Hierarchical Modelling for Univariate Spatial Data

Transcription

1 Hierarchical Modelling for Univariate Spatial Data Sudipto Banerjee 1 and Andrew O. Finley 2 1 Biostatistics, School of Public Health, University of Minnesota, Minneapolis, Minnesota, U.S.A. 2 Department of Forestry & Department of Geography, Michigan State University, Lansing Michigan, U.S.A. March 3,

2 Spatial Domain y Frontiers of Statistical Decision Making and Bayesian Analysis x

3 Algorithmic Modelling 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: f(s) = i w i (S ; s)f(s i ) Interpolate by reading off f(s 0 ). Issues: Sensitivity to tessellations Choices of multivariate interpolators Numerical error analysis 3 Frontiers of Statistical Decision Making and Bayesian Analysis

4 What is a spatial process? x x Y(s1) Y(s 2) Y(sn) x x 4 Frontiers of Statistical Decision Making and Bayesian Analysis

5 Simple linear model Simple linear model Y (s) = µ(s) + ɛ(s), Response: Y (s) at location s Mean: µ = x T (s)β Error: ɛ(s) iid N(0, τ 2 ) D Y (s), x(s) 5 Frontiers of Statistical Decision Making and Bayesian Analysis

6 Simple linear model Simple linear model Assumptions regarding ɛ(s): ɛ(s) iid N(0, τ 2 ) Y (s) = µ(s) + ɛ(s), D ɛ(s i) ɛ(s j) 6 Frontiers of Statistical Decision Making and Bayesian Analysis

7 Simple linear model Simple linear model 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 D ɛ(s i) ɛ(s j) 6 Frontiers of Statistical Decision Making and Bayesian Analysis

8 Sources of variation 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 ) D w(s i) w(s j) 7 Frontiers of Statistical Decision Making and Bayesian Analysis

9 Sources of variation 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 i=1, then w N(0, σ 2 R(φ)), where R(φ) = [ρ(φ; s i s j )] n i,j=1 D w(s i) w(s j) 7 Frontiers of Statistical Decision Making and Bayesian Analysis

10 Sources of variation 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 8 Frontiers of Statistical Decision Making and Bayesian Analysis

11 Sources of variation 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., exponential decay ρ(φ; t) = exp( φt) if t > 0. 8 Frontiers of Statistical Decision Making and Bayesian Analysis

12 Sources of variation 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., exponential decay ρ(φ; t) = exp( 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 Decision Making and Bayesian Analysis

13 Sources of variation w N(0, σ 2 wr(φ)) defines complex 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). Simulated Predicted 9 Frontiers of Statistical Decision Making and Bayesian Analysis

14 Univariate spatial regression Simple linear model + random spatial effects Y (s) = µ(s) + w(s) + ɛ(s), Response: Y (s) at some site Mean: µ = x 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 ) 10 Frontiers of Statistical Decision Making and Bayesian Analysis

15 Univariate spatial regression Hierarchical modelling First stage: n y β, w, τ 2 N(Y (s i ) x T (s i )β + w(s i ), τ 2 ) i=1 11 Frontiers of Statistical Decision Making and Bayesian Analysis

16 Univariate spatial regression Hierarchical modelling First stage: n y β, w, τ 2 N(Y (s i ) x T (s i )β + w(s i ), τ 2 ) i=1 Second stage: w σ 2, φ N(0, σ 2 R(φ)) 11 Frontiers of Statistical Decision Making and Bayesian Analysis

17 Univariate spatial regression Hierarchical modelling First stage: n y β, w, τ 2 N(Y (s i ) x T (s i )β + w(s i ), τ 2 ) i=1 Second stage: w σ 2, φ N(0, σ 2 R(φ)) Third stage: Priors on Ω = (β, τ 2, σ 2, φ) 11 Frontiers of Statistical Decision Making and Bayesian Analysis

