Julian Besag FRS, 1945–2010 Peter Green School of Mathematics University of Bristol 23 August 2011 / ISI World Statistics Congress Green (Bristol) Julian Besag FRS, 1945–2010 Dublin, August 2011 1 / 57 Outline 1 Biography 2 Methodology 3 Applications 4 Last seminars Green (Bristol) Julian Besag FRS, 1945–2010 Dublin, August 2011 2 / 57 Biography Julian Besag as a young boy Green (Bristol) Julian Besag FRS, 1945–2010 Dublin, August 2011 3 / 57 Biography A brief biography 1945 Born in Loughborough 1965–68 BSc Mathematical Statistics, Birmingham 1968–69 Research Assistant to Maurice Bartlett, Oxford 1970–75 Lecturer in Statistics, Liverpool 1975–89 Reader (from 1985, Professor), Durham 1989–90 Visiting professor, U Washington 1990–91 Professor, Newcastle-upon-Tyne 1991–2007 Professor, U Washington 2007–09 Visiting professor, Bath 2010 Died in Bristol Visiting appointments in Oxford, Princeton, Western Australia, ISI New Delhi, PBI Cambridge, Carnegie-Mellon, Stanford, Newcastle-u-Tyne, Washington, CWI Amsterdam, Bristol Green (Bristol) Julian Besag FRS, 1945–2010 Dublin, August 2011 4 / 57 Biography Julian Besag in 1976 Green (Bristol) Julian Besag FRS, 1945–2010 Dublin, August 2011 5 / 57 Methodology Methodology Spatial statistics Modelling, conditional formulations Frequentist and Bayesian inference Algebra of interacting systems Digital image analysis Monte Carlo computation and hypothesis testing Markov chain Monte Carlo methods Exploratory data analysis Green (Bristol) Julian Besag FRS, 1945–2010 Dublin, August 2011 6 / 57 Methodology Spatial statistics: Modelling, conditional formulations Modelling, conditional formulations: key papers Nearest-neighbour systems and the auto-logistic model for binary data. Journal of the Royal Statistical Society B (1972). Spatial interaction and the statistical analysis of lattice systems (with Discussion). Journal of the Royal Statistical Society B (1974). On spatial-temporal models and Markov fields. Proceedings of the 10th (1974) European Meeting of Statisticians (1977). Green (Bristol) Julian Besag FRS, 1945–2010 Dublin, August 2011 7 / 57 Methodology Spatial statistics: Modelling, conditional formulations The 1974 paper in JRSS(B) 192 [No.2, Spatial Interactionand the StatisticalAnalysis of LatticeSystems ByJULIAN BESAG UniversityofLiverpool [Read beforethe ROYAL STATISTICAL SOCIETY at a meetingorganized by the RESEARCH SECTION on Wednesday,March 13th,1974, Professor J. DURBIN in theChair] SUMMARY The formulationof conditionalprobability models for finitesystems of spatiallyinteracting random variables is examined.A simplealternative proofof theHammersley-Clifford theorem is presentedand thetheorem is then used to constructspecific spatial schemeson and offthe lattice. Particularemphasis is placed upon practicalapplications of themodels in plantecology when the variatesare binaryor Gaussian. Some aspectsof infinitelattice Gaussian processesare discussed. Methodsof statistical analysisfor lattice schemes are proposed,including a veryflexible coding technique.The methodsare illustratedby two numericalexamples. It is maintainedthroughout that the conditionalprobability approach to the specificationand analysisof spatialinteraction is moreattractive than the alternativejoint probability approach. Keywords:MARKOV FIELDS; SPATIAL INTERACTION; AUTO-MODELS; NEAREST-NEIGHBOUR SCHEMES; STATISTICAL ANALYSIS OF LATTICE SCHEMES; CODING TECHNIQUES; SIMULTANEOUSBILATERAL AUTOREGRESSIONS; CONDITIONAL PROBABILITY MODELS Green (Bristol) Julian1. BesagINTRODUCTION FRS, 1945–2010 Dublin, August 2011 8 / 57 IN this paper, we examinesome stochasticmodels whichmay be used to describe certaintypes of spatial processes. Potential applications of the models occur in plant ecologyand the paper concludeswith two detailednumerical examples in this area. At a formallevel, we shall largelybe concernedwith a ratherarbitrary system, consistingof a finiteset of sites, each site having associated with it a univariate randomvariable. In most ecological applications,the sites will representpoints or regionsin the Euclideanplane and will oftenbe subjectto a rigidlattice structure. For example,Cochran (1936) discussesthe incidence of spottedwilt over a rectangular array of tomato plants. The disease is transmittedby insectsand, afteran initial period of time,we should clearlyexpect to observeclusters of infectedplants. The formulationof spatial stochasticmodels will be consideredin Sections2-5 of the paper. Once havingset up a modelto describea particularsituation, we shouldthen hope to be able to estimateany unknownparameters and to testthe goodness-of-fit of the model on the basis of observation.We shall discussthe statisticalanalysis of latticeschemes in Sections6 and 7. We begin by makingsome generalcomments