Departement of Mathematics Markov Random Fields An extension from a temporal to a spatial setting Cornelia Richter STA381: Stochastic Simulation Seminar 03 June 2021 Departement of Mathematics Outline I 1. Spatial Models for Lattice Data 2. Conditionally Autoregressive Models 3. Graphs and Gaussian Markov Random Fields 4. Random Walk Models: Intrinsic Gaussian Markov Random Fields 5. Extension: A way of predicting in statistical models for random fields 6. Summary 7. Outlook 03 June 2021 Markov Random Fields Page 2 Departement of Mathematics Motivation – very important branch of probability theory – promising results in both theory and application 03 June 2021 Markov Random Fields Page 3 Departement of Mathematics Spatial Models for Lattice Data I – so far: Markov property based on time (continuous and discrete) – now: Markov property based on space (naturally continuous) ! typically called lattice or areal data 03 June 2021 Markov Random Fields Page 4 Departement of Mathematics Spatial Models for Lattice Data II Main difference between time series and lattice data: – time series: temporal direction forecasting (prediction) is very natural concept – lattice data: prediction very unusual, ! focus on smoothing (filtering) ! used to separate signal from noise or to fill in missing values 03 June 2021 Markov Random Fields Page 5 Departement of Mathematics Spatial Models for Lattice Data: Example – Partition of lower 48 US states into counties ! set of representative location of the block defines a lattice ! specific lattice has a varying density because of different county sizes 03 June 2021 Markov Random Fields Page 6 Departement of Mathematics Example Lattice Data US Counties I Figure: 48 lower US States 03 June 2021 Markov Random Fields Page 7 Departement of Mathematics Example Lattice Data US Counties II Figure: 3082 US Counties 03 June 2021 Markov Random Fields Page 8 Departement of Mathematics Example Lattice Data US Counties III Figure: Histogram of number of neighbors 03 June 2021 Markov Random Fields Page 9 Departement of Mathematics Conditionally Autoregressive Models I – basic idea: construct densities based on neighbors – random vector Y, density fY (y) ! f (yi j yj ; i 6= j); i = 1;::: n – approach: specify the joint density through n full conditional densities f (yi j y−i ) – assume Gaussian full conditionals, likely to get a valid joint density 03 June 2021 Markov Random Fields Page 10 Departement of Mathematics Conditionally Autoregressive Models II simple model for location i, given by P 2 Yi j yj ; j 6= i ∼ N( i;j6=i bij yj ; τi ); i = 1; :::; n ! with Brooks Lemma we conclude −1 Y ∼ Nn(0; (I − B) T), 2 B = bij , bii = 0, T = diag(τi ) 03 June 2021 Markov Random Fields Page 11 Departement of Mathematics Conditionally Autoregressive Models II Continued – covariance matrix is symmetric bij bji ! we require 2 = 2 τi τj – (I − B) has to be positive definite 03 June 2021 Markov Random Fields Page 12 Departement of Mathematics Conditionally Autoregressive Models III Neighbors – i and j are neighbors, denoted by i ∼ j – temporal neighbors: previous/next value – spatial neighbors: sharing a common edge or boundary on a Graph 03 June 2021 Markov Random Fields Page 13 Departement of Mathematics Conditionally Autoregressive Models III Neighbors - Example I Figure: Neighbours: left - 4 neighbors of s, right - 8 neighbors of s 03 June 2021 Markov Random Fields Page 14 Departement of Mathematics Conditionally Autoregressive Models III Neighbors - Example II Figure: Neighbours: middle - von Neumann neighborhood, right - Moore neighborhood 03 June 2021 Markov Random Fields Page 15 Departement of Mathematics Sudden Infant Death Syndrome Dataset (SIDS) – sudden, unexplained death of a less than one year old child – SIDS dataset has been analysed a lot in literature, e.g. in Cressie (1993) – dataset: ! birth counts and SIDS cases ! periods: 1974 - 1978 and 1979 - 1984 ! two ethnicities ! 100 counties of North Carolina (NC) – our analysis: neighbors, fitting an conditionally autoregressive model, Morans test, kriging 03 June 2021 Markov Random Fields Page 16 Departement of Mathematics Application CAR: SIDS I Figure: SIDS Data - Area 03 June 2021 Markov Random Fields Page 17 Departement of Mathematics Application CAR: SIDS I Figure: Neighbors: SIDS Data 03 June 2021 Markov Random Fields Page 18 Departement of Mathematics Application CAR: SIDS II Figure: SIDS Data - Rates 03 June 2021 Markov Random Fields Page 19 Departement of Mathematics Application CAR: SIDS II Figure: SIDS Data - Rates without outlier 03 June 2021 Markov Random Fields Page 20 Departement of Mathematics Application CAR: SIDS III Figure: SIDS Data - Summary CAR Fit 03 June 2021 Markov Random Fields Page 21 Departement of Mathematics Application CAR: SIDS IV