A Complete Spatial Downscaler

Transcription

Yen-Ning Huang, Brian J Reich, Montserrat Fuentes
Sankar Arumugam 2
1 Department
2 Department
1
of Statistics, NC State University
of Civil, Construction, and Environmental Engineering,
NC State University
April 28th , 2016
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Outline
Introduction
Complete spatial downscaler
Marginal distributions
Spatial correlation
Calibration
Application
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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
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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
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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.
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Monitor observations (top) and model output (bottom) means
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Monitor observations (top) and model output (bottom) standard deviations
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Variograms for monitor observations and model outputs
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Semi−variogram
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Monitor observations
Model outputs
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500
1000
1500
2000
Distance (km)
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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
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Marginal distributions
Xt (s): computer model output
Yt (s): monitor observations
Let Fs (x) and Gs (y ) be the CDF of Xt (s) and Yt (s),
respectively
We can transform both processes to standard normal
distribution:
ZXt (s) = Φ−1 {Fs [Xt (s)]} and ZYt (s) = Φ−1 {Gs [Yt (s)]}
Similarly,
Xt (s) = Fs−1 {Φ[ZXt (s)]}
and
Yt (s) = Gs−1 {Φ[ZYt (s)]}
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Assuming normality
When both processes and are Gaussian,
Xt (s) = µX (s)+σX (s)ZXt (s)
and
Yt (s) = µY (s)+σY (s)ZYt (s)
where
µX (s) and σX (s) > 0 are spatially-varying mean and standard
deviation of Xt (s)
µY (s) and σY (s) > 0 are spatially-varying mean and standard
deviation of Yt (s)
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Normal Q-Q plots for monitor observations
( −82.5, 32.5)
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(−111.5, 31.5)
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(−103.5, 46.5)
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(−120.5, 40.5)
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Normal Q-Q plots for model outputs
( −82.5, 32.5)
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(−111.5, 31.5)
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(−103.5, 46.5)
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(−120.5, 40.5)
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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):
Xt (s) = µX (s) + ξX |rXt | + σXt σX (s)ZXt (s)
indep
(1)
iid
2 ), σ 2 ∼ InvGamma(ν /2, ν /2)
where rXt ∼ N(0, σXt
X
X
Xt
In this setting the marginal distribution of Xt (s) follow skew-t
distribution with parameters {µX (s), σX (s), νX , ξX }
A skew-t model also permits asymptotic spatial dependence
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Skew t Q-Q plots for monitor observations
( −82.5, 32.5)
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(−111.5, 31.5)
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(−103.5, 46.5)
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(−120.5, 40.5)
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Skew t Q-Q plots for model outputs
( −82.5, 32.5)
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(−111.5, 31.5)
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(−103.5, 46.5)
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(−120.5, 40.5)
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Spatial correlation
Assume ZXt (s) and ZYt (s) have mean zero and variance one
Expand the two processes using Karhunen-Loéve
decomposition:
ZXt (s) =
L
X
φl (s)xtl
and
l=1
ZYt (s) =
L
X
φl (s)ytl
(2)
l=1
where
φl (s) are orthonormal functions
xtl and ytl are pairwise independence with mean zero and
Var(xtl ) = fl and Var(ytl ) = gl
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Spatial covariance
The covariance functions can be written as:
Cov[ZXt (si ), ZXt (sj )] =
L
X
φl (si )φl (sj )fl
l=1
Cov[ZYt (si ), ZYt (sj )] =
L
X
φl (si )φl (sj )gl
l=1
In particular, the basis functions we consider in data analysis
are
cos(s0 ωl/2 ) l is even
φl (s) =
sin(s0 ωdl/2e ) l is odd.
where ω = (ω1 , . . . , ωn ) is the set of Fourier frequencies
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Matérn spectral density function
For the covariances fl and gl we consider the spectral density of
Matérn covariance on lattice (Guinness et al. 2014):
fl (ω) = fl (ω1 , ω2 ) = 1+
αf 2
δ
sin2
σf2
δω1
2
+ sin2
δω2
2
νf +1
evaluated over the set of Fourier frequencies ω
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Calibration
Conditional on the parameters in the marginal distributions Fs
and Gs , the covariances f and g , and the latent xtl and ytl ,
the standardized computer model output is calibrated as
Z̃Xt (s) =
L
X
wl φl (s)xtl
l=1
p
wl = gl /fl is the ratio of spectral densities which is used to
calibrate the spatial correlation of the model output
The covariance of Z̃Xt (s) matches with ZYt (s) since
Cov[Z̃Xt (si ), Z̃Xt (sj )] =
L
X
wl2 φl (si )φl (sj )fl
l=1
= Cov[ZYt (si ), ZYt (sj )]
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Calibration
The calibrated computer model output
X̃t (s) = Gs−1 {Φ[Z̃Xt (s)]}
has Yt (s)0 s marginal distribution and spatial correlation
If both data sources follow skew-t distribution, then the
calibrated computer model output is
X̃t (s) = µY (s) + ξY |rYt | + σYt σY (s)
L
X
wl φl (s)xtl
(3)
l=1
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Calibration for skew-t model
If skew-t assumption is appropriate, then we can have
2
Xt ∼ N(µX + ξX |rXt |, σXt
(ΦF Φ0 + τX2 In ))
and
2
Yt ∼ N(µY + ξY |rYt |, σYt
(ΦG Φ0 + τY2 In ))
2 ) and σ 2 ∼ InvGamma( νX , νX ).
where rXt ∼ N(0, σXt
Xt
2
2
Each MCMC iteration gives a sample for all model parameters
(µX , σX2 , etc), as well as random effects xtl
This gives a posterior predictive distribution of X̃t (s) in (3)
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Monitor observations (top) and calibrated model output (bottom) means
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Monitor observations (top) and calibrated model output (bottom) standard deviations
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1
0
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0
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Calibrated quantiles (skew t)
(−111.5, 31.5)
(−82.5, 32.5)
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Model outputs
Calibrated model outputs
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probs
probs
(−120.5, 40.5)
( −103.5, 46.5)
1.0
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probs
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probs
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Calibrated variograms (skew t)
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Semi−variogram
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Model outputs
Calibrated model outputs
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Distance (km)
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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)
.
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Future RCP8.5 means before and after calibration
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Acknowledgment
This work was supported by grants DOI (14-1-04-9),
NIH (R21ES022795-01A1, 5R01ES014843-02),
and NSF (DMS1107046).
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Thank you!
For comments or questions, please contact me at
yehuang@ncsu.edu
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A Complete Spatial Downscaler

Transcription

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