Modeling spatially-correlated data of sensor networks with irregular topologies

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1 This full text pape was pee eviewed at the diection of IEEE Communications Society subject matte expets fo publication in the IEEE SECON 25 poceedings Modeling spatially-coelated data of senso netwoks with iegula topologies Apoova Jindal Depatment of Electical Engineeing - Systems Univesity of Southen Califonia Los Angeles, CA apoovaj@uscedu Konstantinos Psounis Depatment of Electical Engineeing - Systems Univesity of Southen Califonia Los Angeles, CA kpsounis@uscedu Abstact The physical phenomena monitoed by senso netwoks, eg foest tempeatue, usually yield sensed data that ae stongly coelated in space We have ecently intoduced a mathematical model fo such data, and used it to geneate synthetic taces and study the pefomance of algoithms whose behavio depends on this spatial coelation [] That wok studied senso netwoks with gid topologies This wok extends ou modeling methodology to senso netwoks with iegula topologies We descibe a igoous mathematical pocedue and a simple pactical method to extact the model paametes fom eal taces We also show how to efficiently geneate synthetic taces that coespond to senso netwoks with abitay topologies using the poposed model The coectness of the model is veified by statistically compaing synthetic and eal data Futhe, the model is validated by compaing the pefomance of algoithms whose behavio depends on the degee of spatial coelation in data, unde eal and synthetic taces The eal taces ae obtained fom both publically available senso data, and senso netwoks that we deploy Finally, we augment ou existing tace-geneation tool with new functionality suited fo senso netwoks with iegula topologies I INTRODUCTION The wieless senso netwoks of the nea futue ae envisioned to consist of a lage numbe of inexpensive wieless nodes These nodes will opeate unde significant powe constaints, which pecludes them fom using lage tansmission anges This, togethe with the low cost of individual sensos, implies that sensos will be densely deployed As a esult, it is expected that a high degee of spatial coelation will exist in the senso netwok data Many algoithms have been poposed that exploit this coelation Fo example, spatial coelation has been used in data aggegation and outing algoithms [2] [5], data stoage and queying [6], [7], [8], mac potocol design [9], data compession and encoding [], and calibation [] The evaluation of potocols that ae sensitive to the spatial featues of input data equies data epesenting a wide ange of ealistic conditions Howeve, since vey few eal systems have been deployed, thee is hadly any expeimental data available to test the poposed algoithms As a esult, senso netwok eseaches make diffeent assumptions when geneating data inputs to evaluate systems; some assume the data to be jointly Gaussian with the coelation being a function of the distance [9], some assume that the data follows the diffusion popety [8], and some assume a function fo the joint entopy of the data [3] Othe eseaches popose the use of envionmental monitoing data obtained fom emote sensing [6], howeve the ganulaity of these data sets do not match the expected ganulaity of senso netwoks data With this in mind, the authos in [2] poposed a method to intepolate existing expeimental data to suppot iegula topologies and incease ganulaity The goal of this pape and its pedecesso [] is to come up with a pasimonious mathematical model that can captue spatial coelation of any degee iespective of the ganulaity, density, numbe of souce nodes o topology Thee ae many benefits fom such an endeavo Fist, the model will povide a pocedue to synthetically geneate senso netwoks data exhibiting vaious degees of coelation, enabling a meticulous study of the pefomance of poposed algoithms Second, it will enable diffeent eseaches to evaluate diffeent algoithms using a common tace geneation method, which, in tun, will make compaisons between diffeent algoithms meaningful In othe wods, the model can seve as a benchmak Thid, it will povide guidelines fo designing algoithms that exploit coelations in an optimal manne In [] we intoduced such a model and used it to geneate synthetic taces We established the validity of the model by compaing the statistical popeties of oiginal (envionmental) and synthetic data, and the pefomance of senso netwok algoithms, whose behavio depends on the spatial coelation, unde oiginal and synthetic data The main dawback of this fist attempt is that it equies the senso netwok to have a gid topology Unfotunately, fixing this poblem is not staightfowad The simple appoach of geneating a huge numbe of gid-based data and then keeping only the values that coespond to senso locations is computationally vey wasteful Futhe, when this appoach is used, the model paametes depend on the specific node locations; diffeent locations would yield diffeent paametes to descibe the same coelation stuctue Hence, the cuent model is unsuitable fo iegula topologies The main contibution of this wok is the intoduction of a new, moe geneal flavo of the model in [] that can be efficiently used fo abitay topologies The model comes with a igoous mathematical pocedue and a simple pactical /5/$2 25 IEEE 35

