Spatio-Temporal Statistics With R

397 Pages·2019·5.813 MB·English

by  Christopher Wikle, Andrew Zammit Mangion, Noel Cressie

Spatio-Temporal
Statistics with R
Chapman & Hall/CRC
The R Series
Series Editors
John M. Chambers, Department of Statistics, Stanford University Stanford, California, USA
Torsten Hothorn, Division of Biostatistics, University of Zurich, Switzerland
Duncan Temple Lang, Department of Statistics, University of California, Davis, California, USA
Hadley Wickham, RStudio, Boston, Massachusetts, USA
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Spatio-Temporal Statistics with R
Christopher K. Wikle, Andrew Zammit-Mangion, and Noel Cressie
For more information about this series, please visit: https://www.crcpress.com/go/the-r-series
Spatio-Temporal
Statistics with R
CHRISTOPHER K. WIKLE
ANDREW ZAMMIT-MANGION
NOEL CRESSIE
Cover Illustration: Julinu (Julian Mallia)
www.julinu.com
CRC Press
Taylor & Francis Group
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Library of Congress Cataloging-in-Publication Data
Names: Wikle, Christopher K., 1963- author. | Zammit-Mangion, Andrew, author.
| Cressie, Noel A. C., author.
Title: Spatio-temporal statistics with R / Christopher K. Wikle, Andrew
Zammit-Mangion, Noel Cressie.
Description: Boca Raton, Florida : CRC Press, [2019] | Includes
bibliographical references and index.
Identifiers: LCCN 2018048440| ISBN 9781138711136 (hardback : alk. paper) |
ISBN 9781351769723 (e-book : alk. paper)
Subjects: LCSH: Spatial analysis (Statistics) | Statistics. | R (Computer
program language)
Classification: LCC QA278.2 .W55 2019 | DDC 519.5/37--dc23
LC record available at https://lccn.loc.gov/2018048440
Visit the Taylor & Francis Web site at
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Contents
Acknowledgements ix
Preface xiii
1 IntroductiontoSpatio-TemporalStatistics 1
1.1 WhyShouldSpatio-TemporalModelsBeStatistical? . . . . . . . . . . . . 6
1.2 GoalsofSpatio-TemporalStatistics . . . . . . . . . . . . . . . . . . . . . 7
1.2.1 TheTwoDsofSpatio-TemporalStatisticalModeling . . . . . . . . 7
1.2.2 DescriptiveModeling . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.3 DynamicModeling . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 HierarchicalStatisticalModels . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4 StructureoftheBook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 ExploringSpatio-TemporalData 17
2.1 Spatio-TemporalData . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 RepresentationofSpatio-TemporalDatainR . . . . . . . . . . . . . . . . 22
2.3 VisualizationofSpatio-TemporalData . . . . . . . . . . . . . . . . . . . . 24
2.3.1 SpatialPlots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3.2 Time-SeriesPlots . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3.3 HovmöllerPlots . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3.4 InteractivePlots. . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3.5 Animations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3.6 Trelliscope: VisualizingLargeSpatio-TemporalDataSets . . . . . 29
2.3.7 VisualizingUncertainty . . . . . . . . . . . . . . . . . . . . . . . 31
2.4 ExploratoryAnalysisofSpatio-TemporalData . . . . . . . . . . . . . . . 32
2.4.1 EmpiricalSpatialMeansandCovariances . . . . . . . . . . . . . . 33
2.4.2 Spatio-TemporalCovariogramsandSemivariograms . . . . . . . . 36
2.4.3 EmpiricalOrthogonalFunctions(EOFs) . . . . . . . . . . . . . . . 39
2.4.4 Spatio-TemporalCanonicalCorrelationAnalysis . . . . . . . . . . 47
2.5 Chapter2Wrap-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Lab2.1: DataWrangling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
v
vi Contents
Lab2.2: Visualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Lab2.3: ExploratoryDataAnalysis . . . . . . . . . . . . . . . . . . . . . . . . 67
3 Spatio-TemporalStatisticalModels 77
3.1 Spatio-TemporalPrediction . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.2 Regression(Trend-Surface)Estimation . . . . . . . . . . . . . . . . . . . . 84
3.2.1 ModelDiagnostics: DependentErrors . . . . . . . . . . . . . . . . 88
3.2.2 ParameterInferenceforSpatio-TemporalData . . . . . . . . . . . 93
