised May 14, 2012; acceptedMay 15, 2012. Date of publication June 15, 2012; date of current versionJuly 16, 2012. This work was supported in part by the EU FP7 Program,ICT-231646, and SHOAL.
The authors are with the School of Computer Science and ElectronicEngineering, University of Essex, Essex CO4 3SQ, U.K. (e-mail:[email protected]; [email protected]).
Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TNNLS.2012.2200694
The strategies of cooperative modeling and coordinated motionare the key techniques to be developed.
Gaussian process regression (GPR), also known asthe Kriging filter, is a popular regression technique for dataassimilation. In recent years, there has been a growing interestin Gaussian processes for regression and classification tasks[4]–[6]. A GPR is specified by a predictive mean function, apredictive covariance function, and a set of hyperparameterswhich can be determined from sampled data [7]– [9]. Themain advantage of the GPR over other regression techniquesis its ability to predict not only the mean function of theGaussian process (GP), but also the covariance function ofthe GP. The predicted uncertainty is valuable informationfor further decision-making in environmental surveillanceapplications. In [10], GPR was applied for monitoring theecological condition of a river for individual sensors. Thesensor placement was determined by maximizing a mutualinformation gain. In [11], the computation complexity ofthe GPR was reduced by a Bayesian Monte Carlo approach,and an information entropy was used to allocate individualsensors. In [12], a mixture of GPs was applied for buildinga gas distribution system, with the aim of reducing thecomputation complexity. In [13], a Kalman filter was builton the top of a GP model to characterize spatiotemporalfunctions. A path planning problem was solved by optimizinga mutual information gain. In [14], a Kriged Kalman filterwas developed to reconstruct spatiotemporal functions and acentroidal Voronoi tessellation (CVT) algorithm was employedto allocate mobile sensor nodes. The Kriged Kalman filter andswarm control were used in [15] to reconstruct spatiotemporalfunctions. In [16], several nonseparable spatiotemporal covari-ance functions were proposed for modeling spatiotemporalfunctions.
Environmental spatial functions have been modeled inmobile sensor networks by using radial basis function (RBF)networks in [17] where a CVT coverage control was used toallocate mobile sensor nodes, and in [18] where a flockingcontrol was used to move sensor nodes. The environmentalspatial function was approximated by using an inverse dis-tance weighting interpolation method and updated by using aKalman filter in [19]. In the function space view of GPR [7],RBF is regarded as a truncated GP model where a limitednumber of basis functions are used.
A centralized GPR can be implemented in a sensor networkby sending data from all sensor nodes to a centralized unitwhere the computation of the predictive mean function and thecovariance function is conducted. For making effective motion
2162–237X/$31.00 © 2012 IEEE
1280 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 23, NO. 8, AUGUST 2012
decisions, the predictive results need to be sent back to eachsensor node. Thus this version of GPR could cause a severeproblem for wireless communication. It is also not robustto the failure of individual nodes. Further, it does not scalewell with the size of networks, as GPR is a nonparametricregression technique and its computational complexity growsin the order of O(N3) with the size N of sampled data. Inthe GPR research community, various sparse Gaussian processregression (SGPR) techniques have been proposed to alleviatethe computation complexity problem [9], [20]. They are basedon a small set of active points which are projected from thesampled data according to variant projection strategies underdistribution approximation assumptions.
In this paper, an SGPR strategy, called the projected processapproximation in [21] and [22], is used to develop a distributedGaussian process regression (DGPR). In this proposed DGPRapproach, the active point set of the SGPR corresponds tothe neighbor set definition of sensor networks. It also usesa compactly supported covariance function to decouple thecomputation in a local node from the contribution of remotenodes. A compactly supported covariance function, called theWu’s polynomial positive-definite function [23], is adopted inthis paper, as its analytic derivative form is available for thehyperparameter learning. The hyperparameters of covariancefunction are learned by maximizing the marginal likelihoodof the sampled data. The proposed DGPR approach canmake this learning algorithm a distributed algorithm so thata mobile sensor network can update the predictive mean func-tion and covariance function online. This feature extends theproposed DGPR approach from modeling spatial functions tomodeling spatiotemporal functions. Because of its distributednature, it is robust against the failure of individual nodes andscales well with the size of networks. The collective motionstrategy is based on the predictive result of the GP. The CVTalgorithm is adopted for motion coordination. However, theutility function of CVT algorithm is not defined as a meanfunction as others. It is defined as an information entropy. Useof the predictive mean utility function results in the sensornodes moving to the locations where high mean values arepredicted. This could lead to the sensor nodes clustering onsome locations where the predictions are very certain whilesome locations where the predictions are poor are unobserved.With the information entropy utility function, the sensor nodespotentially move to the locations where high uncertaintiesare predicted and then make observations there to reduce theuncertainties.
In summary, this paper contributes to the development of anovel DGPR algorithm which is built on an SGPR strategycombined with a compactly supported covariance function.Given the distributed implementation of GPR, informationentropy is available for the decision-making in collectivemotion control process for each sensor node by using informa-tion entropy as the utility function in the CVT algorithm. In thefollowing, Section II revisits some basics of GPR and presentsthe SGPR strategy within the framework of a mobile sensornetwork. The compactly supported covariance function and theproposed DGPR are detailed in Section III. The informationentropy-based coverage control is introduced in Section IV.
Section V provides the simulation results. Our conclusion andfuture work are given in Section VI.
II. GPR
A mobile sensor network with N sensors is to be deployedin a 2-D area Q to model a scalar environmental spatialfunction in that area. The sensor node i is located at a 2-Dposition xi,t and it is assumed that the position xi,t canbe found by itself with self-localization techniques at timestep t . Each sensor node i can make a point observation zi,t
of an environmental spatial function f (xi,t ) at time step t .The sensory observation distribution is assumed to be Gaussian
zi,t = f (xi,t ) + εi,t
where εi,t is a Gaussian noise with zero mean and covarianceσ 2
i,t , noted as εi,t = N (0, σ 2i,t ).
In this section, it is assumed that each sensor node cancollect the location information x j,t and the correspondingobservation z j,t from all the other sensor nodes via wire-less communication. The distributed implementation will bedetailed in the next section.
