Embed Size (px)
Citation preview
1
Spatial Field Reconstruction and SensorSelection in Heterogeneous Sensor Networks
with Stochastic Energy HarvestingPengfei Zhang1, Ido Nevat2, Gareth W. Peters3, Francois Septier4 and Michael A. Osborne1
1 Department of Engineering Science, University of Oxford, UK2 TUM CREATE, Singapore
3 Department of Statistical Science, University College London, UK4 IMT Lille Douai, Univ. Lille, CNRS, UMR 9189 - CRIStAL, F-59000 Lille, France
Abstract—We address the two fundamental problemsof spatial field reconstruction and sensor selection inheterogeneous sensor networks. We consider the casewhere two types of sensors are deployed: the first consistsof expensive, high quality sensors; and the second, ofcheap low quality sensors, which are activated only ifthe intensity of the spatial field exceeds a pre-definedactivation threshold (eg. wind sensors). In addition, thesesensors are powered by means of energy harvesting andtheir time varying energy status impacts on the accuracyof the measurement that may be obtained. We accountfor this phenomenon by encoding the energy harvestingprocess into the second moment properties of the additivenoise, resulting in a spatial heteroscedastic process. Wethen address the following two important problems: (i)how to efficiently perform spatial field reconstructionbased on measurements obtained simultaneously fromboth networks; and (ii) how to perform query based sensorset selection with predictive MSE performance guarantee.We first show that the resulting predictive posterior distri-bution, which is key in fusing such disparate observations,involves solving intractable integrals. To overcome thisproblem, we solve the first problem by developing a lowcomplexity algorithm based on the spatial best linearunbiased estimator (S-BLUE). Next, building on the S-BLUE, we address the second problem, and develop anefficient algorithm for query based sensor set selectionwith performance guarantee. Our algorithm is based onthe Cross Entropy method which solves the combinatorialoptimization problem in an efficient manner. We presenta comprehensive study of the performance gain thatcan be obtained by augmenting the high-quality sensorswith low-quality sensors using both synthetic and realinsurance storm surge database known as the ExtremeWind Storms Catalogue.Keywords: Internet of Things, Sensor Networks, GaussianProcess, Energy harvesting, Cross Entropy method, Sensorselection
I. INTRODUCTION
Wireless Sensor Networks (WSN) have attractedconsiderable attention due to the large number ofapplications, such as environmental monitoring [1],weather forecasts [2]–[4], surveillance [5], health care[6], structural safety and building monitoring [7] andhome automation [4], [8]. We consider a WSN whichconsists of a set of spatially distributed sensors that
may have limited resources, such as energy and com-munication bandwidth. These sensors monitor a spa-tial physical phenomenon containing some desiredattributes (e.g pressure, temperature, concentrationsof substance, sound intensity, radiation levels, pollu-tion concentrations, seismic activity etc.) and regularlycommunicate their observations to a Fusion Centre(FC) [9]–[11]. The FC collects these observations andfuses them in order to reconstruct the spatial field [8].
In many cases these WSN use a small set of high-quality and expensive sensors (such as weather sta-tions) [12]–[14]. While these sensors are capable ofreliably measuring the environmental physical phe-nomenon, the low spatial deployment resolution pro-hibits their use in spatial field reconstruction tasks. Toovercome this problem, sparse high-quality sensor de-ployment can be augmented by the use of complemen-tary cheap low-quality sensors that can be deployedmore densely due to their low costs [2], [15]. This typeof heterogeneous sensor networks approach has gainedattention in the last few years due to the vision of theInternet of Things (IoT) where networks may sharetheir data over the internet [16], [17]. This couplingenables the concept of Collaborative Wireless SensorNetwork (CWSN), in which networks with differentcapabilities are deployed in the same physical regionand collaborate in order to optimize various designcriteria and processes [18]. The incentive to developsuch heterogeneous sensor network and associatedsignal processing was further strengthened when theUS Environmental Protection Agency (EPA) publishedits shift in the paradigm of data collection which pro-motes the notion of augmenting sparse deploymentsof high-quality sensors with dense deployment of low-quality and inaccurate sensors [19].
Two practical scenarios that are of importance are:
1) High-quality sensors may be deployed by gov-ernment agencies (eg. weather stations). These aresparsely deployed due to their high costs, limitedspace constraints, high power consumption etc.To improve the coverage of the WSN, low-quality
cheap sensors can be deployed to augment thehigh-quality sensor network [15].
2) High-quality sensors cannot be easily deployedin remote locations, for example in oceans, lakes,mountains and volcanoes. In these cases, energyharvesting based battery operated low-qualitycheap sensors can be deployed [20].
The main goal of this paper is to develop lowcomplexity algorithms to solve the problems of spatialfield reconstruction and query based sensor set selectionwith performance guarantee of spatial random fieldsin WSN under practical scenarios of high and lowquality sensors. More specifically, the following twofundamental problems are the focus of this paper:
1) Spatial field reconstruction: the task is to ac-curately estimate and predict the intensity of aspatial random field, not only at the locations ofthe sensors, but at all locations [13], [21], [22],given heterogeneous observations from both sen-sor networks.
2) Query based sensor set selection with perfor-mance guarantee: the task is to perform on-linesensor set selection which meets the QoS criterionimposed by the user, as well as minimises thecosts of activating the sensors of these networks[23]–[25].
A. Related work on spatial field reconstruction in sensornetworks:
It is common in the literature to model the physicalphenomenon being monitored by the WSN accordingto a Gaussian random field (GRF) with a spatial cor-relation structure [26]–[33]. More generally, examplesof GRFs include wireless channels [34], speech pro-cessing [35], natural phenomena (temperature, rainfallintensity etc.) [3], [36], and recently in [21], [22].
The simplest form of Gaussian process model wouldtypically assume that the spatial field observed at theFC is only corrupted by additive Gaussian noise. Forexample, in [27] a linear regression algorithm for GRFreconstruction in mobile wireless sensor networkswas presented, but relied on the assumption of onlyAdditive White Gaussian Noise (AWGN); in [37] analgorithm was developed to learn the parameters ofnon-stationary spatio-temporal GRFs again assumingAWGN; and in [38] an algorithm for choosing sensorlocations in GRF assuming AWGN was developed.
In practical WSN deployments, two deviations fromthese simplified modelling assumptions arise and areimportant to consider: the use of heterogeneous sen-sor types i.e. sensors may have different degrees ofaccuracy throughout the field of spatial monitoring[21], [22]; and secondly quite often the sensors maybe powered by means of energy harvesting whichimpacts their reading accuracy and lifetime duration[32], [39].
In [23], the authors developed an algorithm forsensor selection and power allocation in energy har-vesting wireless sensor networks. They extended themodel proposed by Joshi and Boyd [24] and incorpo-rated the energy harvesting process into the problemformulation. However, they assumed that the energyharvesting process is fully observed, which is tooan optimistic assumption in practice. In [25], [40],a sparsity-promoting penalty function to discouragerepeated selection of any sensor node was proposed.
All of these works did not attempt to solve thesensor selection problem in heterogeneous sensor net-works. Nor did they assume that the energy harvest-ing process is not fully observed and impacts thereading accuracy of the sensors. Our paper considersthese important aspects and provides both holistic andpractical solution for those problems.
B. Contributions:
1) We develop a novel statistical model to accountfor high and low quality sensors (see A3-A4 inSection II-B).
2) We model the practical scenario of spatially corre-lated additive noise due to energy harvesting viaa spatially correlated heteroscedastic process (seeA5-A6 in Section II-B).
3) We develop a point-wise estimation algorithm forspatial field reconstruction which is based on theSpatial Best Linear Unbiased Estimator (S-BLUE)(Corollary 1 in Section III-A).
4) We develop an efficient algorithm to performquery based sensor set selection with performanceguarantee which solves the combinatorial op-timization problem using the Cross Entropymethod (Section IV).
II. SENSOR NETWORK MODEL AND DEFINITIONS
We begin by presenting the statistical model for thespatial physical phenomena, followed by the systemmodel.
A. Spatial Gaussian Random Fields Background
We model the physical phenomena (both monitoredand energy harvesting phenomena) as spatially de-pendent continuous processes with a spatial correla-tion structure and are independent from each other.Such models have recently become popular due totheir mathematical tractability and accuracy [30], [41],[42]. The degree of the spatial correlation in the pro-cess increases with the decrease of the separationbetween two observing locations and can be accu-rately modelled as a Gaussian random field 1 [3],[21], [22], [26], [28], [34]. A Gaussian process (GP)defines a distribution over a space of functions and it
1We use Gaussian Process and Gaussian random field inter-changeably.
is completely specified by the equivalent of sufficientstatistics for such a process, and is formally definedas follows.Definition 1. (Gaussian process [43], [44]): Let X ⊂Rd be some bounded domain of a d-dimensional realvalued vector space. Denote by f(x) : X 7→ R astochastic process parametrized by x ∈ X . Then, therandom function f(x) is a Gaussian process if all itsfinite dimensional distributions are Gaussian, wherefor anym ∈ N, the random vectors (f (x1) , · · · , f (xm))are normally distributed.
