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

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Motivation

– very important branch of probability theory – promising results in both theory and application

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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

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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

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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

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Example Lattice Data US Counties I

Figure: 48 lower US States

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Example Lattice Data US Counties II

Figure: 3082 US Counties

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Example Lattice Data US Counties III

Figure: Histogram of number of neighbors

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Conditionally Autoregressive Models I

– basic idea: construct densities based on neighbors

– random vector Y, density fY (y) → f (yi | yj , i 6= j), i = 1,... n – approach: specify the joint density through n full conditional densities

f (yi | y−i ) – assume Gaussian full conditionals, likely to get a valid joint density

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Conditionally Autoregressive Models II simple model for location i, given by

P 2 Yi | 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 )

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Conditionally Autoregressive Models II Continued

– covariance matrix is symmetric bij bji → we require 2 = 2 τi τj – (I − B) has to be positive definite

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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

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Conditionally Autoregressive Models III Neighbors - Example I

Figure: Neighbours: left - 4 neighbors of s, right - 8 neighbors of s

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Conditionally Autoregressive Models III Neighbors - Example II

Figure: Neighbours: middle - von Neumann neighborhood, right - Moore neighborhood

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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

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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

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Application CAR: SIDS II

Figure: SIDS Data - Rates without outlier

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Application CAR: SIDS III

Figure: SIDS Data - Summary CAR Fit

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Application CAR: SIDS IV

Figure: SIDS Data - Summary CAR Fit

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Application CAR: SIDS V

Figure: SIDS Data - Summary CAR Fit

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Graphs and (Gaussian) Markov Random Fields Graphs I

– graph G = (V, E), V...nodes, E... edges – graphs can be directed or undirected

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Graphs and (Gaussian) Markov Random Fields Graphs II

Figure: Graphs: left - undirected graph, right - directed graph

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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.

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Graphs and (Gaussian) Markov Random Fields Definitions II

Definition: Markov Random Field (MRF)

Let X = (Xs, s ∈ 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 ∈ S), if P(Xs = x | Xt , t 6= s) = P(Xs = x | Xt , t ∈ Ns), x ∈ V, s ∈ S

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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 | Z = z denotes conditional independence of X and Y

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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 ∈ E∀i 6= j

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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 | Y−ij = y−ij ⇔ Qij = 0 1 P 2. E[Y | Y− = y− ] = µ − Q (y − µ ) i i i i Qii j:j∼i ij j j 3. Prec(Yi | Y−i = y−i ) = Qii Qij 4. Cor(Yi Yj | Y−ij = y−ij ) = −√ , i 6= j Qii Qjj

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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

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Simulation of Markov Random Fields: Binary Simulation II

Figure: Binary Simulation: Threshold 1, low sequence

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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

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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)

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Random Walk Models: Intrinsic Gaussian Markov Random Fields II cont.   1 −1 0 ··· 0  . .   −1 2 −1 .. .     . . .  T Q = κ  0 ...... 0  = κD D,    . .   . .. −1 2 −1    0 ··· 0 −1 1

 −1 1 0 ···   . . .  D =  0 ......  ∈ R(n−1)xn  . .  . .. −1 1

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Random Walk Models: Intrinsic Gaussian Markov Random Fields III

Comments: – f is an improper density, thus IGMRF of first order – also Random Walk model of first order RW1 – matrix Q has rank n-1, hence rank deficient

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Random Walk Models: Intrinsic Gaussian Markov Random Fields Definitons

Definition – Q (nxn) symmetric, positive semi-definite matrix, rank n − k > 0 – Y is an improper GMRF (IGMRF) of rank n − k, parameters (µ, Q) with respect to a labeled graph G = (V, E), if its density is f ∗(y) = (2π)−(n−k)/2det∗(Q)1/2exp(−0.5 ∗ (y − µ)T Q(y − µ))

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Random Walk Models: Intrinsic Gaussian Markov Random Fields Definitons cont.

