Correlation Latitude...Longitude Latitude-20-10 0 10 E ^(a^) Posterior mean of teleconnection...
1
Modeling climate teleconnections Joshua Hewitt 1 , Jennifer A. Hoeting 1 , and James Done 2 1 Department of Statistics, Colorado State University; 2 National Center for Atmospheric Research Overview We begin to develop a regional geostatistical model to study teleconnections—a climate phenomenon in which geographically distant areas influence regional climate patterns—at decadal time scales. Such a model could ultimately help regional planners use climate forecasts to study and prepare for local impacts of climate change. 20 40 -150 -120 Longitude Latitude 50 100 150 Precipitation (mm) 280 285 290 295 Sea surface temp. (K) Teleconnections occur when remote covariates, like Pacific Ocean sea surface temperatures, influence regional climate variables, like average Spring precipitation in Oklahoma. Model We modify the spatially-varying intercept model to model teleconnection with Y t |{z} Local response = X t β | {z } Local effects + ( z T t α ) 1 n s | {z } Teleconnection effects + ω t |{z} Local spatial noise where Y t = ( y s 1 ,t ,...,y s n s ,t ) T and for timepoints t ∈T = {t 1 ,...,t n t } and Data Y t ∈ R n s X t ∈ R n s ×p z t ∈ R n r Spatial effects α ∼ N ( 0 n r ×1 ,σ 2 y σ 2 α R(λ r ) ) ω t iid ∼ N ( 0 n s ,σ 2 y ( H (λ l )+ σ 2 ε I n s )) Parameters β ∼ N (0 p×1 , Λ) σ 2 ∼ Inv-Gamma(k, θ ) λ ∼ Uniform(a, b) Key observations/notes: 1. We model teleconnection effects as a random intercept controlled by coefficients, α, that vary spatially over remote covariates, z t . Assumptions: time-constant assumption justified by climate system stability within decadal scales; local space-constant assumption justified by only applying model to regional data, within which effects are assumed constant. 2. Time independence of {ω t } justified by lack of large temporal trends, but could be modeled with standard spatio-temporal techniques. 3. Bounded spatial range parameters, λ, and nugget effect, σ 2 ε , ensure the model’s numerical stability. 4. Scale parameters, σ 2 , are parameterized for identifiability and decreased correlation during estimation. Results Posterior mean of α|Y when fixing σ 2 y = 10, 000 and placing an informative prior on β . 20 40 -150 -120 Longitude Latitude -20 -10 0 10 E ^ (α ^ 29 Posterior mean of teleconnection coefficients. Compared to pointwise correlation maps, estimated teleconnection coefficients may yield results more consistent with known El-Niño effects, in which tropical sea surface temperatures tend to be positively correlated with spring Oklahoma precipitation. Posterior mean 95% HPD Interval β 0 119 (98.3, 146) β 1 -2.86 (-3.19, -2.52) λ l 7.6 (7.42, 7.74) λ r 1.9 (0.134, 4.17) σ 2 α 0.311 (0.121, 0.537) σ 2 ε 0.00424 (0.00415, 0.00432) Posterior parameter estimates generally appear consistent with exploratory analysis. Exploratory analysis Data: (1981-2013) Average Oklahoma and sea surface temperatures in spring from ERA-Interim Reanalysis data; PRISM precipitation (total rain and melted snow). Time series plots of response variable over Oklahoma show no consistent, large temporal dependence. The time series for each spatial location is plotted in a different color. 0 100 200 1980 1990 2000 2010 Time Precipitation (mm) Temperature is approximately linearly related to precipitation at each location in Oklahoma. Spatial locations are plotted in different colors and the linear relationship at each location is overlaid. 