Post on 19-Dec-2015
A Bayesian hierarchical modeling approach to
reconstructing past climates
David Hirst
Norwegian Computing Center
Temperature data
• Many locations
• Direct measure of temperature
• Annual or better resolution
• small (known?) error
• Not too many missing values
• Short series
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
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
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
Consequences:
• Underestimation of past climate variability
• Wrong uncertainty
An alternative approach
• Regard both instruments and proxies as observations of an underlying temperature process.
• Model all observations including appropriate error terms
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
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
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)
For comparison, regression estimator
• Find first pc of proxies
• Regress thermometer mean on pc
• predict ”temperature” (actually thermometer mean) using regression
time before present
pro
xy v
alu
e
0 50 100 150 200 250
-50
51
0 true
no.therm = 10no.prox = 5prox error sd = 1
0 50 100 150 200 250
-6-4
-20
24
68
time before present
tem
pe
ratu
reBayesianregressiontrue
no.therm = 10no.prox = 5prox error sd = 1
Point estimates
0 50 100 150 200 250
-6-4
-20
24
68
time before present
tem
pe
ratu
rermse = 0.76coverage = 0.82int.width = 2.03
Bayesian
0 50 100 150 200 250
-6-4
-20
24
68
time before present
tem
pe
ratu
rermse = 1.26coverage = 0.38int.width = 1.36
Regression
Add uncertainty to proxies
• Only 2 proxies
• error sd = 2
time before present
pro
xy v
alu
e
0 50 100 150 200 250
-50
51
0
true
no.therm = 10no.prox = 2prox error sd = 2
0 50 100 150 200 250
-50
5
time before present
tem
pe
ratu
reBayesianregressiontrue
no.therm = 10no.prox = 2prox error sd = 2
Point estimates
0 50 100 150 200 250
-50
5
time before present
tem
pe
ratu
rermse = 1.53coverage = 0.82int.width = 4.09
Bayesian
0 50 100 150 200 250
-50
5
time before present
tem
pe
ratu
rermse = 2.44coverage = 0.42int.width = 3.34
Regression
The effect of missing data
• 5 proxies, error sd = 1
• 50% proxy data missing at random
time before present
pro
xy v
alu
e
0 50 100 150 200 250
-50
51
0 true
no.therm = 10no.prox = 5prox error sd = 1
50% missing
0 50 100 150 200 250
-6-4
-20
24
68
time before present
tem
pe
ratu
reBayesianregressiontrue
no.therm = 10no.prox = 5prox error sd = 1
Point estimates, 50% missing
0 50 100 150 200 250
-6-4
-20
24
68
time before present
tem
pe
ratu
reBayesianregressiontrue
no.therm = 10no.prox = 5prox error sd = 1
Point estimates
0 50 100 150 200 250
-6-4
-20
24
68
time before present
tem
pe
ratu
rermse = 0.81coverage = 0.9int.width = 2.6
Bayesian, 50% missing
0 50 100 150 200 250
-6-4
-20
24
68
time before present
tem
pe
ratu
rermse = 0.76coverage = 0.82int.width = 2.03
Bayesian
0 50 100 150 200 250
-6-4
-20
24
68
time before present
tem
pe
ratu
rermse = 1.61coverage = 0.56int.width = 2.67
Regression, 50% missing
0 50 100 150 200 250
-6-4
-20
24
68
time before present
tem
pe
ratu
rermse = 1.26coverage = 0.38int.width = 1.36
Regression
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
time before present
pro
xy v
alu
e
0 50 100 150
-50
5
true
no.therm = 10no.prox = 5prox error sd = 1
0 50 100 150
-6-4
-20
24
time before present
tem
pe
ratu
reBayesianregressiontrue
no.therm = 10no.prox = 5prox error sd = 1
Point estimates
0 50 100 150
-6-4
-20
24
time before present
tem
pe
ratu
rermse = 0.79coverage = 0.81int.width = 1.98
Bayesian
0 50 100 150
-6-4
-20
24
time before present
tem
pe
ratu
rermse = 1.06coverage = 0.54int.width = 1.74
Regression
Add lots of ”bad” proxies
• 2 proxies linearly related to temperture
• 20 proxies unrelated to temperature
0 50 100 150 200 250
-6-4
-20
24
68
time before present
tem
pe
ratu
re
Bayesianregressiontrue
no.therm = 10no.good prox = 2no.bad prox = 20prox error sd = 2
Point estimates
Some data from China
• Two proxies used in Moberg et at 2005
• 10 closest instrumental data sets
1000 1200 1400 1600 1800 2000
56
78
91
01
1
year
tem
pe
ratu
re
BeijingChina
Chinese Proxies
1850 1900 1950 2000
51
01
5
year
tem
pe
ratu
re
Instrumental Beijing China
0 200 400 600 800 1000
-4-3
-2-1
01
time before present
tem
pe
ratu
re
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
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