A. GPR
In a sensor node, the Gaussian inference is conducted ateach time step based on the information available at thatmoment. The given information in a sensor node includes theinput vectors X N,t = [x1,t , . . . , xN,t ]T and the correspondingobservations zN,t = [z1,t , . . . , zN,t ]T .
In a GP model, the prior distribution of latent variable fi,t =f (xi,t ) is modeled as a Gaussian. Its mean value is assumedto be zero because offsets and simple trends can be subtractedout first. The prior knowledge about multiple latent variables ismodeled by a covariance matrix KN N = [k(xi,t , x j,t )], wherek(xi,t , x j,t ) is a covariance function. With a positive-definitecovariance matrix KN N , the GP prior distribution of full latentvariables fN,t = [ f1,t , . . . , fN,t ]T is represented as
p(fN,t ) = N (0, KN N ).
The likelihood distribution of the observation vector zN,t isrepresented as
p(zN,t |fN,t ) = N (fN,t , RN,t ) (1)
where RN,t = diag(σ 21,t , . . . , σ
2N,t ).
GPR can infer f∗,t = f (x∗,t) for a test point x∗,t ∈ Qusing p( f∗,t |zN,t ) given a training data set (X N,t , zN,t ). Thelatent predictive distribution of a given test point is obtainedby solving the maximum a posteriori problem and is given by
p( f∗,t |zN,t ) = N (μ∗,t ,�∗,t ) (2)
where the predictive mean function and the predictive covari-ance function are
μ∗,t = K∗N (KN N + RN,t )−1zN,t
�∗,t = K∗∗ − K∗N (KN N + RN,t )−1KN∗ (3)
where K∗N = [k(x∗,t , x1,t ), . . . , k(x∗,t , xN,t )] andK∗∗ = k(x∗,t , x∗,t ). KN∗ = K T∗N for symmetrical covariancefunctions.
GU AND HU: SPATIAL GAUSSIAN PROCESS REGRESSION WITH MOBILE SENSOR NETWORKS 1281
B. SGPR
For agent i , the full dataset is separated into a neighborset Ni which includes i itself, and a nonneighbor set N̄i . Theneighbor set Ni is viewed as the active set and the nonneighborset N̄i is viewed as the remaining set of the SGPR withprojected process approximation [22].
In a GPR, the prior distribution of full latent variables canbe rewritten into a form with the separate sets as follows:
p(fNi ,t , fN̄i ,t) = N
(0,
[KNi Ni KNi N̄i
KN̄i NiKN̄i N̄i
]). (4)
From above, the conditional distribution can be found as
p(fN̄i ,t|fNi ,t ) = N
(KN̄i Ni
K −1Ni Ni
fNi ,t
× KN̄i N̄i− KN̄i Ni
K −1Ni Ni
KNi N̄i
). (5)
However in an SGPR, the prior distribution of the full latentvariables in (4) is not available. Only the prior distribution oflatent variables in the active set Ni is assumed to be Gaussian
p(fNi ,t ) = N (0, KNi ,Ni ). (6)
Without the prior distribution of full latent variables in (4),the conditional distribution in (5) cannot be directly obtained.There are several approximation approaches available. Amongthem, the projected process approximation approach is aneffective way (see details in [9]).
The SGPR with projected process approximation isrecovered in a unifying framework of SGPR in [20], wherethe SGPR is deduced by using two GPRs. In the first GPR,the conditional distribution in (5) is approximated by a deltafunction which keeps the mean unchanged
p(fN̄i ,t|fNi ,t ) = N (KN̄i Ni
K −1Ni Ni
fNi ,t , 0). (7)
This approximation, combined with the fact that
p(fNi ,t |fNi ,t ) = N (fNi ,t , 0)
= N (KNi Ni K −1Ni Ni
fNi ,t , 0) (8)
contributes a conditional distribution of full latent variables
p(fN,t |fNi ,t ) = N (KN Ni K −1Ni Ni
fNi ,t , 0) (9)
where KNi N = [KNi Ni KNi N̄i] and KN Ni = K T
Ni N .Given the prior distribution of latent variables in neighbor
set (6), the joint distribution of latent variable with a test pointf∗i,t is known as follows:
p(fNi ,t , f∗i,t ) = N(
0,
[KNi Ni KNi ∗
K∗Ni K∗∗
]). (10)
From above joint distribution in (10), the conditional distrib-ution of a testing point in the first GPR is obtained as
p( f∗i,t |fNi ,t ) = N(
K∗Ni K −1Ni Ni
fNi ,t , K∗∗
− K∗Ni K −1Ni Ni
KNi ∗). (11)
Given the latent variables of a neighbor set, it is assumedthat fN,t and f∗i,t are independent. Then the effective prior
distribution is given by marginalizing out the neighbor latentvariables
p(fN,t , f∗i,t ) =∫
p(fN,t |fNi ,t )p( f∗i,t |fNi ,t )p(fNi ,t )dfNi ,t
= N(
0,
[QN N QN∗Q∗N K∗∗
])(12)
where
QN N = KN Ni K −1Ni Ni
KNi N
QN∗ = KN Ni K −1Ni Ni
KNi ∗Q∗N = K∗Ni K −1
Ni NiKNi N . (13)
In the second GPR, by combining the effective priordistribution in (12) with the likelihood distribution in (1), thejoint distribution between observations and a test point latentvariable is given as follows:
p(zN,t , f∗i,t ) = N(
0,
[QN N + RN,t QN∗
Q∗N K∗∗
]). (14)
From the joint distribution in (14), the posterior conditionaldistribution is obtained as follows:
p( f∗i,t |zN,t ) = N (μ∗i,t ,�∗i,t ) (15)
where
μ∗i, = Q∗N (QN N + RN,t )−1zN,t
�∗i,t = K∗∗ − Q∗N (QN N + RN,t )−1 QN∗ . (16)
Remark 1: Although the active set or the neighbor set isused in SGPR and the computation complexity can be reducedby using the matrix inversion lemma for (QN N + RN,t )
−1,the predictive mean function and the predictive covariancefunction in a node i still require all the other nodes to sendtheir data to it. To develop a DGPR, a further requirement,for instance a compactly supported covariance function, isnecessary.
