Stat 598 Presentation on Nonseperable, Stationary Covariance Functions for Space-Time Data Xiongzhi Chen Department of Statistics November 16, 2010 Xiongzhi Chen (Department of Statistics) Stat 598 Presentation November 16, 2010 1 / 22
Outline Positive Definite Functions Main Mathematical Results Application to Irish Wind Data Why Such A Correlation Model Other Strategies and Wrong Claims Proofs Acknowledgements Xiongzhi Chen (Department of Statistics) Stat 598 Presentation November 16, 2010 2 / 22
Key Component of Spatial-Temporal Modelling Observations generated from a square-integrable stochastic process � � Z ( s ; t ) , ( s ; t ) ∈ R d × R Z = Process Z induces a family of functions � � C s , t : R d × R → R | ( s ; t ) ∈ R d × R C = , where C s , t ( h ; u ) = Cov (( Z ( s ; t ) , Z ( s + h ; t + u )) at each ( s ; t ) . Covariance stationarity is equivalent to C = { C ( h ; u ) } Then C ( · ; · ) is called the covariance function of Z. Xiongzhi Chen (Department of Statistics) Stat 598 Presentation November 16, 2010 3 / 22
Positive-Definite (PSD) Functions A function f : R d + 1 → K is positive definite iff k f ( w i − w j ) a i a ∗ ∑ j ≥ 0 i , j = 1 First equivalence f being a covariance function ⇔ f positive definite Second equivalence (Bochner’s Theorem) : A continuous function f on R d + 1 is positive definite iff � � √ � f ( w ) = − 1 � η , w � dF ( η ) exp (1) Equivalences and Spectral Representation Proof of Bochner’s Theorem Xiongzhi Chen (Department of Statistics) Stat 598 Presentation November 16, 2010 4 / 22
Construction of PSD Functions Main method: via Laplace transform and Bochner’s theorem . Cressie and Huang (1999) [1] based on closed-form F -inverse of � e − ih � ω C ( h ; u ) dh C ω ( u ) = (2) where C ( h ; u ) ∈ C b ∩ L 1 and C and C ω both are covariance functions or neither are (3) Obviously C ≡ C ω mod ( F ) but C might be un-usable. Completely Monotone Functions Xiongzhi Chen (Department of Statistics) Stat 598 Presentation November 16, 2010 5 / 22
Construction of PSD Functions Main method: via Laplace transform and Bochner’s theorem . Cressie and Huang (1999) [1] based on closed-form F -inverse of � e − ih � ω C ( h ; u ) dh C ω ( u ) = (2) where C ( h ; u ) ∈ C b ∩ L 1 and C and C ω both are covariance functions or neither are (3) Obviously C ≡ C ω mod ( F ) but C might be un-usable. Gneiting (2002) [2] (i) avoids explicit F - inverse of C ω and uses L -representation of completely monotone (cm) functions and the Bochner’s theorem; (ii) produces explicit product form of some PSD functions Completely Monotone Functions Xiongzhi Chen (Department of Statistics) Stat 598 Presentation November 16, 2010 5 / 22
