Fast Dimension-Reduced Climate Model Calibration and the Effect of Data Aggregation

Transcription

1 Fast Dimension-Reduced Climate Model Calibration and the Effect of Data Aggregation Won Chang Post Doctoral Scholar, Department of Statistics, University of Chicago Oct 15, 2014 Thesis Advisors: Murali Haran, Klaus Keller Coauthors: Murali Haran, Klaus Keller (Penn State), Roman Olson (U of New South Wales) Won Chang (U of Chicago) Climate Model Calibration Oct 15, / 21

2 Overview Climate models are often used to make projections about future climate. A major source of uncertainty about these projections is due to uncertainty about climate model input parameters. We propose methods for learning about climate model parameters from climate model outputs and observations. Challenges: Data in the form of high-dimensional spatial fields. Complicated error structures. I will describe novel computationally efficient calibration approach. Won Chang (U of Chicago) Climate Model Calibration Oct 15, / 21

3 Scientific Motivation Scientific Goal: Making projections for AMOC using climate model. Atlantic Meridional Overturning Circulation (AMOC): Large scale ocean circulation, important for equilibrium climate in Europe Rahmstorf (1997) Background vertical diffusivity (K bg ): Model parameter that quantifies intensity of vertical mixing in ocean. Source of uncertainty in AMOC projections. Won Chang (U of Chicago) Climate Model Calibration Oct 15, / 21

4 Parametric Uncertainty in AMOC Projections Bhat et al. (2012) Won Chang (U of Chicago) Climate Model Calibration Oct 15, / 21

5 Calibration Problem Which parameter settings best match observations? Won Chang (U of Chicago) Climate Model Calibration Oct 15, / 21

6 Emulation Step Computer model output (y-axis) vs. input (x-axis) Emulation (approximation) of computer model using GP Won Chang (U of Chicago) Climate Model Calibration Oct 15, / 21

7 Calibration Step Combining observation and emulator Posterior PDF of θ given model output and observations Won Chang (U of Chicago) Climate Model Calibration Oct 15, / 21

8 Summary of Statistical Problem Goal: Learning about θ based on two sources of information: Observations*: Mean potential ocean temperature, Z = (Z(s 1 ),..., Z(s n )) T, where s 1,..., s n are 3D locations. Climate model output** for mean potential temperature Y(θ 1 ),..., Y(θ p ), where each Y(θ i ) = (Y (s 1, θ i ),..., Y (s n, θ i )) T is spatial field (Sriver et al., 2012). Z and Y(θ i ) s are n-dimensional vectors Important: output at each θ i is a high-dimensional spatial field. n = 61, 051 locations, p = 250 runs. *World Ocean Atlas 2009 **University of Victoria (UVic) Earth System Climate Model Averaged over Won Chang (U of Chicago) Climate Model Calibration Oct 15, / 21

9 GP for Computer Model Emulation Fit GP to np-dimensional data Y = ( Y(θ 1 ) T,..., Y(θ p ) T ) T by finding MLE ˆξ of emulator parameter ξ. Covariance used for non-linear relationship between parameter and model output (model output as a function of parameter) non-linear spatial surface (model output as a function of location) Covariance function example: Cov ( Y (s, θ), Y (s, θ ); ξ ) ( =κ exp g (s, ) ( s ) θ θ ) exp φ s φ θ + ζi (θ = θ )I (s = s ) where g is geodesic distance, and ξ = (κ, φ s, φ θ, ζ) is covariance parameter. Won Chang (U of Chicago) Climate Model Calibration Oct 15, / 21

10 Step 1: Emulation (Approximating Computer Model) Get η(θ NEW, Y) for prediction at any θ NEW Θ: GP gives ( Y Y(θ NEW ) ) N ( ( 0 0 ), n(p+1) 1 ( ) Σ11 Σ 12 Σ 21 Σ 22 n(p+1) n(p+1) Emulator: η (θ NEW, Y) = Y(θ NEW ) Y N ( Σ 21 Σ 1 11 Y, Σ 22 Σ 21 Σ 1 11 Σ ) 12 ) Won Chang (U of Chicago) Climate Model Calibration Oct 15, / 21

11 Calibration Model based on GP Emulator Model observational data by Z = η(θ, Y) + δ, where n-dimensional spatial field δ is model-observation discrepancy with covariance parameter ξ δ. Corresponding probability model: Z Y, θ, ξ δ N ( µ η, Σ η + Σ d ) where µ η = Σ 12 Σ 1 22 Y and Σ η = Σ 22 Σ 21 Σ 1 11 Σ 12. Σ d is covariance matrix for discrepancy process such that ( {Σ d } ij = κ d exp g(s ) i, s j ) + ζ d I (s i = s j ), φ d where κ d > 0, φ d > 0, ζ d > 0, and ξ δ = (κ d, φ d, ζ d ). Won Chang (U of Chicago) Climate Model Calibration Oct 15, / 21

