SpaceTime Dynamic Design of Environmental Monitoring Networks ChristopherK. WIKLEand J. Andrew ROYLE Methods for constructing optimal spatial sampling designs for environmental monitoring networks, widely applied in a large number of disciplines, generally produce static designs that are optimal under models with no explicit temporal structure. However, environmental processes tend to exhibit both spatial and temporal variability; hence, static networks may not capture the essential spatiotemporal variability of the process. Static designs, often necessary due to geopolitical and economic considerations, could be supplemented with mobile monitoring devices. The design problem is to decide where mobile monitors should be located at time t + 1 based on observations through time t. We propose a simple, general, dynamical space-time model that allows estimation of prediction error covariance at time t + 1, given information up to time t. We then seek the optimal spatial locations at time t + 1 that satisfy some design criterion. Several experiments show the importance of spatial and temporal structure in the selection of optimal designs. Data from the Chicago area ozone monitoring network are used to demonstrate potential dynamical designs under realistic space-time dependence assumptions. Key Words: Environment, Kalman filter, Network design, Optimal design, Ozone, Sampling, Space-time modeling, Spatial design, Spatial sampling, Spatial statistics.
1. INTRODUCTION Many environmental processes include variability over both space and time. In practice, it is never possible to completely sample a spatiotemporal environmental process. Traditionally, political, geographic, and/or economic reasons have determined the design of monitoring networks. However, it is intuitive that if the process has significant spatiotemporal interaction, such designs will not be optimal at any particular time. Given the opportunity to move monitors or to sample over different spatial locations through time, it is possible that the design efficiency can be improved by allowing for time-varying designs. An obvious real-world example of maximizing environmental information by time-varying sampling ChristopherWikle is AssistantProfessor,Departmentof Statistics,Universityof Missouri,Columbia,MO 65211 (E-mail:
[email protected]).J. Andrew Royle is Biological Statistician,U.S. Fish and Wildlife Service AdaptiveManagementand Assessment Team,Laurel,MD 20708-4017 (E-mail:
[email protected]). ( 1999 AmericanStatisticalAssociation and the InternationalBiometric Society JournalofAgricultural,Biological, and EnvironmentalStatistics, Volume4, Number4, Pages 489-507 489
490
C. K. WIKLEANDJ. A. ROYLE
occurs with the monitoringof hurricanesvia aircraft.In this case, specially equippedaircraftfly into hurricanesat specifiedintervalsto collect observations.Clearly,over time, the location of such monitoringchanges accordingto the evolutionof the storm. The notionof optimalmonitoringnetworkdesignis intuitive.Let Y be a spatiotemporal process, such as ozone, sea surface temperature,or some naturalresourcevariable(e.g., habitatcondition), and suppose that the process can be monitoredat a set of m spatial locations (the "design") denoted as D = (uli, u2, ... ,
Um)
where ui E JZ C S, and S is
some geographicregion. The design objective is to locate these m points in an optimal fashion, meaningthe design minimizes some variancecriterion.Commoncriteriainclude the averagepredictionvariance,maximumpredictionvariance,or the varianceof regression parameterestimates. In recentyears,substantialeffortin the statisticalcommunityhas been devotedto examining optimalspatial designs for environmentalprocesses (e.g., FederovandMueller 1989; Haas 1992; Guttorp,Le, Sampson, and Zidek 1993; Cox, Cox, and Ensor 1995; Oehlert 1996; Nychka,Yang,andRoyle 1997; Bueso, Angulo, andAlonso 1998). Variousnonparametric space-fillingcriteriacan also be considered(see Atkinson and Federov [1988] and Federovand Hackl [1997] for generalreviews of optimal spatialdesign). However,given that most environmentalprocesses are temporallydynamic, static