2 method to extact the model paametes fom eal taces It also comes with an efficient method to geneate synthetic taces that coespond to iegula topologies Futhe, it is shown that pio appoaches to model spatial coelations, eg assuming that data ae jointly Gaussian, ae subcases of ou moe geneal model Finally, publicly available tace-geneation tools ae ceated as pat of this wok The pape is oganized as follows Section II intoduces the vaiogam, which is a handy metic to chaacteize spatial coelation in data Section III summaizes ou model fo senso netwoks with gid topologies and motivates the need fo a new vesion The new model is pesented in Section IV, followed by a mathematically igoous pocedue and a simple, pactical method to infe the model paametes in Section V In Section VI we veify the coectness of the model by statistically compaing synthetic and eal data, and validate it by compaing the pefomance of a well know algoithm, CMAC [9], unde the expeimental and the synthetic data The eal taces ae obtained fom both publically available stuctual health monitoing data, and senso netwoks that we deploy Finally, Section VII descibes the tace-geneation tools and Section VIII concludes the wok II VARIOGRAM: ASTATISTIC TO MEASURE CORRELATION IN DATA A statistic often used to chaacteize spatial coelation in data is the vaiogam [2] [4] Given a two dimensional stationay pocess V (x, y), the vaiogam (also called semivaiance) is defined as γ(d,d 2 )= 2 E[(V (x, y) V (x + d,y+ d 2 )) 2 ] Fo isotopic andom pocesses [5] the vaiogam depends only on the distance d = d 2 + d2 2 between two nodes In this case, if (x d,y d ) denotes a point which is d distance away fom (x, y), γ(d) = 2 E[(V (x, y) V (x d,y d )) 2 ], () whee (x x d ) 2 +(y y d ) 2 = d 2 Fo a set of samples v(x i,y i ), i =, 2, on a egula gid, γ(d) can be estimated as follows, γ (d) = m(d) [v(x i,y i ) v(x j,y j )] 2, 2m(d) whee m(d) is the numbe of points at a distance d within each othe, ie the sum is ove all points fo which (x i x j ) 2 +(y i y j ) 2 = d 2 A staightfowad method to estimate the vaiogam fo a set of samples on an iegula gid consists of the following steps: (i) fo evey pai of samples, compute the distance between them and the squaed diffeence between thei data values, (ii) make a scatte plot of the vaiogam values against the We will use the Euclidean distance to measue distances between two points distance, and (iii) cuve fit the scatte plot to obtain an estimate of the vaiogam A moe statistically obust method, taditionally used in Geostatistics [5], [6], consists of the following steps: (i) as befoe, fo evey pai of samples compute the distance between them and the squaed diffeence between thei data values, (ii) divide the entie ange of distance into discete intevals with an inteval size being equal to the aveage distance to the neaest neighbo, (iii) assign each of the pai of samples to one of the distance intevals and compute the aveage vaiance in each inteval by dividing the sum of the squaed-diffeences between data-values by the total numbe of pais lying in that distance inteval, and (iv) assign the aveage vaiance to the mid point of each inteval and cuve fit these points to one of the standad vaiogam models [5], [6] In this pape, we will use the second method to estimate the vaiogam fom the expeimental taces A Relationship between the vaiogam and the covaiance Anothe vey commonly used statistic to measue coelation in data is the covaiance Fo a two dimensional isotopic stationay pocess V (x, y), the covaiance is defined as C(d) =E [(V (x, y) µ)(v (x d,y d ) µ)], whee (x d,y d ) is denotes a point d distance away fom (x, y) and µ = E[V (x, y)] Since both the vaiogam and the covaiance ae measues of the coelation in data, we deive the elationship between them and veify that both of them can be used intechangeably Fom Equation (), γ(d) = [ 2 E (V (x, y) V (x d,y d )) 2] = 2 E [ ((V (x, y) µ) (V (x d,y d ) µ)) 2] γ(d) =σ 2 V C(d), (2) whee σ 2 V = E [ (V (x, y) µ) 2] is the vaiance of the pocess V (x, y) Equation (2) implies that a lowe (highe) value of the vaiogam implies a highe (lowe) value of the covaiance and coelation Figue plots the vaiogam and the covaiance fo a tace geneated by assuming a jointly Gaussian model fo the spatial data [7] A chaacteistic of the vaiogam which can be infeed fom the plot is that it levels off (becomes paallel to the x-axis) at a distance beyond which the covaiance o the coelation between the samples go to zeo Futhe, the constant value to which the vaiogam satuates is equal to the vaiance of the pocess Since both metics can be intechangeably used, in this pape we will only pesent vaiogam plots 36

3 covaiance vaiogam d Fig Vaiogam and Covaiance plots fo a tace geneated by assuming jointly Gaussian model fo the spatial data III MODEL FOR SPATIALLY CORRELATED DATA FOR A NETWORK WITH GRID TOPOLOGY In this section, we summaize the model poposed in [] Conside a senso netwok whose nodes eside on a gid topology Let V (x, y) be the data value at node (x, y) Itis assumed that V (x, y) is a stationay pocess that has a unique fist ode distibution whose pobability density function (pdf) is denoted by f V (v) (This is efeed to as the long tem distibution of the data) Let N(d) denote the numbe of nodes at a distance d fom (x, y) LetV d denote the data value at a node which is d distance away fom (x, y), and Vd k denote the data value at the k th node ( k N(d)) at a distance d fom (x, y) Following is the model fo geneating the data values, V (x, y) = V + Z with pobability α N() V N() α + Z wp V2 + Z wp V N(2) 2 + Z wp N() α 2 N(2) α 2 N(2) V h + Z wp α h N(h) V N(h) α h N(h) h + Z wp Y wp β, (3) whee Z and Y ae andom vaiables independent of each othe as well as V, and thei pdf s ae denoted by f Z (z) and f Y (y) espectively Both Y and Z detemine the long tem distibution of V, and Z captues the small diffeences between neighboing data values The above equation simply says that the pobability that V (x, y) is deived fom the value of any node which is d distance away fom (x, y) is α d Futhe, the pobability that V (x, y) is deived fom the α d N(d) value of a paticula such node is The paametes of the model ae h, theα i s, β, f Y (y) and f Z (z) Since the sum of pobabilities should equal one, β + h α i = i= Remak: Fo a gid, the L o manhattan distance is a meaningful way to measue distances between two nodes (The L distance between two nodes (x,y ) and (x 2,y 2 ) is given by d = x x 2 + y y 2 ) Thus, the distances between nodes on a gid ae in multiples of the minimum inte-senso distance which is equal to the size of the gid This simplifies the model stuctue as both the vaiogam and the α s can be viewed as a discete function of distance A Motivation fo a new model One way to extend the model in [] to an iegula topology is to convet the iegula topology to a gid topology by adding moe nodes, geneate data at all these nodes, and then discad the additional nodes The poblem with this appoach is that it is computationally vey expensive We pefom a simple calculation to obtain at how many additional nodes we have to geneate data Let us andomly distibute n nodes in a squae of side Let the locations of each node (x i,y i ), be chosen unifomly and independently of each othe Let p denote the minimum coodinate distance between the nodes, that is p = min i n, j n [ min [ x i x j, y i y j ]] It is easy to see that we need to geneate data fo at least p nodes 2 The pobability that k out of n nodes distibuted andomly ( ) n in an inteval of size lie in an inteval of size d equals k (d) k ( d) n k which is Binomial (n, d) Fo a vey lage n, Binomial(n, d) can be appoximated by Poisson(nd) So, the inte senso distance is distibuted as Exponential(nd) The minimum inte senso distance in one dimension will be the minimum of n exponentials, and hence is distibuted as Exponential (n 2 d) Since the x coodinate and the y coodinate of each node is chosen independently of each othe, the minimum coodinate distance p is distibuted as Exponential(2n 2 d) Hence, on aveage, we will have to geneate at least O(n 4 ) additional nodes to get the data values on n nodes This simple calculation shows that it is computationally vey expensive to populate data on an iegula topology by conveting it to a gid topology fist Even if we have the capability to pefom these calculations, a change in the node locations will change the model paametes as the value of p depends on the actual node locations This is poblematic, since the model paametes should only depend on the physical phenomenon being monitoed, and not on the actual node locations Hence, modeling data on an iegula topology by conveting the iegula topology to a gid topology is not appopiate, which motivates the need to exploe moe well-suited models IV MODEL FOR AN IRREGULAR GRID In this section we intoduce ou model fo captuing the statistical popeties of senso netwoks data Fo ease of notation, we use pola coodinates to define node locations We assume that nodes ae distibuted in a cicle of unit adius Let V (, θ) be the data value at node (, θ), whee << and 37

4 <θ<2π We assume that V (, θ) is a stationay isotopic pocess that has a unique fist ode distibution denoted by f V (v) Without loss of geneality and to simplify exposition, we assume that we want to geneate the data value at the oigin We popose the following model to do so: V (, ) = I (U=T ) Y + I (U=H) (V (, θ)+z) θ δ(r = )dδ(θ = θ R = )dθ, (4) whee: a) U epesents a coin that when it lands heads (H), with pobability β, the oigin s data value is obtained fom neighboing nodes, and when it lands tails (T), with pobability β, it is obtained fom a andom vaiable Y (I A denotes an indicato function that equals one when event A occus and equals zeo othewise) b) Similaly to the egula topology model, Y and Z ae andom vaiables independent of each othe as well as V, with pdf s f Y (y) and f Z (z) espectively Y models the situation whee the oigin s data value is not obtained fom neighboing nodes, Z captues the small diffeences between neighboing data values, and both of them detemine the long tem distibution of V c) R is a andom vaiable with pdf f R () When R =, the oigin s data value is obtained fom locations at distance fom the oigin (δ(r = ) denotes a δ-function of R that is non-zeo when R = ) f R ()d is the pobability of this event f R () is a paamete of ou model and fom now on we efe to it as α(), since it coesponds to the α i s of the gid topology model d) Θ is a andom vaiable with pdf f Θ (θ) When Θ=θ R =, the oigin s data value is obtained fom locations at angle θ given that thei distance fom the oigin is f Θ R (θ )dθ is the pobability of this event We assume that θ is unifomly distibuted between angles θ and θ 2 Thus, { f Θ R (θ ) = (θ 2 θ ) θ <θ<θ 2 othewise Given the above, the cdf of V (, ) can be expessed as follows, P (V (, ) v) =βp(y v)+( β) α() P (V (, θ)+z v) ddθ (5) (θ 2 θ ) θ Equation (4) and (5) simply say that the the pobability that the data value at a node is diectly deived fom a node lying α() in the shaded egion A in Figue 2 is (θ 2 θ ) ddθ α()d is the pobability that a node s data value is deived fom any node at a distance away fom it The numbe of nodes distance away and lying in an ac of (θ 2 θ ) is popotional to (θ 2 θ ) Now, given that the node s data value is deived fom a node distance away, the pobability that it is deived dθ fom a node in an ac of dθ is (θ 2 θ ) The paametes of the model ae α(), β, f Y (y) and f Z (z) The values of θ and θ 2 depend on the method used to populate data We will explain thei ole in moe detail in Section IV-A α() will be a deceasing function of as the coelation between nodes deceases as thei distance inceases Thoughout this pape, we assume that α() is zeo fo max fo some value of max Now, since the pdf s should integate out to, we get the following equation, max θ2 θ α() ddθ = (θ 2 θ ) max α() = (6) A Instantiation of the model In a eal life scenaio, the exact node