Figure: SIDS Data - Summary CAR Fit 03 June 2021 Markov Random Fields Page 22 Departement of Mathematics Application CAR: SIDS V Figure: SIDS Data - Summary CAR Fit 03 June 2021 Markov Random Fields Page 23 Departement of Mathematics Graphs and (Gaussian) Markov Random Fields Graphs I – graph G = (V; E), V...nodes, E... edges – graphs can be directed or undirected 03 June 2021 Markov Random Fields Page 24 Departement of Mathematics Graphs and (Gaussian) Markov Random Fields Graphs II Figure: Graphs: left - undirected graph, right - directed graph 03 June 2021 Markov Random Fields Page 25 Departement of Mathematics Graphs and (Gaussian) Markov Random Fields Definitions I Definition: Precision The Precision of a random variable is the inverse of the variance. In case of random vectors the inverse of a covariance function is called precision matrix. Note: High precision (i.e. low variance) implies a lot of knowledge about the variable. 03 June 2021 Markov Random Fields Page 26 Departement of Mathematics Graphs and (Gaussian) Markov Random Fields Definitions II Definition: Markov Random Field (MRF) Let X = (Xs; s 2 S) be a set of discrete random variables taking values in set V. X is a Markov random field with respect to the neighborhood system (Ns; s 2 S), if P(Xs = x j Xt ; t 6= s) = P(Xs = x j Xt ; t 2 Ns), x 2 V; s 2 S 03 June 2021 Markov Random Fields Page 27 Departement of Mathematics Graphs and (Gaussian) Markov Random Fields Connection ! a random field is a MRF if it satisfies the Markov property ! a MRF is a set of random variables having a Markov property described by an undirected graph ! X ? Y denotes X and Y are independent ! X ? Y j Z = z denotes conditional independence of X and Y 03 June 2021 Markov Random Fields Page 28 Departement of Mathematics Graphs and Gaussian Markov Random Fields Definitions III Definition: Gaussian Markov Random Field (GMRF) T A random vector Y = (Y1; :::; Yn) is a Gaussian Markov Random Field with respect to a labeled graph G = (V; E) with mean µ and a symmetric and positive definite precision matrix Q, if its density is given by −n=2 1=2 1 T f (y) = (2π) det(Q) exp(− 2 (y − µ) Q(y − µ)) Qij 6= 0 , i; j 2 E8i 6= j 03 June 2021 Markov Random Fields Page 29 Departement of Mathematics Graphs and Gaussian Markov Random Fields Properties Property Let Y be a GMRF with respect to G = (V; E), with mean µ and precision matrix Q, then 1. Yi ? Yj j Y−ij = y−ij , Qij = 0 1 P 2. E[Y j Y− = y− ] = µ − Q (y − µ ) i i i i Qii j:j∼i ij j j 3. Prec(Yi j Y−i = y−i ) = Qii Qij 4. Cor(Yi Yj j Y−ij = y−ij ) = −p , i 6= j Qii Qjj 03 June 2021 Markov Random Fields Page 30 Departement of Mathematics Simulation of Markov Random Fields: Binary Simulation I Figure: Binary Simulation: Threshold 0, high sequence 03 June 2021 Markov Random Fields Page 31 Departement of Mathematics Simulation of Markov Random Fields: Binary Simulation I Figure: Binary Simulation: Threshold 0, low sequence 03 June 2021 Markov Random Fields Page 32 Departement of Mathematics Simulation of Markov Random Fields: Binary Simulation II Figure: Binary Simulation: Threshold 1, high sequence 03 June 2021 Markov Random Fields Page 33 Departement of Mathematics Simulation of Markov Random Fields: Binary Simulation II Figure: Binary Simulation: Threshold 1, low sequence 03 June 2021 Markov Random Fields Page 34 Departement of Mathematics Simulation of Markov Random Fields: Circular Embedding Figure: Circular Embedding: high sequence 03 June 2021 Markov Random Fields Page 35 Departement of Mathematics Simulation of Markov Random Fields: Circular Embedding Figure: Circular Embedding: low sequence 03 June 2021 Markov Random Fields Page 36 Departement of Mathematics Random Walk Models: Intrinsic Gaussian Markov Random Fields I – improper densities not integrating to one often used as priors in Bayesian frameworks ! resulting posterior may be appropriate – consider e.g. a regression setting −1 relax condition of constant µ, Yi+1 − Yi ∼ N(0; κ ), instead of Yi+1 − Yi ≡ 0 – NOW locations of n random variables are equispaced observations on the transection 03 June 2021 Markov Random Fields Page 37 Departement of Mathematics Random Walk Models: Intrinsic Gaussian Markov Random Fields II iid −1 ∆Yi = Yi+1 − Yi , i = 1;:::; n − 1, ∆Yi ∼ N(0; κ ) n−1 n−1 n−1 2 κ P 2 2 1 T ) f (∆y) / κ exp(− 2 i=1 (∆yi )) = κ exp(− 2 y Qy) 03 June 2021 Markov Random Fields Page 38 Departement of Mathematics Random Walk Models: Intrinsic Gaussian Markov Random Fields II cont. 0 1 1 −1 0 ··· 0 B . C B −1 2 −1 .. C B C B . C T Q = κ B 0 .. .. .. 0 C = κD D, B C B . C B . .. −1 2 −1 C @ A 0 ··· 0 −1 1 0 −1 1 0 ··· 1 B . C D = B 0 .. .. .. C 2 R(n−1)xn @ . A . .. −1 1 03 June 2021 Markov Random Fields Page 39 Departement of Mathematics Random Walk Models: Intrinsic Gaussian Markov