We can therefore interpret a GP as formally definedby the following class of random functions:
F := f (·) : X 7→ R s.t. f (·) ∼ GP (µ (·;θ) , C (·, ·; Ψ)) ,
with µ (x;θ) := E [f (x)] : X 7→ R,C (xi,xj ; Ψ) := E [(f (xi)− µ (xi;θ)) (f (xj)− µ (xj ;θ))] ,
: X × X 7→ R+
where at each point the mean of the function is µ(·;θ),parameterised by θ, and the spatial dependence be-tween any two points is given by the covariancefunction (Mercer kernel) C (·, ·; Ψ), parameterised byΨ, see detailed discussion in [43].It will be useful to make the following notationaldefinitions for the cross correlation vector and auto-correlation matrix, respectively:
k (x∗,x1:N ) := E [f (x∗) f (x1:N )] ∈ R1×N
K (x1:N ,x1:N ) :=
C (x1,x1) · · · C (x1,xN )...
. . ....
C (xN ,x1) · · · C (xN ,xN )
,with S+ (Rn) is the manifold of symmetric positivedefinite matrices. k (xi,xj) is the correlation functionand C (xi,xj) is the covariance function.
We define the Log Gaussian Process (LGP) in thefollowing definition:Definition 2. (Log Gaussian process): A Log GaussianProcess (LGP) defines a stochastic process whose logorithmfollows Gaussian Process. In mathematics, given a
g (x) ∼ LGP (µg (x;θg) , Cg (x1,x2; Ψg)) . (1)
whereE[g (x)] = eµg(x;θg)+Cg(x1,x2;Ψg)
2/2
Having formally specified the semi-parametric classof Gaussian process models and Log Gaussian processmodel, we proceed with presenting the system model.
B. Heterogeneous Sensor Network System Model
We now present the system model for the physicalphenomenon observed by two types of networks andthe energy harvesting model (See Fig. 1).
A1 Consider a random spatial phenomenon (eg.wind) to be monitored defined over a 2-dimensional space X ∈ R2. The mean responseof the physical process is a smooth continuous
Fig. 1: System model of heterogeneous sensor net-works: one (green) which contains high-quality sen-sors and connected to mains power, and second (blue)which contains low-quality sensors and rechargeablebatteries with solar irradiance energy harvesting ca-pabilities.
spatial function f (·) : X 7→ R, and is modelled asa Gaussian Process (GP) according to
f (x) ∼ GP (µf (x;θf ) , Cf (x1,x2; Ψf )) , (2)
where the mean and covariance functionsµf (x;θf ) , Cf (x1,x2; Ψf ) are assumed to beknown.
A2 Let N be the total number of sensors that aredeployed over a 2-D region X ⊆ R2, with xn ∈X , n = 1, · · · , N being the physical location ofthe n-th sensor, assumed known by the FC. Thenumber of sensors deployed by Network 1 andNetwork 2 are NH and NL, respectively, so thatN = NH +NL .
A3 Sensor network 1: High Quality SensorsThe sensors have a 0-threshold activation andeach of the sensors collects a noisy observation ofthe spatial phenomenon f (·). At the n-th sensor,located at xn, the observation is given by:
Y H (xn) = f (xn) +W (xn) , n = 1, · · · , NH(3)
where W (xn) is i.i.d Gaussian noise W (xn) ∼N(0, σ2
W
).
A4 Sensor network 2: Low Quality SensorsThe sensors have a T -threshold activation andeach of the sensors collects a noisy observation ofthe spatial phenomenon f (·), only if the intensityof the field at that location exceeds the pre-definedthreshold T , (eg. anemometer sensors for windmonitoring [45], [46]). At the n-th sensor, locatedat xn, the observation is given by:
Y L (xn) =
f (xn) + V (xn) , f (xn) ≥ TV (xn) , f (xn) < T
(4)
The statistical properties of the additive noiseV (xn) are detailed in A6.
A5 Energy harvesting model:The energy harvesting process (eg. solar irradi-ance) is modelled as a spatial phenomenon de-fined over a 2-dimensional space X ∈ R2. Themean response of the physical process is a smoothcontinuous spatial function g (·) : X 7→ R, and in
order to ensure positivity it is modelled as a log-Gaussian Process (LGP), similar as [29], [30], [32],[33], given by
g (x) ∼ LGP (µg (x;θg) , Cg (x1,x2; Ψg)) . (5)
We assume that the energy harvesting processg(x) is independent of the monitored physicalphenomenon f(x).2
A6 Spatial noise process model:Since all sensors in Network 2 use energy harvest-ing techniques from a spatially correlated energyfield g (x), this will have an impact on the per-formance of the electronic circuits (eg. amplifiers,voltage and frequency biases). These impact thethermal noise (V (x)) and its characteristics, see[47]. As a result of the variations and fluctuationsof the energy field, the additive thermal noiseis now also spatially correlated across sensorsexposed to common environmental energy har-vesting conditions, such that E[V (x1)V (x2)] 6= 0.The spatial noise can be modelled as a spatialstochastic volatility model in which the varianceof the additive noise is itself a random processwhich is spatially correlated. This means that ifthe energy harvested by a particular sensor ishigh, the noise variance should be small andvice versa. A common approach to modelling thisimpact is via a link function as follows:
σ2V (x) = ψ (g (x)) , (6)
where ψ (α) : R+ → R+ is a deterministic knownmapping. The choice of ψ(x) can be flexible inpractice, but it needs to satisfy the constraints thatthe larger the value of x, the smaller ψ(x) is. Somechoice of ψ(x) includes 1/x, exp(−x), 1/x2 and etc.In this paper, without loss of generality and fornotational simplicity, in the paper we assume thatψ (α) = 1/α.
In Table I we present the notations which will be usedthroughout the paper.
III. FIELD RECONSTRUCTION VIA SPATIAL BESTLINEAR UNBIASED ESTIMATOR (S-BLUE)
To perform inference in our Bayesian framework,one would typically be interested in computing thepredictive posterior density at any location in space,x∗ ∈ X , denoted p (f∗|YN ). Based on this quan-tity various point estimators, like the Minimum
2The energy harvesting and the physical phenomenon beingmeasured would not be independent processes in some cases. Inthis paper however, we only consider the case where the energyharvesting model and the physical phenomenon to be independentprocess, which is of practical importance in many cases. Someexamples include energy harvesting via solar power and a physicalphenomenon which is precipitation or pollution.
Mean Squared Error (MMSE) and the Maximum A-Posteriori (MAP) estimators can be derived:
f∗MMSE
=
∞∫−∞
p (f∗|YN ,x1:N ,x∗) f∗df∗,
f∗MAP
= arg maxf∗
p (f∗|YN ,x1:N ,x∗) .
These estimators provide a pointwise estimator of theintensity of the spatial field, f∗ at location x∗. Thisenables us to reconstruct the whole spatial field byevaluating f∗ on a fine grid of points. The predictiveposterior density is given by:
p (f∗|YN ,x1:N ,x∗) =
∫RN
p (f∗|fN ,x1:N ,x∗) p (fN |YN ,x1:N ,x∗) dfN
=
∫RN
p (f∗|fN ,x1:N ,x∗) p (fH|YH,YL, fL,x1:N ,gL)
×∫
RNL
p (YN |fL,x1:N ) p(fL|x1:NL
)∫RNL
p (YN |fL,x1:N ,gL) p(fL|x1:NL
)dfL
p (gL|YH,YL,x1:N )
dgLdfN .(7)
where RNL defines the domain for gL, which is theNL dimensional LGP specified in A5 in Section II-B.Unfortunately, the predictive posterior density cannotbe calculated analytically in closed form, prohibitinga direct calculation of any Bayesian estimator. Oneapproach to approximating p (f∗|YN ) is the Laplaceapproximation [48], which was used in our previ-ous works [21], [22]. A different approach is basedon Markov Chain Monte Carlo (MCMC) methodswhich generates samples from the target distributionp (f∗|YN ,x1:N ,x∗) [49]. These methods are not suitablefor our problem as we require low complexity algo-rithm which is suitable for reconstructing the wholespatial field as well as for selecting the optimal subsetof sensors in real-time. In addition, expectation prop-agation and variational inference are unsuitable dueto computational constraints. To achieve these goals,we develop a low-complexity linear estimator for f∗,presented next.
