Bayesian spatial hierarchical modeling for temperature extremes

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Transcription:

Bayesian spatial hierarchical modeling for temperature extremes Indriati Bisono Dr. Andrew Robinson Dr. Aloke Phatak Mathematics and Statistics Department The University of Melbourne Maths, Informatics and Statistics, CSIRO WA October 6-7 th, 2011

Outline 1 Problem Discussion 2 The Three Stages Model 3 Implementation 4 Simulation Results 5 Conclusion and Future Works

Problem Discussion What we observe? Downscaled annual maximum temperatures, z it, for grid cells i = 1,..., 2856 covering all Tasmania, during year t = 1,..., 49 (1961 to 2009).

Problem Discussion What we observe? Downscaled annual maximum temperatures, z it, for grid cells i = 1,..., 2856 covering all Tasmania, during year t = 1,..., 49 (1961 to 2009). What we do? Developing a three stage Bayesian spatial hierarchical model, following Schliep E. and Dan Cooley s (2010) model. The spatial structure is depicted by means of the random effect which is modelled using conditional autoregressive (CAR).

Problem Discussion Aim of study Constructing a return level map of weather extremes.

Problem Discussion Aim of study Constructing a return level map of weather extremes. p-year Return Level is an expected value to be exceeded once every 1/p years with probability p. In other words, a p-quantile that associated with return period 1/p.

Problem Discussion Aim of study Constructing a return level map of weather extremes. p-year Return Level is an expected value to be exceeded once every 1/p years with probability p. In other words, a p-quantile that associated with return period 1/p. can be find simply by inverted the following equation;

Problem Discussion Aim of study Constructing a return level map of weather extremes. p-year Return Level is an expected value to be exceeded once every 1/p years with probability p. In other words, a p-quantile that associated with return period 1/p. can be find simply by inverted the following equation; { ( P(Z z) = exp 1 + ξ x µ ) } 1/ξ = p σ

Problem Discussion Aim of study Constructing a return level map of weather extremes. p-year Return Level is an expected value to be exceeded once every 1/p years with probability p. In other words, a p-quantile that associated with return period 1/p. can be find simply by inverted the following equation; { ( P(Z z) = exp 1 + ξ x µ ) } 1/ξ = p σ For p = 1, return level has 1% chance of being exceeded during return period of 1/p = 100 years.

Stage 1 of 3 Data level Assume z it follows GEV distribution [ P(Z it z µ i, σ i, ξ i ) = exp ( ) ] z µ 1/ξi i 1 + ξ i σ i ( ) z µ provided 1 + ξ i i σ i > 0 for each i, where µ i, σ i and ξ i are unknown location, scale and shape parameters at grid cell i.

Stage 2 of 3 Process level where Let θ = (µ, log(σ), ξ) Assume θ follows Normal distribution ( θ N X β + U, 1 ) τ 2 X is a 2856 3 matrix of covariates (latitude and longitude) β is a 3 3 matrix of regression coefficients U is a 2856 3 matrix of random effect τ 2 is a fixed precision matrix

Stage 2 of 3 Process level where Let θ = (µ, log(σ), ξ) Assume θ follows Normal distribution ( θ N X β + U, 1 ) τ 2 X is a 2856 3 matrix of covariates (latitude and longitude) β is a 3 3 matrix of regression coefficients U is a 2856 3 matrix of random effect τ 2 is a fixed precision matrix U is modelled spatially using conditional autoregressive model

Conditional autoregressive model for random effects U Assume random effect U is a Gaussian Markov random field (GMRF) that satisfies conditional independence assumptions. U i u j, j i N b ij u j, ti 2, i = 1,..., n j where b ij is a spatial dependence parameter, i.e. b ij = w ij w i+ and w i+ = P j w ij.

Conditional autoregressive model for random effects U Assume random effect U is a Gaussian Markov random field (GMRF) that satisfies conditional independence assumptions. U i u j, j i N b ij u j, ti 2, i = 1,..., n j where b ij is a spatial dependence parameter, i.e. b ij = w ij w i+ and w i+ = P j w ij. w ij = 1 if node i and j share the same boundary, and w ij = 0 otherwise.

Conditional autoregressive model for random effects U Assume random effect U is a Gaussian Markov random field (GMRF) that satisfies conditional independence assumptions. U i u j, j i N b ij u j, ti 2, i = 1,..., n j where b ij is a spatial dependence parameter, i.e. b ij = w ij w i+ and w i+ = P j w ij. w ij = 1 if node i and j share the same boundary, and w ij = 0 otherwise. t 2 i is conditional precision; set t 2 i = T 2 w i+.