designs will not be as efficient as designs thatare allowed to evolve over time. We referto the problemof constructingtime-varyingdesigns as dynamic(or adaptive) design, which is not to be confused with adaptivesampling (e.g., Thompson and Seber 1996), althoughthe notions are similar.The primarydifferenceis thatthe formeris model based, whereasthe latteroccurs in a traditionalsample surveyframework,but with sample unit selection being modified as observationsare made. The term space-time design has been used to describe the design problem for environmentalmonitoringnetworks, but generally these approacheshave not accountedfor the temporallydynamic natureof the process andhavebeen static.Forexample,ArbiaandLafratta(1997) proposedan approach to space-time design thatproducesa staticspatialdesign for minimizingthe varianceof the mean. They used the temporalinformationfor estimatingnonstationaryspatial variances but otherwise ignored the possibility of temporalevolution of the process under study. Similarly,Le and Zidek (1994) took a Bayesian approachbut again ignoredthe possibility of a dynamicdesign. Otherpapers,includingFederovand Nachtsheim(1995), considered spatiotemporalmodels wherethe optimaldesign was independentof time. In this paper,we will refer to designs that consider spatiotemporalinformationbut that do not change over time as static space-time designs. The dynamicdesign problemis difficult.The chief difficultyis that model-basedapproachesrequirethe specificationof therelativelycomplicatedspace-timeinteractioninherent in environmentalprocesses.Manyof these models areoversimplified(e.g., they assume space and time are separable).Othermodels tend to be complicatedand are custom developed for specific problems,makinggeneralizationto other space-time problemsdifficult. For example, Haas (1992) and Oehlert(1996) both addressedthe problemof design with regardto an acid depositionnetworkusing complicatedand finely tuned procedures(i.e.,
DYNAMIC DESIGN OF MONITORING NETWORKS
491
Haas 1990; Oehlert 1993). Berliner,Lu, and Snyder (1998) considered adaptiveweather observations,allowing the design to change over time accordingto explicit atmospheric dynamics. Alternatively,Titterington(1980) discussed a general Kalman filter approach to dynamicdesign, seeking to optimize networksfor estimationof regressionparameters. His paper introducedmuch of the optimal dynamic design work from the control theory literatureto the generalstatisticscommunity. The purposeof this paperis to discuss issues associatedwith the problemof designing a dynamicnetworkfor spatiotemporalprocesses and to providea method,based on a relatively simple and generalizablestatisticalmodel, for designing such a monitoringnetwork. Specifically,we areinterestedin the relativeimportanceof spatialandtemporaldependence (and their interaction)on potential dynamic design improvementsover static space-time designs. Section 2 discusses our space-time dynamicstatisticalmodel. Section 3 describes some experimentsandresultsrelatedto the importanceof temporaland spatialdependence structures.An example based on Chicago ozone datais presentedin Section 4 to illustrate the effect of realistic spatiotemporalcovariancestructureon dynamic design. Finally, we discuss the results and some relatedissues in Section 5.
2. SPACE-TIME DYNAMIC MODEL In orderto evaluatepotentialdynamicdesigns, we mustspecify a reasonablespace-time model. In principle,assumingthat one knows the spatiotemporalcovariancestructureof a process, it is simple to develop the best linearunbiasedpredictorand associatedprediction variancefor some location and time given a sample of observations.Typically,one does not know the full joint spatiotemporalcovariance structureand, necessarily,must make simplifying assumptions.Traditionalapproachesto such problemsinclude extending the geostatisticalparadigmso that time is treatedas anotherspatialdimension (e.g., Bilonick 1983; Cressie and Majure 1997), multivariatetime series methods where the processes are viewed as a set of spatially correlatedtime series (e.g., Bennett 1979; Rouhani and