A. Spatial Best Linear Unbiased Estimator (S-BLUE) FieldReconstruction Algorithm
We develop the spatial field reconstruction via BestLinear Unbiased Estimator (S-BLUE), which enjoys alow computational complexity [50]. The S-BLUE doesnot require calculating the predictive posterior density,but only the first two cross moments of the model. TheS-BLUE is the optimal (in terms of minimizing MeanSquared Error (MSE)) of all linear estimators and isgiven by the solution to the following optimizationproblem:
f∗ := a+BYN = arg mina,B
E[(f∗ − (a+ BYN ))
2], (8)
where a ∈ R and B ∈ R1×N .
TABLE I: Table of NotationsVariable Meaningx1:N physical locations in terms of [x, y] coordinates of the N sensors deployed in the field.YN = Y1, . . . , YN ∈ R1×N collection of observations from all sensors (both Network 1 and Network 2) at the fusion center.YH ∈ R1×NH ⊆ YN collection of observations from all sensors in Network 1 at the fusion center.YL ∈ R1×NL ⊆ YN collection of observations from all sensors in Network 2, at the fusion center.fN = f1, . . . , fN ∈ RN×1 realisation of f at the sensors located at x1:N .fH ⊆ fN realisation of f at the sensors of Network 1, located at x1:NH ⊆ x1:N .fL ⊆ fN realisation of f at the sensors of Network 2, located at x1:NL ⊆ x1:N .gL =
g1, . . . , gNL
∈ RNL×1 realisation of the energy field G at the sensors of in Network 2 located at x1:NL .
The optimal linear estimator that solves (8) is givenby
f∗ = Ef∗ YN [f∗ YN ]EYN [YN YN ]−1
(YN − E [YN ]) ,(9)
and the Mean Squared Error (MSE) is given by
σ2∗ = k (x∗,x∗)− Ef∗ YN [f∗ YN ]EYN [YN YN ]
−1
× EYN f∗ [YN f∗] .(10)
Remark 1. Non-zero mean random spatial phenomenon:in order to handle the practical case where µf (x;θf ) 6= 0,we first subtract this known value from our observationsY H (xn) , and Y L (xn). We then apply our algorithm, andfinally shift back our results by µf (x;θf ) .
To evaluate (9-10) we need to calculate thecross-correlation Ef∗,YN [f∗ YN ], auto-correlationEYN
[YN YT
N]
and E [YN ].
B. Cross-correlation between a test point and sensors ob-servations Ef∗,YN [f∗ YN ]
To calculate the cross-correlation vector, we decom-pose the observation vector into its high and lowquality observations, given by:
YN = [YH, YL] , (11)
and express the cross-correlation vectorEf∗,YN [f∗ YN ] in (9-10) as:
Ef∗,YN [f∗ YN ] =[Ef∗,YH [f∗,YH]︸ ︷︷ ︸
q1
,Ef∗,YL [f∗,YL]︸ ︷︷ ︸q2
](12)
We can now derive separately the cross-correlationbetween the test point x∗ and a sensor observationfrom Network 1 and Network 2. We begin with thederivation of the cross correlation between a test pointf∗ and the observations from sensors in Network 1,YH, presented in Lemma 1 followed by the crosscorrelation between a test point f∗ and observationsfrom sensors in Network 2, YL, presented in Lemma2.Lemma 1. (Calculating Ef∗,YH [f∗ YH]):The cross correlation between a test point f∗ and the k-
th (k = 1, . . . , NH) sensor observation in Network 1 isgiven by:
[q1]k := Ef∗,YH(xk) [f∗ YH (xk)] = kf (x∗,xk) .
Proof. See Appendix A
Lemma 2. (Calculating Ef∗,YL [f∗ YL]):The cross correlation between a test point f∗ and the k-
th (k = 1, . . . , NL) sensor observation in Network 2 isgiven by:
[q2]k := Ef∗,YL(xk) [f∗ YL (xk)]
= Cf (x∗,xk)
×
(1− Φ
(T√
Cf (xk,xk)
)+
T√Cf (xk,xk)
φ
(T√
Cf (xk,xk)
)).
Proof. See Appendix B
Using Lemmas 1-2, the cross-correlation vector q1,and q2 in (12) is derived.
C. Correlation of sensors observations EYN
[YN YT
N]
We now derive EYN
[YN YT
N], which is the cor-
relation matrix of all sensors observations from bothNetwork 1 and Network 2. This involves the correla-tion within Network 1 and Network 2 and across thenetworks, given by
EYN
[YN YT
N
]= EYH,YL
[[YH YT
L
], [YH, YL]
]=
[EYH
[YH YT
H]
EYH,YL
[YH YT
L]
EYH,YL
[YL YT
H]
EYL
[YL YT
L] ]
:=
Q1 Q2
QT2 Q4
.(13)
Lemma 3. (Calculating EYH
[YH YT
H]):
The correlation between a sensor observation in Network1, YH (xk) ∈ YH and a sensor observation in Network 1,YH (xj) ∈ YH is given by
[Q1]k,j := EYH(xk),YH(xj) [YH (xk)YH (xj)]
= kf (xk,xj) + 1 (k = j)σ2W.
Proof. See Appendix C
Remark 2. Sensors of high quality have spatially uncor-related thermal noise as no energy harvesting is required,and the Dirac measure on the diagonal when k = j.Lemma 4. (Calculating EYH,YL [YH,YL]):The correlation between a sensor observation in Network
1, YH (xk) ∈ YH and a sensor observation in Network 2,denoted YL (xj) ∈ YL is given by
[Q2]k,j := EYH (xk),YL(xj) [YH (xk) YL (xj)]
= Cf (xk,xj)
(1− Φ
(T√
Cf (xj ,xj)
)
+
(T√
Cf (xj ,xj)
)φ
(T√
Cf (xj ,xj)
)).
Proof. See Appendix D
Next we derive Q4, where we separate this calcu-lation into two cases: the diagonal elements of Q4 arecalculated in Lemma 5, and the non-diagonal elementsin Lemma 6.Lemma 5. (Calculating diagonal elements ofEYL
[YLYT
L]):
The auto-correlation of sensor observations in Network 2,Y (xk) ∈ YL is given by
[Q4]k,k := EYL(xk)[YL (xk) YL (xk)]
= Cf (xk,xk)
(1− Φ
(T√
Cf (xk,xk)
)+
(T√
Cf (xk,xk)
)
× φ(
T√Cf (xk,xk)
))+ exp
(µg (xk) +
Cg (xk, xk)
2
).
Proof. See Appendix E.
We now consider the calculation of the correlationbetween a single sensor observation in Network 2,YL (xk) ∈ YL and a sensor observation in Network 2,YL (xj) ∈ YL at different locations, (ie. non-diagonalelements xk 6= xj). This is given by
[Q4]k,j := EYL(xk),YL(xj) [YL (xk) YL (xj)] .
To obtain this result, we first present the followingTheorem which states useful results regarding thecorrelation of bi-variate truncated Normal randomvariables.Theorem 1 (Correlation of Bivariate Truncated NormalRandom Variables [51]).
Given the standardized bivariate Gaussian distributionZ = [Z1, Z2], where E [Z1 Z2] = ρ, and observationsavailable only inside the region [a ≤ z1 <∞, b ≤ z2 <∞],then the cross correlation E[Z1Z2] is given by:
E [Z1 Z2] =
∞∫a
∞∫b
p (z1, z2) z1z2dz1 dz2 = ρ(aφ (a) (1− Φ (A))
+ bφ (b) (1− Φ (B)) + Ωa,b
)+ (1− ρ2)fZ([a, b]; ρ),
where A = (b−ρa)√1−ρ2
, B = (a−ρb)√1−ρ2
, and Ωa,b :=
P (Z1 ≥ a ∩ Z2 ≥ b) is the joint complementary cumula-tive distribution function (CCDF), given by:
Ωa,b := P (Z1 ≥ a ∩ Z2 ≥ b)
=
∞∫a
∞∫b
fZ(z; ρ)dz1dz2
= (1− Φ (a)) (1− Φ (b)) + φ (a)φ (b)
∞∑n=1
ρn
n!Hn−1 (a)Hn−1 (b) ,
where Φ (·) the distribution function of a standard Gaus-sian, and Hn (z) are the Hermite-Chebyshev polynomials
orthogonal to the standardized normal distribution suchthat
∞∫−∞
Hem(z)Hen(z) e−z2
2 dz =√
2πn!δnm,
and
Hen(z) = (−1)nez2
2dn
dzne−
z2
2
=
(z − d
dz
)n· 1,
or explicitly as
Hen(z) = n!
bn2 c∑m=0
(−1)m
m!(n− 2m)!
zn−2m
2m.