Conditional autoregressive model for random effects U The setup suggests a joint multivariate normal distribution for U = (U 1,..., U n ) with mean 0 and precision Q = (D w W ) where D w is diagonal with (D w ) ii = w i+

Conditional autoregressive model for random effects U The setup suggests a joint multivariate normal distribution for U = (U 1,..., U n ) with mean 0 and precision Q = (D w W ) where D w is diagonal with (D w ) ii = w i+ But, (D w W )1 = 0, i.e. Σ 1 U is singular so that Σ U does not exist

Conditional autoregressive model for random effects U The setup suggests a joint multivariate normal distribution for U = (U 1,..., U n ) with mean 0 and precision Q = (D w W ) where D w is diagonal with (D w ) ii = w i+ But, (D w W )1 = 0, i.e. Σ 1 U is singular so that Σ U does not exist Replacing each element in W with w ij w i+ would restrict each row sum to one, (D w W ) would not be singular and Σ U does exist. This is often referred to as an intrinsically autoregressive (IAR) model [Banerjee et al., 2004]

Stage 3 of 3 Parameter level The hyperparameters are β and T Choose conjugate priors for the two hyperparameters, i.e. a normal distribution for β priors, β N(β 0, κ 1 ) a Wishart prior with 3 d.f. for precision matrix T

MCMC implementation Using hybrid Monte Carlo; combining Metropolis and Gibbs sampler algorithms Metropolis algorithm used for generating GEV parameters posterior distribution where θ represents µ, σ and ξ. π(θ z) = π(z θ)π(θ) Gibbs sampler employed for generating posterior distributions for U, β and T U θ, β N C ( τ 2 (θ X β), T + τ 2) β θ, U N C ( τ 2 (θ U) + κβ 0, τ 2 + κ ) T β, U W 1 (Ψ, 3 + k), whereψ = U T WU + T 0, k = 2856

Metropolis Algorithm 1 Start with MLE estimates of corresponding parameters θ (0). Set k = 1 2 Generate a proposal θ from proposal distribution as follow µ = µ (k 1) + scale rt(1, 2) σ = σ (k 1) + scale (runif ) ξ = ξ (k 1) + scale rt(1, 5) 3 Set θ (k) = θ with probability { π(θ } ) α = min 1, π(θ (k 1) ) Otherwise set θ (k) = θ (k 1) 4 Set k = k + 1 and return to 2.

Prior and posterior GEV parameters µ mle σ mle ξ mle µ posterior σ posterior ξ posterior 20 25 30 35 1.5 2.0 2.5 0.2 0.1 0.1

Diagnostic plots for GEV parameters at one grid cell µ Density µ of var1 36.5 36.0 35.5 35.0 34.5 1.2 0.8 0.6 0.4 0.2 36.4 36.2 36.0 35.8 35.6 35.4 35.2 35.0 0 500 1000 1500 2000 2500 34.5 35.0 35.5 36.0 36.5 37.0 0 500 1000 1500 2000 2500 0 5 10 15 20 25 30 Iterations N = 2500 Bandwidth = 7247

Diagnostic plots for GEV parameters at one grid cell µ Density µ of var1 36.5 36.0 35.5 35.0 34.5 1.2 0.8 0.6 0.4 0.2 36.4 36.2 36.0 35.8 35.6 35.4 35.2 35.0 0 500 1000 1500 2000 2500 34.5 35.0 35.5 36.0 36.5 37.0 0 500 1000 1500 2000 2500 0 5 10 15 20 25 30 Iterations N = 2500 Bandwidth = 7247 σ Density σof var1 3.0 2.8 2.6 2.4 2.2 2.0 1.8 1.5 2.8 2.6 2.4 2.2 2.0 0 500 1000 1500 2000 2500 2.0 2.5 3.0 0 500 1000 1500 2000 2500 0 5 10 15 20 25 30 Iterations N = 2500 Bandwidth = 4652

Diagnostic plots for GEV parameters at one grid cell µ Density µ of var1 36.5 36.0 35.5 35.0 34.5 1.2 0.8 0.6 0.4 0.2 36.4 36.2 36.0 35.8 35.6 35.4 35.2 35.0 0 500 1000 1500 2000 2500 34.5 35.0 35.5 36.0 36.5 37.0 0 500 1000 1500 2000 2500 0 5 10 15 20 25 30 Iterations N = 2500 Bandwidth = 7247 σ Density σof var1 3.0 2.8 2.6 2.4 2.2 2.0 1.8 1.5 2.8 2.6 2.4 2.2 2.0 0 500 1000 1500 2000 2500 2.0 2.5 3.0 0 500 1000 1500 2000 2500 0 5 10 15 20 25 30 Iterations N = 2500 Bandwidth = 4652 ξ Density ξof var1 0 5 0.10 0.15 0.20 10 8 6 4 2 5 0.10 0.15 0.25 0 0.20 0 500 1000 1500 2000 2500 0.25 0.15 5 0 500 1000 1500 2000 2500 0 5 10 15 20 25 30 Iterations N = 2500 Bandwidth = 08155