III. DGPR
The prior knowledge about k(xi,t , x j,t ) is the crucial ingre-dient in a GPR. It determines the properties of underlyingprocess. We start this section with a general covariancefunction. Then a compactly supported covariance functionis introduced. In the end, the core of this paper, DGPR, isformulated.
A. Matern Family of Covariance Functions
The Matern family of covariance functions is a class of func-tions that are flexible and commonly used in data assimilation.It is defined as follows:
k(xi,t , x j,t ) = k(rt )
= at
2ν−1�(ν)
(√2νrt
lt
)ν
Kν
(√2νrt
lt
)(17)
where � is the Gamma function and Kν is the modified Besselfunction of the second-order ν [24, pp. 374–379]. rt = ‖xi,t −x j,t‖. at is the amplitude. lt is the effective range. A smaller
1282 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 23, NO. 8, AUGUST 2012
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Distance r
Cov
aria
nce
func
tion
k(r)
Compact support functionν=3/2 Matern functionν=5/2 Matern functionGaussian function
Fig. 1. Covariance functions.
effective range implies the sample function of GP model variesmore rapidly and a larger effective range implies the samplefunction of GP model varies more slowly.
ν is a smoothness parameter and its integer part determinesthe number of times the underlying spatial process is mean-square differentiable [9]. When ν = 1/2, the function definedin (17) is an exponential function. When ν → ∞, thefunction defined in (17) converges to the squared exponentialcovariance function or the Gaussian covariance function
k(xi,t , x j,t ) = at e− r2
t2l2t . (18)
The Gaussian covariance function is infinite-differentiable,which means that the GP with this covariance function hasmean square derivatives of all orders, and is thus very smooth.However, such a strong smoothness assumption is unrealisticfor modeling some physical processes [25].
When ν is of the form m + 1/2 with m a nonnegativeinteger, Kν takes an explicit form. In this case, the covariancefunction is a product of an exponential and a polynomial oforder m. When ν < 1, the covariance function is continuous,but nondifferential. When ν ≥ 7/2, the covariance functionsare all very similar to the Gaussian covariance function. Thusthere are two values of ν that are useful, i.e., ν = 3/2 orν = 5/2. For ν = 3/2
k(xi,t , x j,t ) = at e−
√3rtlt
(√3rt
lt+ 1
). (19)
The infinite support of these covariance functions leads tocomputational complexity because of the demand of comput-ing the inverse of covariance matrix in the order of O(N3).
B. Compactly Supported Covariance Functions
The sparse covariance matrix is favorable to reducing thecomputational complexity of the GPR. Covariance taperingis one way to taper a covariance matrix with an appropriatecompactly supported covariance function, which can reducethe computational burden significantly and still lead to asymp-totically optimal mean-squared error [26]. Another option is
Fig. 2. Neighbor sets.
to directly use a compactly supported covariance function.Our proposed DGPR is built on the use of a compactlysupported covariance function so that only local communica-tion is required. There are several positive-definite covariancefunctions with compact support reported in [27]. In this paper,we use Wu’s polynomial function [23], [27] with the followingexpression:
k(xi,t , x j,t) = k(rt )
= at
(1 − rt
lt
)4
+
(1 + 4rt
lt+ 3r2
t
l2t
+ 3r3t
4l3t
)
(20)
where (1 − rt
lt
)+
={(
1 − rtlt
), if rt < lt
0, otherwise.
This polynomial function is with C2 smoothness. The smooth-ness is a guarantee for good regression results and the compactsupport is a requirement for the distributed DGR. Other high-order Wu’s polynomial functions look similar to this one, buttake more time to compute. The comparisons with infinitelysupported covariance functions in (18) and (19) are made inFig. 1, where at = 1 and lt = 0.3 are used for the infinitelysupported covariance functions and at = 1 and lt = 1.0are used for the compactly supported covariance function in(20). It can be seen that the compactly supported covariancefunction preserves some shape of the infinitely supportedcovariance functions.
Remark 2: In general, a compactly supported covariancefunction means that the distant observation will have zeroeffect on the prediction result. In some cases where a covari-ance with long-range correlation is effective, it should still beuseful qualitatively. There is an unproved argument in geosta-tistics that the prediction result can be nearly independent ofdistant observation conditional on the neighbors even if it maybe highly correlated with distant observation [26].
C. DGPR
Let Di denote the communication range for node i .The neighbor set Ni used in the SGPR is termed as the
GU AND HU: SPATIAL GAUSSIAN PROCESS REGRESSION WITH MOBILE SENSOR NETWORKS 1283
communication range neighbor set Ni and is defined asfollows:
Ni = { j ∈ N | ‖xi,t − x j,t‖ ≤ Di } ∪ i.
For each node i , the compactly supported covariance functionis defined within the local region and therefore has localhyperparameters li,t , ai,t . An effective range neighbor set Li
can be defined for node i as follows:
Li = { j ∈ N | ‖xi,t − x j,t‖ ≤ li,t } ∪ i.
A prediction neighbor set Mi of node i will be useful forDGPR presentation and is defined as follows:
Mi =|Ni |⋃j=1
L j
where |Ni | is the cardinality of set Ni . Let x∗i,t be atesting point of node i . It is within the area defined by{x ∈ Q | ‖xi,t − x‖ ≤ li,t }. It should be noted that Di ≥ 2li,t
is required so that k(x∗i,t , x j,t) = 0 for j /∈ Ni .The neighbor sets of sensor nodes (black dots) defined
above are depicted in Fig. 2. L1 is the effective range neighborset and N1 is the communication range neighbor set ofnode 1. L1 = {1, 2}. L5 = {5, 7}. N1 = {1, 2, 3, 4, 5, 6}.M1 = {1, 2, 3, 4, 5, 6, 7}.
Now we are in a position to deduce the DGPR from theSGPR in (16). Given the neighbor set definitions and thecompactly supported covariance function in (20), the followingresults are obtained.
1) The computation of KNi Ni for node i only requires thatits communication range neighbor j ( j ∈ Ni ) sends theposition x j,t and observation z j,t to it.