0 100 200 275 280 285 290 295 Average Spring Temperature (K) Precipitation (mm) 20 40 -150 -120 Longitude Latitude -0.3 -0.2 -0.1 0.0 0.1 0.2 Correlation Pointwise correlations of sea surface temp. with average Oklahoma spring precipitation. This correlation map plots the correlation between the overall average spring precipitation in Oklahoma with sea surface temperature (SST) at remote locations in the Pacific Ocean. The correlation map suggests clear, but weak teleconnection effects. Parameter estimation • Estimate parameters with hybrid Gibbs algorithm applied to the model marginalized over spatial effects. – Adaptive random walk updates computed for: • β , log(σ 2 ), logit(λ) • Estimate teleconnection coefficients, α, by using posterior parameter samples to draw (in parallel) composition samples of α. • Marginal likelihood complexity: – Likelihood evaluation in O (n 3 s ∨ n 3 r ) by applying Kronecker product properties and Sherman-Morrison-Woodbury formula. Marginal model: ( let Σ y = σ 2 y ( H (λ l )+ σ 2 ε I n s ) and R α = σ 2 y σ 2 α R(λ r ) ) Y = Y t 1 . . . Y t n t ∼ N X t 1 . . . X t n t β . . . β , Σ y . . . Σ y + (z T t 1 R α z t 1 )J n s ··· (z T t 1 R α z t n t )J n s . . . (z T t n t R α z t 1 )J n s ··· (z T t n t R α z t n t )J n s Composition-sample distribution: ( let R T = R -1 α + ( 1 T n s Σ -1 y 1 n s )( ∑ t∈T z t z T t )) α|Y ∼ N X t∈T ( 1 T n s Σ -1 y (Y t - X t β ) ) R -1 T z t ,R -1 T ! Future work • If our preliminary results are validated, our flexible model may produce more informative and reliable inferences than classical tools for studying teleconnections, such as correlation maps. • Improve estimation for σ 2 y . • Allow α to vary locally as well as remotely. Challenge is to efficiently composition sample n s × n r teleconnection coefficients. References Banerjee, S., Carlin, B.P., and Gelfand, A.E., 2004, Hierarchical Modeling and Analysis for Spatial Data: Boca Raton, FL, Chapman and Hall/CRC. Givens, G.H., and Hoeting, J., 2013, Computational Statistics: Hoboken, NJ, John Wiley & Sons, Inc. Towler, E., Pai Mazumder, D., and Holland, G. A framework for investigating large-scale patterns as an alternative to precipitation for downscaling to local drought: Climate Dynamics (in review). Acknowledgements This material is based upon work supported by the National Science Foundation under Grant No. AGS - 1419558.
Transcript of Correlation Latitude...Longitude Latitude-20-10 0 10 E ^(a^) Posterior mean of teleconnection...
Modeling climate teleconnectionsJoshua Hewitt1, Jennifer A. Hoeting1, and James Done21Department of Statistics, Colorado State University; 2National Center for Atmospheric Research
OverviewWe begin to develop a regional geostatistical model tostudy teleconnections—a climate phenomenon in whichgeographically distant areas influence regional climatepatterns—at decadal time scales. Such a model couldultimately help regional planners use climate forecaststo study and prepare for local impacts of climate change.
20
40
−150 −120Longitude
Latit
ude
50
100
150
Precipitation (mm)
280
285
290
295
Sea surface temp. (K)
Teleconnections occur when remote covariates, like Pacific Ocean sea surface temperatures,influence regional climate variables, like average Spring precipitation in Oklahoma.
ModelWe modify the spatially-varying intercept model to model teleconnection with
Yt︸︷︷︸Local response
= Xtβ︸︷︷︸Local effects
+(zTt α
)1ns︸ ︷︷ ︸
Teleconnection effects
+ ωt︸︷︷︸Local spatial noise
where Yt =(ys1,t, . . . , ysns ,t
)T and for timepoints t ∈ T = {t1, . . . , tnt} and
DataYt ∈ Rns
Xt ∈ Rns×p
zt ∈ Rnr
Spatial effects
α ∼ N(0nr×1, σ
2yσ
2αR(λr)
)ωt
iid∼ N(0ns
, σ2y
(H(λl) + σ2
εIns
))Parameters
β ∼ N (0p×1, Λ)σ2 ∼ Inv-Gamma(k, θ)λ ∼ Uniform(a, b)
Key observations/notes:
1. We model teleconnection effects as a random intercept controlled by coefficients, α, that vary spatiallyover remote covariates, zt.Assumptions: time-constant assumption justified by climate system stability within decadal scales;local space-constant assumption justified by only applying model to regional data, within which effectsare assumed constant.
2. Time independence of {ωt} justified by lack of large temporal trends, but could be modeled withstandard spatio-temporal techniques.