Wackernagel1990), and space-time autoregressivemoving averagemethods on a lattice (e.g., Cliff and Ord 1975; Deutsch and Pfeifer 1981). In recent years, more interest has been given to a hybridspace-time dynamicmodeling approach,in which explicit temporal structureis prescribed(e.g., Markovianevolution), yet the process is assumed to have descriptivespatialstructure(e.g., Guttorp,Meiring,andSampson 1994; HuangandCressie 1996; Wikle and Cressie 1997). One advantageof this approachis that the models can be implementedeasily via an empiricalBayesian or spatiotemporalKalmanfilterprocedure. We will focus on such an approachin this paper. 2.1 MODEL FORMULATION Let the spatial-temporalprocess of interestbe denotedby Y(s; t), where s C S, with S some continuousspatialdomainin two-dimensionalEuclideanspace, and t C {1, 2, ... ., } a discreteindex of times. We furtherassume that at each time t the process is observedat
492
C. K. WIKLEANDJ. A. ROYLE
some finitesubsetof S andthatthe dataaregiven by Zt = (Z(si; t),.* , Z(smt; t))'. Now, assumingthatobservationsoccur with error,we can write a measurementequationas Zt = KtYt + Et,
(2.1)
where Yt is an n x 1 vector of n prediction locations at time t, Et is the zero mean measurementerrorprocess with covariancematrix E,, and Kt is an mt x n matrixthat maps the trueprocess (Y) at predictionlocations to the data (Z) at observationlocations. Kt simply determineswhich of the spatiallocations in Yt are observedand so is a sparse matrixof O'sand l's (Wikle, Berliner,and Cressie 1998). The form of this matrixcan, in general,be muchmorecomplicated.It is criticalthatwe let Kt varyin time to accommodate differentpotentialobservationnetworks(i.e., designs) at each time. A model is specified for the spatiotemporalprocess (Y) at the n predictionlocations accordingto a first-ordervectorMarkovprocess: Yt = HtYt-,
+ 77t
(2.2)
where Ht is the first-orderMarkovparametermatrix, and T7t is the conditional spatiotemporalnoise processwith covariancematrixE. Note thatHt representsthe dynamicsof the process and can be timevarying.This is particularlyrelevantfor linearapproximations to nonlinear systems (e.g., Berliner et al. 1998). Such a formulationtypically requires substantialknowledge of the underlyingprocess (e.g., the governing partialdifferential equations).Similarly,one could allow the conditionalcovarianceto change with time, with similarcomplications.Furthermore,we could allow additionalautoregressivelags in (2.2) as well as moving averagestructures.In the currentcase, the problemis well definedif we know or can estimateKt, Se, Ht, and E,,Given the measurementequation(2.1) and state equation(2.2), a space-time Kalman filtercan be derivedeitherby Bayesianarguments(e.g., Meinholdand Singpurwalla1983; West and Harrison1997) or projectionarguments(e.g., Hamilton1994). In eithercase, one gets the following recursiveequationsfor the predictionerrorcovariance:
I Zt,
At-var(Yt -
[KtEe-Kt
...
,
(2.3)
+ B-1]-1
BtK/[KtBtK/
Bt-
Zi) + E,] 1KtBt
(2.4)
where var(Yt I Zt_,X .** Zi)
Bt =
HtAt1H
+ E*
(2.5) (2.6)
To start the recursion, Ao must be specified and is typically chosen to be the unconditional(i.e., marginal)variance-covariancematrixof the Y processat the prediction locations(e.g., Harvey1993, p. 88). Furthermore,to obtainoptimalpredictionsin a Kalman filter,the parametermatricesHt, Kt, Se, and Er1 must be known. In practice,we seldom know these and must either specify or estimate them. In this case, we no longer obtain exactly the conditionalvariance.However,our approachis analogousto Kalmanfiltering
DYNAMICDESIGNOFMONITORING NETWORKS
493
in time and kriging in space, where the covarianceor variogramparametersmust also be estimated. For a discussion of the difficulties in treatingestimatedparametersas known when formingpredictivedistributions,see Cressie and Zimmerman(1992). Althoughone could employ a fully Bayesian model (e.g., Wikle et al. 1998) to obtain more realistic estimatesof precision,one loses the computationalefficiencyand simplicityrealizedby the Kalmanfilterapproach.Suchtrade-offsmustalwaysbe consideredwhenbuildingstochastic models.
