Some Statistical Problems in Climate Reconstruction Dan Cervone - PowerPoint PPT Presentation

Some Statistical Problems in Climate Reconstruction Dan Cervone April 15, 2014 Dan Cervone () STAT 300: Research in Statistics April 15, 2014

Historical Global Temperature Reconstruction Data: CRUTEMv3 Northern hemisphere temperature anomolies 1.0 Temp ( ° C) 0.0 −1.0 1850 1900 1950 2000 Year Dan Cervone () STAT 300: Research in Statistics April 15, 2014

Historical Global Temperature Reconstruction Data: CRUTEMv3 Northern hemisphere temperature anomolies 1.0 Temp ( ° C) 0.0 −1.0 1850 1900 1950 2000 Year Dan Cervone () STAT 300: Research in Statistics April 15, 2014

Historical Global Temperature Reconstruction Data: CRUTEMv3 Northern hemisphere temperature anomolies 1.0 Temp ( ° C) 0.0 −1.0 1850 1900 1950 2000 Year Northern hemisphere temperature sites 250 Number of sites 150 50 1850 1900 1950 2000 Year Dan Cervone () STAT 300: Research in Statistics April 15, 2014

Historical Global Temperature Reconstruction Data: CRUTEMv3 Northern hemisphere temperature anomolies 1.0 Temp ( ° C) 0.0 −1.0 1850 1900 1950 2000 Year What is the estimand? Interpolate gaps in Northern hemisphere temperature sites observational record 250 Number of sites Extrapolate before 1850 150 50 1850 1900 1950 2000 Year Dan Cervone () STAT 300: Research in Statistics April 15, 2014

Historical Global Temperature Reconstruction Data: CRUTEMv3 Northern hemisphere temperature anomolies 1.0 Temp ( ° C) 0.0 −1.0 1850 1900 1950 2000 Year What is the estimand? Interpolate gaps in Northern hemisphere temperature sites observational record 250 Number of sites Extrapolate before 1850 150 1.0 0.0 50 −1.0 1400 1500 1600 1700 1800 1900 2000 1850 1900 1950 2000 Year Dan Cervone () STAT 300: Research in Statistics April 15, 2014

All the moments each moment Image: NASA/GISS Dan Cervone () STAT 300: Research in Statistics April 15, 2014

[image source: wikimedia commons] Proxies: 18 O / 16 O, ocean sediment Ice cores Varves (rock sediment) Tree rings Dan Cervone () STAT 300: Research in Statistics April 15, 2014

Example: BARCAST Tingley & Huybers 2010, 2013 Spatiotemporal temperature reconstruction using temperature record and proxies: � T o,t � � S o � T t = at locations S = T p,t S p Dan Cervone () STAT 300: Research in Statistics April 15, 2014

Example: BARCAST Tingley & Huybers 2010, 2013 Spatiotemporal temperature reconstruction using temperature record and proxies: � T o,t � � S o � T t = at locations S = T p,t S p T o are temperatures at locations of temperature records S o . T p are temperatures at locations of proxy records S p . Dan Cervone () STAT 300: Research in Statistics April 15, 2014

Example: BARCAST Tingley & Huybers 2010, 2013 Spatiotemporal temperature reconstruction using temperature record and proxies: � T o,t � � S o � T t = at locations S = T p,t S p T o are temperatures at locations of temperature records S o . T p are temperatures at locations of proxy records S p . With t indexing years, T t − µ 1 = α ( T t − 1 − µ 1 ) + ǫ t iid ∼ N ( 0 , K ( S , S )) ǫ t K ( s , s ∗ ) = τ 2 exp( − γ || s − s ∗ || 2 ) Dan Cervone () STAT 300: Research in Statistics April 15, 2014

Example: BARCAST Tingley & Huybers 2010, 2013 “Errors in variables”: True temperatures T are not observed. Measurement error for temperature sites W o,t ∼ N ( T o,t , σ 2 o I ). Linear model for proxies W p,t ∼ N ( µ p 1 + T p,t β p , σ 2 p I ). Dan Cervone () STAT 300: Research in Statistics April 15, 2014

Example: BARCAST Tingley & Huybers 2010, 2013 “Errors in variables”: True temperatures T are not observed. Measurement error for temperature sites W o,t ∼ N ( T o,t , σ 2 o I ). Linear model for proxies W p,t ∼ N ( µ p 1 + T p,t β p , σ 2 p I ). ( W T ) ′ is just a huge multivariate normal! Dan Cervone () STAT 300: Research in Statistics April 15, 2014

Example: BARCAST Tingley & Huybers 2010, 2013 “Errors in variables”: True temperatures T are not observed. Measurement error for temperature sites W o,t ∼ N ( T o,t , σ 2 o I ). Linear model for proxies W p,t ∼ N ( µ p 1 + T p,t β p , σ 2 p I ). ( W T ) ′ is just a huge multivariate normal! Inference with Gibbs sampling or EM: Update latent T . Update parameters τ 2 , γ, µ, α, µ p , β p , σ 2 o , σ 2 p . Dan Cervone () STAT 300: Research in Statistics April 15, 2014

Example: BARCAST Tingley & Huybers 2010, 2013 Difficulties: Spatiotemporal nonstationarity and anisotropy. Dan Cervone () STAT 300: Research in Statistics April 15, 2014

Example: BARCAST Tingley & Huybers 2010, 2013 Difficulties: Spatiotemporal nonstationarity and anisotropy. Model inhomogeneity. Dan Cervone () STAT 300: Research in Statistics April 15, 2014

Example: BARCAST Tingley & Huybers 2010, 2013 Difficulties: Spatiotemporal nonstationarity and anisotropy. Model inhomogeneity. Uncertainty in spatial referencing. Dan Cervone () STAT 300: Research in Statistics April 15, 2014