locations detemined though some location distibution will be given as an input and the use should be able to geneate data values at these nodes using the model In this section we descibe how to geneate the data using an instantiation of the model (a) (b) (,) (,) R (,) d dθ Fig 3 Two methods to populate data (a) Semi Cicula Dependence: The data value at node (, ) can be diectly deived fom any node lying in the semi cicula egion (b) Quate Cicula Dependence: The data value at node (, ) can be diectly deived fom any node lying in the quate cicula egion A Aea A = d dθ Fig 2 The pobability that the data value at (, ) is deived fom a node α() in egion A is (θ 2 θ ) ddθ Befoe we poceed, we look at how the values of θ and θ 2 effect the population of data A couple of examples ae given in Figue 3 The fist method coesponds to population of data using a semi cicula data dependence while the second method coesponds to a quate cicula data dependence Quate cicula data dependence implies that a node s data value can be diectly deived fom only those nodes which lie 38

5 in the shaded egion R which is quate of a cicle centeed at the node The values of θ and θ 2 ae π and 3π 2 fo quate cicula dependence and π and 2π espectively fo semi cicula dependence Which method to choose will depend on the physical phenomenon being modeled The default data population method in the est of the pape is going to be the quate cicula data dependence (θ = π and θ 2 = 3π 2 ) 3 Fig An example topology As an example, conside the node locations given by Figue 4 Let the node location of node i be ( i,θ i ), i, and let the data values at these nodes be denoted by V ( i,θ i )The instantiation of the model fo node (, ) fo a quate cicula data dependence is as follows: V (,θ )+Z V ( 2,θ 2 )+Z V (, ) = V ( 3,θ 3 )+Z V ( 4,θ 4 )+Z Y with pobability c α() 9 wp c α(2) 2 wp c α(3) 3 wp c α(4) 4 wp cβ (7) Equation (7) is vey simila to the model fo gid topologies in spiit Both Equations (7) and (3) ae instantiations of the model descibed by Equation (4) Howeve, thee ae some diffeences that it is woth mentioning: Fist, α() in Equation (7) is no longe a discete function of distance Second, since the numbe of nodes at a distance is popotional to, instead of N() (see Equation (3)) we have in the denominato of the tems of Equation (7) Thid, note the pesence of the scaling constant c in Equation (7) which is pesent to make the sum of pobabilities go to one (c will depend on the exact node locations and will change when these change, but the model paametes emain the same) Equation 7 assumes that the data values at nodes lying in the data dependence egion of V (, ) have aleady been populated Thus, an ode of populating data has implicitly been assumed A valid odeing to populate data will ensue that when a data value at a node is populated, the data value at all the nodes lying in its data dependence egion have aleady been populated Also, befoe stating to populate the data we andomly initalize the values that ae within the data dependence egion of the fist node we populate B How the model paametes affect coelation The pesence of many paametes in the model gives us geat flexibility to model pocesses having diffeent coelation popeties In this section, we study how diffeent paametes affect the coelation popeties of the geneated data We use the simple linea topology shown in Figue 5 Synthetic taces ae geneated using the model unde a 2 node scenaio We assume Y N(, ) and Z N(,σ z ) d= N- N Fig 5 Simple linea topology used to study the effect of model paametes on the coelation in data ) Effect of β: Since β govens the pobability with which a node will choose a andom value independent of evey othe node, it is expected that a lowe value of β will lead to a highe value of coelation Also, a vaiation in β will change the distibution of V The exact elationship between the two will be deived in Section V-A Vaiogam Values β = β = 2 β = 4 β = 6 β = 8 β = Fig 6 Effect on the coelation stuctue of the data when β is vaied keeping all the othe paametes constant Figue 6 plots the vaiogam fo taces geneated using diffeent values of β The othe paametes ae: max =2, α() =λ2 fo << max and othewise, and σ z =3 The plots show that as the value of β deceases, not only does the distance at which the vaiogam levels off (the distance beyond which the nodes ae uncoelated) incease, but also the y-value to which it levels off inceases 2) Effect of max : If the distance between the nodes is moe than max, then they cannot be diectly deived fom each othe Hence, we expect that inceasing max will incease the distance at which the vaiogam levels off Figue 7 plots the vaiogam fo taces geneated using diffeent values of max The othe paametes ae: α() = λ2 fo << max and othewise, β =4 and σ z =4 A look at the vaiogams tells us that coelation between the data values is independent of the value of max This obsevation is contay to ou initial intuition and hence equies a moe detailed explanation We take this oppotunity to highlight a key chaacteistic of ou model If node 2 is deived fom node, and node 3 is deived fom node 2, then node and node 3 will show a stong coelation too So, even if max is small, when β is small, nodes having distances much lage than max will have high coelation 39