2) The computation of K∗Ni for node i only requires thatits communication range neighbor j ( j ∈ Ni ) sends theposition x j,t and observation z j,t to it. This is possibledue to the requirement of Di ≥ 2li,t .
3) The computation of KNi N for node i requires that itsprediction range neighbor j ( j ∈ Mi ) sends the positionx j,t and observation z j,t to it. This becomes clear byrearranging the matrix rows and columns so that its firstMi columns are from the set Mi and all the remainingcolumns have k(xi,t , x j,t ) = 0 for j /∈ Mi , i.e., KNi N =[KNi Mi 0].
Next the following matrices are defined:QMi Mi = KMi Ni K −1
Ni NiKNi Mi
QMi ∗ = KMi Ni K −1Ni Ni
KNi ∗ (21)
Q∗Mi = K∗Ni K −1Ni Ni
KNi Mi .
Using the above matrices, the SGPR matrices defined in (13)are changed into
Q∗N = K∗Ni K −1Ni Ni
[KNi Mi 0] = [Q∗Mi 0]QN∗ =
[KMi Ni
0
]K −1
Ni NiKNi ∗ =
[QMi ∗
0
]
QN N =[
KMi Ni
0
]K −1
Ni Ni[KNi Mi 0]
=[
QMi Mi 00 0
].
Finally, the predictive mean function and the predictivecovariance function of DGPR are found as follows:
μ∗i,t = Q∗N (QN N + RN,t )−1zN,t
= [Q∗Mi 0]([QMi Mi 0
0 0
]+
[RMi ,t 0
0 ∗])−1 [
zMi ,t
∗]
= Q∗Mi (QMi Mi + RMi ,t )−1zMi ,t
�∗i,t = K∗∗ − Q∗N (QN N + RN,t )−1 QN∗
= K∗∗ − [Q∗Mi 0]([QMi Mi 0
0 0
]+
[RMi ,t 0
0 ∗])−1 [
QMi ∗0
]
= K∗∗ − Q∗Mi (QMi Mi + RMi ,t )−1 QMi ∗. (22)
Remark 3: The computation of DGPR asks for informationfrom the nodes in the prediction range neighbor set Mi . Thiscan be achieved by communicating with the nodes in thecommunication range neighbor set Ni because any neighborj in Ni has information from the nodes in its effective rangeneighbor set L j .
D. Hyperparameter Learning
The hyperparameter set of GP in node i is defined asθi,t = [σi,t , ai,t , li,t ]T . Given a hyperparameter set θi,t , thelog marginal likelihood is
Li,t = − log p(zMi ,t |θi,t )
= 1
2zT
Mi ,t C−1i,t zMi ,t + 1
2log det(Ci,t ) + |Mi |
2log(2π)
where Ci,t = QMi Mi + RMi ,t . The partial derivative is
∂Li,t
∂θi,t= −1
2zT
Mi ,t C−1i,t
∂Ci,t
∂θi,tC−1
i,t zMi ,t + 1
2tr
(C−1
i,t∂Ci,t
∂θi,t
)
= −1
2tr
((αi,t α
Ti,t − C−1
i,t
) ∂Ci,t
∂θi,t
)
where αi,t = C−1i,t zMi ,t .
Although Ci,t = QMi Mi + RMi ,t is analytically available,its partial derivatives with respect to the parameters ai,t , li,t
are complex and its computation is time consuming. Here,Ci,t is approximated by using Ci,t = KMi Mi + RMi ,t . Thisapproximation means that a GPR with Mi data for node i isused to learn the hyperparameters via the maximum likelihoodlearning approach.
With Ci,t = KMi Mi + RMi ,t , the following results areobtained:
∂Ci,t
∂σi,t=
[∂k(rt )
∂σi,t
]= [
2σi,t]
∂Ci,t
∂ai,t=
[∂k(rt )
∂ai,t
]=
[2k(rt )
ai,t
]
∂Ci,t
∂li,t=
[∂k(rt )
∂li,t
].
For the Gaussian covariance function (18), it can be found that
∂k(rt )
∂li,t= ai,t r2
t
l3i,t
e− r2
t2l2i,t .
1284 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 23, NO. 8, AUGUST 2012
For the Matern covariance function with ν = 3/2 (19), it canbe found that
∂k(rt )
∂li,t= 3ai,t r2
t
l3i,t
e−
√3rt
li,t .
For the compactly supported covariance function (20), it canbe found that
∂k(rt )
∂li,t= 4rt k(rt )
li,t (li,t − rt )+
−ai,t
(1 − rt
li,t
)4
+
(4rt
l2i,t
+ 6r2t
l3i,t
+ 9r3t
4l4i,t
).
The constraint on θi,t should be imposed with θmin ≤ θi,t ≤θmax, especially for li,t and lmax = Di/2.
The maximum likelihood learning algorithm is based onthe gradient formulation given above. At time step t , theparameter vector θi,t = [σi,t , ai,t , li,t ]T is the initial valuefor the maximum likelihood learning algorithm. The estimatedresult is denoted as θ∗
i,t = [σ ∗i,t , a∗
i,t , l∗i,t ]T . In order to smooththe transition from one GP model to another, θ∗
i,t is not directlyused for regression. Instead, it is fed into a low-pass filter tofilter out the high-frequency noise. The output θi,t+1 of thelow-pass filter is used for regression at time step t . The low-pass filter is expressed as follows:
σi,t+1 = λσ σi,t + (1 − λσ )σ ∗i,t
ai,t+1 = λaai,t + (1 − λa)a∗i,t (23)
li,t+1 = λl li,t + (1 − λl )l∗i,t
where λσ , λa, and λl are updating constants in the range of(0, 1).
Remark 4: The hyperparameter θi,t is updated accordingto (23). It means that the covariance function k(xi,t , x j,t ) isnonstationary in both spatial and temporal domains.
IV. INFORMATION ENTROPY-BASED COVERAGE CONTROL
A sensor node i moves from current position xi,t to nextposition xi,t+1 according to the point motion model
xi,t+1 = xi,t + ui,t
where ui,t is the control signal to be found. In this section, thestrategy of finding the control signal in order to gain a betterregression result is sought.