Using Theorem 1, we now derive the non-diagonalelements of Q4, presented in the following Lemma:Lemma 6. (Calculating non-diagonal elements ofEYL [YL,YL]):
The correlation between a single sensor observation inNetwork 2, YL (xk) ∈ YL and a sensor observationin Network 2, YL (xj) ∈ YL at different locations, (ie.xk 6= xj) is given by:
[Q4]k,j = EYL(xk),YL(xj) [YL (xk) YL (xj)]
=√Cf (xk,xk) Cf (xj ,xj)Cf (xk,xj)
× (Tkφ (Tk) (1− Φ (A)) + Tjφ (Tj) (1− Φ (B)) + Ω)
+ (1− C2f (xk,xj))fZ([Tk, Tj ]; C2f (xk,xj)).
where A =(Tj−Cf (xk,xj)Tk)√
1−C2f (xk,xj)
, B =(Tk−Cf (xk,xj)Tj)√
1−C2f (xk,xj)
.
Proof. See Appendix F
D. Expected value of the observations EYN [YN ]
Finally, we need to derive the expected value ofthe observations for the high and low quality sensors.The expected value of the k-th observations for a highquality sensor is given by:
E [Y (xk)] = E [Y (xk) +Wxk] = 0. (14)
Lemma 7. The expected value of the k-th observations fora low quality sensor is presented in the following Lemma.
E[Y (xk)] = Eσ2V
[E[Y (xk)|σ2
V
]]=√Cf (xk,xk)φ (Tk) .
Proof. See Appendix G
We now use the results we derived to express theS-BLUE for the spatial field reconstruction in (9) andthe associated MSE in (10):Corollary 1. The S-BLUE spatial field reconstruction atlocation x∗ is given by:
f∗ =[q1 q2
]Q1 Q2
QT2 Q4
−1
YHYL
− E[YH]E[YL]
.
The predictive variance is given by
σ2∗ = C(x∗,x∗)− [q1 q2]
Q1 Q2
QT2 Q4
−1 [q1 q2]T,
(15)
where q1, q2, Q1, Q2, Q4 are given in Lemma 1, Lemma 2,Lemma 3, Lemma 4, Lemma 5 and , Lemma 6, and E[YH]and E[YL] are given in Eq. (14) and Lemma 7.
There are several quantities in this expression, q1,q2, Q1, Q2, Q4. Each of these quantity will playimportant role in determining the f∗. q1 specifies thecross correlation between a test point f∗ and the k-th (k = 1, . . . , NH) sensor observation in Network1, the higher this value, the larger effect high qualitysensor observation have on f∗. Similarly as q2, itspecifies the cross correlation between a test pointand sensor observation in Network 2. Specifically, insummary, the close the test point to the sensor locationin either Network 1 or Network 2, the more effect willthe sensor have on the estimated quantity at the testlocation.
IV. QUERY BASED SENSOR SET SELECTION WITHPERFORMANCE GUARANTEE
In this Section we develop an algorithm to performon-line sensor set selection in order to meet the re-quirements of a query made by users of the system.In this scenario users can prompt the system andrequest the system to provide an estimated value ofthe spatial random field at a location of interest x∗.The user also provides the required allowed error,quantified by the Mean Squared Error (MSE) of theS-BLUE in Eq. (15). This means that the input tothe system is a pair of values indicating the locationof interest, denoted by x∗ and the maximal alloweduncertainty, denoted by σ2
q . Our algorithm will thenchoose a subset of sensors from both networks toactivate in such a way that meets the QoS criterion(maximal allowed uncertainty) as well as minimisesthe costs of activating the sensors of these networks.It is important to note that the MSE at any locationcan be evaluated without taking any measurements,see (15). This means that our algorithm for choosingwhich sensors to activate does not require the sensorsto be activated beforehand. We now formulate thegeneric sensor set selection problem where the sensorsfrom both Network 1 and Network 2 are candidatesfor activation. We first define the user’s query:Definition 3. (User’s Query):A User’s Query consists of a 2-tuple Q := (x∗, ε), where
1) x∗ ∈ X ⊆ R2 represents the location at whichthe user is interested in estimating the quantity ofinterest, denoted f (x∗).
2) ε ∈ R+ represents the maximum statistical errorwhich the user is willing to allow for the estima-
tion of the quantity of interest at x∗, quantified by
the MSE: E[(f (x∗)− f (x∗)
)2], given in (15).
Based on the query Q, the network outputs a reportR :=
(f (x∗) , σ∗
), with the constraint σ∗ ≤ ε. If this
condition cannot be met, the network reports a Nullvalue and does not activate any sensor configuration.
We defined the activation sets of the sensors in bothnetworks by S1 ∈ 0, 1|NH| ,S2 ∈ 0, 1|NL|. Then thesensor selection problem can be formulated as follows:
S = arg minS1∈0,1|NH|
S2∈0,1|NL|
wh |S1|+ wl |S2| ,
s.t. σ2∗ < σ2
q ,
(16)
where σ2q is the maximal allowed uncertainty at the
query location x∗, and wh and wl are the knowncosts of activating a sensor from Network 1 andNetwork 2, respectively. This optimization problem isnot convex, due to the non-convex Boolean constraintsS1 ∈ 0, 1|NH| ,S2 ∈ 0, 1|NL|. Solving this optimiza-tion problem involves exhaustive evaluation of allpossible combinations of sensor selections which isimpractical for real-time applications. Previous meth-ods to solve such optimization problems in sensorselection involved a relaxation of the non-convex con-straint, see for example [23]–[25]. These approachesprovide sub-optimal solutions and their theoreticalproperties are not well understood. We take a dif-ferent approach for solving the non-convex problemwhich does not involve relaxation, but instead utilizea stochastic optimization technique, known as theCross Entropy Method (CEM). The CEM was firstproposed by Rubinstein in 1999 [52] for rare eventsimulation, but was adapted for solving estimationand optimization problems see [52], [53]. We nowpresent a short overview of the CEM, for more detailssee [52], [53]. We then develop the algorithm to solvethe optimization problem in (16).
A. Cross Entropy Method for Optimization problems
Suppose we wish to maximize a function U (x) oversome set X . Let us denote the maximum by γ∗; thus,
γ∗ = maxx∈X
U(x). (17)
The CEM solves this optimization problem by castingthe original problem (17) into an estimation problemof rare-event probabilities. By doing so, the CEMaims to locate an optimal parametric sampling dis-tribution, that is, a probability distribution on X ,rather than locating the optimal solution directly. Tothis end, we define a collection of indicator functions1S(x)≥γ on X for various levels γ ∈ R. Next, letf(·; V),V ∈ V be a family of probability densitieson X parametrized by a real-valued parameter vector
v. For a fixed u ∈ V we associate with (17) the problemof estimating the rare-event probability
l(γ) = Pu(U(x) ≥ γ) = Eu[1U(x)≥γ
], (18)
where Pu is the probability measure under which therandom state x has a discrete pdf f (·; V) and Eudenotes the corresponding expectation operator. Fora detailed exposition of the CEM, see [52], [53]. TheCE method involves the following iterative procedureshown in Algorithm 1:
Algorithm 1 CE Method
1: Initialization: Choose an initial parameter vectorV
2: while stopping criterion do3: Generate K samples:Γi, where Γi ∼ f (·; Vt);
1 ≤ i ≤ K.4: Evaluate U (Γi) for all the K samples.5: Calculate βt = (1− ρ) quantile of U1, . . . , UK6: Solve the stochastic program to update the pa-
rameter vector V:
Vt = arg maxV
1
K
K∑i=1
1 (U (Γi) ≥ βt) ln (f (Γi; Vt))
7: end while
The most challenging aspect in applying the CEMis the selection of an appropriate class of parametricsampling densities f (·; V) ,V ∈ V . There is not aunique parametric family and the selection is guidedby competing objectives. The class f (·; V) ,V ∈ V hasto be flexible enough to include a reasonable paramet-ric approximation to the optimal importance samplingdensity. The density f (·; V) ,V ∈ V has to be simpleenough to allow fast random variable generation andclosed-form solutions to the optimization problem. Inaddition, to be able to analytically solve the stochasticprogram, then f (·; V) should be a member of theNatural Exponential Families (NEF) of distributions.Under NEFs, the optimization problem can be solvedanalytically in closed form making the CE very easyto implement [53].
B. Cross Entropy Method for Sensor Set Selection
To apply the CEM to solve our optimization prob-lem in (16), we need to choose a parametric distri-bution. Since the activation of the sensors is a bi-nary variable (eg. 0 → don’t activate, 1 → activate),we choose an independent Bernoulli variable as ourparametric distribution, with a single parameter p (ie.V = p). The Bernoulli distribution is a member of theNEF of distributions, hence, an analytical solution of
the stochastic program is available in closed form asfollows:
pt,j =
K∑i=1
1(ΓH
i,j = 1)1 (U (k) ≥ βt)
K∑i=1
1 (U (k) ≥ βt).
Since the optimization problem in Eq. (16) is aconstrained optimization problem, we introduce anAccept\Reject step which rejects samples which do notmeet the QoS criterion σ2
∗ < σ2q , as follows
U (k) =
−(wh∣∣SH ∣∣+ wl
∣∣SL∣∣) , σ2∗ (k) < ε
−∞, Otherwise
The resulting algorithm is presented in Algorithm 2.