6 Vaiogam Values max = 3 max = 7 max = max = 5 max = 9 max = 23 Fig 7 Effect on the coelation stuctue of the data when max is vaied keeping all the othe paametes constant Thus, we infe that the distance at which the vaiogam levels off depends pimaily on β 3) Effect of σ z : Finally, we study whethe changing f Z (z) will effect the coelation in data We had assumed f Z (z) to be N(,σ z ) Taces fo diffeent values of σ z ae geneated and thei vaiogams ae plotted in Figue 8 The othe paametes ae: max = 2,α() = λ2 fo < < max and othewise, and β =4 Vaiogam Values σ z = σ z = 2 σ z = 3 σ z = 4 σ z = 5 σ z = Fig 8 Effect on the coelation stuctue of the data when σ z is vaied keeping all the othe paametes constant It can be easily seen fom the plots that the σ z does not effect the coelation stuctue of the data, though it has a significant effect on the distibution of V The value at which the vaiogam satuates to, which is the vaiance of V, inceases as σ z inceases Remak: While evaluating the pefomance of algoithms fo diffeent coelation stuctues, it is useful to have a single tunable paamete whose value detemines the level of coelation in data Fo ou model, this tunable paamete is β Taces with diffeent coelation stuctues can be geneated by tuning β fom to and the pefomance of the algoithm can be plotted against β V INFERRING MODEL PARAMETERS In this section, we pesent techniques fo infeing model paametes fom eal taces The model paametes to be infeed ae α(), β, f Y (y), max and f Z (z) Without loss of geneality, fom now onwads we will assume that Z is a nomal andom vaiable with zeo mean and standad deviation σ = σ z Note that the distibution of Z need not necessaily be Gaussian; any othe distibution will not effect the model, though the analysis pesented in this section will be modified Fist, in Section V-A we deive the elationship between f V (v), f Y (y) and σ z Note that f V (v) can be easily estimated by its empiical distibution Then, in Section V-B we pesent a igoous pocedue to infe α(), β, max and σ z fom a eal tace But this pocedue involves solving integal equations [8], [9] and hence, it is not always possible to obtain a closed fom expession fo the model paametes So, in Section V- C, we pesent a simple, pactical method that uses both the discete and the continuous models A Relationship between the distibutions of V, Y and Z In this section, we deive the elationship between the distibutions of V, Y and Z Fom Equation (5) we get, f V (v) =( β)f V +Z (v)+βf Y (v) (8) Using chaacteistic functions and since V independent, Equation (8) can be ewitten as, and Z ae Φ V (jω)=( β)φ V (jω)φ Z (jω)+βφ Y (jω) (9) Since Z is a zeo mean nomal andom vaiable, its chaacteistic function is given by e [ σz 2 ω 2 2 ] Equation (9) finally educes to β Φ V (jω)= Φ ( β)e [ σz 2 ω 2 Y (jω) () 2 ] Fo mathematical convenience, we define a new andom vaiable having a chaacteistic function given by β Φ L (jω)= ( β)e [ σz 2 ω 2 2 ] Equation () can now be ewitten as V = Y + L () B A igoous pocedue to infe the model paametes In this section, we pesent a igoous method to infe the model paametes Fist we compute the vaiogam γ() using the model and then equate it with its estimate γ () obtained fom the eal tace Using Equation (), 2π γ() = 2 2π E [ (V (, ) V (, θ)) 2] dθ = 2π 2 ( β) max θ2 2π θ E [ (V (,θ )+Z V (, θ)) 2] α( ) d dθ dθ θ 2 θ + 2π β 2 2π E[(Y V (, θ))2 ]dθ (2) 3

7 The tem E [ (V (,θ )+Z V (, θ)) 2] in the above equation can be expanded as, E [ (V (,θ )+Z V (, θ)) 2] = E [ (V (,θ ) V (, θ)) 2] ( ) +E[Z 2 ]=2γ cos(θ θ ) + σz 2 The second tem in Equation (2) E[(Y V (, θ)) 2 ] is equal to E[L 2 ] Using Equation (), E[L 2 ] is evaluated to be ( β)σ2 z β Substituting all of the above in Equation (2), γ() =( β)σ 2 z +( β) max θ2 2π θ 2π α( ) ( γ 2 + θ 2 θ 2 2 cos(θ θ )) dθdθ d (3) Equation (3) gives the elationship between the vaiogam and the model paametes α(), β, σ z and max Substituting γ() with its estimation γ () in Equation (3) gives us an integal equation of the fist kind [8], [9], which along with the conditions max α()d =and α( max )=fom a system of equations with one unknown function α() and thee unknown constants β,σ z and max Solving these equations will give us the model paametes Afte obtaining σ z and β, f Y (y) is obtained though Equation () In Equation (3), the unknown function α() is inside an integal In geneal, it is not possible to find closed fom solutions fo α() fo evey vaiogam function In the next section, we assume a specific vaiogam function that coesponds to a covaiance function commonly used in the senso netwoks liteatue, and solve fo α() ) Case Study: In this section, we will the find model paametes fo a tace having the following vaiogam, γ() = c( e 2 ) <<5 (4) The coesponding covaiance is C() =ce 2 which is a vey commonly assumed coelation stuctue fo spatially coelated data in the senso netwoks liteatue, see, fo example, [7], [2] Note that these papes also assume the data to be jointly Gaussian, wheeas we don t make any such assumption hee Actually, the jointly Gaussian scenaio is a subcase of ou model, as discussed in Section VIII To find the model paametes, we have to solve the following integal equation: c max θ2 2π α( ) c( e 2 )=( β) θ 2π θ 2 θ ( ) e ( cos(θ θ )) dθdθ d +( β)σz 2 (5) Befoe ventuing into the solution of the above equation, we fist integate out θ and θ, θ2 2π θ 2π θ 2 θ c ( e 2 e 2 e 2 cos(θ θ )) ) dθdθ (6) Since <, < 5 2 cos(θ θ ) <, by neglecting the squae tems and beyond, the last tem in the above equation can be appoximated by, e 2 cos(θ θ )) =+2 cos(θ θ ) Note that the assumption on the size of the sample aea < <5 has been made to enable the above appoximation, othewise it is not possible to find a closed fom solution fo α() We assume the semi cicula data dependence to populate data, hence θ = π and θ 2 =2π ( With the above appoximation, Equation (6) educes to c e 2 e 2) Substituting in Equation (5), max c( e 2 )=c( β) α( ) ( e 2 e 2) d +( β)σ 2 z (7) Using the method descibed in [8] to solve fo integal equations, we detemine that α() has the fom a + be 2 whee a and b ae constants to be detemined by the bounday conditions α()d = max ( and α( max ) = Solving them yields b = πef(max) 2 max e max) 2 and a = be 2 max, whee