A. Information Entropy
Mobile sensor node i makes observation zi,t at its locationxi,t , and obtains observation z j,t and position x j,t from all theneighbors j ∈ Mi via wireless communication. It then updatesthe hyperparameter θi,t . With the DGPR, the evaluations ofμ∗i,t and �∗i,t are available.
One motion control strategy is to move sensor nodes towardthe locations with high mean values μ∗i,t . The consequenceis a concentration of sensor nodes on the area with high meanvalues. However, this result cannot reduce the uncertainty ofthe GP model in other areas. Another strategy is to move thesensor nodes toward the locations with high uncertainty �∗i,t .The consequence is a concentration of sensor nodes on the
area with high covariance values. Then sensor nodes can makeobservations there and reduce the uncertainty at next step.The latter strategy is adopted in this paper. The differentialinformation entropy is a measure of uncertainty. It is optimizedto compute the control signal ui,t .
The information entropy of a Gaussian random variable f∗i,t
conditioned on observation zN,t is a monotonic function of itsvariance
H ( f∗i,t |zN,t ) = 1
2log
(2πe det(�∗i,t )
)
= 1
2log
(2πe det(K∗∗ − Q∗Mi C
−1i,t QMi ∗)
).
Remark 5: From the above equation, it looks like that theobservation is not related to the information entropy. However,the observation affects the hyperparameters because of theonline learning algorithm. Thus it is related to the covariancefunction. As the observation is dependent on position x∗i,t ,the information entropy can be interpreted as a function ofposition x∗,t , as
H (x∗i,t) = H ( f∗i,t |zN,t ).
B. CVT
It is necessary for multiple mobile sensor nodes to movewith coordination when they are exploring the area of interest.The CVT approach proposed in [28] is an effective wayfor motion coordination. The CVT algorithm is a locationaloptimization approach based on a utility function. Here wepropose to use the information entropy H (x∗i,t ) as the utilityfunction for node i in the CVT algorithm.
A Voronoi tessellation consists of multiple Voronoi cellsVi,t , each of which is occupied by a sensor node at time step t .A Voronoi cell Vi,t is defined as follows:
Vi,t = {x∗i,t ∈ Q | ∥∥x∗i,t − xi,t
∥∥ ≤ ∥∥x∗i,t − x j,t∥∥ ,∀i �= j
}.
CVT is a special Voronoi tessellation that requires eachsensor node to move forward to the mass center of its Voronoicell. The utility function is defined as
U(X N,t ) =N∑
i=1
∫Vi,t
1
2
∥∥x∗i,t − xi,t∥∥2
H (x∗i,t)dx∗i,t .
To compute the mass center of each Vi,t , the followingdefinitions are used:
MVi,t =∫
Vi,t
H (x∗i,t )dx∗i,t
LVi,t =∫
Vi,t
x∗i,t H (x∗,t)dx∗i,t
CVi,t = LVi,t
MVi,t
.
The partial derivative of the utility function with respect toposition xi,t is
∂U(X N,t )
∂xi,t= −
∫Vi,t
(x∗i,t − xi,t )H (x∗i,t)dx∗i,t
= −MVi,t (CVi,t − xi,t ).
GU AND HU: SPATIAL GAUSSIAN PROCESS REGRESSION WITH MOBILE SENSOR NETWORKS 1285
(a)
(b)
(c)
Fig. 3. Predictive mean function recovered by (a) GPR with Gaussiancovariance function, (b) DGPR with Wu’s compactly supported covariancefunction, and (c) DGPR with Matern covariance function.
The control input signal of node i is
ui,t = ku∂U(X N,t )
∂xi,t
where ku is the control gain.
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
y
Fig. 4. Mobile sensor trajectories produced by DGPR.
0 10 20 30 40 50
0.2
0.25
0.3
0.35
0.4
loops
effe
ctiv
e ra
nge
(a)
0 10 20 30 40 500
0.5
1
1.5
2
2.5
3
loops
effe
ctiv
e ra
nge
(b)
Fig. 5. Effective range of covariance function for (a) GPR and (b) DGPR.
V. SIMULATIONS
A. Stationary Spatial Functions
A mobile sensor network with N = 30 sensor nodes wasto be deployed to a 1 × 1 area to model stationary spatialfunctions. Initially they were randomly placed in a small areawith a size of 0.2 × 0.2. The hyperparameters to be learnedwere the effective range li,t and the amplitude ai,t , while thesensor variance σi,t = 0.01 was taken as constant. A Gaussian-like spatial function
z(x, y) = e− (x−0.5)2+(y−0.5)2
0.07
was simulated first by using the GPR with the Gaussiancovariance function, then with the DGPR with the Wu’scompactly supported covariance function, and, finally, withthe DGPR with the Matern covariance function (ν = 3/2).The control loop of the CVT algorithm was selected as 50.
After the simulation, the predictive mean functions areshown in Fig. 3(a) for the GPR with the Gaussian covari-ance function, Fig. 3(b) for DGPR with the Wu’s compactlysupported covariance function, and Fig. 3(c) for DGPR withthe Matern covariance function (ν = 3/2). Visually, all ofthem can reconstruct the Gaussian-like function. The resultof the DGPR with the Wu’s compactly supported covariancefunction was very close to that of the GPR with the Gaussiancovariance function. The result of the DGPR with the Materncovariance function suffered from discontinuities at the cornersof the area.
1286 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 23, NO. 8, AUGUST 2012
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
loops
ampl
itude
(a)
0 10 20 30 40 500
0.05
0.1
0.15
0.2
0.25
0.3
0.35
loops
ampl
itude
(b)
Fig. 6. Amplitude of the covariance function for (a) GPR and (b) DGPR.
(a)
(b)
Fig. 7. (a) Ground-truth function and (b) predictive mean function of DGPR.