2.2 MODEL-BASED DYNAMIC DESIGN The recursiveformulationof equations(2.3)-(2.6) providesthe mechanismfor dynamic design of monitoringnetworks.The two features of the model that allow the prediction varianceto change over time arethe time-varyingparametermatricesHt and Kt. Of these, the Markov parametermatrix (Ht) is most clearly related to changes in the dynamical natureof the process with time. As mentionedpreviously,withoutfundamentalknowledge of the underlyingprocess, it is not always beneficialto allow this parametermatrix(or the conditionalcovariancematrix)to varywith time. Forthe remainderof this paperwe assume that Ht = H is fixed in time. Thus, the predictionvarianceAt changes throughtime only due to the effect of Kt; thatis, it changes only as a functionof the observationlocations at each time. Algorithmically,one obtains the optimal design for time t given informationup to time t - 1 by (a) calculatingBt based on At-, as in Equation(2.6) and (b) minimizing some functionof At, the design criterion,over all potentialdesigns D (e.g., one common criterionis the averagespatialpredictionerrorvariance). This procedureis clearly adaptiveif H, e, Er,, and Ao have been specified. Note that if we are interestedin the one-step-aheadpredictionvariance,then the appropriate minimizationin Step b would be some function of Bt, ratherthan At. The minimization in Step b can be carriedout using a simple exchange algorithm.The basic idea is that the criterionis evaluatedsuccessively for differentdesigns, and the design is updatedby exchanging bad points for betterpoints. Such algorithmsare widely used in practice and many variationson the basic theme exist (see Cook and Nachtsheim 1980; Atkinson and Federov 1988; Nychka et al. 1997). Although these algorithmsare somewhatgreedy and tend to find local optima, for relatively small problems experience indicates that they do find the global optimum.For largerproblems,the solutions tend to be arbitrarilyclose to the global optimumdependingon how long the algorithmis allowed to run.
3. THE ROLE OF SPATIAL AND TEMPORAL DEPENDENCE In order to gain some understandingof the relative role of spatial and temporal correlation in space-time dynamic design, we conducted some simple experiments. Specifically, we considered a spatial domain with 49 potential sampling locations on a 7 x 7 unit spacingregulargrid.We assumethatthe conditionalcovariancebetween Y(si; t)
C. K. WIKLEAND J. A. ROYLE
494
and Y(sj; t) given Y at all prediction sites at the previous time [i.e., the (i, j) element of Er'] has a stationary isotropic exponential structure c (II si -sj I1) = 2 exp[l I si -sj IIlog(p)], = 1. For the where p is the spatial dependence between locations si and sj when si -Sj = o2 19. Furthermore, we let Kt be a simple experiments presented here, we assume that incidence matrix (i.e., potential observation locations coincide with prediction locations).
We further assume that the measurement error covariance structure is spatial white noise, e=
2I, where U2
=
1. Five monitors are assumed to be available for our 7 x 7 grid. In
one set of experiments, we fix four monitors and allow one of them to move freely about the domain. In a second set of experiments, we allow all five monitor locations to move freely. For these experiments, our design criterion is the average prediction variance (APV), which is the mean of the diagonal elements of At.
3.1 SIMPLE DYNAMICAL STRUCTURE
We first consider a simple structure on the Markov parameter matrix, H = hI, in which h is an autoregressive temporal dependence parameter. This model is essentially the separable spatiotemporal model described in Huang and Cressie (1996). Three values are considered for the temporal dependence parameter (h = {.9, .75, .5}) and three values for the spatial dependence parameter (p = {.95, .90, .80}). Note in the case of the exponential spatial model described in the previous section that these values of p correspond to high, medium, and low spatial dependence, respectively. This characterization is based on our subjective assessment of the strength of spatial correlation in air quality monitoring networks. For example, in the Chicago ozone data discussed in Section 4, we find h to vary from 0.4 to 0.7 across the network, while p is approximately 0.9. Although we feel that the values of h and p used in this experiment are representative of many atmospheric processes on local scales, these parameters only have meaning relative to the units of measurement. For example, in the urban ozone data case, p
=
0.9 may be realistic when distance is measured in
kilometers, but may not be realistic when distance is measured in hundreds of kilometers. If one were studying soil nitrogen levels in an agricultural field, distance might be measured in meters and one might expect large values of p on this scale, but small values of p if distance were measured in kilometers. Similarly, the temporal correlation parameter h is scale dependent. Larger values are appropriate for high frequency measurements (minutes, hours, days), depending on the temporal smoothness of the process under consideration. Two ozone measurements separated by one day might be highly correlated, but two precipitation measurements separated by one day are likely to be much less correlated. The effect of high temporal correlation (h = 0.9) on dynamic design is well illustrated in Figure 1. The first plot in this figure shows the optimal static space-time design under the case of moderate spatial dependence (p = 0.9), which is the starting design. In this case, the monitor in the middle of the domain is allowed to move, and the other four are held fixed. Other plots in this figure show how the optimal design changes with time. Clearly, under this strong temporal dependence, the information from a monitor at time t - 1 is still very much available to reduce the prediction variance in that region during the next few periods,
495
DYNAmic DESIGN OF MONITORING NETWORKS
rove=7-1 h=.9 rho .90 tinie =1 APV - 17.34
rove=1 h .9rho .90time
2APV
I rove
1h
rove
1h
Q9rho
.9rho
p
.90 tim-e 3 APV
15.79
rove
1 hi
5 APV
14.76
rove
1h
.90time
16.48
9 rhio .90 time
.9rho
4 APV
15 22
90Otime 6 APV
14.51
Figure 1. Change in Optimal Design Over Time. Optimal design locations (X) and prediction variances for the first six times under the model with moderate spatial correlation (p = 0.9), high temporal correlation (hi = 0.9), and one roving monitor. The APV is averaged over space.