Key Mathematical Result Theorem 2 (Gneiting (2002) [2]): Let k , l ∈ N + , σ 2 > 0 . Let ϕ ( t ) , t ≥ 0 be a cm function, and let ψ ( t ) , t ≥ 0, be a positive function with a cm derivative. Then � h � 2 σ 2 , ( h ; u ) ∈ R k × R l � � u � 2 � C ( h ; u ) = � u � 2 � k / 2 ϕ (4) � ψ ψ is a space-time covariance function. The family includes almost all examples given in Cressie and Huang (1999) [1] Xiongzhi Chen (Department of Statistics) Stat 598 Presentation November 16, 2010 6 / 22
Key Mathematical Result Theorem 2 (Gneiting (2002) [2]): Let k , l ∈ N + , σ 2 > 0 . Let ϕ ( t ) , t ≥ 0 be a cm function, and let ψ ( t ) , t ≥ 0, be a positive function with a cm derivative. Then � h � 2 σ 2 , ( h ; u ) ∈ R k × R l � � u � 2 � C ( h ; u ) = � u � 2 � k / 2 ϕ (4) � ψ ψ is a space-time covariance function. The family includes almost all examples given in Cressie and Huang (1999) [1] In application, ϕ , ψ can be associated with the data’s spatial and temporal structures, respectively Xiongzhi Chen (Department of Statistics) Stat 598 Presentation November 16, 2010 6 / 22
Key Mathematical Result Theorem 2 (Gneiting (2002) [2]): Let k , l ∈ N + , σ 2 > 0 . Let ϕ ( t ) , t ≥ 0 be a cm function, and let ψ ( t ) , t ≥ 0, be a positive function with a cm derivative. Then � h � 2 σ 2 , ( h ; u ) ∈ R k × R l � � u � 2 � C ( h ; u ) = � u � 2 � k / 2 ϕ (4) � ψ ψ is a space-time covariance function. The family includes almost all examples given in Cressie and Huang (1999) [1] In application, ϕ , ψ can be associated with the data’s spatial and temporal structures, respectively Note how the temporal correlation is "scaled" by spatial correlation Xiongzhi Chen (Department of Statistics) Stat 598 Presentation November 16, 2010 6 / 22
Examples of Building Functions ϕ 1 ( t ) = exp ( − ct γ ) c > 0 , γ ∈ ( 0 , 1 ] � � − 1 � ct 1 / 2 � ν K ν � ct 1 / 2 � 2 ν − 1 Γ ( ν ) ϕ 2 ( t ) = c > 0 , ν > 0 ϕ 3 ( t ) = ( 1 + ct γ ) − ν c > 0 , ν > 0 , γ ∈ ( 0 , 1 ] ϕ 4 ( t ) = 2 ν �� � ct 1 / 2 �� + exp � − ct 1 / 2 �� − ν c > 0 , ν > 0 exp ψ 1 ( t ) = ( at α + 1 ) β a > 0 , α ∈ ( 0 , 1 ] , β ∈ [ 0 , 1 ] ψ 1 ( t ) = ln ( at α + b ) / ln ( b ) a > 0 , b > 1 , α ∈ ( 0 , 1 ] ψ 1 ( t ) = ( at α + b ) / ( b ( at α + 1 )) a > 0 ; b , α ∈ ( 0 , 1 ] Table 1: Building Functions Fitted Model Why This Model Xiongzhi Chen (Department of Statistics) Stat 598 Presentation November 16, 2010 7 / 22
Correlation Model for Irish Wind Data For this data set, the estimated correlation model is � � − 1 . 901 | u | 1 . 544 + 1 ψ − 1 ˆ ( u ) = if h = 0 1 � � C ( h ; u | β ) = (5) − . 00134 � h � . 968 ˆ ψ 1 ( u ) exp otherwise ψ β / 2 ( u ) ˆ where ψ 1 and ϕ 1 in Table 1 are chosen and � h � ≤ 450 km , | u | ≤ 3 � � − 1 . 901 | u | 1 . 544 + 1 ψ − 1 ˆ ( u ) = , temporal correlation, estimated 1 from data Table of Functions Why This Model Xiongzhi Chen (Department of Statistics) Stat 598 Presentation November 16, 2010 8 / 22
