Temporal GIS: Advanced Functions for Field-Based Applications

224 Pages·2001·4.031 MB·English

by  George Christakos Ph.D., P.Eng., Patrick Bogaert, Marc L. Serre Ph.D. (auth.)

TemporalGIS:AdvancedFunctions forField-BasedApplications
Springer-Verlag Berlin Heidelberg GmbH
George Christakos · Patrick Bogaert · Marc L.Serre
Temporal GIS:
Advanced Functions for
Field-Based Applications
With 73 Figures and 23 Tables
Authors
George Christakos, Ph.D., P.Eng.
Professor & Director
Center for the Advanced Study
of the Environment -CASE
Department of Environmental
Sciences & Engineering
School of Public Health
University of North Carolina
Chapel Hill, NC 27599-7431, USA
Additional material to this book can be downloaded from http://extras.springer.com.
Library of Congress Cataloging-in-Publication Data applied for
Die Deutsche Bibliothek - CIP Einheitsaufnahme
Christakos, George: Advaneed funetions of temporal GIS : with 23 tables / George Christakos ; Patriek
Bogaert ; Mare L. Serre. --Berlin; Heidelberg ; New York; Bareelona ; Hong Kong ; London ; Milan
; Paris; Tokyo :Springer, 2002
ISBN 978-3-642-62563-3 ISBN 978-3-642-56540-3 (eBook)
DOI 10.1007/978-3-642-56540-3
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Preface
Trustonlymovement.
Life happensat the level ofevents not ofwords.
Trust movement.
A.Adler
Asits title suggests,the main goal ofthis bookis the development ofadvancedfunc-
tionsfor field-based TemporalGeographical Information Systems (TGIS).These fields
may describe a variety of natural,epidemiological,economical,andsocial phenom-
ena distributed across space and time.Within such aframework,the book makesan
attempt to establish links between,(a) the currently conceived TGIS techniques,and
(b) the Bayesian maximum entropy (BME) techniques ofModernSpatiotemporal Geo-
statistics.This link could be vital for offeringsignificantimprovements in the advanced
functionsofTGIS analysis and modelling, as well as generating useful information in
a variety ofreal-world decisionmaking and planning situations.
Toachievethe above goals,the eight Chapters of the bookareorganized around
four main themes:
Concepts,mathematical tools,computer programs,and applications.
Infact,the focus is mainly on the step-by-step implementationof the computa-
tional BME approach and the extensive use of illustrative examples andreal-world
applications.Indeed,because ofthe applied character ofthe present book,no detailed
theoretical explanationsor mathematical derivations are included.Instead,the reader
is referredto the earlier bookbyChristakos (Modern Spatiotemporal Geostatistics,
OxfordUniv.Press,New York,N.Y.,2000) for a comprehensive presentationof these
BMEaspects.With this in mind,the chapter-by-chapterorganizationof the book is
described next.
Chapter 1 reviews the conceptual framework of the TGISconsidered in thisbook
fromaBMEpointof view.Chapter 2 introduces the readerto the fundamental ideas
and methods ofspatiotemporal analysis,including space/time geometriesandrandom
field theory.Chapter 3 discusses issues of physical knowledge synthesis.Chapter 4
presents the reader with the state-of-the-art mathematical tools of spatiotemporal
mapping with emphasis on formal BMEtechniques.Comparisons are made with other
techniques,likestatistical regression,Kriging,neural networks,and Kalman-Bucy filters.
Chapter 5 analyzes the interpretivefeatures ofBME,establishingcorrespondencerela-
tionships betweenthe natural system and the formal mathematics which describe it,
measuring and testing the formal structure,or justifyingthe methodological steps of
spatiotemporal mapping.InChapters 4 and 5 the readerwill also find some interest-
ing non-Bayesian and non-entropic extensions ofthe BME approach.The purpose of
Chapters 6 and 7 is to make the readerfamiliar with the BMEtoolboxand the associ-
ated libraryofcomprehensive computerprograms(BMElib) currently used in Modern
Spatiotemporal Geostatisticsapplications.This familiarity is achieved by means ofana-
VI Preface
lytical examples and numerical experiments,as well as real-world applications.The
reader can reconstruct many ofthe numerical experiments with the help ofthe BMElib,
as isindicatedby the computermouse logo ( ) at the beginning of the relevantex-
amples.Wehave spent a considerable amount oftime onthese two chapters and tried
to make the material as easy as possible tograsp.Finally,Chapter 8 discusses impor-
tant uses of the TGIS analysis and modelling in the context of scientific hypothesis
testing,explanation,anddecision making.