496
C. K. WIKLEANDJ. A. ROYLE
so we wantto move sites at the next time to reducethe redundancyin the observations.For the most part,the roving monitorcycles aroundthe outside of the domain.Figure2 shows the identicalexperimentexcept that all five monitoringlocations can move. As expected, thereis a substantialreductionin APV when all monitorsareallowed to move. By contrast, if we have relatively low temporaldependence(e.g., h = 0.5), then the dynamic design does not change from the optimal static space-time design if only one monitorcan rove. However,if all five monitorsare free to rove, then there is a differencebetween the static and the dynamicdesigns, and the APV is reduced,althoughonly slightly. A summaryof the decrease in APV (averagedover time steps 10-20) relative to the optimal static design (i.e., the time 1 design) for each simulation is shown in Table 1. Furthermore,Figures3 and 4 show plots of the APV decreasefrom the optimalstatic (i.e., starting)design as a functionof time for the one andfive rovingmonitorcases, respectively. From these figures and Table 1, the dynamic design experimentsrelative to APV can be summarizedby the following intuitiveresults. * All otherfactorsbeing constant,the improvementof a dynamicdesign over a static design is largestwhen temporalstructureis strong. * All otherfactorsbeing constant,the improvementof a dynamicdesign over a static design is largestwhen more monitorsare allowed to rove. * When only one monitorcan rove, the improvementof a dynamic design over a staticdesign is generallynot dependenton the strengthof the spatialcorrelationin the process.However,when all five areallowedto rove, processeswith low spatial dependencetypically show dynamic designs with relatively large improvements over staticdesigns. 3.2 MORE COMPLICATED DYNAMICS
The parametermatrix H in the previous section was assumed to be diagonal and constant across all spatial locations. Although such a model can be useful (e.g., Huang and Cressie 1996), it is inherentlyseparableand does not capturecomplicateddynamics. A simple extension is to let neighboringspatial locations at the previoustime contribute to the process at the currenttime. In that case, H has a lagged nearestneighborstructure, with diagonalsof H correspondingto each neighbor(e.g., Wikle et al. 1998). As a simple example,we considerthe case wherethe elementsof H correspondingto the east and west neighborsarebothequalto 0.25, theelementscorrespondingto thenorthandsouthneighbors are assumedto be zero, andthe same-site(i.e., main-diagonal)parametersof H aredefined to be 0.5. Given a moderatespatialdependenceof 0.9 andthe conditionalcovariancemodel describedin Section 3, dynamicallyvaryingdesigns were examined.Figure 5 shows the firstsix of these designs where five monitorsareallowed to rove. Thus,even with relatively low same-sitetemporalcorrelation,the simple off-site (lagged neighbor)structurein H still allows nonintuitivedynamicallyvaryingdesigns of some complexity. Finally,considera very simplenonseparablespace-time modelin which H = diag(h), wherethe elementsof h varywith location.Althoughnot shownhere,sucha modelproduces
NETWORKS DYNAmicDESIGN OFMONITORING
rove
5 h .9 rho .90Otime 1 APV
17.34
rove5
497
h .9 rho =.90time
2 APV
14.51
x~~~~~~
x
~~~~~~~~~~~~~~~~~~easting
