Hierarchical Modeling for Univariate Spatial Data

Transcription

1 Hierarchical Modeling for Univariate Spatial Data Geography 890, Hierarchical Bayesian Models for Environmental Spatial Data Analysis February 15,

2 Spatial Domain 2 Geography 890

3 Spatial Domain This is a bad example because it s not a convex hull, but just go with it 2 Geography 890

4 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) 3 Geography 890

5 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) 4 Geography 890

6 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) 4 Geography 890

7 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) 5 Geography 890

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 ) 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) 5 Geography 890

9 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 6 Geography 890

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 Correlation model for R(φ): e.g., exponential decay ρ(φ; t) = exp( φt) if t > 0. 6 Geography 890

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. Other valid models e.g., Gaussian, Spherical, Matérn. Effective range, t 0 = ln(0.05)/φ 3/φ 6 Geography 890

12 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 7 Geography 890

13 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 ) 8 Geography 890

14 Univariate spatial regression Hierarchical modeling First stage: n y β, w, τ 2 N(Y (s i ) x T (s i )β + w(s i ), τ 2 ) i=1 9 Geography 890

15 Univariate spatial regression Hierarchical modeling 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)) 9 Geography 890

16 Univariate spatial regression Hierarchical modeling 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) 9 Geography 890

17 Univariate spatial regression Hierarchical modeling 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) 9 Geography 890

18 Univariate spatial regression Hierarchical modeling 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 9 Geography 890

19 Univariate spatial regression Bayesian Computations Choice: Fit [y Ω] [Ω] or [y β, w, τ 2 ] [w σ 2, φ] [Ω]. 10 Geography 890

20 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. 10 Geography 890

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. 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). 10 Geography 890

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). But what about R 1 (φ)?? EXPENSIVE! 10 Geography 890

23 Univariate spatial regression Where are the w s? Interest often lies in the spatial surface w y. 11 Geography 890

24 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: 11 Geography 890

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: Obtain Ω (1),..., Ω (G) [Ω y, X] For each Ω (g), draw w (g) [w Ω (g), y, X] 11 Geography 890

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] 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. 11 Geography 890

27 Univariate spatial regression Residual plot: [w(s) y] w mean of Y Easting Northing 12 Geography 890

28 Univariate spatial regression Another look: [w(s) y] w Easting Northing 13 Geography 890

29 Univariate spatial regression Another look: [w(s) y] w Easting Northing 14 Geography 890

30 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]. 15 Geography 890

31 Easting Spatial Prediction Prediction: Summary of [Y (s) y] y Northing 16 Geography 890

32 Illustration Colorado data illustration Modeling 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) 17 Geography 890

33 Illustration Temperature residual map Northing Easting 18 Geography 890

34 Illustration Elevation map Latitude Longitude 19 Geography 890

35 Illustration Residual map with elev. as covariate Northing Easting 20 Geography 890