Fig. 3(b) and (c) present similar shapes in most of thearea except at the corners. The similarity in most of thearea confirms the equivalence between the Wu’s compactlysupported covariance function and the Matern covariancefunction (ν = 3/2) for regression purpose in this scenario.The differences at the corners of the area are caused bythe finite and infinite supported property. Wu’s compactlysupported covariance function converges to zero at theboundary of communication limit due to the finite supportedproperty, which leads to a smooth result. In contrast, theMatern covariance function does not converge to zero at the
0 10 20 30 40 500
1
2
3
4
5
6
7
8
loops
RS
S
(a)
0 10 20 30 40 500
1
2
3
4
5
6
7x 104
loops
RS
S
(b)
Fig. 8. RSS for (a) GPR and (b) DGPR.
boundary of communication limit due to the infinite supportedproperty, which leads to the discontinuities at the corners.The discontinuity can be reflected by a first-order gradientmeasure, which is defined as the maximum value of the squareroot of the first-order gradients in the x and y directions
max(x,y)∈Q
√(∂μ
∂x
)2
+(
∂μ
∂y
)2
where Q is the area where the mobile sensor network isdeployed and μ is the predictive mean function. The maximumvalue is used here to show the worst case scenario. Thegradient measure is 0.0330 and 0.0334 in Fig. 3(a) and (b),respectively. They are very close. However, the gradientmeasure is 0.0648 in Fig. 3(c). It nearly doubles the values ofthe smooth regression results and thus reflects the existenceof discontinuities.
Mobile sensor trajectories produced by the DGPR with theWu’s compactly supported covariance function are shown inFig. 4. All the sensor nodes were initially placed at the bottomleft corner of the area. They were able to move to cover aslarge an area as possible. A similar result was obtained fromthe GPR and hence is omitted here.
GU AND HU: SPATIAL GAUSSIAN PROCESS REGRESSION WITH MOBILE SENSOR NETWORKS 1287
00.2
0.40.6
0.81
0
0.5
10
1
2
3
4
5
6
xy
Mea
n
(a)
0 10 20 30 40 500
1
2
3
4
5
6
7x 104
loops
RS
S
(b)
Fig. 9. (a) Predictive mean function. (b) RSS with fixed effective rangeunder DGPR.
The effective range and the amplitude were learned onlineusing the maximum likelihood algorithm. The learned resultsare shown in Fig. 5 for the effective range and in Fig. 6 forthe amplitude. These parameters experienced some changesat the early stage of the process. After about 25 loops, theygradually reached stable values.
In the second simulation, a complex 2-D spatial function
z(x, y) = 1.9(1.35 + ex sin
(13(x − 0.6)2)e−y sin(7y)
)was simulated. All the parameters were the same as in the firstsimulation. The ground truth function is shown Fig. 7(a). In thefollowing presentation, the DGPR is referred to as the DGPRwith the Wu’s compactly supported covariance function.
The predictive mean function of DGPR is shown inFig. 7(b). Visually, their shapes are very similar although thereare some errors found at the boundaries. By dividing the areainto 100 × 100 cells, the residual sum of squares (RSS) wascalculated at the center of cells at the end of each loop. Theresults are shown in Fig. 8(a) for GPR and 8(b) for DGPR.Both of them demonstrated an error-reducing behavior andbecame very small after about 35 loops. However, a less stable
0 10 20 30 40 500
1
2
3
4
5
6
7x 104
loops
RS
S
(a)
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
y
(b)
Fig. 10. (a) RSS with faulty nodes. (b) Mobile node trajectories with faultynodes.
behavior of GPR compared to DGPR was observed becauseof the difference of their covariance functions.
To demonstrate the effect of limited communication on theresult, the constraint li,t = Di = 0.4 was used while theamplitude was still learned. The predictive mean function andthe RSS of DGPR are shown in Fig. 9. The shape of thepredictive mean function was still similar to the ground-truthfunction, but there were some minor changes. This is due tothe fact that a short effective range was used. The RSS resultstill had a stable behavior, but it had a larger RSS than theresult in Fig. 8(b).
The algorithm robustness and scalability were tested forthis 2-D spatial function. First, the algorithm robustness wastested against node failure. All the simulation parameters werethe same as before. Five nodes failed at the 20th loop andanother five failed at the 30th loop. The RSS result is shownin Fig. 10(a). It can be seen that the RSS value was increasedat the 20th and 30th loops due to the failure occurrences, but itstill decreased after the interruptions. The circles and squaresshown in Fig. 10(b) are the faulty nodes at the 20th and 30th
1288 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 23, NO. 8, AUGUST 2012
0 10 20 30 40 500
1
2
3
4
5
6
7x 10
4
loops
RS
S
(a)
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
y
(b)
Fig. 11. (a) RSS with the joining of more nodes. (b) Mobile node trajectorieswith the joining of more nodes.
loop, respectively. All the other nodes could still successfullycover the area.
Second, the algorithm scalability was tested against thejoining of more nodes. Twenty nodes were used initially andthey were spread out for 20 loops. At the 20th loop, another10 nodes located at the top right corner of the area joinedthe team. All 30 nodes worked together after the 20th loop.The RSS result is shown in Fig. 11(a). The RSS value wasreduced at the 20th loop because of the joining of the 10 nodes.Eventually, all the nodes worked as a team to cover the area.Fig. 11(b) shows their trajectories, with the circles representingthe nodes joining the team at the 20th loop.
B. Light Intensity Regression
To evaluate the DGPR algorithm in a physical environment,a test on light intensity was conducted. All the parameters ofthe algorithm were the same as in the previous simulationsand the Wu’s compactly supported covariance function wasused. The environment was an area with the size of 4000mm by 6000 mm. A light source was placed on one sideof the area and the light intensity was sampled by using thelight sensor of a LEGO NXT robots. The testing area wasalso equipped with a 3-D tracking system VICON, which
(a)
(b)
Fig. 12. (a) DGPR result for light intensity test, including the trainingsamples (blue crosses) and evaluation samples (red dots). (b) Regression errors(red dots) on the evaluation positions (black dots).
provided the position information of the samples. In total,88 samples were collected and they were separated into twogroups. One group was used to train the DGPR model, and theother group was used to evaluate the prediction result. Boththe training and evaluation procedure were conducted offline.The result is shown in Fig. 12(a). The training samples (blackcrosses) just got stuck on the surface of the regression model,while the evaluation samples (red dots) were slightly awayfrom the surface. The errors (red dots) between the evaluationsamples and the model prediction results were calculated andare shown in Fig. 12(b). The evaluation samples (black dots)were evenly distributed in the area. The average absolute erroron the evaluation samples was 3.5%. The maximum error forthis test was 15.5%. In general, the prediction result fit wellwith the actual measurement.
