Hierarchical Modelling for Univariate Spatial Data

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1 Spatial omain Hierarchical Modelling for Univariate Spatial ata Sudipto Banerjee 1 and Andrew O. Finley 2 1 Biostatistics, School of Public Health, University of Minnesota, Minneapolis, Minnesota, U.S.A. 2 epartment of Forestry & epartment of Geography, Michigan State University, Lansing Michigan, U.S.A. March 3, 2010 y Frontiers of Statistical ecision Making and Bayesian Analysis Algorithmic Modelling What is a spatial process? Spatial surface observed at finite set of locations S = {s 1, s 2,..., s n } Tessellate the spatial domain (usually with data locations as vertices) Fit an interpolating polynomial: Y(s1) Y(s2) f(s) = i w i (S ; s)f(s i ) Y(sn) Interpolate by reading off f(s 0 ). Issues: Sensitivity to tessellations Choices of multivariate interpolators Numerical error analysis 3 Frontiers of Statistical ecision Making and Bayesian Analysis 4 Frontiers of Statistical ecision Making and Bayesian Analysis Y (s) = µ(s) + ɛ(s), Response: Y (s) at location s Mean: µ = T (s)β Error: ɛ(s) iid N(0, τ 2 ) Assumptions regarding ɛ(s): ɛ(s) iid N(0, τ 2 ) Y (s) = µ(s) + ɛ(s), Y (s), (s) ɛ(s i) ɛ(s j) 5 Frontiers of Statistical ecision Making and Bayesian Analysis 6 Frontiers of Statistical ecision Making and Bayesian Analysis

2 Assumptions regarding ɛ(s): ɛ(s) iid N(0, τ 2 ) Y (s) = µ(s) + ɛ(s), ɛ(s i ) and ɛ(s j ) are uncorrelated for all i j Spatial Gaussian processes (GP ): Say w(s) GP (0, σ 2 ρ( )) and Cov(w(s 1 ), w(s 2 )) = σ 2 ρ (φ; s 1 s 2 ) ɛ(s i) ɛ(s j) w(s i) w(s j) 6 Frontiers of Statistical ecision Making and Bayesian Analysis 7 Frontiers of Statistical ecision Making and Bayesian Analysis Spatial Gaussian processes (GP ): Say w(s) GP (0, σ 2 ρ( )) and Cov(w(s 1 ), w(s 2 )) = σ 2 ρ (φ; s 1 s 2 ) Let w = [w(s i )] n, then Realization of a Gaussian process: Changing φ and holding σ 2 = 1: w N(0, σ 2 R(φ)), where R(φ) = [ρ(φ; s i s j )] n i,j=1 w N(0, σ 2 R(φ)), where R(φ) = [ρ(φ; s i s j )] n i,j=1 w(s i) w(s j) 7 Frontiers of Statistical ecision Making and Bayesian Analysis 8 Frontiers of Statistical ecision Making and Bayesian Analysis Realization of a Gaussian process: Changing φ and holding σ 2 = 1: w N(0, σ 2 R(φ)), where R(φ) = [ρ(φ; s i s j )] n i,j=1 Realization of a Gaussian process: Changing φ and holding σ 2 = 1: w N(0, σ 2 R(φ)), where R(φ) = [ρ(φ; s i s j )] n i,j=1 Correlation model for R(φ): e.g., eponential decay ρ(φ; t) = ep( φt) if t > 0. Correlation model for R(φ): e.g., eponential decay ρ(φ; t) = ep( φt) if t > 0. Other valid models e.g., Gaussian, Spherical, Matérn. Effective range, t 0 = ln(0.05)/φ 3/φ 8 Frontiers of Statistical ecision Making and Bayesian Analysis 8 Frontiers of Statistical ecision Making and Bayesian Analysis

3 w N(0, σ 2 wr(φ)) defines comple spatial dependence structures. E.g., anisotropic Matérn correlation function: ρ(s i, s j; φ) = `1/Γ(ν)2 ν 1 2 p νd ij) ν κ p ν(2 νd ij, where d ij = (s i s j) Σ 1 (s i s j), Σ = G(ψ)Λ 2 G(ψ). Thus, φ = (ν, ψ, Λ). + random spatial effects Y (s) = µ(s) + w(s) + ɛ(s), Simulated Predicted Response: Y (s) at some site Mean: µ = T (s)β Spatial random effects: w(s) GP (0, σ 2 ρ(φ; s 1 s 2 )) Non-spatial variance: ɛ(s) iid N(0, τ 2 ) 9 Frontiers of Statistical ecision Making and Bayesian Analysis 10 Frontiers of Statistical ecision Making and Bayesian Analysis 11 Frontiers of Statistical ecision Making and Bayesian Analysis 11 Frontiers of Statistical ecision Making and Bayesian Analysis Third stage: Priors on Ω = (β, τ 2, σ 2, φ) Third stage: Priors on Ω = (β, τ 2, σ 2, φ) Marginalized likelihood: y Ω N(Xβ, σ 2 R(φ) + τ 2 I) 11 Frontiers of Statistical ecision Making and Bayesian Analysis 11 Frontiers of Statistical ecision Making and Bayesian Analysis