V. SIMULATIONS
In this section, we present extensive simulationsto evaluate the performance of the system. First, inSection V-A we present the accuracy of the fieldreconstruction using our proposed S-BLUE algorithmfor synthetic data. Then in Section V-B we presentresults for the field reconstruction of real data set inthe form of wind storm. Finally, in Section V-C wepresent the effectiveness of using Cross Entropy basedalgorithm for sensor selection and activation.
A. Field Reconstruction of Synthetic Data
To generate synthetic data, we used a SquaredExponential kernel Cf
(x1,x2; Ψ :=
σ2, l
)=
σ2 exp(‖x1−x2‖2
2l
)for both spatial random fields,
f (·) and g (·). The hyper-parameters for the randomspatial phenomenon f (·) are: Ψf = 10, 1 andfor the energy harvesting field Ψg = 0.3, 1. Theadditive noise standard deviation is σw = 1, the meanis µf = 8. In Fig. 2, we present a single realisationof the field intensity. In this example we deployed 4high quality and 64 low quality sensors uniformlyin the rectangular region. In Fig. 3 we present thespatial field reconstruction for various of activationspeeds T = 8, 10, 13, 15.
In Fig. 4 we present the point-wise Root SquaredError (RSE) for these T values. Fig. 4 illustrates thatthe estimated wind field closely matches the true windfield for small values of T and does not match the truewind field when T values are high. We also observe
Wind field
0 2 4 6 8
x
0
1
2
3
4
5
6
7
8
9
y
0
2
4
6
8
10
12
14
16
Fig. 2: Wind field intensity
Algorithm 2 Sensor Selection in Heterogeneous SensorNetworks via Cross Entropy method
Require: User’s query Q := (x∗, ε), α, wh, wl and Ψ0. Initialization at iteration t = 0: set pH0 =pH0,1, pH0,2, · · · , pH0,NH
such that pH0,j = 0.5, and setpL0 = pL0,1, pL0,2, · · · , pL0,NL
such that pL0,j = 0.5.while stopping criterion do
1. Generate K independent samples of binary setsΓHi = γHi,1, γHi,2 · · · , γHi,NH
, where γHi,j ∼Ber
(pHt,j); 1 ≤ i ≤ K and
ΓLi = γLi,1, γLi,2 · · · , γLi,NL, where γLi,j ∼ Ber
(pLt,j);
1 ≤ i ≤ K.2. Calculate the MSE values σ2
∗ (k) , k =1, . . . ,K, which would be obtained by activat-ing the corresponding sensors to each of the Ksamples, according to (15).3. Evaluate for each of the K samples the perfor-mance metric
U (k) =
−(wh∣∣SH ∣∣+ wl
∣∣SL∣∣) , σ2∗ (k) < ε
∞, Otherwise
where
SHj =
1, γHj,1 = 1
0, Otherwise
and SLj =
1, γLj,1 = 1
0, Otherwise
4. Calculate the βt = (1−ρ) quantile level of U1:K .5. Update pH as follows:
pHt,j = α
K∑i=1
1(ΓH
i,j = 1) Choose elite samples︷ ︸︸ ︷1 (U (k) ≥ βt)
K∑i=1
1 (U (k) ≥ βt)
+ (1− α)pHt−1,j ,
6. Update pL as follows:
pLt,j = α
K∑i=1
1(ΓL
i,j = 1) Choose elite samples︷ ︸︸ ︷1 (U (k) ≥ βt)
K∑i=1
1 (U (k) ≥ βt)
+ (1− α)pLt−1,j ,
end while7. For each element in pH and pL make the finalbinary activation decision as follows:
SHj =
1, pHt,j ≥ Ψ
0, Otherwiseand SLj =
1, pLt,j ≥ Ψ
0, Otherwise
where Ψ is a pre-defined threshold.8. Evaluate the objective function in (16) withoutthe sensors which do not have enough energy. Ifthe QoS constraint is met, then no further steps arerequired; If the QoS constraint is not met, solvethe optimisation problem again, excluding thosesensors which were not able to transmit.
T=8
0 2 4 6 8
x
0
2
4
6
8
y
0
5
10
15
T=10
0 2 4 6 8
x
0
2
4
6
8
y
0
5
10
15
T=13
0 2 4 6 8
x
0
2
4
6
8
y
0
5
10
15
T=15
0 2 4 6 8
x
0
2
4
6
8
y
0
5
10
15
Fig. 3: S-BLUE wind field reconstruction of Corol-lary 1 for different activation speed thresholds T =0, 2, 5, 7
RSE T=8
0 2 4 6 8
x
0
2
4
6
8
y
1
1.5
2
2.5
3
RSE T=10
0 2 4 6 8
x
0
2
4
6
8
y
1
1.5
2
2.5
3
RSE T=13
0 2 4 6 8
x
0
2
4
6
8
y
1
1.5
2
2.5
3
RSE T=15
0 2 4 6 8
x
0
2
4
6
8y
1
1.5
2
2.5
3
Fig. 4: Root Mean Squared Error (RMSE) estimation ofthe wind field intensity
that RSE decreases very fast with respect to increasingvalues of T . In addition, we observe that the RSEvalues are low in the region where many high andlow quality sensors are distributed and high in theregion where few high and low quality sensors aredistributed.
In Fig. 5 we present a quantitative comparison ofRSE over 100 realizations from the spatial field withrespect of different number of high quality and lowquality sensors when T = 8, as a function of thenumber of low quality sensors. We set the numberof high quality sensors to 4, 9, 16, 25 and vary thenumber of low quality sensors from 4 to 250. Thefigure shows how adding low quality sensors aids inreducing the overall RSE.
B. Field Reconstruction of Storm Surge Data Set
In order to test our algorithm on real data sets,we use a publicly available insurance storm surgedatabase known as the Extreme Wind Storms Cat-
5 50 100 150 200 250
Number of low quality sensors
1.5
2
2.5
3
RS
E
Number of high quality sensors =4
Number of high quality sensors =9
Number of high quality sensors =16
Number of high quality sensors=25
Fig. 5: RSE as a function different configurations ofnumber of high and low quality sensors
alogue 3. The data is available for research as theXWS Datasets: (c) Copyright Met Office, Universityof Reading and University of Exeter. Licensed underCreative Commons CC BY 4.0 International License.This database is comprised of 23 storms which causedhigh insurance losses known as ‘insurance storms’ and27 storms which were selected because they are the top‘non-insurance’ storms as ranked by the storm severityindex, see details on the website. The data providedis comprehensive and provides features such as thefootprint of the observations on a location grid witha rotated pole at longitude = 177.5 degrees, latitude =37.5 degrees. As discussed in the data description pro-vided with the data-set, this is a standard techniqueused to ensure that the spacing in km between gridpoints remains relatively consistent. The footprints areon a regular grid in the rotated coordinate system,with horizontal grid spacing 0.22 degrees. The data foreach of the storms provides a list of grid number andmaximum 3-second gust speed in meters per second.The true locations (longitude and latitude) of the gridpoints are given in grid locations file. We selected onestorm to analyse, known as Dagmar which took placeon 26/12/2011 and affected Finland and Norway. Tocalibrate the model we first fit the hyperparameters ofthe model via Maximum Likelihood Estimation (MLE)procedure. We used a 2-D radial basis function, of thefollowing form
C (xi,xj ; Ψ) := σ2x exp
(−|xi − xj |
lx
)exp
(−|yi − yj |
ly
),
thus decomposing the kernel into orthogonal coordi-nates which we found provided a much more accuratefit. The reason for this is it allows for inhomogeneitythrough differences in spatial dependence in verticaland horizontal directions, which is highly likely tooccur in the types of wind speed data studied. TheMLE of the length and scale parameters obtained aregiven by σ2
x = 0.1, lx = 0.5 and σ2y = 10, ly = 0.1. De-
tails on how to estimate the GP hyperparameters canbe found in [Chapter 5] [43]. In our model, historicaldata is used in order to estimate the hyperparametersof the model at the current time. Then, using theseparameters we perform all the inferential tasks.
The left panel of Fig. 6 shows the region of intereston the map. Both high and low quality sensors are se-
3http://www.met.reading.ac.uk/ extws/database/dataDesc
lected randomly within the region. In this experimentwe uniformly deployed 50 high quality and 250 lowquality sensors. The right panel of Fig. 6 shows theDagmar storm wind speed intensity. The left columnof Fig. 7 presents the estimated wind speed intensitywith varying activation speed thresholds T and theright column presents the spatial RSE values. Thefigure shows that the true wind intensity field can berecovered for low activation speed, but as the thresh-old increases, the performance deteriorates. The RSEis lower at the points where sensors are deployed, andgrows with the increase of activation speed threshold.The standard deviation is very low in the middleregion, close to a value of 2 and a bit high in theboundary region where fewer sensors are deployed.