C. Spatiotemporal Function
A 2-D Sine–Gordon equation is a spatiotemporal functionthat has an exact analytic solution [29]. The Sine–Gordonequation has the following expression:∂2z(t, x, y)
∂x2 + ∂2z(t, x, y)
∂y2 − ∂2z(t, x, y)
∂ t2 = sin(z(t, x, y)).
GU AND HU: SPATIAL GAUSSIAN PROCESS REGRESSION WITH MOBILE SENSOR NETWORKS 1289
0
1
2
3
0
1
2
30
1
2
3
4
5
6
x
t=1
y
z
(a)
0
1
2
3
0
1
2
30
1
2
3
4
5
6
x
t=3
y
z
(b)
0
1
2
3
0
1
2
30
1
2
3
4
5
6
x
t=5
y
z
(c)
0
1
2
3
0
1
2
30
1
2
3
4
5
6
x
t=7
y
z
(d)
Fig. 13. Sine–Gordon equation (a) t = 1, (b) t = 3, (c) t = 5, and (d) t = 7.
(a) (b)
(d)(c)
Fig. 14. Predictive mean function reconstructed by DGPR (a) t = 1,(b) t = 3, (c) t = 5, and (d) t = 7.
Its solution has the following form:z(t, x, y) = 4 tan−1
(g(t, x, y)
f (t, x, y)
)
where
f (t, x, y) = 1 + a12eτ1+τ2 + a13eτ1+τ3 + a23eτ2+τ3
g(t, x, y) = eτ1 + eτ2 + eτ3 + a12a13a23eτ1+τ2+τ3
ai j = (Pi − Pj )2 + (pi − p j )
2 − (wi − w j )2
(Pi + Pj )2 + (pi + p j )2 − (wi + w j )2
τi = Pi x + p j y − wi t
provided that the following conditions are satisfied:P2
i + p2j − w2
i = 1 for i = 1, 2, 3
det
⎛⎝ P1 p1 w1
P2 p2 w2P3 p3 w3
⎞⎠ = 0.
The parameters used in the simulation were as follows:P1 = 1.1, P2 = 0.3, P3 = 0.3, p1 = 0, p2 = 1.2, p3 = 1.2,w1 = 0.4583, w2 = 0.6633, and w3 = 0.6633. Theground-truth solution with an area of 3 × 3 is shown inFig. 13 for t = 1, 3, 5, and 7.
A mobile sensor network with N = 30 sensor nodes wasused. Initially they were randomly placed in the area. Thehyperparameters to be learned were the effective range li,t
and the amplitude ai,t . The Sine–Gordon equation was testedwith 10 time steps starting with t = 0 and ending with t = 9.Between each two consecutive time steps, DGPR and CVTalgorithms were executed iteratively and the number of loopswas selected as 5. The predictive mean function is shown inFig. 14 for t = 1, 3, 5, and 7. By comparing with the ground-truth solutions in Fig. 13, it can be seen that the DGPR canreconstruct the Sine–Gordon function very well.
VI. CONCLUSION
This paper presented a DGPR for mobile sensor networksto model spatial functions. The proposed DGPR was deducedfrom the projected process approximation the SGPR com-bined with a compactly supported covariance function. Withthe proposed DGPR, the hyperparameter learning was alsoimplemented in a distributed manner and the hyperparameterscould be updated online. This paper also made use of theadvantage of the GPR over other regression techniques forcollective motion decision-making. To this end, an informationentropy-based CVT approach was used, which allowed themobile sensor network to reduce the uncertainty of the GPmodel. The proposed strategies of cooperative modeling andcoordinated motion provided satisfactory results for stationaryspatial functions and spatiotemporal functions. As our nextstep, we would like to test our proposed algorithm in morerealistic simulations or real experiments.
REFERENCES
[1] L. F. Merino, J. R. Caballero, J. M. de Dios, and A. O. Ferruz,“A cooperative perception system for multiple UAVs: Application toautomatic detection of forest fires,” J. Field Robot., vol. 23, no. 3, pp.165–184, 2006.
[2] C. E. Corrigan, G. C. Roberts, M. V. Ramana, D. Kim, andV. Ramanathan, “Capturing vertical profiles of aerosols and blackcarbon over the Indian ocean using autonomous unmanned aerialvehicles,” Atmosp. Chem. Phys. Discuss., vol. 7, no. 4, pp. 11429–11463, 2007.
[3] N. E. Leonard, D. Paley, F. Lekien, R. Sepulchre, D. M. Fratantoni, andR. Davis, “Collective motion, sensor networks and ocean sampling,”Proc. IEEE, vol. 95, no. 1, pp. 48–74, Jan. 2007.
[4] M. Lazaro-Gredilla and A. R. Figueiras-Vidal, “Marginalized neuralnetwork mixtures for large-scale regression,” IEEE Trans. NeuralNetw., vol. 21, no. 8, pp. 1345–1351, Aug. 2010.
[5] G. Skolidis and G. Sanguinetti, “Bayesian multitask classification withGaussian process priors,” IEEE Trans. Neural Netw., vol. 22, no. 12,pp. 2011–2021, Dec. 2011.
[6] Y. Miche, A. Sorjamaa, P. Bas, O. Simula, C. Jutten, and A. Lendasse,“OP-ELM: Optimally pruned extreme learning machine,” IEEE Trans.Neural Netw., vol. 21, no. 1, pp. 158–162, Jan. 2010.
[7] C. K. I. Williams and C. E. Rasmussen, “Gaussian processes forregression,” in Advances in Neural Information Processing Systems 8,D. S. Touretzky, M. C. Mozer, and M. E. Hasselmo, Eds. Cambridge,MA: MIT Press, 1996, pp. 514–520.
[8] D. J. C. MacKay, “Introduction to Gaussian processes,” in NeuralNetworks and Machine Learning, vol. 168, C. M. Bishop, Ed. Berlin,Germany: Springer-Verlag, 1998, pp. 133–165.
1290 IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 23, NO. 8, AUGUST 2012
[9] C. E. Rasmussen and C. Williams, Gaussian Processes for MachineLearning. Cambridge MA: MIT Press, 2006.