4 Third stage: Priors on Ω = (β, τ 2, σ 2, φ) Marginalized likelihood: y Ω N(Xβ, σ 2 R(φ) + τ 2 I) Note: Spatial process parametrizes Σ: y = Xβ + ɛ, ɛ N (0, Σ), Σ = σ 2 R(φ) + τ 2 I 11 Frontiers of Statistical ecision Making and Bayesian Analysis 12 Frontiers of Statistical ecision Making and Bayesian Analysis Conditional model: conjugate full conditionals for σ 2, τ 2 and w easier to program. Conditional model: conjugate full conditionals for σ 2, τ 2 and w easier to program. Marginalized model: need Metropolis or Slice sampling for σ 2, τ 2 and φ. Harder to program. But, reduced parameter space faster convergence σ 2 R(φ) + τ 2 I is more stable than σ 2 R(φ). 12 Frontiers of Statistical ecision Making and Bayesian Analysis 12 Frontiers of Statistical ecision Making and Bayesian Analysis Conditional model: conjugate full conditionals for σ 2, τ 2 and w easier to program. Marginalized model: need Metropolis or Slice sampling for σ 2, τ 2 and φ. Harder to program. But, reduced parameter space faster convergence σ 2 R(φ) + τ 2 I is more stable than σ 2 R(φ). But what about R 1 (φ)?? EXPENSIVE! 12 Frontiers of Statistical ecision Making and Bayesian Analysis 13 Frontiers of Statistical ecision Making and Bayesian Analysis

5 They are recovered from [w y, X] = [w Ω, y, X] [Ω y, X]dΩ using posterior samples: They are recovered from [w y, X] = [w Ω, y, X] [Ω y, X]dΩ using posterior samples: Obtain Ω (1),..., Ω (G) [Ω y, X] For each Ω (g), draw w (g) [w Ω (g), y, X] 13 Frontiers of Statistical ecision Making and Bayesian Analysis 13 Frontiers of Statistical ecision Making and Bayesian Analysis Residual plot: [w(s) y] They are recovered from [w y, X] = [w Ω, y, X] [Ω y, X]dΩ using posterior samples: Obtain Ω (1),..., Ω (G) [Ω y, X] For each Ω (g), draw w (g) [w Ω (g), y, X] w mean of Y NOTE: With Gaussian likelihoods [w Ω, y, X] is also Gaussian. With other likelihoods this may not be easy and often the conditional updating scheme is preferred. 13 Frontiers of Statistical ecision Making and Bayesian Analysis 14 Frontiers of Statistical ecision Making and Bayesian Analysis Another look: [w(s) y] Another look: [w(s) y] w w 15 Frontiers of Statistical ecision Making and Bayesian Analysis 16 Frontiers of Statistical ecision Making and Bayesian Analysis

6 Spatial Prediction Spatial Prediction Prediction: Summary of [Y (s) y] Often we need to predict Y (s) at a new set of locations { s 0,..., s m } with associated predictor matri X. Sample from predictive distribution: [ỹ y, X, X] = [ỹ, Ω y, X, X]dΩ = [ỹ y, Ω, X, X] [Ω y, X]dΩ, y [ỹ y, Ω, X, X] is multivariate normal. Sampling scheme: Obtain Ω (1),..., Ω (G) [Ω y, X] For each Ω (g), draw ỹ (g) [ỹ y, Ω (g), X, X]. 17 Frontiers of Statistical ecision Making and Bayesian Analysis 18 Frontiers of Statistical ecision Making and Bayesian Analysis Colorado data illustration Temperature residual map Modelling temperature: 507 locations in Colorado. Simple spatial regression model: Y (s) = T (s)β + w(s) + ɛ(s) w(s) GP (0, σ 2 ρ( ; φ, ν)); ɛ(s) iid N(0, τ 2 ) Parameters 50% (2.5%,97.5%) Intercept (2.131,3.866) [Elevation] (-0.527,-0.333) Precipitation (0.002,0.072) σ (0.051, 1.245) φ 7.39E-3 (4.71E-3, 51.21E-3) Range (38.8, 476.3) τ (0.022, 0.092) Frontiers of Statistical ecision Making and Bayesian Analysis 20 Frontiers of Statistical ecision Making and Bayesian Analysis Elevation map Residual map with elev. as covariate Latitude Longitude Frontiers of Statistical ecision Making and Bayesian Analysis 22 Frontiers of Statistical ecision Making and Bayesian Analysis