Ef(x) is the eo function defined as Ef(x) = 2 x π dt e t2 Now substituting α() =a + be 2 in Equation (7) gives β = 4 π (2aEf( max )+ 2bEf( 2 max )) and σ 2 z = cβ β We still need to detemine the value of max Anyvalueof max would do, as long as the esulting β is between and -since it is a pobability-, and the esulting α() is positive fo all -since it is a pdf- In this example, we choose the lagest max value that satisfies both constaints In paticula, we stat with max =5, and keep on educing its value till we obtain a positive value of β (α() is positive fo all fo this value of max ) The model paametes tun out to be, max = 5, β =23 6 and σ 2 z =23 Vaiogam Values Fig 9 The given vaiogam γ() = ( e 2 ) Lets see if these values make sense To do so, fist we plot the vaiogam of Equation (4) in Figue 9 Dawing fom the discussion in Section IV-B, we can easily ague that fo the 3

8 given coelation stuctue, the values of β and max should be vey small which is consistent with the values we obtain C A simple method to infe the model paametes In this section we pesent a simple pocedue to infe the model paametes Seveal numeical techniques to solve integal equations exist in the liteatue But integal equations of the fist kind ae inheently ill posed poblems [2] and hence, thei solutions ae geneally unstable This ill-posedness makes numeical solutions vey difficult, as a small eo can lead to an unbounded eo So, we pesent a pactical simple method which uses the discete model [] to infe the model paametes and the continuous model to populate the data and geneate taces The fist step is to use the method descibed in Section II to obtain a discetized vaiogam, which coesponds to the continuous vaiogam sampled at multiples of the aveage neaest neighbo distance The second step is to use the method descibed in [] to obtain a discete vesion α[] of α(), which coesponds to the continuous α() sampled at multiples of the aveage neighbo distance Note that the method in [] is simila to the method descibed in Section V-B Except now, instead of integating ove infinitesimally small aeas each having at most one node in them, like aea A in Figue 2, we sum ove squae egions as shown in Figue, assuming one node esides in the cente of each squae Avg neaest neighbo distance (,) Fig In the appoximate method, instead of integating ove the egion A, we sum ove the squae egions Since the squae aea is no longe infinitesimally small, the integals in Equation (3) ae eplaced by sums The esultant system of linea equations can be easily solved to obtain the model paametes, β, σ z and α[] Finally, the α[] s ae intepolated o cuve fitted to obtain the continuous α() Afte obtaining the model paametes, we use the model descibed in Section IV to geneate synthetic taces This pocedue fomulates the poblem in continuous domain, convets it to the discete domain by sampling, solves it in the discete domain and tansfoms the solution back to the continuous domain by intepolation Intuitively, this pocedue is vey simila to seveal signal pocessing techniques, fo example using the FFT to find the fouie tansfom of a continuous signal Obviously, as in the signal pocessing techniques, the distance between the two neighboing samples (which is the aveage neaest neighbo distance fo the given pocedue) has an impotant ole to play The lage the numbe of samples in an aea, the smalle the aveage neaest-neighbo distance, and the moe accuate is the estimation of the model paametes VI MODEL VERIFICATION AND VALIDATION In this section, the model paametes fo expeimental taces ae infeed using the method descibed in Section V-C Then these model paametes ae used to geneate synthetic taces We veify ou model by compaing the vaiogams of the oiginal expeimental taces and the coesponding synthetic taces, and then we validate it by compaing the pefomance of an algoithm which exploits spatial coelation, against both the taces A Data Set Desciption We need actual senso netwok taces to be able to veify and validate ou model In this section, we descibe the taces which we have used fo veification and validation puposes ) SHM Tace: One of the eal wold expeiments whee eal senso netwok taces have been collected afte deploying a senso netwok is epoted in [22] A 4 MicaZ node senso netwok was deployed in a lage seismic test stuctue used by civil enginees to study stuctual health monitoing (SHM) Acceleometes on the sensos collected vibation samples fom the stuctue and send them to a base station using a data acquisition system called Wisden We use a time snapshot of this tace to veify and validate ou model We ae not awae of othe simila senso netwok taces So, we collected ou own taces using MICA2 motes with MTS3CA senso boads attached to them We used the light sensos on the senso boad to take light intensity measuements Two taces in two diffeently lighted envionments wee collected using these motes 2) Tace : 44 senso nodes ae deployed in a feet squae aea The location of each node is andomly chosen accoding to a unifom location distibution We use a maste mote to send a message to evey mote When a senso node eceives the message, it samples the light intensity of the envionment Thus all sensos take the eadings at the same time Thus, we get a spatially coelated tace of 44 samples The expeiment is pefomed in an outdoo envionment unde stong sunlight with a few nodes in a shaded aea caused by the pesence of tees in the envionment Thus, the eadings of the sensos will be close to each othe, but the eadings fom the sensos in the shaded aea will be lowe than those in diect sunlight 3) Tace 2: 3 senso nodes ae deployed in a 4 2 feet squae aea The location of each node is andomly chosen accoding to a unifom location distibution As befoe, all sensos take eadings at the same time when they eceive a message fom the maste mote 32