Cumulative and AR plots for β 24.50 24.45 24.40 1.30 1.25 1.20 1.15 1.10 5 0 0.8 0.6 0.4 0.2 0.2 0.4 0.6 75 70 65 60 45 40 35 30 25 20 15 5 0 5 0.10 0.15 82 83 84 85 86 87 88 89 04 06 08 10 12 6 5 4 3 2 1

Cumulative and AR plots for β 24.50 24.45 24.40 1.30 1.25 1.20 1.15 1.10 5 0 0.8 0.6 0.4 0.2 0.2 0.4 0.6 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 75 70 65 60 45 40 35 30 25 20 15 5 0 5 0.10 0.15 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 82 83 84 85 86 87 88 89 04 06 08 10 12 6 5 4 3 2 1 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20

Cumulative and AR plots for T 5 0 0.95 0.90 0.85 0.80 0.75 0.70 8 9 10 11 6 4 2 0 8 9 10 11 130 120 110 100 90 80 0 20 40 60 80 6 4 2 0 0 20 40 60 80 700 650 600 550 500 450

Cumulative and AR plots for T 5 0 0.95 0.90 0.85 0.80 0.75 0.70 8 9 10 11 6 4 2 0 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 8 9 10 11 130 120 110 100 90 80 0 20 40 60 80 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 6 4 2 0 0 20 40 60 80 700 650 600 550 500 450 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20

Maps of U-posterior U µ U σ U ξ 10 5 0 0.8 0.6 0.4 0.2 0.2 0.4 5 0 5

Return level maps 25 years (Gev) 25 years (Poste) Poste Gev 45 40 35 30 25 50 years (Gev) 50 years (Poste) Poste Gev 45 40 35 30 25 1.5 1.5 100 years (Gev) 100 years (Poste) Poste Gev 45 40 35 30 25 2 1 0 1 2

Conclusion We used a hierarchical model with three stages.

Conclusion We used a hierarchical model with three stages. The spatial patterns are not directly modeled from the data but through parameters of the assume data distribution.

Conclusion We used a hierarchical model with three stages. The spatial patterns are not directly modeled from the data but through parameters of the assume data distribution. Conjugate priors were chosen for hyperparameters to ease the computation.

Conclusion We used a hierarchical model with three stages. The spatial patterns are not directly modeled from the data but through parameters of the assume data distribution. Conjugate priors were chosen for hyperparameters to ease the computation. Bayesian inference was carried out by hybrid MCMC, and following canonical parameterization of Rue and Held [Rue and Held, 2005].

Conclusion We used a hierarchical model with three stages. The spatial patterns are not directly modeled from the data but through parameters of the assume data distribution. Conjugate priors were chosen for hyperparameters to ease the computation. Bayesian inference was carried out by hybrid MCMC, and following canonical parameterization of Rue and Held [Rue and Held, 2005]. Determination of τ 2 ; a fixed precision matrix for GEV parameters, considerably affects the rate of convergence.

Conclusion We used a hierarchical model with three stages. The spatial patterns are not directly modeled from the data but through parameters of the assume data distribution. Conjugate priors were chosen for hyperparameters to ease the computation. Bayesian inference was carried out by hybrid MCMC, and following canonical parameterization of Rue and Held [Rue and Held, 2005]. Determination of τ 2 ; a fixed precision matrix for GEV parameters, considerably affects the rate of convergence. Other variables that greatly improved the produced chains are the choice of proposal distribution for Metropolis algorithm and the jump of for proposed parameters; too small or too big jump results in slower convergence and higher autocorrelation.

Future Works Estimate the best proposal distribution for Metropolis algorithm using an MCMC algorithm; MCMC within MCMC.

Future Works Estimate the best proposal distribution for Metropolis algorithm using an MCMC algorithm; MCMC within MCMC. Model the weather extremes based on observed data, and compare it to that of downscaled data.

Future Works Estimate the best proposal distribution for Metropolis algorithm using an MCMC algorithm; MCMC within MCMC. Model the weather extremes based on observed data, and compare it to that of downscaled data. Develop a joint distribution of temperature and wind extremes as multivariate spatial hierarchical model, possibly using copulas.

References Schliep, Erin M., Dan Cooley, Stephan R. Sain and Jennifer A. Hoeting (2010). A comparison study of extreme precipitation from six different regional climate models via spatial hierarchical modeling. Extremes, 13:219-239. Havard Rue and Leonhard Held (2005). Gaussian Markov Random Fields: Theory and Applications, Chapman & Hall/CRC. Sudipto Banerjee, Bradley P. Carlin, and Alan E. Gelfand, 2004. Hierarchical Modeling and Analysis for Spatial Data, Chapman & Hall/CRC.

Acknowledgements Special thanks for Dan Cooley who kindly provides his R-script for this research. Thanks to my supervisor and Department of Mathematics and Statistics, The University of Melbourne for providing financial support for attending this conference. I would also like to convey thanks to the Indonesia Higher Education Ministry for providing a scholarship.

Thank you!

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