eLastirg
rove 5 h = 9rho
90Otime 35APV 12561
5
rove 5 h = 9rho
90Otime 46APV 1240
6~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~------
---
4
5
eastin~g
Figure
Change rhon OtmaDegnOrTime 9 1.6
Sam oeas F rgore91 . exetim
tfie rovAPV12 g
morntors~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ..... complicatd asymmencaldesgns Mostenvironmntal proesses exibit subtantiall morecomlictednonepaabl stuctresforH tan he impe srucure prsened ere Efficientdesignsare likely to be more complicated designs~~~~~~~~~~~~~~~~........ and asymmetricalthan spatial obtainedith separble and/o sainrcorIanemdl.WkeadCese(97
C. K. WIKLEANDJ. A. ROYLE
498
Table 1. Results from Simple-Structure Dynamic Design Experiments
Experiment
Number of roving monitors
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
1 1 1 1 1 1 1 1 1 5 5 5 5 5 5 5 5 5
Temporal structure (h) 0.90 0.90 0.90 0.75 0.75 0.75 0.50 0.50 0.50 0.90 0.90 0.90 0.75 0.75 0.75 0.50 0.50 0.50
Spatial structure (p) 0.95 0.90 0.80 0.95 0.90 0.80 0.95 0.90 0.80 0.95 0.90 0.80 0.95 0.90 0.80 0.95 0.90 0.80
Average predicted variance decrease (?%)a -16.1 -16.7 -16.7 -5.7 -5.5 -5.4 -0.6 -0.2 -0.1 -29.6 -31.5 -35.0 -11.0 -12.5 -13.3 -3.1 -3.3 -2.3
a
Prediction variance decrease relative to the optimal static space-time design, averaged over time steps 10-20.
providea dimensionreductionspace-time Kalmanfiltermechanismwherebynonseparable and spatiallynonstationaryprocessescan be modeled. Such approacheswould be a natural choice for examiningdynamicaldesigns for complicatedprocesses.
4. CHICAGO AREA OZONE MONITORING NETWORK To examine the potential for dynamic design of monitoringnetworks with realistic and complicatedcovarianceand parametermatrices, our methodology was applied to a collection of 21 stationsused to monitorambientozone in the Chicago area(see Figure6). The data consist of eight-houraverageozone (from 9 A.M.-5 P.M.) measurementstaken each day from the periodJune 3 to August 21, 1987. These data are availablein AIRS, the EPA air qualitydatabase,and have been examinedin numerousstudies (e.g., Bloomfield, Royle, Steinberg,andYang 1996; Nychkaet al. 1997). We considerthe existing monitoring networkto be ouruniverseof potentialsite locations.Underthe conditionthatthereareonly five monitorsavailableandthatthese monitorscan be moved on a daily basis, ourgoal is to examinehow the optimalspatialdesign might change with time. Thatis, we are interested in the optimal "thinning"of an existing networkthat finds the most efficient designs for a five-point subset of that network. For this example, we focus on the minimizationof the maximumpredictionvariance(MPV) as our design criterion.This is an appropriate design criterionfor ozone, as we are interestedin preventingan exceedanceof the ambient standardsat the locations in the existing network.Thus, we do not want to predictat any
499
DYNAMIC DESIGN OF MONITORING NETWORKS
rove =1 h =.9 rho=.95
9
rove = 1 h = .9 rho = .80
rove =1 h =.9 rho=.90
.9
.9
9
I o
___D_r o ________0
_______
5
10
15
20
:_:1_v_r
5
10
Time
I
0
o9
_
_
5
...
_
_
10
9
_
10
15
_
9
_
_
5
20
10
20
rove =1 h =.75 rho=.80
9]*@e@@e
_
15
Time
_
_
_
15
_
_
_
5
20
10
_
_
15
Time
Time
rove =1 h = .5 rho = .95
rove =1 h = .5 rho = .90
rove =1 h = .5 rho= .80
...
O -.....................
.. .