12 Step 2: Calibration (Inferring Input Parameter) Inference for θ based on posterior distribution π(θ, ξ δ Z, Y, ˆξ) L(Z Y, θ, ξ δ, ˆξ) p(θ) p(ξ }{{} δ ) }{{} likelihood priors for θ and ξ δ with emulator parameter ˆξ fixed at value estimated in emulation step. Won Chang (U of Chicago) Climate Model Calibration Oct 15, / 21

13 Computational Challenges Emulation Step: Computational Challenges for np np covariance matrix: Cholesky decomposition costs 1 3 n3 p 3 = flops. Storing covariance matrix requires = 1, 735, 624 Gb 3 memory space. Calibration Step: Similar challenges for dealing with n n covariance matrix. Won Chang (U of Chicago) Climate Model Calibration Oct 15, / 21

14 Our Solution: Dimension Reduction Fast computation using PC and Kernel Convolution (Chang et al. 2014a, the Annals of Applied Statistics) Consider model outputs at θ 1,..., θ p as replicates and obtain PCs Y (s 1, θ 1 )... Y (s n, θ 1 )..... Y (s 1, θ p )... Y (s n, θ p ) p n Y1 R(θ 1)... Y R J y (θ 1 )..... Y1 R(θ p)... YJ R y (θ p ) p J y PCs pick up characteristics of model output that vary most across input parameters θ 1,..., θ p. Won Chang (U of Chicago) Climate Model Calibration Oct 15, / 21

15 Emulation Using PCs Fit 1-dimensional GP for each series Yj R (θ 1 ),..., Yj R (θ p ) η(θ, Y R ): J y -dimensional emulation process for PCs, Y R is collection of PCs Computation reduces from O(n 3 p 3 ) to O(J y p 3 ) ( to flops). Emulation for original output: compute K y η(θ, Y R ) where K y is matrix of scaled eigenvectors Covariance Structure Won Chang (U of Chicago) Climate Model Calibration Oct 15, / 21

16 Cross-validation for Emulator Check emulation performance: Won Chang (U of Chicago) Climate Model Calibration Oct 15, / 21

17 Calibration in Reduced Dimensions Probability model for dimension-reduced observation Z R : Z = K y η(θ, Y R ) + }{{} K d ν }{{} emulator discrepancy + ɛ }{{} ( Z R = (K T K) 1 K T η(θ, Y Z = R ) ν observation error, ) + (K T K) 1 K T ɛ, with combined basis K = [K y K d ], knot process ν N(0, κ d I), and observational error ɛ N(0, σ 2 I). Infer θ through posterior distribution π(θ, κ d, σ 2 Z R, Y R ) L(Z R Y R, θ, κ d, σ 2 ) p(θ)p(κ }{{} d )p(σ 2 ) }{{} likelihood given by above priors Won Chang (U of Chicago) Climate Model Calibration Oct 15, / 21

18 Modeling Discrepancy with Kernel Convolution Kernel convolution: Specifying n-dimensional discrepancy process δ using J d -dimensional knot process ν (J d < n) and n J d kernel basis matrix K d. Let δ = K d ν. Won Chang (U of Chicago) Climate Model Calibration Oct 15, / 21

19 Scientific Question: Effect of Data Aggregation Data aggregation is commonly used to avoid computational challenges Its effect is previously unknown due to computational challenges for unaggregated data Our approach allows us to calibrate using unaggregated data. We compare results based on 1D (depth profile, n=13) 2D (zonal average, n=984) 3D (unaggregated, n=61051) Inference using 3D data: shaper inference for input parameter θ more robust to changes in prior specifications for discrepancy parameters. Won Chang (U of Chicago) Climate Model Calibration Oct 15, / 21

20 Projection Results Using unaggregated spatial patterns (3D) leads to sharper projections Won Chang (U of Chicago) Climate Model Calibration Oct 15, / 21

21 Discussion Dimension reduction-based approach: Very fast, scales well with n, number of spatial locations Very easy to use: Automatic emulation step Scientific finding: Using full spatial pattern without data aggregation leads to better results Other application: Greenland ice sheet model calibration (Chang 2014b) Ongoing Work: Calibration using binary spatial data: First extension to non-gaussian data Easily extended to one parametric exponential family Possible application: Ice sheet model calibration, ecological model, infectious disease model Won Chang (U of Chicago) Climate Model Calibration Oct 15, / 21