9 The expeiment is pefomed in an indoo envionment with just one light souce This coesponds to a single souce scenaio whee the eadings go on deceasing as the distance fom the light souce inceases So, the sensos fa away fom the light souce have much lowe eadings than the sensos close to the light souce This geneates a stongly coelated data tace Note that all the above taces ae not vey big, which is a esult of the difficulty in deploying vey lage senso netwoks Due to statistical consideations, we want to veify and validate ou model against a tace having thousands of spatial samples So, we use the S-Pol Rada Data Tace 2, which was obtained fom emote sensing studies and has been used in the senso netwoks liteatue as an expeimental tace, see, fo example [2] Though the distance between the sensing nodes fo this tace is hundeds of metes which is not epesentative of actual senso netwoks in which the inte senso distance is a few metes, we use it fo veification because of the absence of lage senso netwok taces 4) S-Pol Rada Data Set: The esampled S-Pol ada data, povided by NCAR, ecods the intensity of eflectivity of atmosphee in dbz, whee Z is popotional to the etuned powe fo a paticula ada and a paticula ange The oiginal data wee ecoded in the pola coodinate system Samples wee taken at evey 7 degees in azimuth and 8 sample locations (appoximately 5 metes between neighboing samples) in ange, esulting in a total of 5 8 samples fo each 36 degee azimuthal sweep They wee conveted to the catesian gid using the neaest neighboing esampling method [23] In this pape, we have selected a spatial subset of the oiginal data (496 spatial samples) and 259 time snapshots acoss 2 days in May 22 B Model Veification ) Tace : The method descibed in Section V-C is used to infe the model paametes But befoe, applying the method, we have to estimate the vaiogam fom the given tace We use the second method descibed in Section II to estimate the vaiogam We fitted seveal standad vaiogams on to it [5], [6] and etained the one which had the minimum squae eo Fo the fist tace, the spheical vaiogam given by Equation (8) was the best fit amongst all of them Fo the given tace, c =9,c= 7 and a =9 c + c( γ() ={ 3 2 a 2 ( a )3 ) c + c, a,a (8) Afte infeing the model paametes fom the estimated vaiogam, we geneate a synthetic tace on the same senso node locations as the oiginal tace We compae the distibution of the taces in Figue and thei vaiogams in Figue 2 Both the distibution and the vaiogams match closely 2 S-Pol ada data wee collected duing the IHOP 22 poject (http: //wwwatducaedu/tf/pojects/ihop_22/spol/) S-Pol is fielded by the Atmospheic Technology division of the National Cente fo Atmospheic Reseach We acknowledge NCAR and its sponso, the National Science Foundation, fo povision of the S-Pol data set f V (v) Synthetic Tace Oiginal Tace v Fig Tace : Compaison of the distibution of the oiginal and the synthetic taces Vaiogam Values Synthetic Tace Oiginal Tace Fitted Vaiogam (spheical model) Fig 2 Tace : Compaison of the vaiogam of the oiginal and the synthetic taces 2) Tace 2: The vaiogam of the second tace is best estimated by the powe semi vaiogam model (Equation (9)) with paametes c = 45 and c = 45 γ() =c + c 2 (9) We use the estimated vaiogam to obtain the model paametes and then geneate a synthetic tace on the same senso node locations as the oiginal tace We plot the vaiogams of both the taces in Figue 3 Once again, the vaiogams match closely Vaiogam Values 8 x Synthetic Tace Oiginal Tace Fitted Vaiogam ( * 2 ) Fig 3 Tace 2: Compaison of the vaiogam of the oiginal and the synthetic taces 3) SHM Tace: The vaiogam of the SHM tace is best estimated by the spheical semi vaiogam model (Equation (8)) with paametes c = 6,c= and a =27 We use the estimated vaiogam to obtain the model paametes and then geneate a synthetic tace on the same senso node locations as the oiginal tace We plot the vaiogams of both 33

10 the taces in Figue 4 Once again, the vaiogams match closely Vaiogam Values x 4 4 Synthetic Tace 2 Oiginal Tace Fitted Vaiogam (Spheical Model) Fig 4 SHM Tace: Compaison of the vaiogam of the oiginal and the synthetic taces 4) S-Pol Rada data set: We choose a snapshot in time of the S-Pol Rada data as the expeimental data tace Figue 5 shows the compaison of the distibution of the synthetic and oiginal taces and Figue 6 shows thei espective vaiogams Both the distibution and the vaiogam of the two taces match closely one of these algoithms on both the oiginal and the synthetic taces and compae its pefomance Amongst the many such algoithms mentioned in the intoduction, we selected CMAC [9] to evaluate ou model The undelying idea behind CMAC is that due to the pesence of spatial coelation between senso obsevations, it is not necessay fo evey node to tansmit its data Amongst a cluste of senso nodes, one of them can act as a epesentative of all othe nodes We efe to the node that sends infomation to the sink as the epesentative node of the cluste Thus, a smalle numbe of senso measuements ae adequate to communicate the event featues to the sink within a cetain acceptable eo aveage eo Synthetic Tace Oiginal Tace d distibution of the synthetic tace distibution of the oiginal tace Fig 7 Tace : Compaison of the pefomance of CMAC on oiginal and synthetic tace: Vaiation of eo with d 35 f V (v) v aveage eo 8 6 Fig 5 S-Pol Rada data tace: Compaison of the distibution of the oiginal and the synthetic taces 4 2 Synthetic Tace Oiginal Tace d vaiogam of the oiginal tace vaiogam of the synthetic tace Fig 8 Tace 2: Compaison of the pefomance of CMAC on oiginal and synthetic tace: Vaiation of eo with d Vaiogam Values aveage eo Fig 6 S-Pol Rada data tace: Compaison of the vaiogam of the oiginal and the synthetic taces 5 Synthetic Tace Oiginal Tace d C Model Validation The poposed model will be used to compae and evaluate diffeent algoithms which exploit the pesence of spatial coelation in data To validate that ou model can be used to evaluate the pefomance of diffeent algoithms, we un Fig 9 SHM Tace: Compaison of the pefomance of CMAC on oiginal and synthetic tace: Vaiation of eo with d In ou simulations, we assumed the cluste stuctue to be a squae of side d Amongst all the nodes within this squae, the epesentative node is selected andomly Only one node 34