18 Univariate spatial regression Hierarchical modelling First stage: n y β, w, τ 2 N(Y (s i ) x T (s i )β + w(s i ), τ 2 ) i=1 Second stage: w σ 2, φ N(0, σ 2 R(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 Decision Making and Bayesian Analysis

19 Univariate spatial regression Hierarchical modelling First stage: n y β, w, τ 2 N(Y (s i ) x T (s i )β + w(s i ), τ 2 ) i=1 Second stage: w σ 2, φ N(0, σ 2 R(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) Note: Spatial process parametrizes Σ: y = Xβ + ɛ, ɛ N (0, Σ), Σ = σ 2 R(φ) + τ 2 I 11 Frontiers of Statistical Decision Making and Bayesian Analysis

20 Univariate spatial regression Bayesian Computations Choice: Fit [y Ω] [Ω] or [y β, w, τ 2 ] [w σ 2, φ] [Ω]. 12 Frontiers of Statistical Decision Making and Bayesian Analysis

21 Univariate spatial regression Bayesian Computations Choice: Fit [y Ω] [Ω] or [y β, w, τ 2 ] [w σ 2, φ] [Ω]. Conditional model: conjugate full conditionals for σ 2, τ 2 and w easier to program. 12 Frontiers of Statistical Decision Making and Bayesian Analysis

22 Univariate spatial regression Bayesian Computations Choice: Fit [y Ω] [Ω] or [y β, w, τ 2 ] [w σ 2, φ] [Ω]. 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 Decision Making and Bayesian Analysis

23 Univariate spatial regression Bayesian Computations Choice: Fit [y Ω] [Ω] or [y β, w, τ 2 ] [w σ 2, φ] [Ω]. 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 Decision Making and Bayesian Analysis

24 Univariate spatial regression Where are the w s? Interest often lies in the spatial surface w y. 13 Frontiers of Statistical Decision Making and Bayesian Analysis

25 Univariate spatial regression Where are the w s? Interest often lies in the spatial surface w y. They are recovered from [w y, X] = [w Ω, y, X] [Ω y, X]dΩ using posterior samples: 13 Frontiers of Statistical Decision Making and Bayesian Analysis

26 Univariate spatial regression Where are the w s? Interest often lies in the spatial surface w 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] 13 Frontiers of Statistical Decision Making and Bayesian Analysis

27 Univariate spatial regression Where are the w s? Interest often lies in the spatial surface w 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] 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 Decision Making and Bayesian Analysis

28 Univariate spatial regression Residual plot: [w(s) y] w mean of Y Easting Northing 14 Frontiers of Statistical Decision Making and Bayesian Analysis

29 Univariate spatial regression Another look: [w(s) y] w Easting Northing 15 Frontiers of Statistical Decision Making and Bayesian Analysis

30 Univariate spatial regression Another look: [w(s) y] w Easting Northing 16 Frontiers of Statistical Decision Making and Bayesian Analysis

31 Spatial Prediction Often we need to predict Y (s) at a new set of locations { s 0,..., s m } with associated predictor matrix X. Sample from predictive distribution: [ỹ y, X, X] = [ỹ, Ω y, X, X]dΩ = [ỹ y, Ω, X, X] [Ω y, X]dΩ, [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 Decision Making and Bayesian Analysis

32 Spatial Prediction Prediction: Summary of [Y (s) y] y Easting Northing 18 Frontiers of Statistical Decision Making and Bayesian Analysis

33 Illustration Colorado data illustration Modelling temperature: 507 locations in Colorado. Simple spatial regression model: Y (s) = x 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) 19 Frontiers of Statistical Decision Making and Bayesian Analysis

34 Illustration Temperature residual map Northing Easting 20 Frontiers of Statistical Decision Making and Bayesian Analysis

35 Illustration Elevation map Latitude Longitude 21 Frontiers of Statistical Decision Making and Bayesian Analysis

36 Illustration Residual map with elev. as covariate Northing Easting 22 Frontiers of Statistical Decision Making and Bayesian Analysis