![Page 1: A Bayesian hierarchical modeling approach to reconstructing past climates David Hirst Norwegian Computing Center](https://reader030.vdocuments.site/reader030/viewer/2022032703/56649d395503460f94a131ba/html5/thumbnails/1.jpg)
A Bayesian hierarchical modeling approach to
reconstructing past climates
David Hirst
Norwegian Computing Center
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Temperature data
• Many locations
• Direct measure of temperature
• Annual or better resolution
• small (known?) error
• Not too many missing values
• Short series
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Proxy data
• Long series
• Few (”strange”) locations• Relationship with temperature unclear, may
change over time• Often coarse resolution• Large (unknown) error• Lots of missing values• Pre-processing critical
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Current reconstruction methods:
1) Choose proxies
2) Create matrix X of pre-processed proxy by time
3) Create matrix Y of instrumental temperatures.
4) Relate X to Y (by PCA of one or both, then regression of X on Y or Y on X)
5) Use X to predict Y back in time
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Difficulties with existing methods:
• Missing data
• Spatial association between proxies and instruments lost
• PCA of proxy data dangerous
• Uncertainty in temperature data ignored
• Difficult to include proxies at different resolutions
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Consequences:
• Underestimation of past climate variability
• Wrong uncertainty
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An alternative approach
• Regard both instruments and proxies as observations of an underlying temperature process.
• Model all observations including appropriate error terms
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In general:
• Model temperature as an underlying space-time field
• Model data (proxies and thermometers) as observations of this field
• Use appropriate functional relationship between proxies and temperature
• Use appropriate error terms
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Specifically:
True temperature T(t) an AR(1) process:
21 ,~ TtTt TNT
Observations O = linear function of T plus AR(1) error E + measurement error
tititiiti ETO ,,,
For low resolution proxy replace T by mean over appropriate period
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A simulation study
• 50 years of thermometer data
• 250 years of proxies
• True temperature AR1, coefficient=0.95, sd =1
• 10 thermometers, small AR1 error (coef=0.7, sd=0.1)
• 5 proxies, (coef=0.7, sd=1)
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For comparison, regression estimator
• Find first pc of proxies
• Regress thermometer mean on pc
• predict ”temperature” (actually thermometer mean) using regression
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time before present
pro
xy v
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no.therm = 10no.prox = 5prox error sd = 1
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Point estimates
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Bayesian
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Regression
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Add uncertainty to proxies
• Only 2 proxies
• error sd = 2
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time before present
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true
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Point estimates
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rermse = 1.53coverage = 0.82int.width = 4.09
Bayesian
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rermse = 2.44coverage = 0.42int.width = 3.34
Regression
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The effect of missing data
• 5 proxies, error sd = 1
• 50% proxy data missing at random
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time before present
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50% missing
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Point estimates, 50% missing
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Point estimates
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Bayesian, 50% missing
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Bayesian
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Regression, 50% missing
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Regression
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Add a trend
• Only 150 years for proxies
• cosine trend, cycle 50 years, amplitude 4 (first 50 years) 8 (next 50) and 12 (last 50)
• AR1 model for temperature no longer correct
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time before present
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Point estimates
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Bayesian
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rermse = 1.06coverage = 0.54int.width = 1.74
Regression
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Add lots of ”bad” proxies
• 2 proxies linearly related to temperture
• 20 proxies unrelated to temperature
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Bayesianregressiontrue
no.therm = 10no.good prox = 2no.bad prox = 20prox error sd = 2
Point estimates
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Some data from China
• Two proxies used in Moberg et at 2005
• 10 closest instrumental data sets
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1000 1200 1400 1600 1800 2000
56
78
91
01
1
year
tem
pe
ratu
re
BeijingChina
Chinese Proxies
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1850 1900 1950 2000
51
01
5
year
tem
pe
ratu
re
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Instrumental Beijing China
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0 200 400 600 800 1000
-4-3
-2-1
01
time before present
tem
pe
ratu
re
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Modelling conclusions
• A flexible model which can take account of many sources of uncertainty
• Theoretically easy to include spatial correlations• Can include proxies at different resolutions• Missing data not a problem• Avoids underestimation of variability if model
correct• Functional form of temperature and error series
very important
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Other conclusions
• Impossible to work with proxies without help from appropriate scientists (preferably those who collected the data)
• Pre-processing crucial
• Selection of proxies important
• Some assumptions impossible to verify