A Complete Spatial Downscaler

A Complete Spatial Downscaler Yen-Ning Huang, Brian J Reich, Montserrat Fuentes 1 Sankar Arumugam 2 1 Department of Statistics, NC State University 2 Department of Civil, Construction, and Environmental Engineering, NC State University April 28 th, 2016 A Complete Spatial Downscaler 1 / 30

Outline Introduction Complete spatial downscaler Marginal distributions Spatial correlation Calibration Application A Complete Spatial Downscaler 2 / 30

Introduction Computer models are essential for environmental science to evaluate climate/emission scenarios and make predictions where data are not available. Calibration Evaluation of the performance of physical models is needed to obtain reliable forecasts Model calibration is challenging when the process of interest is dynamic across space and time Reproducing the spatial behavior of the physical process of interest is important Downscaler Combine disparate spatial data sources to improve the spatial prediction A Complete Spatial Downscaler 3 / 30

Challenges Comparing monitor data and computer output is difficult because they often measure different spatial scales Computer models notoriously struggle to explain extreme behaviour Models run at coarse scales have trouble capturing local phenomenon, as thus may have different spatial correlation from monitor data A Complete Spatial Downscaler 4 / 30

Data sources Our motivating example is a study of summer temperatures in the US. Climate model output: a 30-year Hindcast model of monthly average temperature in summer (Jun-Aug) from 1981 to 2010 Monitor data: monthly average temperature in summer from 1950 to 1999 (interpolated to the same grid points as model output) Our goal is not to match data sources for individual months, but rather to match their distributions. A Complete Spatial Downscaler 5 / 30

Monitor observations (top) and model output (bottom) means 35 30 25 20 15 10 35 30 25 20 15 10 A Complete Spatial Downscaler 6 / 30

Monitor observations (top) and model output (bottom) standard deviations 4 3 2 1 0 4 3 2 1 0 A Complete Spatial Downscaler 7 / 30

Variograms for monitor observations and model outputs Semi variogram 5 10 15 20 25 Monitor observations Model outputs 0 500 1000 1500 2000 Distance (km) A Complete Spatial Downscaler 8 / 30

Complete spatial downscaler We propose a Bayesian approach to adjust for biases of the marginal distribution as well as the spatial correlation of computer model outputs Our approach has the following three components: (1) Model the spatially varying and potentially non-gaussian marginal distributions of the model outputs and monitor data (2) Model the spatial covariance of both data sources (3) Perform calibration to rectify differences between the two data sources These three components are implemented simultaneously using Bayesian hierarchical modeling A Complete Spatial Downscaler 9 / 30

Marginal distributions X t (s): computer model output Y t (s): monitor observations Let Fs(x) and Gs(y) be the CDF of X t (s) and Y t (s), respectively We can transform both processes to standard normal distribution: Z Xt (s) = Φ 1 {Fs[X t (s)]} and Z Yt (s) = Φ 1 {Gs[Y t (s)]} Similarly, X t (s) = Fs 1 Xt(s)]} and Y t (s) = Gs 1 Yt(s)]} A Complete Spatial Downscaler 10 / 30

Assuming normality When both processes and are Gaussian, X t (s) = µ X (s)+σ X (s)z Xt (s) and Y t (s) = µ Y (s)+σ Y (s)z Yt (s) where µ X (s) and σ X (s) > 0 are spatially-varying mean and standard deviation of X t (s) µ Y (s) and σ Y (s) > 0 are spatially-varying mean and standard deviation of Y t (s) A Complete Spatial Downscaler 11 / 30

Normal Q-Q plots for monitor observations ( 111.5, 31.5) ( 82.5, 32.5) 23 25 27 29 24 26 28 30 23 24 25 26 27 28 29 ( 120.5, 40.5) 24 26 28 30 ( 103.5, 46.5) 10 14 18 10 12 14 16 18 20 14 18 22 14 16 18 20 22 24 A Complete Spatial Downscaler 12 / 30

Normal Q-Q plots for model outputs ( 111.5, 31.5) ( 82.5, 32.5) 22 26 30 22 26 30 22 24 26 28 30 ( 120.5, 40.5) 22 24 26 28 30 ( 103.5, 46.5) 10 14 18 22 16 20 24 28 10 12 14 16 18 20 22 24 16 18 20 22 24 26 28 A Complete Spatial Downscaler 13 / 30

Skew-t model The Gaussian model fails to fit the tails For a richer class of marginal distributions we also consider the spatial skew-t process (Jones and Faddy 2003; Azzalini and Capitanio 2003): X t (s) = µ X (s) + ξ X r Xt + σ Xt σ X (s)z Xt (s) (1) indep where r Xt N(0, σ 2 Xt ), σ2 Xt iid InvGamma(νX /2, ν X /2) In this setting the marginal distribution of X t (s) follow skew-t distribution with parameters {µ X (s), σ X (s), ν X, ξ X } A skew-t model also permits asymptotic spatial dependence A Complete Spatial Downscaler 14 / 30

Skew t Q-Q plots for monitor observations ( 111.5, 31.5) ( 82.5, 32.5) 23 25 27 29 24 26 28 30 24 25 26 27 28 29 ( 120.5, 40.5) 24 25 26 27 28 29 30 ( 103.5, 46.5) 10 14 18 12 14 16 18 20 14 18 22 14 16 18 20 22 24 A Complete Spatial Downscaler 15 / 30