3.2.3 VariableSelection . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.3 Spatio-TemporalForecasting . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.4 Non-GaussianErrors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.4.1 GeneralizedLinearModelsandGeneralizedAdditiveModels . . . 101
3.5 HierarchicalSpatio-TemporalStatisticalModels . . . . . . . . . . . . . . . 104
3.6 Chapter3Wrap-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Lab3.1: DeterministicPredictionMethods . . . . . . . . . . . . . . . . . . . . . 106
Lab3.2: TrendPrediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Lab3.3: RegressionModelsforForecasting . . . . . . . . . . . . . . . . . . . . 125
Lab3.4: GeneralizedLinearSpatio-TemporalRegression . . . . . . . . . . . . . 130
4 DescriptiveSpatio-TemporalStatisticalModels 137
4.1 AdditiveMeasurementErrorandProcessModels . . . . . . . . . . . . . . 138
4.2 PredictionforGaussianDataandProcesses . . . . . . . . . . . . . . . . . 139
4.2.1 Spatio-TemporalCovarianceFunctions . . . . . . . . . . . . . . . 143
4.2.2 Spatio-TemporalSemivariograms . . . . . . . . . . . . . . . . . . 150
4.2.3 GaussianSpatio-TemporalModelEstimation . . . . . . . . . . . . 151
4.3 Random-EffectsParameterizations . . . . . . . . . . . . . . . . . . . . . . 154
4.4 Basis-FunctionRepresentations. . . . . . . . . . . . . . . . . . . . . . . . 157
4.4.1 RandomEffectswithSpatio-TemporalBasisFunctions . . . . . . . 158
4.4.2 RandomEffectswithSpatialBasisFunctions . . . . . . . . . . . . 161
4.4.3 RandomEffectswithTemporalBasisFunctions . . . . . . . . . . . 162
4.4.4 ConfoundingofFixedEffectsandRandomEffects . . . . . . . . . 164
4.5 Non-GaussianDataModelswithLatentGaussianProcesses . . . . . . . . 165
4.5.1 GeneralizedAdditiveModels(GAMs) . . . . . . . . . . . . . . . . 166
4.5.2 InferenceforSpatio-TemporalHierarchicalModels . . . . . . . . . 167
4.6 Chapter4Wrap-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
Lab4.1: Spatio-TemporalKrigingwithgstat . . . . . . . . . . . . . . . . . . . 172
Lab4.2: Spatio-TemporalBasisFunctionswithFRK . . . . . . . . . . . . . . . 175
Lab4.3: TemporalBasisFunctionswithSpatioTemporal . . . . . . . . . . . . . 180
Lab4.4: Non-GaussianSpatio-TemporalGAMswithmgcv . . . . . . . . . . . . 189
Lab4.5: Non-GaussianSpatio-TemporalModelswithINLA . . . . . . . . . . . 192
Contents vii
5 DynamicSpatio-TemporalModels 205
5.1 GeneralDynamicSpatio-TemporalModels . . . . . . . . . . . . . . . . . 206
5.1.1 DataModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
5.1.2 ProcessModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
5.1.3 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
5.2 LatentLinearGaussianDSTMs . . . . . . . . . . . . . . . . . . . . . . . 209
5.2.1 LinearDataModelwithAdditiveGaussianError . . . . . . . . . . 209
5.2.2 Non-GaussianandNonlinearDataModel . . . . . . . . . . . . . . 212
5.2.3 ProcessModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
5.3 ProcessandParameterDimensionReduction . . . . . . . . . . . . . . . . 218
5.3.1 ParameterDimensionReduction . . . . . . . . . . . . . . . . . . . 218
5.3.2 DimensionReductionintheProcessModel . . . . . . . . . . . . . 222
5.4 NonlinearDSTMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
5.5 Chapter5Wrap-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
Lab5.1: ImplementinganIDEModelinOne-DimensionalSpace . . . . . . . . 229
Lab5.2: Spatio-TemporalInferenceusingtheIDEModel . . . . . . . . . . . . . 234
Lab5.3: Spatio-TemporalInferencewithUnknownEvolutionOperator . . . . . 244
6 EvaluatingSpatio-TemporalStatisticalModels 253
6.1 ComparingModelOutputtoData: WhatDoWeCompare? . . . . . . . . . 254
6.1.1 ComparisontoaSimulated“True”Process . . . . . . . . . . . . . 255
6.1.2 PredictiveDistributionsoftheData . . . . . . . . . . . . . . . . . 256
6.1.3 ValidationandCross-Validation . . . . . . . . . . . . . . . . . . . 258
6.2 ModelChecking. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
6.2.1 ExtensionsofRegressionDiagnostics . . . . . . . . . . . . . . . . 260
6.2.2 GraphicalDiagnostics . . . . . . . . . . . . . . . . . . . . . . . . 262
6.2.3 SensitivityAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . 266
6.3 ModelValidation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
6.3.1 PredictiveModelValidation . . . . . . . . . . . . . . . . . . . . . 268