11 quey eo synthetic tace oiginal tace d Fig 2 S-Pol Rada Data Tace: Compaison of the pefomance of CMAC on oiginal and synthetic tace: Vaiation of eo with d in the cluste (the epesentative node) will tansmit its data to the sink The lage the value of d, the smalle is the numbe of nodes tansmitting data to the sink, and hence lage is the eo We plot the aveage eo against the value of d fo the oiginal as well as the synthetic tace fo Tace, Tace 2, the SHM tace and the S-Pol Rada Data Tace in Figues 7, 8, 9 2 espectively It is easy to see fom the plots that the pefomance of the algoithm fo both the taces is simila as the plots match closely Fom the above plots, we conclude that the poposed model is able to captue the spatial coelation in senso netwok data VII TOOLS TO GENERATE LARGE SYNTHETIC TRACES In this section we descibe two tools which we have ceated to help eseaches geneate synthetic taces of any size and degee of coelation These tools can be downloaded fom apoovaj geneatelagetacefomiegula will ceate lage synthetic taces having the same coelation stuctue as the input eal data tace It takes the estimated vaiogam of the eal tace as its input It also equies the use to specify the data dependence patten The use can choose eithe of the methods descibed in Section IV-A geneatesynthetictacesoniegula will ceate lage synthetic taces epesenting a wide ange of conditions by tweaking the model paametes It takes the model paametes, max, α(), β, σ z and f V (v), the location of the nodes and the data dependence patten as its input Data collected fom a testbed having a few senso nodes is not sufficient to evaluate potocols The fist tool can geneate a lage tace having simila coelation popeties as the eal tace, and hence, help eseaches to evaluate potocols with eal data The second tool will enable eseaches to evaluate thei potocols with data having vaied coelation stuctues Hence, these two tools will help eseaches to evaluate thei potocols with data epesenting wide ange of ealistic conditions without the need of actual dense deployment of senso nodes VIII CONCLUSIONS AND DISCUSSION In this pape, we poposed a mathematical model to captue spatial coelation in the data of senso netwoks with iegula topologies The model can geneate synthetic taces epesenting a wide ange of conditions and exhibiting any degee of coelation fo any abitay topology We also descibed a igoous mathematical pocedue and a simple, pactical method to infe the model paametes fom a eal tace Finally, we veified the coectness of the model, and validated its ability to accuately captue coelations by compaing the pefomance of CMAC, an algoithm whose behavio depends on the degee of spatial coelation in data, unde eal and synthetic taces To model spatial coelation in data, most eseaches assume the data to be jointly Gaussian [9], [7], [2], [24] The pimay easons fo this choice is ease of use and analytical tactability, athe than accuacy [25] Ou model is moe geneal and moe ealistic and hence moe complex than the jointly Gaussian model Actually, it is easy to ague that the jointly Gaussian model is a special case of the poposed model The pdf of jointly Gaussian andom vaiables is completely defined by the covaiance matix Each covaiance matix coesponds to a unique vaiogam and each vaiogam coesponds to a unique α(), max, σ z and β f V (v) is Gaussian and f Y (y) can be infeed fom Equation () The chief limitation of the jointly Gaussian model is that it foces the joint pdf s of the data values to be jointly Gaussian, while in most of the expeimental taces, even the fist ode distibution is not Gaussian The poposed model has no such estiction and though a pope choice of f Y (y) and f Z (z), any distibution function of data values can be modeled To summaize, the jointly Gaussian model should be used to pedict tends and get some quick intuition into the behavio of an algoithm But to do a moe thoough study, though analysis o simulations, a moe ealistic model such as the poposed model should be used It is impotant to point out that tace-diven simulations ae the best choice fo simulating an algoithm, but in the absence of lage senso netwok data taces, this model can act as a close substitute ACKNOWLEDGMENT The authos would like to thank D Bhaska Kishnamachai fo helpful suggestions duing the couse of the wok and fo poviding us with the motes fo collecting taces (The motes wee bought unde his gant CNS-34762) REFERENCES [] A Jindal and K Psounis, Modeling spatially-coelated data senso netwok data, in Poceedings of the IEEE Intenational Confeence on Senso and Ad hoc Communications and Netwoks, Oct 24 [2] A Goel and D Estin, Simultaneous optimization fo concave costs: single sink aggegation o single souce buy-at-bulk, in SODA, 23, pp [3] S Pattem, B Kishnmachai, and R Govindan, The impact of spatial coelation on outing with compession in wieless senso netwoks, in Symposium on Infomation Pocessing in Senso Netwoks (IPSN), Ap 24 [4] C Intanagonwiwat, D Estin, R Govindan, and J Heidemann, Impact of netwok density on data aggegation in wieless senso netwoks, in ICDCS, 22 35

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