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1 Spatial processes and statistical modelling Peter Green University of Bristol, UK BCCS GM&CSS 2008/09 Lecture 8

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2 Continuous space Discrete space –lattice –irregular - general graphs –areally aggregated Point processes –other object processes Spatial indexing

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3 Space vs. time apparently slight difference profound implications for mathematical formulation and computational tractability

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4 Requirements of particular application domains agriculture (design) ecology (sparse point pattern, poor data?) environmetrics (space/time) climatology (huge physical models) epidemiology (multiple indexing) image analysis (huge size)

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5 Key themes conditional independence –graphical/hierarchical modelling aggregation –analysing dependence between differently indexed data –opportunities and obstacles literal credibility of models Bayes/non-Bayes distinction blurred

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6 Why build spatial dependence into a model? No more reason to suppose independence in spatially-indexed data than in a time- series However, substantive basis for form of spatial dependent sometimes slight - very often space is a surrogate for missing covariates that are correlated with location

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7 Discretely indexed data

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8 Modelling spatial dependence in discretely-indexed fields Direct Indirect –Hidden Markov models –Hierarchical models

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9 Hierarchical models, using DAGs Variables at several levels - allows modelling of complex systems, borrowing strength, etc.

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10 Modelling with undirected graphs Directed acyclic graphs are a natural representation of the way we usually specify a statistical model - directionally: disease symptom past future parameters data …… whether or not causality is understood. But sometimes (e.g. spatial models) there is no natural direction

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11 Conditional independence In model specification, spatial context often rules out directional dependence (that would have been acceptable in time series context) X0X0 X1X1 X2X2 X3X3 X4X4

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12 Conditional independence In model specification, spatial context often rules out directional dependence X 20 X 21 X 22 X 23 X 24 X 00 X 01 X 02 X 03 X 04 X 10 X 11 X 12 X 13 X 14

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13 Conditional independence In model specification, spatial context often rules out directional dependence X 20 X 21 X 22 X 23 X 24 X 00 X 01 X 02 X 03 X 04 X 10 X 11 X 12 X 13 X 14

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14 Directed acyclic graph in general: for example: ab c d p(a,b,c,d)=p(a)p(b)p(c|a,b)p(d|c) In the RHS, any distributions are legal, and uniquely define joint distribution

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15 Undirected (CI) graph X 20 X 21 X 22 X 00 X 01 X 02 X 10 X 11 X 12 Absence of edge denotes conditional independence given all other variables But now there are non-trivial constraints on conditional distributions Regular lattice, irregular graph, areal data...

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16 Undirected (CI) graph X 20 X 21 X 22 X 00 X 01 X 02 X 10 X 11 X 12 then and so The Hammersley-Clifford theorem says essentially that the converse is also true - the only sure way to get a valid joint distribution is to use ( ) ( ) clique Suppose we assume

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17 Hammersley-Clifford X 20 X 21 X 22 X 00 X 01 X 02 X 10 X 11 X 12 A positive distribution p(X) is a Markov random field if and only if it is a Gibbs distribution - Sum over cliques C (complete subgraphs)

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18 Partition function X 20 X 21 X 22 X 00 X 01 X 02 X 10 X 11 X 12 Almost always, the constant of proportionality in is not available in tractable form: an obstacle to likelihood or Bayesian inference about parameters in the potential functions Physicists call the partition function

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19 Markov properties for undirected graphs The situation is a bit more complicated than it is for DAGs. There are 4 kinds of Markovness: P – pairwise –Non-adjacent pairs of variables are conditionally independent given the rest L – local –Conditional only on adjacent variables (neighbours), each variable is independent of all others

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20 G – global –Any two subsets of variables separated by a third are conditionally independent given the values of the third subset. F – factorisation –the joint distribution factorises as a product of functions of cliques In general these are different, but F G L P always. For a positive distribution, they are all the same.

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21 Gaussian Markov random fields: spatial autoregression is a multivariate Gaussian distribution, and If V C (X C ) is - ij (x i -x j ) 2 /2 for C={i,j} and 0 otherwise, then is the univariate Gaussian distribution where

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22 Inverse of (co)variance matrix: dependent case ABCD non-zero A B C D ABCDABCD Gaussian random fields

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23 Non-Gaussian Markov random fields Pairwise interaction random fields with less smooth realisations obtained by replacing squared differences by a term with smaller tails, e.g.

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24 Discrete-valued Markov random fields Besag (1974) introduced various cases of for discrete variables, e.g. auto-logistic (binary variables), auto-Poisson (local conditionals are Poisson), auto-binomial, etc.

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25 Auto-logistic model is Bernoulli( p i ) with - a very useful model for dependent binary variables ( NB various parameterisations ) (X i = 0 or 1)

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26 Statistical mechanics models The classic Ising model (for ferromagnetism) is the symmetric autologistic model on a square lattice in 2-D or 3-D. The Potts model is the generalisation to more than 2 colours and of course you can usefully un-symmetrise this.

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27 Auto-Poisson model is Poisson For integrability, ij must be 0, so this only models negative dependence: very limited use.

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28 Hierarchical models and hidden Markov processes

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29 Chain graphs If both directed and undirected edges, but no directed loops: can rearrange to form global DAG with undirected edges within blocks

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30 Chain graphs If both directed and undirected edges, but no directed loops: can rearrange to form global DAG with undirected edges within blocks Hammersley-Clifford within blocks

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31 Hidden Markov random fields We have a lot of freedom modelling spatially-dependent continuously- distributed random fields on regular or irregular graphs But very little freedom with discretely distributed variables use hidden random fields, continuous or discrete compatible with introducing covariates, etc.

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32 Hidden Markov models z0z0 z1z1 z2z2 z3z3 z4z4 y1y1 y2y2 y3y3 y4y4 e.g. Hidden Markov chain observed hidden

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33 Hidden Markov random fields Unobserved dependent field Observed conditionally- independent discrete field (a chain graph)

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34 Spatial epidemiology applications independently, for each region i. Options : CAR, CAR+white noise (BYM, 1989) Direct modelling of,e.g. SAR Mixture/allocation/partition models: Covariates, e.g.: cases expected cases relative risk

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35 Spatial epidemiology applications Spatial contiguity is usually somewhat idealised

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36 relative risk parameters Richardson & Green ( JASA, 2002) used a hidden Markov random field model for disease mapping observed incidence expected incidence hidden MRF Spatial epidemiology applications

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37 Chain graph for disease mapping e Y z k based on Potts model

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38 Larynx cancer in females in France SMRs

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