The focus oncriticalpracticalitythroughout the bookis necessary in order tomake
thepointaboutthe usefulnessand relevanceof theBMEtechniques in the study of
TGIS environments.The term“critical”refers to a situation in which practical ap-
proaches and software packages arenot seen merely as “data massaging”tools,but
rather as the means of interpreting and processing knowledge in order to increase
one’s scientific understanding of the phenomenonof interest,provide accuratepre-
dictions,and make useful decisions.We ask the readerto bear with us atsome parts
ofthe bookwhere we philosophize aboutthe TGISprocess.We are convinced that the
reward for such an effort is,indeed,substantial.It is hoped thatby integrating the
currentTGISsoftware with the BMEsoftware presented in this book one could get a
whole thatis greater than the sum of its parts.The mathematics ofthe presentation
have been kept at a level with which most TGIS modellers should feel comfortable
with.Some moreadvanced material,which the readercan omit in a first reading,are
identified bythe water-clock logo ( ).Nevertheless,the bookcontains a numberof
mathematical equations,forwhich we make no apology.Insciences,these equations
correspond to real physical,biological, ecological, and epidemiological situations and
constituteacommon language between the various scientific disciplines involved.If
TGIS specialists really expectto communicate with natural scientists and engineers
and make meaningful contributions to the relevantscientific disciplines,they should
haveconsiderable familiarity with the basic equations ofthese disciplines.
The researchpresentedin the bookhas beenpartiallyfunded bygrants fromthe
National Aeronautics and SpaceAdministration (Grant 60-00RFQ041),the National
Institute ofEnvironmental Health Science (Grant P42 ES05948-02),and the U.S. Civilian
Researchand Development Foundation (Grant RG2-2236).To these financial benefac-
torswe remain grateful.The comments made by Drs. Kyung-mee ChoiandAlexander
Kolovos werevery helpful and appreciated.We are alsoindebted to Lara Freeburg Kees
and Lucinda Thompsonfortheir editorial assistance.
Working onthis bookhas been a very rewarding experienceforus.In fact,the book
was written while each one ofus was working in a different continent:America (George
Christakos),Europe (Patrick Bogaert),andAfrica (Marc L.Serre).This provides the
advancedTGISfunctions of this bookwith a kind of an“inter-continental”flavor.
George Christakos Chapel Hill,USA
Patrick Bogaert Louvain-la-Neuve,Belgium
MarcL.Serre Alexandria,Egypt
Tothe unendedquest.
GC,PB,MLS
Contents
Thoughts without content are empty,
intuitionswithoutconcepts are blind.
I.Kant
1 ABMEView to the New Realities of TGIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Introducing a Temporal Geographical Information System (TGIS) . . . . . . . . . . . . . . 1
1.1.1 Purposefulness,Content,andContext . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Synthesis,Organization,andVisualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.3 Action-Oriented . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Field-BasedTGIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 TGIS Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 NovelContributiontoTGIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4.1 BME-Based AdvancedFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4.2 StochasticModelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4.3 BMElib Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4.4 EpistemicViewpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4.5 Scientific HypothesisTestingand Explanation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4.6 Revisionistic Paradigm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 Spatiotemporal Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1 Spatiotemporal Continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 The Random Field Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3 The Role ofMetaphorsin TGIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4 The ImportanceofPhysical Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.5 Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3 Knowledge Bases Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1 IntegratingKnowledge Bases (KB) intoTGIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 General KBandthe Associated Physical Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2.1 Space/Time CorrelationFunctionsBetween Two orMore Points
(Multiple-PointStatistics) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2.2 PhysicalModels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3 SpecificatoryKB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.3.1 HardandSoftData . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3.2 The EffectofSoft DataonTheCalculation
ofthe Space/Time CorrelationFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.4 Accommodating Knowledge Needs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.4.1 Knowledge Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.4.2 ModelBuilding andRealityCheck . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
X Contents
4 Spatiotemporal Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.1 A Formulationofthe Spatiotemporal Mapping Problem. . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2 Formal BMEAnalysis and Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.2.1 The Basic BME Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2.2 The Advantage ofCompositeSpace/Time Mapping. . . . . . . . . . . . . . . . . . . . . . . . 59
4.2.3 Continuous-Valued Map Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2.4 Modifications ofthe BME Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.2.5 Spatiotemporal Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.2.6 Spatiotemporal Mapping and Change-of-Scale Procedures . . . . . . . . . . . . . . . 68