Comments:

– Qij 6= 0 ⇔ i, j ∈ E∀i 6= j – det∗(Q) ... generalized determinant, i.e. product of all nonzero eigenvalues of Q – ⇒ (µ, Q) no longer represent mean and precision, they formally no longer exist – IGMRF of first order (or (n-1) order) is an improper GMRF of rank (n-1), where Q1 ≡ 0

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Random Walk Models: Intrinsic Gaussian Markov Random Fields Predictions in a RW Model

Prediction in a random walk model (RW1): 1 Yi | Y−i = y−i ∼ N(0.5(yi+1 + yi−1), 2κ ) p Yi+p | Yi = yi , Yi−1 = yi−1, · · · ∼ N(yi , κ ) 0 ≤ i + p ≤ n ⇒ no shrinkage towards the mean similar to a RW1, define RW2: 2 ∆ Yi = ∆Yi+1 − ∆Yi = Yi+2 − 2Yi+1 + Yi , i = 1, ..., n − 2 2 iid −1 ∆ Yi ∼ N(0, κ )

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Sampling from an IGMRF Simulation Setup I

– Two types of singularities in covariance matrix for GMRF and IGMRF → (constrained) GMRF: "variance collapse" → IGMRF: "variance inflation" – Illustration of concept in the following

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Sampling from an IGMRF Simulation Setup I

– same seed for all simulations, s.t. "projections" become clear → covariance matrix of GMRF simulations result in rank deficient covariance matrix (one eigenvalue is zero) same for precision matrix.

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Sampling from an IGMRF Simulation Setup III

– RW1, simulate from Q(γ) = Q + γ11T for small γ – in RW1 model, no "marginal" distributions specified ⇒ no information about individual means known which could spread over the entire real line – eigenvalues of Q(γ): 2 and 2(0.5 + γ) 1 – one eigenvalue of covariance matrix is dependent on γ ⇒ variance inflation as γ tends to 0

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Sampling from an IGMRF Sampling

Figure: Constrained sampling

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Sampling from an IGMRF Sampling

Figure: mimicked RW1, γ = 0.01 (blue), γ = 0.1 (green)

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Statistical Models for observations of random fields

(i) How can we predict an observation at an unobserved

location s0? (ii) How can we estimate parameters in such a model?

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Statistical Models for observations of random fields

– one answer to prediction problem is kriging → after South American mining engineer D.G. Krige

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Kriging - preparation

– fit a simultaneous autoregressive model → alternative to CAR one introduced earlier – we have P 2 Yi = j,j6=i bij Yj + i , with i ∼ N(0, σi ), i = 1,..., n – variogram function measuring dissimilarities, describing behavior of a process, important to analyze spatial variability

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Kriging Process

(1) remove some observations (arbitrarily)

Figure: Missing Observations: red crosses indicate removed observations

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Kriging Process

(2) fit a SAR model (3) compute variogram (4) perform kriging (5) compute prediction error → -0.00047, -0.00063, 0.0011 → very low ⇒ GOOD! (6) further steps: e.g. cross validation

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Moran’s test for autocorrelation

– multivariate random n-vector Y,

– W = (wij ) matrix of spatial weights → often describing a first order neighborhood structure – define Moran’s I as follows, P ¯ ¯ n i,j wij (Yi −Y )(Yj −Y ) I = P w ∗ P ¯ 2 ∈ [−1, 1] i,j ij i (Yi −Y )

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Moran’s test for autocorrelation

Figure: R Output - Moran’s I

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Moran’s test for autocorrelation

– → positive autocorrelation observed!

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Moran’s test for autocorrelation

Figure: Moran Plot for autocorrelation 03 June 2021 Markov Random Fields Page 57

→ SIDS rates (x-axis) against spatially lagged values (y-axis) → showing influence measures for linear relationship between data and lag → Observations influencing this are highlighted with a diamond shape Departement of Mathematics

Summary

– Extension of Markov Property from temporal to spatial setting – Lattice Data: SIDS dataset – CAR models – Introduction to (intrinsic) Gaussian Markov Random Fields and how to simulate them – Random Walk Models on IGMRF – Outlook on Spatial Statistics: Kriging

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Outlook

– many more extensions possible – non Gaussian MRF: Ising Model – extend Spatial Statistics Setting

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Thanks for your attention! Questions?

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Backup Slides Specific Models for GMRFs

– low parametrisations are needed

– assume τi = τ∀i ⇒ conditional precision is constant – bij = bi ⇒ independent coefficients of j (no preference given to neighboring information)

P 2 Yi ∼ Y−i | N( j,j∼i θ1yj , τ ) τ > 0, i = 1,..., n encode neighbor structure in a matrix ⇒ Adjacency matrix A or spatial weight matrix W

wij = 1ifi ∼ j, zero otherwise B = λw for some λ

03 June 2021 Markov Random Fields Page 61 For a legitimate precision matrix we need Q = I − λW to be positive definite, i.e. choose λ ∈ (λmax , λmin), with λ eigenvalues of W Departement of Mathematics

Statistical Models for observations of random fields Variogram

For every intrinsically stationary process the variogram exists and is unique → often parametrized with two more parameters → assume that process has an underlying parametrized variogram (as we rarely know the process explicitly)

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References

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