Finally, in Fig. 8 we present a quantitative com-parison of the RMSE for various values of high andlow quality sensors. The result shows a clear trendof RMSE with the increasing of high and low qualitysensors.
C. Sensor Selection
In this section we illustrate how our sensor selectionalgorithm performs. For comparison, we use a optimalselection method which only selects the sensor setcollections that minimize the U values and ensuresthat the QoS criterion is being met. The simulationparameters we have are: Nh = 5, Nl = 10, T =8, wh = 150, wl = 30, σw = 1, σg = 0.3, kf (x∗, x∗) =10, x∗ = 3.5, y∗ = 3.1, ε = 5.4, 5.6, 5.8, 6, 6.2.
We fix the Nh = 5, Nl = 10. The compari-son is shown in Fig. 9. We change the ε within5.4, 5.6, 5.8, 6, 6.2. We also increase the number ofiterations in CE method from 1 to 10. It shows CEmethod converges quickly to the optimal selectionalgorithm within 10 iterations for all the ε values.
To compare our method with convex optimizationapproach, we followed a similar line of thought whichwas presented in [38]. We compared the performanceof our CEM algorithm with the relaxation-based opti-mization algorithm. The result shows that for differentQoS, ε, our CEM has a significant lower cost comparedto the convex optimization scenario, as shown in thefigure 10.
VI. CONCLUSIONS
We addressed the problem of spatial field recon-struction and sensor selection in heterogeneous sensornetworks, containing two types of sensors: expensive,high quality sensors; and cheap, low quality sensorswhich are activated only if the intensity of the spatialfield exceeds a pre-defined activation threshold. Inaddition, these sensors are powered by means ofenergy harvesting which impacts their accuracy. Wethen addressed the problems of performing spatial fieldreconstruction and query based sensor set selection withperformance guarantee. We solved the first problem by
High quality
Low quality
0
5
10
15
20
25
30
35
40
Fig. 6: Left panel: map of region of interest with sensors locations. Right panel: Dagmar surge storm intensitymap
Estimated field, T=0
0
10
20
30
40
Estimated field, T=15
0
10
20
30
40
Estimated field, T=35
0
10
20
30
40
RMSE T=0
0
2
4
6
RMSE T=15
0
2
4
6
RMSE T=35
0
2
4
6
Fig. 7: True storm field and estimated storm field withvarious activation speed.
50 100 150 200 250
Nl
4
4.2
4.4
4.6
4.8
5
5.2
5.4
5.6
5.8
RM
SE
Nh=10
Nh=20
Nh=30
Nh=40
Nh=50
Fig. 8: RMSE with effect of different number of highand low quality sensors.
developing a low complexity algorithm based on thespatial best linear unbiased estimator (S-BLUE). Next,building on the S-BLUE, developed an efficient algo-rithm for query based sensor set selection with performanceguarantee, based on the Cross Entropy method whichsolves the combinatorial optimization problem in anefficient manner. We presented a comprehensive studyof the performance gain that can be obtained byaugmenting the high-quality sensors with low-qualitysensors using both synthetic and real insurance stormsurge database known as the Extreme Wind StormsCatalogue.
APPENDIX APROOF OF LEMMA 1
Using the the law of total expectation, the propertiesof the GP and the fact that f (xk) and Wk are indepen-
1 2 3 4 5 6 7 8 9 10
Number of Iterations
0
100
200
300
400
500
600
U
CE = 5.4
Opt = 5.4
CE = 5.6
Opt = 5.6
CE = 5.8
Opt = 5.8
CE = 6
Opt = 6
CE = 6.2
Opt = 6.2
Fig. 9: Comparison of U values between optimalscheme and CE method with effect of number ofiterations.
1 2 3 4 5 6 7 8 9 10
Number of Iterations
100
200
300
400
500
600
700
800
U
CE = 5.03
Convex = 5.03
CE = 5.04
Convex = 5.04
CE = 5.05
Convex = 5.05
Fig. 10: Comparison of objective function values be-tween Convex optimization scheme and CE method.
dent, we obtain that:
Ef∗,YH(xk) [f∗YH (xk)] = Ef∗,YH(xk) [f∗ (f (xk) +W (xk))]
= kf (x∗,xk) .(19)
APPENDIX BPROOF OF LEMMA 2
Ef∗,YL(xk)[f∗ YL (xk)]
= Ef(xk),σ2V(xk)
[Ef∗,Y (xk)
[f∗Y (xk) |f (xk) , σ2
V (xk)]]
=Cf (x∗,xk)
Cf (xk,xk)
∞∫0
∞∫T
f2 (xk)N(f (xk) ; 0, Cf (xk,xk)
)df (xk) p(σ2
V (xk))
dσ2V (xk)
We can derive
Ef∗,YL(xk)[f∗ YL (xk)]
= Cf (x∗,xk)
(1− Φ
(T√
Cf (xk,xk)
)
+T√
Cf (xk,xk)φ
(T√
Cf (xk,xk)
)).
APPENDIX CPROOF OF LEMMA 3
EYH (xk),YH(xj) [YH (xk) YH (xj)]
= Ef(xk),f(xj),W (xk),W(xj) [(f (xk) +W (xk)) (f (xj) +W (xj))]
= kf (xk,xj) + 1 (k = j)σ2W.
APPENDIX DPROOF OF LEMMA 4
EYH (xk),YL(xj) [YH (xk) YL (xj)]
= Ef(xj),σ2V(xj)
[Ef(xk),W (xk),V (xj)
[((f (xk) +W (xk)) (f (xj) + V (xj))
|f (xj) , σ2V (xj)
)]]= Ef(xj),σ2
V(xj)
[Cf (xk,xj)
Cf (xj ,xj)f (xj)
2 1 (f (xj) > T )
]
Now we can follow the derivation in Appendix B, andwe can get
EYH (xk),YL(xj) [YH (xk) YL (xj)]
= Cf (xk,xj)
(1− Φ
(T√
Cf (xj ,xj)
)
+
(T√
Cf (xj ,xj)
)φ
(T√
Cf (xj ,xj)
)).
APPENDIX EPROOF OF LEMMA 5
EYL(xk)[YL (xk) YL (xk)]
= EYL(xk),V (xk)
[(f (xk) + V (xk))2 1 (f (xk) > T )
]= Ef(xk)
[f (xk)2 1 (f (xk) > T )
]+ EV (xk)
[V (xk)2
]= Cf (xk,xk)
(1− Φ
(T√
Cf (xk,xk)
)
+
(T√
Cf (xk,xk)
)φ
(T√
Cf (xk,xk)
))
+ exp
(µg (xk) +
Cg (xk, xk)
2
).
APPENDIX FPROOF OF LEMMA 6
[Q4]k,j := EYL(xk),YL(xj) [YL (xk) YL (xj)]
× 1 (f (xk) > T, f (xj) > T ) |σ2V (xk) , σ2
V (xj)] (20)
Note in the above equations, all the cross mo-ments terms relating to the product of V (xk)V (xj),f (xk)V (xj) and f (xj)V (xj) will become zero since
there is independence between these terms. So theequation reduces to:
[Q4]k,j =
∞∫T
∞∫T
p (f (xk) , f (xj)) f (xk) f (xj) df (xk) df (xj)
= Efs(xk)
√Cf (xk,xk),fs(xj)
√Cf (xj ,xj)[(
fs (xk)√Cf (xk,xk)
)(fs (xj)
√Cf (xj ,xj)
)× 1 (fs (xk) > Tk, fs (xj) > Tj)
]=√Cf (xk,xk) Cf (xj ,xj)
∞∫Tk
∞∫Tj
p (fs (xk) , fs (xj)) fs (xk) fs (xj)
dfs (xk) dfs (xj)
=√Cf (xk,xk) Cf (xj ,xj)Efs(xk),fs(xj)
[fs (xk) fs (xj)
].
(21)
Finally, utilising Theorem 1 we obtain the result.
APPENDIX GPROOF OF LEMMA 7
E[Y (xk)] = Eσ2V
[E[Y (xk)|σ2
V
]]= Eσ2
V
[E[(f(xk) + V (xk))1(f(xk) ≥ T )|σ2
V + V (xk)1(f(xk) < T )|σ2V
]]= Eσ2
V[E [f(xk)1(f(xk) ≥ T )]]
= Eσ2V
[∫f(xk)p(f(xk))1(f(xk) ≥ T )df(xk)
]= Eσ2
V
[1√
Cf (xk,xk)
∫ +∞
Tf(xk)φ
(f(xk)√Cf (xk,xk)
)df(xk)
]
= −√Cf (xk,xk)
(φ
(f(xk)√Cf (xk,xk)
))∣∣∞T
=√Cf (xk,xk)φ (Tk) .