on the types of spatial systems whichwe shall,and shall not,be discussing.Firstly, we shall not be concernedhere withany randomdistribution which may be associatedwith the locationsof the sites themselves.Indeed, when settingup models in practice,we shall require quite specificinformation on the relativepositions of sites,in orderto assess the likely interdependencebetween the associatedrandom variables. Secondly,although, as in Methodology Spatial statistics: Modelling, conditional formulations JRSS(B) 1974: the Hammersley–Clifford theorem The theorem states that “Markov random fields are the same as Gibbs distributions”– that is, that a multivariate distribution satisfies the Markov random field property if and only if its log-density is additive over cliques. Besag gave a simple proof of this, assuming “positivity” – essentially that any combination of values realisable locally was realisable globally. Green (Bristol) Julian Besag FRS, 1945–2010 Dublin, August 2011 9 / 57 198 BESAG- StatisticalAnalysis of Lattice Systems [No. 2, Methodology Spatial statistics: Modelling, conditional formulations for any 1< i<j< ... < s < n, thefunction G1 in(3.3) maybe non-nullif and onlyif the sites i,j, ...,s forma clique. Subjectto thisrestriction, the G-functions may be chosen JRSS(B) 1974:arbitrarily. theThus, Hammersley–Clifford given the neighbours of each site, we theoremcan immediatelywrite down themost general form for Q(x) andhence for the conditional distributions. We shall see examplesof this later on. Proofof theorem.It followsfrom equation (3.2) that,for any x e Q, Q(x) - Q(x,) can onlydepend upon xi itselfand thevalues at siteswhich are neighbours of site i. Withoutloss of generality, we shallonly consider site 1 in detail.We thenhave, from equation(3.3), Q(x)- Q(x1) = Xi G1(X1)+ E xi G1J(x1,x1) + XXi Xk GlJ,k(Xl, Xi, XJ) + 2<S<n 2:!5<k<n + X2X3 ... xnG1,2,..n(x, x2,.. Xn), Now suppose site 1 (# 1) is not a neighbourof site 1. Then Q(x) - Q(xl) must be independentof xl forall x e Q. Puttingxi =0 forio 1 or 1,we immediatelysee that G1,(xl, x) = 0 on Q. Similarly,by othersuitable choices of x, it is easily seen succes- sivelythat all 3-,4-, ..., n-variableG-functions involving both xl andx1 must be null. The analogousresult holds for any pair of siteswhich are notneighbours of each otherand hence,in general,G can onlybe non-nullif the sitesi,j, ..., s forma clique. On theother hand, any set of G-functionsgives rise to a validprobability distri- butionP(x) whichsatisfies the positivity condition. Also sinceQ(x) - Q(xj) depends onlyupon x1 if there is a non-nullG-function involving both xi andx1, it follows that the same is trueof P(x*j xl, ..., xi-,, xi+, ..., xn). This completesthe proof. We nowconsider some simple extensions of the theorem. Suppose firstly that the variatescan take a denumerablyinfinite set of values. Then the theorem still holds if, Green (Bristol)in thesecond part, we imposeJulianthe Besagadded FRS,restriction 1945–2010that the G-functions be chosenDublin, August 2011 10 / 57 suchthat E expQ(x) is finite,where the summation is over all x E Q. Similarly,ifthe variateseach have absolutelycontinuous distributions and we interpretP(x) and alliedquantities as probabilitydensities, the theorem holds provided we ensurethat exp Q(x) is integrableover all x. Theseadditional requirements must not be taken lightly,as we shallsee by examples in Section4. Finally,we may consider the case of multivariaterather than univariate site variables. In particular,suppose that the randomvector at sitei has vi components.Then we mayreplace that site by v* notionalsites, each of whichis associatedwith a singlecomponent of therandom vector.An appropriatesystem of neighboursmay thenbe constructedand the univariatetheorem be appliedin theusual way. We shallnot considerthe multi- variatesituation any further in thepresent paper. As a straightforwardcorollary to thetheorem, itmay easily be establishedthat for anygiven Markov field P(X*= xi, Xi = Xi,..., X,= x I all othersite values) dependsonly upon xi,x1,...,x and thevalues at sitesneighbouring sites i,j, ...,s. In theHammersley-Clifford terminology, the local and global Markovian properties are equivalent. In practice,we shallusually find that the sites occur in a finiteregion of Euclidean spaceand thatthey often fall naturally into two sets: those which are internal to the Methodology Spatial statistics: Modelling, conditional formulations Markov random fields Green (Bristol) Julian Besag FRS, 1945–2010 Dublin, August 2011 11 / 57 Methodology Spatial statistics: Modelling, conditional formulations Markov random fields Green (Bristol) Julian Besag FRS, 1945–2010 Dublin, August 2011 12 / 57 Methodology Spatial statistics: Modelling, conditional formulations Markov random