Spatial Statistics and Modeling

308 Pages·2010·6.3 MB·English

by  Carlo Gaetan

Springer Series in Statistics
Advisors:
P.Bickel,P.Diggle,S.Fienberg,
U.Gather,I.Olkin, S.Zeger
Forothertitlespublishedinthisseries,goto
www.springer.com/series/692
·
Carlo Gaetan Xavier Guyon
Spatial Statistics
and Modeling
Translated by Kevin Bleakley
123
CarloGaetan XavierGuyon
DipartimentodiStatistica SAMOS
Universita`Ca’FoscariVenezia Universite´ParisI
SanGiobbe-Cannaregio,873 90ruedeTolbiac
30121VeneziaVE 75634ParisCX13
ltaly France
[email protected] [email protected]
ISBN978-0-387-92256-0 e-ISBN978-0-387-92257-7
DOI10.1007/978-0-387-92257-7
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Preface
Spatialanalysismethodshaveseenarapidriseinpopularityduetodemandfroma
widerangeoffields.Theseinclude,amongothers,biology,spatialeconomics,im-
ageprocessing,environmentalandearthscience,ecology,geography,epidemiology,
agronomy,forestryandmineralprospection.
In spatial problems, observations come from a spatial process X ={X ,s∈S}
s
indexed by a spatial set S, with X taking values in a state space E. The positions
s
ofobservationsitess∈S areeitherfixedinadvanceorrandom.Classically,S isa
2-dimensionalsubset,S⊆R2.However,itcouldalsobe1-dimensional(chromatog-
raphy,croptrialsalongrows)orasubsetofR3 (mineralprospection,earthscience,
3D imaging). Other fields such as Bayesian statistics and simulation may even re-
quirespacesSofdimensiond≥3.Thestudyofspatialdynamicsaddsatemporal
dimension,forexample(s,t)∈R2×R+inthe2-dimensionalcase.
This multitude of situations and applications makes for a very rich subject. To
illustrate, let us give a few examples of the three types of spatial data that will be
studiedinthebook.
Geostatisticaldata
Here,S isacontinuous subspaceofRd andtherandomfield{X ,s∈S}observed
s
at n fixed sites {s ,...,s }⊂S takes values in a real-valued state space E. The
1 n
rainfall data in Figure 0.1-a and soil porosity data in Fig. 0.1-b fall into this cate-
gory. Observation sites may or may not be regularly spaced. Geostatistics tries to
answer questions about modeling, identification and separation of small and large
scalevariations,prediction(orkriging)atunobservedsitesandreconstructionofX
acrossthewholespaceS.
Latticedataanddataonfixednetworks
Here, S is a fixed discrete non-random set, usually S⊂Rd and X is observed at
points in S. Points s might be geographical regions represented as a network with
v
vi Preface
0
5
2
0
0
2
0
d 5
or 1
o w
Y C 100 ro
0
5
0
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5
−
column
0 50 100 150 200 250 300 350
X Coord
Fig.0.1 (a)RainfallovertheSwissmeteorologicalnetworkonMay8,1986(duringthepassage
ofChernobyl’sradioactivecloud.ThisisthesicdatasetfromthegeoRpackageofR(178));(b)
Soilporosity(soildatasetfromthegeoRpackage).Forboth(a)and(b),thesizeofsymbolsare
proportionaltothevalueofXs.
less than 9 1.0
9 − 12.2
12.2 − 19
more than 19
8
0.
6
0.
4
0.
2
0.
0
0.
0.0 0.2 0.4 0.6 0.8 1.0
(a) (b)
Fig.0.2 (a)PercentageofpeoplewithbloodgroupAinthe26countiesofIreland(eiredataset
fromthespdeppackage);(b)ImageofJohnLennon(256×256pixelsina193-levelgrayscale,
lennondatasetfromthefieldspackage).
given adjacency graph G (cf. the 26 counties of Ireland, Fig. 0.2-a) and X some
s
valueofinterestmeasuredats.ThestatespaceE mayormaynotbereal-valued.In
imageanalysis,Sisaregularlyspacedsetofpixels(cf.Fig.0.2-b).Goalsforthese
types of data include constructing and analyzing explicative models, quantifying
spatialcorrelations,predictionandimagerestoration.
Preface vii
Pointdata
Figure0.3-ashowsthelocationofcellcentersinahistologicalsectionseenunder
amicroscopeandFigure0.3-bthelocationandsizeofpinetreesinaforest.Here,
thesetofobservationsitesx={x ,x ,...,x },x ∈S⊂Rd israndom,alongwith
1 2 n i
thenumbern=n(x)ofobservationsites;xistheoutcomeofaspatialpointprocess
(PP) observed in window S. The process X is said to be marked if at each x we
i
record a value, for example the diameter of the pine trees found at x. A central
i
question in the statistical analysis of PPs is to know if the distribution of points is
essentially regular (Figure 0.3-a), completely random (Poisson PP) or aggregated
(Figure0.3-b).