6.3.2 Spatio-TemporalValidationStatistics . . . . . . . . . . . . . . . . 269
6.3.3 Spatio-TemporalCross-ValidationMeasures . . . . . . . . . . . . 272
6.3.4 ScoringRules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
6.3.5 FieldComparison . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
6.4 ModelSelection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
6.4.1 ModelAveraging . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
6.4.2 ModelComparisonviaBayesFactors . . . . . . . . . . . . . . . . 283
6.4.3 ModelComparisonviaValidation . . . . . . . . . . . . . . . . . . 283
6.4.4 InformationCriteria . . . . . . . . . . . . . . . . . . . . . . . . . 284
6.5 Chapter6Wrap-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
Lab6.1: Spatio-TemporalModelValidation . . . . . . . . . . . . . . . . . . . . 289
viii Contents
Pergimus(Epilogue) 303
Appendices 307
A SomeUsefulMatrix-AlgebraDefinitionsandProperties . . . . . . . . . . . 307
B GeneralSmoothingKernels . . . . . . . . . . . . . . . . . . . . . . . . . . 311
C EstimationandPredictionforDynamicSpatio-TemporalModels . . . . . . 312
C.1 Estimation in Vector Autoregressive Spatio-Temporal Models via
theMethodofMoments . . . . . . . . . . . . . . . . . . . . . . . 312
C.2 PredictionandEstimationinFullyParameterizedLinearDSTMs . 313
C.3 EstimationforNon-GaussianandNonlinearDSTMs . . . . . . . . 318
D MechanisticallyMotivatedDynamicSpatio-TemporalModels . . . . . . . 318
D.1 ExampleofaProcessModelMotivatedbyaPDE:FiniteDifferences318
D.2 ExampleofaProcessModelMotivatedbyaPDE:Spectral . . . . . 320
D.3 ExampleofaProcessModelMotivatedbyanIDE . . . . . . . . . 321
E Case Study: Physical-Statistical Bayesian Hierarchical Model for Predict-
ingMediterraneanSurfaceWinds . . . . . . . . . . . . . . . . . . . . . . 323
F Case Study: Quadratic Echo State Networks for Sea Surface Temperature
Long-LeadPrediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340
ListofRPackages 351
References 355
SubjectIndex 367
AuthorIndex 373
RFunctionIndex 377
Acknowledgements
WhenNoelandIfinishedthemulti-yearprojectthatbecameStatisticsforSpatio-Temporal
Data in 2010, I’m pretty sure I didn’t think that I would be writing another book on this
topic! But, it’s eight years later and here we are .... It has been a great pleasure to work
with Andrew and Noel on this project and I thank them deeply for all of the stimulating
discussion,idea-sharing,advice,andhardworktheyputintothisproject. Ilearnedagreat
deal and it could never have happened without them! In particular, Andrew has worked
magictomaketheRLabsintegrateintothemethodologicalcontent,andthisisthefeature
ofthebookthatmakesitunique. Iwanttothankmyspatio-temporalcolleaguesatMizzou
(ScottHolan,SakisMicheas,andErinSchliep)aswellasstudentsandpostdocswhohave
continued to make this an exciting and fun topic in which to work. My eternal thanks
to Olivia, Nathan, and Andrea for their support of this project and all it entailed and for
enrichingmylifealways! Last,andmostimportantly,IwouldliketothankCarolyn,whois
onthe“frontlines”ofdealingwiththeeffectsofthesesortsofprojects,andalwaysprovides
tremendoussupport, sanity, andencouragement alongtheway. Icouldnotdowhat Idoif
itwerenotforher!
C.K.W.
More than ten years have passed since the day when I was sitting opposite my honors
thesis supervisor, Simon Fabri, at the University of Malta with a scholarship offer from
the University of Sheffield in my hand, and a pen in the other. “What is spatio-temporal
modeling,andisthereanyfutureinit?” Imumbledinquisitively. Itislargelythankstohis
reply and my PhD supervisor Visakan Kadirkamanathan that I took an interest in spatio-
temporal modeling, and in the field of statistics in general. Since then, I have had other
mentorsfromnumerousdisciplines,fromstatisticstocomputerscienceandgeography,and
Iwouldliketothankthemallfortheiradviceandfortheopportunitiestheyhaveprovided
me with; they include Guido Sanguinetti, Jonathan Rougier, Jonathan Bamber, and more
recentlyNoelCressie.
InthelasttenyearsIhavehadtheprivilegetoworkandhavediscussionswithseveral
other colleagues with similar interests. Some of these have inspired my work in several
ways;theyincludeTaraBaldacchino,ParhamAram,MichaelDewar,KennethScerri,Sean
ix