Hierarchical Modelling for non-gaussian Spatial Data

Transcription

1 Hierarchical Modelling for non-gaussian Spatial Data Geography 890, Hierarchical Bayesian Models for Environmental Spatial Data Analysis February 15,

2 Spatial Generalized Linear Models Often data sets preclude Gaussian modelling: Y (s) may not even be continuous Example: Y (s) is a binary or count variable species presence or absence at location s species abundance from count at location s continuous forest variable is high or low at location s Replace Gaussian likelihood by exponential family member Diggle Tawn and Moyeed (1998) 2 Geography 890

3 Spatial Generalized Linear Models First stage: Y (s i ) are conditionally independent given β and w(s i ), so f(y(s i ) β, w(s i ), γ) equals h(y(s i ), γ) exp (γ[y(s i )η(s i ) ψ(η(s i ))]) where g(e(y (s i ))) = η(s i ) = x T (s i )β + w(s i ) (canonical link function) and γ is a dispersion parameter. Second stage: Model w(s) as a Gaussian process: w N(0, σ 2 R(φ)) Third stage: Priors and hyperpriors. No process for Y (s), only a valid joint distribution Not sensible to add a pure error term ɛ(s) 3 Geography 890

4 Spatial Generalized Linear Models Comments We are modelling with spatial random effects Introducing these in the transformed mean encourages means of spatial variables at proximate locations to be close to each other Marginal spatial dependence is induced between, say, Y (s) and Y (s ), but observed Y (s) and Y (s ) need not be close to each other Second stage spatial modelling is attractive for spatial explanation in themean First stage spatial modelling more appropriate to encourage proximate observations to be close. 4 Geography 890

5 Illustration from: Finley, A.O., S. Banerjee, and R.E. McRoberts. (2008) A Bayesian approach to quantifying uncertainty in multi-source forest area estimates. Environmental and Ecological Statistics, 15: Geography 890

6 Binary spatial regression: forest/non-forest We illustrate a non-gaussian model for point-referenced spatial data: Objective is to make pixel-level prediction of forest/non-forest across the domain. 6 Geography 890

7 Binary spatial regression: forest/non-forest We illustrate a non-gaussian model for point-referenced spatial data: Objective is to make pixel-level prediction of forest/non-forest across the domain. Data: Observations are from 500 georeferenced USDA Forest Service Forest Inventory and Analysis (FIA) inventory plots within a 32 km radius circle in MN, USA. The response Y (s) is a binary variable, with { 1 if inventory plot is forested Y (s) = 0 if inventory plot is not forested 6 Geography 890

8 Binary spatial regression: forest/non-forest We illustrate a non-gaussian model for point-referenced spatial data: Objective is to make pixel-level prediction of forest/non-forest across the domain. Data: Observations are from 500 georeferenced USDA Forest Service Forest Inventory and Analysis (FIA) inventory plots within a 32 km radius circle in MN, USA. The response Y (s) is a binary variable, with { 1 if inventory plot is forested Y (s) = 0 if inventory plot is not forested Observed covariates include the coinsiding pixel values for 3 dates of m resolution Landsat imagery. 6 Geography 890

9 Binary spatial regression: forest/non-forest We fit a generalized linear model where Y (s i ) Bernoulli(p(s i )), logit(p(s i )) = x T (s i )β + w(s i ). 7 Geography 890

10 Binary spatial regression: forest/non-forest We fit a generalized linear model where Y (s i ) Bernoulli(p(s i )), logit(p(s i )) = x T (s i )β + w(s i ). Assume vague flat for β, a Uniform(3/32, 3/0.5) prior for φ, and an inverse-gamma(2, ) prior for σ 2. 7 Geography 890

11 Binary spatial regression: forest/non-forest We fit a generalized linear model where Y (s i ) Bernoulli(p(s i )), logit(p(s i )) = x T (s i )β + w(s i ). Assume vague flat for β, a Uniform(3/32, 3/0.5) prior for φ, and an inverse-gamma(2, ) prior for σ 2. Parameters updated with Metropolis algorithm using target log density: ln (p(ω Y )) σ a n 2 nx + i=1 ln `σ 2 σ b Y (s i) x T (s i)β + w(s i) + ln(σ 2 ) + ln(φ φ a) + ln(φ b φ). σ 1 1 ln ( R(φ) ) 2 2 2σ 2 wt R(φ) 1 w nx ln 1 + exp(x T (s i)β + w(s i) i=1 7 Geography 890

12 Posterior parameter estimates Parameter estimates (posterior medians and upper and lower 2.5 percentiles): Parameters Estimates: 50% (2.5%, 97.5%) Intercept (θ 0 ) (49.56, ) AprilTC1 (θ 1 ) (-0.45, -0.11) AprilTC2 (θ 2 ) 0.17 (0.07, 0.29) AprilTC3 (θ 3 ) (-0.43, -0.08) JulyTC1 (θ 4 ) (-0.25, 0.17) JulyTC2 (θ 5 ) 0.09 (-0.01, 0.19) JulyTC3 (θ 6 ) 0.01 (-0.15, 0.16) OctTC1 (θ 7 ) (-0.68, -0.22) OctTC2 (θ 8 ) (-0.19, 0.14) OctTC3 (θ 9 ) (-0.46, -0.07) σ (0.39, 2.42) φ ( , ) log(0.05)/φ (meters) (932.33, ) Covariates and proximity to observed FIA plot will contribute to increase precision of prediction. 8 Geography 890

13 Median of posterior predictive distributions 9 Geography 890

14 97.5%-2.5% range of posterior predictive distributions 10 Geography 890

15 CDF of holdout area s posterior predictive distributions F[P(Y(A) = 1)] cut point P(Y(A) = 1) Classification of pixel areas (based on visual inspection of imagery) into non-forest ( ), moderately forest ( ), and forest (no marker). 11 Geography 890