0
0
2
8
0. 0
5
1
6
0.
0
y y 10
4
0.
5
0
2
0.
0.0 0
0.2 0.4 0.6 0.8 1.0 0 0 5 100 150 200
x x
(a) (b)
Fig.0.3 (a)42cellcentersofahistologicalsectionseenunderamicroscope(cellsdatasetfrom
the spatstat package); (b) Position and diameter of 584 pine trees in a forest (longleaf
datasetfromthespatstatpackage).
Asisthecasefortimeseries,spatialstatisticsdifferfromclassicalstatisticsdue
to non-independence of observations; throughout this book, we will generally call
X aspatialprocessorrandomfield.
This dependency structure means there is redundancy in available information
that can be exploited when making predictions, though it also modifies statistical
behavior. Unbiasedness, consistency, efficiency and convergence in distribution of
estimatorsallhavetobereexaminedinthiscontext.Theoriginalityofspatialstatis-
ticsistomakeuseofnon-causalmodeling;inthissense,spatialstatisticsisradically
differenttotimeseriesstatisticswherecausalmodelsusethepassageoftimeanda
notionofthe“past”(modelingriverflows,stockprices,evolutionofunemployment
rates, etc.). Markov spatial modeling works with the idea of the spatial neighbor-
hood of site s “in all directions.” This includes dimension d =1: for example, if
S⊆Z1 andX isthequantityofcornharvestedfromeachcornstalkalongarow,a
s
reasonablemodelwouldcompareX withitstwoneighbors,thestalkstothe“left”
s
Xs−1 and “right” Xs+1. We see that causal autoregressive modeling of Xs based on
viii Preface
Xs−1 has no obvious meaning. If the crop is in a field, we could let the harvested
quantity Xs,t at site (s,t) depend on that of its 4 nearest neighbors Xs−1,t, Xs+1,t,
Xs,t−1andXs,t+1,orevenperhapsits8nearestneighbors.
These three types of spatial structure (cf. Cressie, (48)) provide the framework
tothisbook.Thefirstthreechaptersaredevotedtomodelingeachinturn(Chapter
1:Second-ordermodels,geostatistics,intrinsicmodelsandautoregressivemodels;
Chapter2:Gibbs-Markovrandomfieldsovernetworks;Chapter3:Spatialpointpro-
cesses).Duetotheimportanceofsimulationinspatialstatistics,Chapter4presents
MonteCarloMarkovChain(MCMC)methodsforspatialstatistics.Chapter5then
brings together the most important statistical methods for the various models and
datatypesandinvestigatestheirproperties.Fourappendicesroundthingsoffwitha
presentationofthemostusefulprobabilisticandstatisticaltoolsinspatialstatistics
(simulation, limit theorems and minimum contrast estimation) as well as software
packagesforperforminganalysespresentedinthebook.
Numerous examples, most of them treated with the R software package (178),
shedlightonthetopicsbeingexamined.Whenthedatabeingstudiedarenotdirectly
availableinRorfromsomeotherspecifiedlocation,descriptions,relevantprogram
scriptsandlinkscanbefoundatthewebsiteofthebook:
www.dst.unive.it/∼gaetan/ModStatSpat.
Eachchapterendswithasetofexercises.
The bibliography gives the reader the chance to enrich their knowledge of no-
tions only briefly presented here as well as several technical results whose proofs
have been omitted. We also list reference books that fill gaps remaining after our
intentionallyreducedandnon-exhaustivetreatmentofthismulti-facetedsubjectun-
dergoinggreatdevelopment(69).
Our thanks go to all our colleagues who have given us a taste for spatial anal-
ysis, for their ideas, remarks, contributions and those who have allowed us to use
datacollectedfromtheirownwork.WewouldequallyliketothanktheRDevelop-
ment Core Team and authors of spatial packages for R (178) who have made their
powerful and efficient software freely available to the public, indispensable when
workingwithmethodsandtoolsdescribedhere.Wethankreviewersfortheircare-
fulrereadingofthefirstdraft;theirremarkshavehelpedtosignificantlyimprovethe
presentversion.ThankstoBernardYcartforencouragingustoexpandaninitially
moremodestproject.Ofcourse,wecouldneverhaveundertakenthisworkwithout
thepatienceandsupportofourfamiliesandthebackingofourrespectiveresearch
teams,DipartimentodiStatistica-UniversitàCa’FoscariVeneziaandLaboratoire
SAMOS-UniversitéParis1.Lastly,manythankstoKevinBleakleyforthetransla-
tionandEnglishadaptation,donewithmuchcompetence.Anyremainingerrorsare
ours.