Example: BARCAST Tingley & Huybers 2010, 2013 Difficulties: Spatiotemporal nonstationarity and anisotropy. Model inhomogeneity. Uncertainty in spatial referencing. ... Dan Cervone () STAT 300: Research in Statistics April 15, 2014

Location uncertainty temperature site ice core tree ring varve Tree locations uncertain for many older specimens Ice cores subject to glacial flow Dan Cervone () STAT 300: Research in Statistics April 15, 2014

Gaussian Processes For s ∈ S , X ( s ) ∼ GP (0 , K ( s , s )) means for any s 1 , . . . s p ∈ S ,       X ( s 1 ) K ( s 1 , s 1 ) . . . K ( s 1 , s p ) . . ... . .  ∼ N  0 ,  ,       . .    X ( s p ) K ( s p , s 1 ) K ( s p , s p ) K ( , ) is a covariance function, e.g. K ( s , s ∗ ) = τ 2 exp( − γ || s − s ∗ || 2 ). Dan Cervone () STAT 300: Research in Statistics April 15, 2014

Gaussian Processes For s ∈ S , X ( s ) ∼ GP (0 , K ( s , s )) means for any s 1 , . . . s p ∈ S ,       X ( s 1 ) K ( s 1 , s 1 ) . . . K ( s 1 , s p ) . . ... . .  ∼ N  0 ,  ,       . .    X ( s p ) K ( s p , s 1 ) K ( s p , s p ) K ( , ) is a covariance function, e.g. K ( s , s ∗ ) = τ 2 exp( − γ || s − s ∗ || 2 ). Interpolation of X at unobserved location s ∗ Dan Cervone () STAT 300: Research in Statistics April 15, 2014

Gaussian Processes For s ∈ S , X ( s ) ∼ GP (0 , K ( s , s )) means for any s 1 , . . . s p ∈ S ,       X ( s 1 ) K ( s 1 , s 1 ) . . . K ( s 1 , s p ) . . ... . .  ∼ N  0 ,  ,       . .    X ( s p ) K ( s p , s 1 ) K ( s p , s p ) K ( , ) is a covariance function, e.g. K ( s , s ∗ ) = τ 2 exp( − γ || s − s ∗ || 2 ). Interpolation of X at unobserved location s ∗ X ( s ∗ ) | X ( s ) ∼ N ( K ( s ∗ , s ) K ( s , s ) − 1 X ( s ) , K ( s ∗ , s ∗ ) − K ( s ∗ , s ) K ( s , s ) − 1 K ( s , s ∗ )) Dan Cervone () STAT 300: Research in Statistics April 15, 2014

Gaussian Processes For s ∈ S , X ( s ) ∼ GP (0 , K ( s , s )) means for any s 1 , . . . s p ∈ S ,       X ( s 1 ) K ( s 1 , s 1 ) . . . K ( s 1 , s p ) . . ... . .  ∼ N  0 ,  ,       . .    X ( s p ) K ( s p , s 1 ) K ( s p , s p ) K ( , ) is a covariance function, e.g. K ( s , s ∗ ) = τ 2 exp( − γ || s − s ∗ || 2 ). Interpolation of X at unobserved location s ∗ X ( s ∗ ) | X ( s ) ∼ N ( K ( s ∗ , s ) K ( s , s ) − 1 X ( s ) , K ( s ∗ , s ∗ ) − K ( s ∗ , s ) K ( s , s ) − 1 K ( s , s ∗ )) Kriging: BLUP without normality assumption Dan Cervone () STAT 300: Research in Statistics April 15, 2014

GP interpolation with location error Errors in variables: X ( s ∗ ) | X ( s ) ∼ N ( b ′ X ( s ) , v 2 ) Observe ˜ X ( s ) = X ( s ) + ǫ where ǫ ⊥ X ( s ). Dan Cervone () STAT 300: Research in Statistics April 15, 2014

GP interpolation with location error Errors in variables: X ( s ∗ ) | X ( s ) ∼ N ( b ′ X ( s ) , v 2 ) Observe ˜ X ( s ) = X ( s ) + ǫ where ǫ ⊥ X ( s ). b ′ ˜ Still a regression problem: X ( s ∗ ) | ˜ X ( s ) ∼ N (˜ v 2 ) X ( s ) , ˜ Dan Cervone () STAT 300: Research in Statistics April 15, 2014

GP interpolation with location error Errors in variables: X ( s ∗ ) | X ( s ) ∼ N ( b ′ X ( s ) , v 2 ) Observe ˜ X ( s ) = X ( s ) + ǫ where ǫ ⊥ X ( s ). b ′ ˜ Still a regression problem: X ( s ∗ ) | ˜ X ( s ) ∼ N (˜ v 2 ) X ( s ) , ˜ Berkson errors: ǫ ⊥ ˜ X ( s ) (not satisfied) Dan Cervone () STAT 300: Research in Statistics April 15, 2014

GP interpolation with location error Errors in variables: X ( s ∗ ) | X ( s ) ∼ N ( b ′ X ( s ) , v 2 ) Observe ˜ X ( s ) = X ( s ) + ǫ where ǫ ⊥ X ( s ). b ′ ˜ Still a regression problem: X ( s ∗ ) | ˜ X ( s ) ∼ N (˜ v 2 ) X ( s ) , ˜ Berkson errors: ǫ ⊥ ˜ X ( s ) (not satisfied) Is i.i.d. error in s just i.i.d. error in X ( s )? X ( s + u ) = X ( s ) + ǫ Dan Cervone () STAT 300: Research in Statistics April 15, 2014