ACKNOWLEDGMENT
This work was supported by the Korea Institute of EnergyTechnology Evaluation and Planning (KETEP) and the Ministry ofTrade, Industry & Energy (MOTIE) of the Republic of Korea (No.20148510011150). F. Septier would like to acknowledge support ofthe Institute of Statistical Mathematics, Tokyo, Japan, and the sup-port of the BNPSI ANR project no ANR-13-BS-03-0006-01. This workwas supported by the Korea Institute of Energy Technology Eval-uation and Planning (KETEP) and the Ministry of Trade, Industry& Energy (MOTIE) of the Republic of Korea (No. 20148510011150).F. Septier would like to acknowledge support of the Institute ofStatistical Mathematics, Tokyo, Japan, and the support of the BNPSIANR project no ANR-13-BS-03-0006-01. This work was financiallysupported by the Singapore National Research Foundation underits Campus for Research Excellence And Technological Enterprise(CREATE) program.
REFERENCES
[1] J. K. Hart and K. Martinez, “Environmental Sensor Networks:A revolution in the earth system science?” Earth-Science Re-views, vol. 78, no. 3, pp. 177–191, 2006.
[2] S. Rajasegarar, T. C. Havens, S. Karunasekera, C. Leckie,J. C. Bezdek, M. Jamriska, A. Gunatilaka, A. Skvortsov, andM. Palaniswami, “High-Resolution Monitoring of AtmosphericPollutants Using a System of Low-Cost Sensors,” IEEE Trans-actions on Geoscience and Remote Sensing, vol. 52, pp. 3823–3832,2014.
[3] C. Fonseca and H. Ferreira, “Stability and contagion mea-sures for spatial extreme value analyses,” arXiv preprintarXiv:1206.1228, 2012.
[4] J. P. French and S. R. Sain, “Spatio-Temporal ExceedanceLocations and Confidence Regions,” Annals of Applied Statistics.Prepress, 2013.
[5] K. Sohraby, D. Minoli, and T. Znati, Wireless sensor networks:technology, protocols, and applications. John Wiley & Sons, 2007.
[6] K. Lorincz, D. J. Malan, T. R. F. Fulford-Jones, A. Nawoj,A. Clavel, V. Shnayder, G. Mainland, M. Welsh, and S. Moul-ton, “Sensor networks for emergency response: challenges andopportunities,” IEEE Pervasive Computing, vol. 3, no. 4, pp. 16–23, 2004.
[7] K. Chintalapudi, T. Fu, J. Paek, N. Kothari, S. Rangwala, J. Caf-frey, R. Govindan, E. Johnson, and S. Masri, “Monitoring civilstructures with a wireless sensor network,” Internet Computing,IEEE, vol. 10, no. 2, pp. 26–34, 2006.
[8] I. F. Akyildiz, W. Su, Y. Sankarasubramaniam, and E. Cayirci,“Wireless sensor networks: a survey,” Computer Networks,vol. 38, no. 4, pp. 393–422, 2002.
[9] F. Fazel, M. Fazel, and M. Stojanovic, “Random access sensornetworks: Field reconstruction from incomplete data,” in IEEEInformation Theory and Applications Workshop (ITA), 2012, pp.300–305.
[10] J. Matamoros, F. Fabbri, C. Anton-Haro, and D. Dardari, “Onthe estimation of randomly sampled 2d spatial fields underbandwidth constraints,” IEEE Transactions on Wireless Commu-nications,, vol. 10, no. 12, pp. 4184–4192, 2011.
[11] M. C. Vuran, O. B. Akan, and I. F. Akyildiz, “Spatio-temporalcorrelation: theory and applications for wireless sensor net-works,” Computer Networks Journal, Elsevier, vol. 45, pp. 245–259, 2004.
[12] “Environment Protection Authority Victoria Sen-sor Locations,” 2012. [Online]. Available:http://www.epa.vic.gov.au/air/airmap
[13] G. W. Peters, I. Nevat, and T. Matsui, “How to Utilize SensorNetwork Data to Efficiently Perform Model Calibration andSpatial Field Reconstruction,” in Modern Methodology and Ap-plications in Spatial-Temporal Modeling. Springer, 2015, pp. 25–62.
[14] G. Peters, I. Nevat, S. Lin, and T. Matsui, “Modelling thresholdexceedence levels for spatial stochastic processes observed bysensor networks,” in 2014 IEEE Ninth International Conferenceon Intelligent Sensors, Sensor Networks and Information Processing(ISSNIP). IEEE, 2014, pp. 1–7.
[15] S. Rajasegarar, P. Zhang, Y. Zhou, S. Karunasekera, C. Leckie,and M. Palaniswami, “High resolution spatio-temporal mon-itoring of air pollutants using wireless sensor networks,” in2014 IEEE Ninth International Conference on Intelligent Sensors,Sensor Networks and Information Processing (ISSNIP). IEEE,2014, pp. 1–6.
[16] J. Gubbi, R. Buyya, S. Marusic, and M. Palaniswami, “Internetof Things (IoT): A vision, architectural elements, and futuredirections,” Future Generation Computer Systems, vol. 29, no. 7,pp. 1645–1660, 2013.
[17] O. Vermesan, P. Friess, P. Guillemin, S. Gusmeroli, H. Sund-maeker, A. Bassi, I. S. Jubert, M. Mazura, M. Harrison,M. Eisenhauer, and Others, “Internet of things strategic re-search roadmap,” O. Vermesan, P. Friess, P. Guillemin, S. Gus-meroli, H. Sundmaeker, A. Bassi, et al., Internet of Things: GlobalTechnological and Societal Trends, vol. 1, pp. 9–52, 2011.
[18] C. Perera, A. Zaslavsky, C. H. Liu, M. Compton, P. Christen,and D. Georgakopoulos, “Sensor search techniques for sensingas a service architecture for the internet of things,” IEEE SensorsJournal, vol. 14, no. 2, pp. 406–420, 2014.
[19] T. Watkins, “DRAFT Roadmap for Next Gen-eration Air Monitoring,” 2013. [Online]. Avail-able: http://www.epa.gov/airscience/docs/next-generation-air-monitoring-region4.pdf
[20] G. Werner-Allen, K. Lorincz, M. Ruiz, O. Marcillo, J. Johnson,J. Lees, and M. Welsh, “Deploying a wireless sensor networkon an active volcano,” IEEE Internet Computing, vol. 10, no. 2,pp. 18–25, 2006.
[21] I. Nevat, G. W. Peters, F. Septier, and T. Matsui, “Estimationof Spatially Correlated Random Fields in Heterogeneous Wire-less Sensor Networks,” IEEE Transactions on Signal Processing,vol. 63, no. 10, pp. 2597–2609, 2015.
[22] I. Nevat, G. W. Peters, and I. B. Collings, “Random Field Re-construction With Quantization in Wireless Sensor Networks,”
IEEE Transactions on Signal Processing, vol. 61, pp. 6020–6033,2013.
[23] M. Calvo-Fullana, J. Matamoros, and C. Anton-Haro, “SensorSelection and Power Allocation Strategies for Energy Harvest-ing Wireless Sensor Networks,” arXiv preprint arXiv:1608.03875,2016.
[24] S. Joshi and S. Boyd, “Sensor selection via convex optimiza-tion,” IEEE Transactions on Signal Processing, vol. 57, no. 2, pp.451–462, 2009.
[25] S. P. Chepuri and G. Leus, “Sparsity-promoting sensor selec-tion for non-linear measurement models,” IEEE Transactions onSignal Processing, vol. 63, no. 3, pp. 684–698, 2015.
[26] I. F. Akyildiz, M. C. Vuran, and O. B. Akan, “On exploitingspatial and temporal correlation in wireless sensor networks,”Proceedings of WiOpt04: Modeling and Optimization in Mobile, AdHoc and Wireless Networks, pp. 71–80, 2004.
[27] D. Gu and H. Hu, “Spatial Gaussian Process Regression WithMobile Sensor Networks,” IEEE Transactions on Neural Networksand Learning Systems,, vol. 23, no. 8, pp. 1279–1290, 2012.
[28] I. Nevat, G. W. Peters, and I. B. Collings, “Location-awarecooperative spectrum sensing via Gaussian Processes,” in Com-munications Theory Workshop (AusCTW), 2012 Australian. IEEE,2012, pp. 19–24.
[29] H. Sheng, J. Xiao, Y. Cheng, Q. Ni, and S. Wang, “Short-termsolar power forecasting based on weighted gaussian processregression,” IEEE Transactions on Industrial Electronics, vol. PP,no. 99, pp. 1–1, 2017.
[30] P. A. Plonski, P. Tokekar, and V. Isler, “Energy-efficient PathPlanning for Solar-powered Mobile Robots,” Journal of FieldRobotics, vol. 30, no. 4, pp. 583–601, 2013.