4.3 OtherMappingTechniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.3.1 Wiener-KolmogorovStochasticInterpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.3.2 Geostatistical Kriging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.3.3 Kalman-Bucy Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3.4 SomeComparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5 Interpretive BME . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.1 InterpretiveIssues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.2 An Epistemic Analysis ofthe BME Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.3 Non-Bayesian Conditionalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.3.1 Material Biconditionalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.3.2 Material Conditionalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.4 By Wayofa Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6 The BME Toolbox In Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.1 The Fundamental KBOperators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.2 Step-by-StepBME . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.2.1 The Formal Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.2.2 TheDiagrammaticRepresentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.3 Analytic and Synthetic Case-Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.3.1 SomeCommonlyEncounteredSituations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.3.2 Spatiotemporal Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.3.3 Exogenous Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.3.4 PhysicalLaws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.3.5 UsingSoftDatatoImproveTGIS Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.3.6 Non-Bayesian Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.4 Quantifying the Mapping Efficiency ofSoft Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.5 Numerical Investigations ofPopular Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.5.1 The Use andMisuse ofSoft Data byStatistical Regression-Based
Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.5.2 The InadequacyofIndicatorKriging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.6 Merging Mapswith BME . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
6.7 Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
7 The BMEComputer Library . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
7.1 Computational BMEAnalysis and the BMElib . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
7.2 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
Contents XI
7.2.1 NotationalConvenience. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
7.2.2 Getting Startedwith MatLab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
7.2.3 GettingStarted withBMElib . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
7.3 The iolib Directory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
7.3.1 ThereadGeoEAS.mandwriteGeoEAS.mFunctions . . . . . . . . . . . . . . . . . . . . . 150
7.3.2 The readProba.m and writeProba.m Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.3.3 The readBMEproba.m and writeBMEproba.m Functions . . . . . . . . . . . . . . . 153
7.4 The graphlibDirectory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
7.4.1 The scatterplot.m function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
7.4.2 The colorplot.m function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
7.4.3 The marketplot.m function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
7.4.4 The valplot.m function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
7.4.5 A tutorial Use ofthe graphlibDirectory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
7.5 The modelslib Directory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
7.5.1 The *C.m and *V.m Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
7.5.2 The modelplot.m Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
7.5.3 A Tutorial Use ofthe modelslib Directory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
7.6 The statlib Directory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
7.6.1 Thekerneldensity.mFunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
7.6.2 The pdf2cdf.m Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
7.6.3 Thecovario.mFunction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
7.6.4 Thecrosscovario.m Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
7.6.5 ThecrosscovarioST.m Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
7.6.6 ATutorial Use ofthe statlibDirectory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
7.7 The bmeprobalib Directory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
7.7.1 The proba*.m Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
7.7.2 The BMEprobaMoments.m Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
7.7.3 The BMEprobaMode.m Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
7.7.4 The BMEprobaPdf.m Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
7.7.5 The BMEprobaCI.m Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
7.7.6 The BMEprobaTMode.m,BMEprobaTPdf.m
and BMEprobaTCI.m Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
7.7.7 WorkingWithFiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
7.7.8 A Tutorial Use ofthe bmeprobalib Directory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
7.8 The bmeintlib Directory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
7.8.1 The BMEintervalMode.m Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
7.8.2 The BMEintervalPdf.m Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
7.8.3 The BMEintervalTMode.m Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
7.8.4 TheBMEintervalTPdf.m Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
7.8.5 A Tutorial Use ofthe bmeintlibDirectory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
7.9 The bmehrlibDirectory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
7.9.1 The kriging.m Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
7.9.2 The krigingfilter.m Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
7.9.3 A Tutorial Use ofthe bmehrlibDirectory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
7.10 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
7.10.1 The simuchol.m Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
7.10.2 The simuseq.m Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182