Correlation Model for Irish Wind Data For this data set, the estimated correlation model is � � − 1 . 901 | u | 1 . 544 + 1 ψ − 1 ˆ ( u ) = if h = 0 1 � � C ( h ; u | β ) = (5) − . 00134 � h � . 968 ˆ ψ 1 ( u ) exp otherwise ψ β / 2 ( u ) ˆ where ψ 1 and ϕ 1 in Table 1 are chosen and � h � ≤ 450 km , | u | ≤ 3 � � − 1 . 901 | u | 1 . 544 + 1 ψ − 1 ˆ ( u ) = , temporal correlation, estimated 1 from data ϕ ( h ) = . 968 exp ( − . 00134 � h � ) , spatial correlation, borrowed from ˆ Haslett and Raftery (1989) ([4]) Table of Functions Why This Model Xiongzhi Chen (Department of Statistics) Stat 598 Presentation November 16, 2010 8 / 22
Correlation Model for Irish Wind Data For this data set, the estimated correlation model is � � − 1 . 901 | u | 1 . 544 + 1 ψ − 1 ˆ ( u ) = if h = 0 1 � � C ( h ; u | β ) = (5) − . 00134 � h � . 968 ˆ ψ 1 ( u ) exp otherwise ψ β / 2 ( u ) ˆ where ψ 1 and ϕ 1 in Table 1 are chosen and � h � ≤ 450 km , | u | ≤ 3 � � − 1 . 901 | u | 1 . 544 + 1 ψ − 1 ˆ ( u ) = , temporal correlation, estimated 1 from data ϕ ( h ) = . 968 exp ( − . 00134 � h � ) , spatial correlation, borrowed from ˆ Haslett and Raftery (1989) ([4]) ˆ β = 0 . 61, spatial-temporal interaction, estimated from data; γ = 1 / 2 in ϕ 1 was forced Table of Functions Why This Model Xiongzhi Chen (Department of Statistics) Stat 598 Presentation November 16, 2010 8 / 22
Why the Above Covariance Function? The fitted model is composed of Purely Spatial Correlation with nugget effect C ( h ; 0 | β ) = 1 if ϕ ∈ C ∞ h = 0 ; C ( h ; 0 | β ) = ˆ ϕ ( h ) , otherwise. Note ˆ Some reasons: Table of Functions Xiongzhi Chen (Department of Statistics) Stat 598 Presentation November 16, 2010 9 / 22
Why the Above Covariance Function? The fitted model is composed of Purely Spatial Correlation with nugget effect C ( h ; 0 | β ) = 1 if ϕ ∈ C ∞ h = 0 ; C ( h ; 0 | β ) = ˆ ϕ ( h ) , otherwise. Note ˆ ψ − 1 ∈ C ∞ Purely Temporal Correlation: C ( u ; 0 | β ) = ˆ ( u ) . Note ˆ ψ 1 / 1 Some reasons: Table of Functions Xiongzhi Chen (Department of Statistics) Stat 598 Presentation November 16, 2010 9 / 22
Why the Above Covariance Function? The fitted model is composed of Purely Spatial Correlation with nugget effect C ( h ; 0 | β ) = 1 if ϕ ∈ C ∞ h = 0 ; C ( h ; 0 | β ) = ˆ ϕ ( h ) , otherwise. Note ˆ ψ − 1 ∈ C ∞ Purely Temporal Correlation: C ( u ; 0 | β ) = ˆ ( u ) . Note ˆ ψ 1 / 1 Some reasons: Intrinsic : instrument variations and highly irregular wind speeds Table of Functions Xiongzhi Chen (Department of Statistics) Stat 598 Presentation November 16, 2010 9 / 22
Why the Above Covariance Function? The fitted model is composed of Purely Spatial Correlation with nugget effect C ( h ; 0 | β ) = 1 if ϕ ∈ C ∞ h = 0 ; C ( h ; 0 | β ) = ˆ ϕ ( h ) , otherwise. Note ˆ ψ − 1 ∈ C ∞ Purely Temporal Correlation: C ( u ; 0 | β ) = ˆ ( u ) . Note ˆ ψ 1 / 1 Some reasons: Intrinsic : instrument variations and highly irregular wind speeds Generic : evidence from data (see Graphs) Table of Functions Xiongzhi Chen (Department of Statistics) Stat 598 Presentation November 16, 2010 9 / 22
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