Skew t Q-Q plots for model outputs ( 111.5, 31.5) ( 82.5, 32.5) 22 26 30 22 26 30 22 24 26 28 30 ( 120.5, 40.5) 22 24 26 28 30 ( 103.5, 46.5) 10 14 18 22 16 20 24 28 10 12 14 16 18 20 22 24 16 18 20 22 24 26 28 A Complete Spatial Downscaler 16 / 30

Spatial correlation Assume Z Xt (s) and Z Yt (s) have mean zero and variance one Expand the two processes using Karhunen-Loéve decomposition: Z Xt (s) = L φ l (s)x tl and Z Yt (s) = l=1 L φ l (s)y tl (2) l=1 where φ l (s) are orthonormal functions x tl and y tl are pairwise independence with mean zero and Var(x tl ) = f l and Var(y tl ) = g l A Complete Spatial Downscaler 17 / 30

Spatial covariance The covariance functions can be written as: Cov[Z Xt (s i ), Z Xt (s j )] = L φ l (s i )φ l (s j )f l l=1 L Cov[Z Yt (s i ), Z Yt (s j )] = φ l (s i )φ l (s j )g l In particular, the basis functions we consider in data analysis are { cos(s φ l (s) = ω l/2 ) l is even sin(s ω l/2 ) l is odd. where ω = (ω 1,..., ω n ) is the set of Fourier frequencies l=1 A Complete Spatial Downscaler 18 / 30

Matérn spectral density function For the covariances f l and g l we consider the spectral density of Matérn covariance on lattice (Guinness et al. 2014): f l (ω) = f l (ω 1, ω 2 ) = σ 2 f ( 1 + ( α f ) ( ( 2 δ sin 2 δω 1 2 ) ( ))) + sin 2 νf +1 δω 2 2 evaluated over the set of Fourier frequencies ω A Complete Spatial Downscaler 19 / 30

Calibration Conditional on the parameters in the marginal distributions Fs and Gs, the covariances f and g, and the latent x tl and y tl, the standardized computer model output is calibrated as Z Xt (s) = L w l φ l (s)x tl l=1 w l = g l /f l is the ratio of spectral densities which is used to calibrate the spatial correlation of the model output The covariance of Z Xt (s) matches with Z Yt (s) since Cov[ Z Xt (s i ), Z Xt (s j )] = L l=1 w 2 l φ l(s i )φ l (s j )f l = Cov[Z Yt (s i ), Z Yt (s j )] A Complete Spatial Downscaler 20 / 30

Calibration The calibrated computer model output X t (s) = G 1 s {Φ[ Z Xt (s)]} has Y t (s) s marginal distribution and spatial correlation If both data sources follow skew-t distribution, then the calibrated computer model output is X t (s) = µ Y (s) + ξ Y r Yt + σ Yt σ Y (s) L w l φ l (s)x tl (3) l=1 A Complete Spatial Downscaler 21 / 30

Calibration for skew-t model If skew-t assumption is appropriate, then we can have X t N(µ X + ξ X r Xt, σ 2 Xt (ΦF Φ + τ 2 X I n)) and Y t N(µ Y + ξ Y r Yt, σ 2 Yt (ΦGΦ + τ 2 Y I n)) where r Xt N(0, σ 2 Xt ) and σ2 Xt InvGamma( ν X 2, ν X 2 ). Each MCMC iteration gives a sample for all model parameters (µ X, σ 2 X, etc), as well as random effects x tl This gives a posterior predictive distribution of X t (s) in (3) A Complete Spatial Downscaler 22 / 30

Monitor observations (top) and calibrated model output (bottom) means 35 30 25 20 15 10 35 30 25 20 15 10 A Complete Spatial Downscaler 23 / 30

Monitor observations (top) and calibrated model output (bottom) standard deviations 4 3 2 1 0 4 3 2 1 0 A Complete Spatial Downscaler 24 / 30

Calibrated quantiles (skew t) ( 111.5, 31.5) ( 82.5, 32.5) 22 26 30 Monitor observations Model outputs Calibrated model outputs 0.0 0.2 0.4 0.6 0.8 1.0 probs 22 26 30 0.0 0.2 0.4 0.6 0.8 1.0 probs 10 15 20 25 ( 120.5, 40.5) 0.0 0.2 0.4 0.6 0.8 1.0 probs 15 25 ( 103.5, 46.5) 0.0 0.2 0.4 0.6 0.8 1.0 probs A Complete Spatial Downscaler 25 / 30

Calibrated variograms (skew t) Semi variogram 5 10 15 20 25 Monitor observations Model outputs Calibrated model outputs 0 500 1000 1500 2000 Distance (km) A Complete Spatial Downscaler 26 / 30

Representative Concentration Pathways (RCPs) The RCPs are greenhouse gas concentration trajectories adopted by the IPCC in 2014 for climate modeling and research Depending on how much greenhouse gases are emitted in the years to come, RCPs can be used to describe different possible future climates. We apply our downscaler to summer monthly temperature data from RCP8.5 (2005-2034) A Complete Spatial Downscaler 27 / 30

Future RCP8.5 means before and after calibration 35 30 25 20 15 10 35 30 25 20 15 10 A Complete Spatial Downscaler 28 / 30

Acknowledgment This work was supported by grants DOI (14-1-04-9), NIH (R21ES022795-01A1, 5R01ES014843-02), and NSF (DMS1107046). A Complete Spatial Downscaler 29 / 30

Thank you! For comments or questions, please contact me at yehuang@ncsu.edu A Complete Spatial Downscaler 30 / 30