VeniceandParis, CarloGaetan
August2009 XavierGuyon
Contents
1 Second-orderspatialmodelsandgeostatistics ..................... 1
1.1 Somebackgroundinstochasticprocesses ...................... 2
1.2 Stationaryprocesses ........................................ 3
1.2.1 Definitionsandexamples.............................. 3
1.2.2 Spectralrepresentationofcovariances................... 5
1.3 Intrinsicprocessesandvariograms ............................ 8
1.3.1 Definitions,examplesandproperties .................... 8
1.3.2 Variogramsforstationaryprocesses..................... 10
1.3.3 Examplesofcovariancesandvariograms ................ 11
1.3.4 Anisotropy.......................................... 14
1.4 Geometricproperties:continuity,differentiability................ 15
1.4.1 Continuityanddifferentiability:thestationarycase........ 17
1.5 Spatialmodelingusingconvolutions........................... 19
1.5.1 Continuousmodel ................................... 19
1.5.2 Discreteconvolution ................................. 21
1.6 Spatio-temporalmodels ..................................... 22
1.7 Spatialautoregressivemodels ................................ 25
1.7.1 StationaryMAandARMAmodels ..................... 26
1.7.2 Stationarysimultaneousautoregression.................. 28
1.7.3 Stationaryconditionalautoregression ................... 30
1.7.4 Non-stationaryautoregressivemodelsonfinitenetworksS.. 34
1.7.5 Autoregressivemodelswithcovariates .................. 37
1.8 Spatialregressionmodels.................................... 38
1.9 Predictionwhenthecovarianceisknown....................... 42
1.9.1 Simplekriging ...................................... 43
1.9.2 Universalkriging .................................... 44
1.9.3 Simulatedexperiments................................ 45
Exercises ...................................................... 47
2 Gibbs-Markovrandomfieldsonnetworks ........................ 53
2.1 Compatibilityofconditionaldistributions ...................... 54
ix
x Contents
2.2 GibbsrandomfieldsonS .................................... 55
2.2.1 InteractionpotentialandGibbsspecification.............. 55
2.2.2 ExamplesofGibbsspecifications....................... 57
2.3 MarkovrandomfieldsandGibbsrandomfields ................. 64
2.3.1 Definitions:cliques,Markovrandomfield................ 64
2.3.2 TheHammersley-Cliffordtheorem ..................... 65
2.4 Besagauto-models ......................................... 67
2.4.1 Compatibleconditionaldistributionsandauto-models ..... 67
2.4.2 Examplesofauto-models ............................. 68
2.5 Markovrandomfielddynamics ............................... 73
2.5.1 MarkovchainMarkovrandomfielddynamics ............ 74
2.5.2 Examplesofdynamics................................ 74
Exercises ...................................................... 76
3 Spatialpointprocesses.......................................... 81
3.1 Definitionsandnotation ..................................... 82
3.1.1 Exponentialspaces................................... 83
3.1.2 Momentsofapointprocess............................ 85
3.1.3 Examplesofpointprocesses........................... 87
3.2 Poissonpointprocess ....................................... 89
3.3 Coxpointprocess .......................................... 91
3.3.1 log-GaussianCoxprocess ............................. 91
3.3.2 DoublystochasticPoissonpointprocess ................. 92
3.4 Pointprocessdensity ....................................... 92
3.4.1 Definition .......................................... 93
3.4.2 Gibbspointprocess .................................. 94
3.5 Nearestneighbordistancesforpointprocesses .................. 98
3.5.1 Palmmeasure ....................................... 98
3.5.2 TwonearestneighbordistancesforX ................... 99
3.5.3 Second-orderreducedmoments ........................100
3.6 Markovpointprocess .......................................102
3.6.1 TheRipley-KellyMarkovproperty .....................102
3.6.2 Markovnearestneighborproperty ......................104
3.6.3 GibbspointprocessonRd.............................107
Exercises ......................................................108
4 Simulationofspatialmodels.....................................111
4.1 ConvergenceofMarkovchains ...............................112
4.1.1 Stronglawoflargenumbersandcentrallimittheorem
forahomogeneousMarkovchain.......................117
4.2 TwoMarkovchainsimulationalgorithms ......................118
4.2.1 Gibbssamplingonproductspaces ......................118
4.2.2 TheMetropolis-Hastingsalgorithm .....................120
4.3 SimulatingaMarkovrandomfieldonanetwork.................124
4.3.1 Thetwostandardalgorithms...........................124