[10] A. Krause, A. Singh, and C. Guestrin, “Near-optimal sensor placementsin Gaussian processes: Theory, efficient algorithms and empiricalstudies,” J. Mach. Learn. Res., vol. 9, pp. 235–284, Feb. 2008.
[11] R. Stranders, A. Rogers, and N. Jennings, “A decentralized, on-linecoordination mechanism for monitoring spatial phenomena with mobilesensors,” in Proc. 2nd Int. Workshop Agent Technol. Sensor Netw.,Estoril, Portugal, 2008, pp. 1–7.
[12] C. Stachniss, C. Plagemann, and A. J. Lilienthal, “Learning gasdistribution models using sparse Gaussian process mixtures,” Auton.Robots, vol. 26, nos. 2–3, pp. 187–202, 2009.
[13] J. L. Ny and G. Pappas, “On trajectory optimization for active sensingin Gaussian process models,” in Proc. IEEE Conf. Decision Control,Shanghai, China, Dec. 2009, pp. 6286–6292.
[14] J. Cortes, “Distributed Kriged Kalman filter for spatial estimation,”IEEE Trans. Autom. Control, vol. 54, no. 12, pp. 2816–2827, Dec.2009.
[15] J. Choi, J. Lee, and S. Oh, “Swarm intelligence for achieving the globalmaximum using spatio-temporal Gaussian processes,” in Proc. Amer.Control Conf., Seattle, WA, 2008, pp. 1–6.
[16] A. Singh, F. Ramos, H. D. Whyte, and W. J. Kaiser, “Modelingand decision making in spatio-temporal processes for environmentalsurveillance,” in Proc. IEEE Int. Conf. Robot. Autom., Anchorage, AK,May 2010, pp. 5490–5497.
[17] M. Schwager, D. Rus, and J. Slotine, “Decentralized, adaptive conver-age control for networked robots,” Int. J. Robot. Res., vol. 28, no. 3,pp. 357–375, 2009.
[18] K. M. Lynch, I. B. Schwartz, P. Yang, and R. A. Freeman, “Decentral-ized environmental modeling by mobile sensor networks,” IEEE Trans.Robot., vol. 24, no. 3, pp. 710–724, Jun. 2008.
[19] S. Martinez, “Distributed interpolation schemes for field estimation bymobile sensor networks,” IEEE Trans. Control Syst. Technol., vol. 18,no. 2, pp. 491–500, Mar. 2010.
[20] J. Quinonero-Candela and C. E. Rasmussen, “A unifying view of sparseapproximate Gaussian process regression,” J. Mach. Learn. Res., vol. 6,pp. 1935–1959, Dec. 2005.
[21] L. Csato and M. Opper, “Sparse online Gaussian processes,” NeuralComput., vol. 14, no. 2, pp. 641–669, 2002.
[22] M. Seeger, C. K. I. Williams, and N. Lawrence, “Fast forward selectionto speed up sparse Gaussian process regression,” in Proc. 9th Int.Workshop Artif. Intell. Stat., 2003, pp. 1–8.
[23] Z. Wu, “Compactly supported positive definite radial functions,” Adv.Comput. Math., vol. 4, no. 1, pp. 283–292, 1995.
[24] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Func-tions. New York: Dover, 1965.
[25] M. L. Stein, Interpolation of Spatial Data. New York: Springer-Verlag,1999.
[26] R. Furrer, M. G. Genton, and D. Nychka, “Covariance tapering forinterpolation of large spatial datasets,” J. Comput. Graph. Stat., vol. 15,no. 3, pp. 502–523, 2006.
[27] R. Schaback, “Creating surfaces from scattered data using radial basisfunctions,” in Mathematical Methods in Computer Aided GeometricDesign III, M. Daehlen, T. Lyche, and L. Schumaker, Eds. Nashville,TN: Vanderbilt Univ. Press, 1995, pp. 477–496.
[28] J. Cortes, S. Martinez, T. Karatas, and F. Bullo, “Coverage control formobile sensing networks,” IEEE Trans. Robot. Auton., vol. 20, no. 2,pp. 243–255, Apr. 2004.
[29] L. Guo and S. A. Billings, “State-space reconstruction and spatio-temporal prediction of lattice dynamical systems,” IEEE Trans. Autom.Control, vol. 52, no. 4, pp. 622–632, Apr. 2007.
Dongbing Gu (SM’07) received the B.Sc. and M.Sc.degrees in control engineering from the BeijingInstitute of Technology, Beijing, China, and thePh.D. degree in robotics from the University ofEssex, Essex, U.K.
He was an Academic Visiting Scholar withthe Department of Engineering Science, Universityof Oxford, Oxford, U.K., from October 1996 toOctober 1997. In 2000, he joined the Universityof Essex as a Lecturer. Currently, he is a Readerwith the School of Computer Science and Electronic
Engineering, University of Essex. His current research interests includemultiagent systems, wireless sensor networks, distributed control algorithms,distributed information fusion, cooperative control, reinforcement learning,fuzzy logic and neural network-based motion control, and model predictivecontrol.
Huosheng Hu (SM’01) received the M.Sc. degree inindustrial automation from Central South University,Beijing, China, and the Ph.D. degree in roboticsfrom the University of Oxford, Oxford, U.K.
He is currently a Professor of computer sciencewith the University of Essex, Essex, U.K., and leadsthe Human Centered Robotics Group. He has pub-lished over 180 research papers in journals, books,and conference proceedings in his areas of exper-tise. His current research interests include mobilerobotics, sensors integration, data fusion, distributed
computing, intelligent control, behavior and hybrid control, cooperative robot-ics, telerobotics, and service robots.
Prof. Hu is currently the Editor-in-Chief of the International Journal ofAutomation and Computing. He is also a reviewer for a number of interna-tional journals, such as the IEEE TRANSACTIONS ON ROBOTICS, the IEEETRANSACTIONS ON AUTOMATIC CONTROL, the IEEE TRANSACTIONS ON
NEURAL NETWORKS, and the International Journal of Robotics Research.He is a member of IEE, AAAI, the Association of Computing Machinery(ACM), IAS, and IASTED, and is a Chartered Engineer.