[31] R. G. Cid-Fuentes, A. Cabellos-Aparicio, and E. Alarcn, “En-ergy buffer dimensioning through energy-erlangs in spatio-temporal-correlated energy-harvesting-enabled wireless sensornetworks,” IEEE Journal on Emerging and Selected Topics inCircuits and Systems, vol. 4, no. 3, pp. 301–312, Sept 2014.
[32] M. Y. Naderi, K. R. Chowdhury, and S. Basagni, “Wireless sen-sor networks with RF energy harvesting: Energy models andanalysis,” in 2015 IEEE Wireless Communications and NetworkingConference (WCNC). IEEE, 2015, pp. 1494–1499.
[33] D. Oliveira and R. Oliveira, “Characterization of energy avail-ability in rf energy harvesting networks,” Mathematical Prob-lems in Engineering, vol. 2016, 2016.
[34] P. Agrawal and N. Patwari, “Correlated link shadow fadingin multi-hop wireless networks,” IEEE Transactions on WirelessCommunications, vol. 8, no. 8, pp. 4024–4036, 2009.
[35] S. Park and S. Choi, “Gaussian processes for source separa-tion,” in IEEE International Conference on Acoustics, Speech andSignal Processing (ICASSP), 2008, pp. 1909–1912.
[36] A. Kottas, Z. Wang, and A. Rodrguez, “Spatial modeling forrisk assessment of extreme values from environmental timeseries: a Bayesian nonparametric approach,” Environmetrics,vol. 23, no. 8, pp. 649–662, 2012.
[37] Y. Xu and J. Choi, “Adaptive sampling for learning Gaussianprocesses using mobile sensor networks,” International Journalon Sensors, vol. 11, no. 3, pp. 3051–3066, 2011.
[38] A. Krause, A. Singh, and C. Guestrin, “Near-optimal sensorplacements in Gaussian processes: Theory, efficient algorithmsand empirical studies,” The Journal of Machine Learning Research,vol. 9, pp. 235–284, 2008.
[39] S. Basagni, M. Y. Naderi, C. Petrioli, and D. Spenza, “Wire-less sensor networks with energy harvesting,” Mobile Ad HocNetworking: The Cutting Edge Directions, pp. 701–736, 2013.
[40] S. Liu, A. Vempaty, M. Fardad, E. Masazade, and P. K. Varsh-ney, “Energy-aware sensor selection in field reconstruction,”IEEE Signal Processing Letters, vol. 21, no. 12, pp. 1476–1480,2014.
[41] Y. Zhang, T. N. Hoang, K. H. Low, and M. Kankanhalli, “Near-optimal active learning of multi-output Gaussian processes,”arXiv preprint arXiv:1511.06891, 2015.
[42] C. K. Ling, K. H. Low, and P. Jaillet, “Gaussian processplanning with Lipschitz continuous reward functions: Towardsunifying Bayesian optimization, active learning, and beyond,”arXiv preprint arXiv:1511.06890, 2015.
[43] C. E. Rasmussen and C. K. I. Williams, Gaussian Processes forMachine Learning (Adaptive Computation and Machine Learning).The MIT Press, 2005.
[44] R. J. Adler and J. E. Taylor, Random fields and geometry. SpringerVerlag, 2007, vol. 115.
[45] “Anemometer Wind Speed Sensor w/Analog Volt-age Output,” Tech. Rep., 2015. [Online]. Available:https://www.adafruit.com/product/1733
[46] “ANEMO 4403 RF WINDSPEED METER (ANEMOMETER)WITH WM44 P RF DISPLAY UNIT WIRELESS WINDSPEED METER,” Tech. Rep., 2015. [Online]. Available:http://cranesafety.co.za/products
[47] Y. Zhang and A. Srivastava, “Accurate temperature estimationusing noisy thermal sensors for Gaussian and non-Gaussiancases,” Very Large Scale Integration (VLSI) Systems, IEEE Trans-actions on, vol. 19, no. 9, pp. 1617–1626, 2011.
[48] H. E. Daniels, “Saddlepoint approximations in statistics,” TheAnnals of Mathematical Statistics, vol. 25, no. 4, pp. 631–650,1954.
[49] C. Wang and R. M. Neal, “Gaussian process regressionwith heteroscedastic or non-gaussian residuals,” arXiv preprintarXiv:1212.6246, 2012.
[50] S. M. Kay, Fundamentals of Statistical Signal Processing, Volume2: Detection Theory. Prentice Hall PTR, 1998.
[51] M. H. Begier and M. A. Hamdan, “Correlation in a bivariatenormal distribution with truncation in both variables,” Aus-tralian Journal of Statistics, vol. 13, no. 2, pp. 77–82, 1971.
[52] R. Rubinstein, “The cross-entropy method for combinatorialand continuous optimization,” Methodology and computing inapplied probability, vol. 1, no. 2, pp. 127–190, 1999.
[53] P.-T. de Boer, D. P. Kroese, S. Mannor, and R. Y. Rubinstein, “ATutorial on the Cross-Entropy Method,” Annals of OperationsResearch, vol. 134, no. 1, pp. 19–67, feb 2005. [Online]. Available:http://link.springer.com/10.1007/s10479-005-5724-z
Pengfei Zhang received his Bachelor of En-gineering in Electrical & Electronic Engineer-ing and Ph.D from Nanyang TechnologicalUniversity in 2010 and 2015 respectively. Heis now working in department of engineer-ing science in Oxford University as post-doctoral researcher. The topic he is workingon is machine learning in wireless sensornetworks. Before that, he worked as researchscientist in Institute for Infocomm Research,under Sense & Sense-abilities Programme
from 2014 to 2016. His research interests include energy efficientclustering algorithms, energy harvesting wireless sensor networks(EH-WSN), and statistical modeling in WSN.
Ido Nevat received the B.Sc. degree in elec-trical engineering from the Technion-IsraelInstitute of Technology, Haifa, Israel, in 1998and the Ph.D. degree in electrical engineer-ing from the University of New South Wales,Sydney, NSW, Australia, in 2010. Between2010 and 2013, he was a Postdoctoral Re-search Fellow with the Wireless and Net-working Technologies Laboratory at CSIRO,Australia. Between 2013 and 2016, he wasa Scientist with the Institute for Infocomm
Research, Singapore. Since 2017, he has been a team leader at TUM-CREATE, Singapore. His main research interests include statisticalsignal processing, machine learning, and Bayesian statistics.
Dr. Gareth W. Peters is the chair Profes-sor for risk and insurance modelling in theDepartment of Actuarial Mathematics andStatistics, in Heriot-Watt University in Edin-burgh. Previously he held tenured positionsin the Department of Statistical Sciences,University College London, UK and the De-partment of Mathematics and Statistics inUniversity of New South Wales, Sydney,Australia. Dr. Peters is the incomming Direc-tor of the Scottish Financial Risk Association.
In addition he continues to hold positions as a Principle Investigatorin CSML , University College London (UCL) and an AcademicMember of the UK PhD Center in Financial Computing (UCL).He has published in excess of 100 peer reviewed articles on riskand insurance modelling, 2 research text books on Operational Riskand Insurance as well as being the editor and contributor to 3edited text books on spatial statistics and Monte Carlo methods. Heholds positions as an Honorary Professor of Statistics at UniversityCollege London, Adjunct Scientist in the Mathematics, Informaticsand Statistics, Commonwealth Scientific and Industrial ResearchOrganisation (CSIRO) since 2009 as well being an Associate MemberOxford-Man Institute in Oxford University since 2012, an AssociateMember Systemic Risk Center in London School of Economics since2014; an Affiliated Prof. School of Earth and Space Sciences, PekingUniversity PKU,Beijing, China since 2015 and a Visiting Prof. in theInstitute of Statistical Mathematics, Tokyo, Japan each year since2010.
Francois Septier received the Engineer De-gree in electrical engineering and signal pro-cessing in 2004 from Telecom Lille (France),and a Ph.D. in Electrical Engineering fromthe University of Valenciennes (France) in2008. From March 2008 to August 2009,he was a Research Associate in the SignalProcessing and Communications Laboratory,Cambridge University, Engineering Depart-ment, UK. Since August 2009, he is an As-sociate Professor with the IMT Lille Douai
/ CRIStAL UMR CNRS 9189, France. His research focuses onBayesian computational methodology with a particular emphasison the development of Monte Carlo based approaches for complexand high-dimensional problems.
Mike Osborne , Dyson Associate Profes-sor in Machine Learning in the Departmentof Engineering Science, co-director of theOxford Martin programme on Technologyand Employment, and Faculty Member ofthe Oxford-Man Institute for QuantitativeFinance. He has expertise in active learn-ing, Gaussian processes, Bayesian optimisa-tion, and Bayesian quadrature, and is a co-founder of the emerging field of probabilisticnumerics. His algorithms have been applied
in fields as diverse as astrostatistics, ornithology, and sensor net-works.
