Hierarchical Modeling for Spatial Data

Transcription

1 Bayesian Spatial Modelling Spatial model specifications: P(y X, θ). Prior specifications: P(θ). Posterior inference of model parameters: P(θ y). Predictions at new locations: P(y 0 y). Model comparisons.

2 Bayesian Spatial Modelling Spatial model specifications: P(y X, θ). Prior specifications: P(θ). Posterior inference of model parameters: P(θ y). Predictions at new locations: P(y 0 y). Model comparisons.

3 Bayesian Spatial Modelling Spatial model specifications: P(y X, θ). Prior specifications: P(θ). Posterior inference of model parameters: P(θ y). Predictions at new locations: P(y 0 y). Model comparisons.

4 Bayesian Spatial Modelling Spatial model specifications: P(y X, θ). Prior specifications: P(θ). Posterior inference of model parameters: P(θ y). Predictions at new locations: P(y 0 y). Model comparisons.

5 Bayesian Spatial Modelling Spatial model specifications: P(y X, θ). Prior specifications: P(θ). Posterior inference of model parameters: P(θ y). Predictions at new locations: P(y 0 y). Model comparisons.

6 Spatial model specifications A spatial Gaussian regression model for point referenced data Y(s) = x T (s)β + w(s) + ǫ(s) (1) w(s) GP(0, σ 2 ρ( ; φ)). ǫ(s) N(0, τ 2 ). θ = (β, σ 2, φ, τ 2 ). Given a set of observed locations s 1,..., s n, f(y X, θ) = N(Xβ, σ 2 H(φ) + τ 2 I)

7 House price data Data description: 74 houses in Baton Rouge, LA. Response: (log) house price. Explanatory variables: LivingArea, OtherArea, Age, Bedrooms,Baths, HalfBaths. Locations: Easting, Northing. We fit a spatial Gaussian regression model for the house price data. Choose X = (1, LivingArea, Age). Use Matérn covariance function.

8 Bayesian Spatial Modelling Spatial model specifications: P(y X, θ). P(y X, θ) = MVN(Xβ, σ 2 H(φ, ν) + τ 2 I). where H i,j = Matérn(s i, s j ; φ, ν) Prior specifications for θ = (β, σ 2, τ 2, φ, ν). π(β) Flat prior orn(µ β, σ β ) π(σ 2 ) IG(2, 1) π(τ 2 ) IG(2, 1) π(φ) U(a φ, b φ ) π(ν) U(a ν, b ν ) Posterior inference P(θ y).

9 Bayesian Spatial Modelling Spatial model specifications: P(y X, θ). P(y X, θ) = MVN(Xβ, σ 2 H(φ, ν) + τ 2 I). where H i,j = Matérn(s i, s j ; φ, ν) Prior specifications for θ = (β, σ 2, τ 2, φ, ν). π(β) Flat prior orn(µ β, σ β ) π(σ 2 ) IG(2, 1) π(τ 2 ) IG(2, 1) π(φ) U(a φ, b φ ) π(ν) U(a ν, b ν ) Posterior inference P(θ y).

10 Bayesian Spatial Modelling Spatial model specifications: P(y X, θ). P(y X, θ) = MVN(Xβ, σ 2 H(φ, ν) + τ 2 I). where H i,j = Matérn(s i, s j ; φ, ν) Prior specifications for θ = (β, σ 2, τ 2, φ, ν). π(β) Flat prior orn(µ β, σ β ) π(σ 2 ) IG(2, 1) π(τ 2 ) IG(2, 1) π(φ) U(a φ, b φ ) π(ν) U(a ν, b ν ) Posterior inference P(θ y).

11 Bayesian Spatial Modelling Spatial model specifications: P(y X, θ). P(y X, θ) = MVN(Xβ, σ 2 H(φ, ν) + τ 2 I). where H i,j = Matérn(s i, s j ; φ, ν) Prior specifications for θ = (β, σ 2, τ 2, φ, ν). π(β) Flat prior orn(µ β, σ β ) π(σ 2 ) IG(2, 1) π(τ 2 ) IG(2, 1) π(φ) U(a φ, b φ ) π(ν) U(a ν, b ν ) Posterior inference P(θ y).

12 Posterior inference We seek the posterior P(θ y): use MCMC algorithm to draw samples from P(θ y) P(y θ)p(θ). Step 0: give initial values for θ. Step t + 1: Sample P(β θ (t), y, x) Sample P(ν θ (t), y, x) Sample P(σ 2 θ (t), y, x) Sample P(τ 2 θ (t), y, x) Sample P(φ θ (t), y, x) This is called Gibbs sampling. Repeat until MCMC converges.