3. Bounded spatial range parameters, λ, and nugget effect, σ2ε , ensure the model’s numerical stability.
4. Scale parameters, σ2, are parameterized for identifiability and decreased correlation during estimation.
ResultsPosterior mean of α|Y when fixing σ2
y = 10, 000and placing an informative prior on β.
20
40
−150 −120Longitude
Latit
ude
−20
−10
0
10
E(α)
Posterior mean of teleconnection coefficients.
Compared to pointwise correlation maps, estimatedteleconnection coefficients may yield results moreconsistent with known El-Niño effects, in which tropicalsea surface temperatures tend to be positively correlatedwith spring Oklahoma precipitation.
Posterior mean 95% HPD Interval
β0 119 (98.3, 146)β1 -2.86 (-3.19, -2.52)λl 7.6 (7.42, 7.74)λr 1.9 (0.134, 4.17)σ2α 0.311 (0.121, 0.537)σ2ε 0.00424 (0.00415, 0.00432)
Posterior parameter estimates generally appearconsistent with exploratory analysis.
Exploratory analysisData: (1981-2013) Average Oklahoma and sea surface temperatures in spring from ERA-Interim
Reanalysis data; PRISM precipitation (total rain and melted snow).
Time series plots of response variable over Oklahoma show no consistent, large temporaldependence. The time series for each spatial location is plotted in a different color.
0
100
200
1980 1990 2000 2010Time
Pre
cipi
tatio
n (m
m)
Temperature is approximately linearly related to precipitation at each location in Oklahoma. Spatiallocations are plotted in different colors and the linear relationship at each location is overlaid.
0
100
200
275 280 285 290 295Average Spring Temperature (K)
Pre
cipi
tatio
n (m
m)
20
40
−150 −120Longitude
Latit
ude
−0.3
−0.2
−0.1
0.0
0.1
0.2
Correlation
Pointwise correlations of sea surface temp. with average Oklahoma spring precipitation.
This correlation map plots the correlation between the overall average spring precipitation inOklahoma with sea surface temperature (SST) at remote locations in the Pacific Ocean. Thecorrelation map suggests clear, but weak teleconnection effects.
Parameter estimation• Estimate parameters with hybrid Gibbs algorithm
applied to the model marginalized over spatial effects.
– Adaptive random walk updates computed for:• β, log(σ2), logit(λ)
• Estimate teleconnection coefficients, α, by usingposterior parameter samples to draw (in parallel)composition samples of α.
• Marginal likelihood complexity:
– Likelihood evaluation in O(n3s ∨ n3
r) byapplying Kronecker product properties andSherman-Morrison-Woodbury formula.
Marginal model:(let Σy = σ2
y
(H(λl) + σ2
εIns
)and Rα = σ2
yσ2αR(λr)
)
Y =
Yt1...Ytnt
∼ N
Xt1
. . .Xtnt
β...β
,Σy
. . .Σy
+
(zTt1Rαzt1)Jns · · · (zTt1Rαztnt)Jns
...(zTtnt
Rαzt1)Jns · · · (zTtntRαztnt
)Jns
Composition-sample distribution:(let RT = R−1α +
(1Tns
Σ−1y 1ns
) (∑t∈T ztz
Tt
))α|Y ∼ N
(∑t∈T
(1Tns
Σ−1y (Yt −Xtβ))R−1T zt, R
−1T
)
Future work• If our preliminary results are validated, our
flexible model may produce more informativeand reliable inferences than classical tools forstudying teleconnections, such as correlationmaps.
• Improve estimation for σ2y.
• Allow α to vary locally as well as remotely.Challenge is to efficiently composition samplens × nr teleconnection coefficients.
ReferencesBanerjee, S., Carlin, B.P., and Gelfand, A.E., 2004,
Hierarchical Modeling and Analysis for Spatial Data:Boca Raton, FL, Chapman and Hall/CRC.
Givens, G.H., and Hoeting, J., 2013, ComputationalStatistics: Hoboken, NJ, John Wiley & Sons, Inc.
Towler, E., Pai Mazumder, D., and Holland, G. Aframework for investigating large-scale patterns as analternative to precipitation for downscaling to localdrought: Climate Dynamics (in review).
Acknowledgements
This material is based upon work supported by the National Science Foundation under Grant No. AGS - 1419558.