13 The Metropolis-Hastings Algorithm Our goal is to sample from a distribution π, e.g., π (ν) P(y ν, θ )π(ν). We start with ν (t) Draw a candidate value ν from a proposal distribution q(ν ν (t) ) With probability α(ν, ν (t) π (ν)q(ν (t) ν) ) = min{1, π (ν (t) )q(ν ν (t) ) }, accept ν (t+1) = ν. Otherwise, ν (t+1) = ν (t).

14 Posterior inference We seek the posterior P(θ y): use MCMC algorithm to draw samples from P(θ y) P(y θ)p(θ). Step 0: give initial values for θ. Step t + 1: Sample P(β θ (t), y, x) from a multivariate normal. Sample P(ν θ (t), y, x) using MH. Sample P(σ 2 θ (t), y, x) using MH. Sample P(τ 2 θ (t), y, x) using MH Sample P(φ θ (t), y, x) using MH This is called Gibbs+Metropolis Hastings. Repeat until MCMC converges.

15 R package: spbayes Input: model, prior. Outout: posterior samples for each parameter. m.1 <- splm(logsellingprice LivingArea+Age, coords=cbind(easting,northing), starting=...,sp.tuning=..., priors=list("phi.unif"=3/c(15,.5),"nu.unif"=c(0.3,4), "sigma.sq.ig"=c(2, 1),"tau.sq.IG"=c(2, 1)), cov.model="matern",...) print(summary(m.1$p.samples)) plot(m.1$p.samples)

16 Posterior estimates of model parameters Mean 2.5% 97.5% (Intercept) LivingArea Age sigma.sq tau.sq phi nu

17 Spatial random effects 0.3 Random effects

18 Bayesian Spatial Modelling Spatial model specifications: P(y X, θ). Prior specifications: P(θ). Posterior inference of model parameters: P(θ y). Predictions at new locations: P(y 0 y).

19 Bayesian Spatial Modelling Spatial model specifications: P(y X, θ). Prior specifications: P(θ). Posterior inference of model parameters: P(θ y). Predictions at new locations: P(y 0 y).

20 Spatial Prediction Predictive distribution: P(y 0 y, X, X 0 ) = P(y 0 y, θ, X, X 0 )P(θ y, X)dθ Recall that we already obtained draws from θ. For each sample θ, draw y 0 from P(y 0 y, θ, X, X 0 ).

21 Spatial Prediction [Y(s 0 ) θ, Y] is a Gaussian distribution with the mean and variance given by and E[Y(s 0 ) θ, Y] = x T (s 0 )β + h T (s 0 )(σ 2 H + τ 2 I n ) 1 (Y β) (2) Var[Y(s 0 ) Ω, Y] = σ 2 h T (s 0 )(σ 2 H + τ 2 I n ) 1 h(s 0 ) + τ 2. (3) From {θ (t) }, we generate {y (t) }. Summarize {y (t) } to obtain posterior inference for predictions.

22 Spatial prediction in spbayes Prediction at two new locations: ( 5068, 1691) and ( 5064, 1692) new.coords=rbind(c(-5068,1691),c(-5064,1692)) new.x=rbind(c(1,2000,5),c(1,2000,5)) pred = sppredict(m.1, pred.coords=new.coords, pred.covars=new.x) y.hat = apply(exp(pred$y.pred),1,mean) >

23 Bayesian Spatial Modelling Spatial model specifications: f(y X, θ). Prior specifications: f(θ). Posterior inference of model parameters: f(θ y). Predictions at new locations: f(y 0 y). Model comparisons.

24 Bayesian Spatial Modelling Spatial model specifications: f(y X, θ). Prior specifications: f(θ). Posterior inference of model parameters: f(θ y). Predictions at new locations: f(y 0 y). Model comparisons.

25 Model comparisons Model fitting: (Spiegelhalter et.al, 2002) DIC = p D + D D = E[ 2 log(p(y θ))], and p D = D D( θ). spdic(m.1) Add # of Bedrooms in the regression model spdic(m.2) D m.1 m p D DIC Conclusion: the model without Bedrooms as covariate fits the data better. Prediction performance for a hold out dataset: MSPE = E[ m j=1 (y true(s j ) y predict (s j )) 2 ]

26 Other extensions Spatial generalized linear models: spglm Multivariate spatial models (coregionalization model): spmvlm Spatial temporal models, non-stationary covariance models and other non-gaussian spatial models: Gibbs+Metropolis Hasting. Thank you!

27 Other extensions Spatial generalized linear models: spglm Multivariate spatial models (coregionalization model): spmvlm Spatial temporal models, non-stationary covariance models and other non-gaussian spatial models: Gibbs+Metropolis Hasting. Thank you!