Covariances for four reference points t + 2 s ,t + m s ,t M - PowerPoint PPT Presentation

Recent, current, and future issues for large scale space-time and nonlinear INLA Finn Lindgren ( finn.lindgren@ed.ac.uk ) Avignon 2018-11-08

GMRFs based on SPDEs (Lindgren et al., 2011) GMRF representations of SPDEs can be constructed for oscillating, anisotropic, non-stationary, non-separable spatio-temporal, and multivariate fields on manifolds. ( κ 2 − ∆)( τ x ( s )) = W ( s ) , s ∈ R d

GMRFs based on SPDEs (Lindgren et al., 2011) GMRF representations of SPDEs can be constructed for oscillating, anisotropic, non-stationary, non-separable spatio-temporal, and multivariate fields on manifolds. ( κ 2 − ∆)( τ x ( s )) = W ( s ) , s ∈ Ω

GMRFs based on SPDEs (Lindgren et al., 2011) GMRF representations of SPDEs can be constructed for oscillating, anisotropic, non-stationary, non-separable spatio-temporal, and multivariate fields on manifolds. � ∂ � ∂t + κ 2 s ,t + ∇ · m s ,t − ∇ · M s ,t ∇ ( τ s ,t x ( s , t )) = E ( s , t ) , ( s , t ) ∈ Ω × R

Covariances for four reference points � ∂ � ∂t + κ 2 s ,t + ∇ · m s ,t − ∇ · M s ,t ∇ ( τ s ,t x ( s , t )) = E ( s , t ) , ( s , t ) ∈ Ω × R

Stochastic Green’s first identity On any sufficiently smooth manifold domain D , � f, −∇ · ∇ g � D = �∇ f, ∇ g � D − � f, ∂ n g � ∂D holds, even if either ∇ f or −∇ · ∇ g are as generalised as white noise. For α = 2 in the Matérn SPDE, �� � � � � ψ i , ( κ 2 − ∇ · ∇ ) � �� � κ 2 � ψ i , ψ j � D + �∇ ψ i , ∇ ψ j � D x j j ψ j x j = j D = ( κ 2 C + G ) x The covariance for the RHS of the SPDE is � Cov ( � ψ i , W� D , � ψ j , W� D � � � ψ i , ψ j � D � = = C by the definition of W . Matching the LHS and RHS distributions leads to the finite element approximation x ∼ N ( 0 , Q = κ 4 C + 2 κ 2 G + GC − 1 G )

EUSTACE EU Surface Temperatures for All Corners of Earth EUSTACE will give publicly available daily estimates of surface air temperature since 1850 across the globe for the first time by combining surface and satellite data using novel statistical techniques.

Matérn driven heat equation on the sphere The iterated heat equation is a simple non-separable space-time SPDE family: � � β φ ∂ ( κ 2 − ∆) γ/ 2 ∂t + ( κ 2 − ∆) α/ 2 x ( s , t ) = W ( s , t ) /τ Fourier spectra are based on eigenfunctions e ω ( s ) of − ∆ . On R 2 , − ∆ e ω ( s ) = � ω � 2 e ω ( s ) , and e ω are harmonic functions. On S 2 , − ∆ e k ( s ) = λ k e k ( s ) = k ( k + 1) e k ( s ) , and e k are spherical harmonics. The isotropic spectrum on S 2 × R is 2 k + 1 � R ( k, ω ) ∝ τ 2 ( κ 2 + λ k ) γ [ φ 2 ω 2 + ( κ 2 + λ k ) α ] β The finite element approximation has precision matrix structure α + β + γ � M [ t ] i ⊗ M [ s ] Q = i i =0 even, e.g., if κ is spatially varying.

Partial hierarchical representation Observations of mean , max , and min . Model mean and range . Q 2 Q 1 Q 0 Q 0 Q 1 Q 2 m m m r r r T 2 T 1 T 0 T 0 T 1 T 2 m m m r r r y Q β m Q β r β m β r ǫ 0 ǫ 1 ǫ 2 Q 0 Q 1 Q 2 ǫ ǫ ǫ Data sources Conditional specifications, e.g. � − 1 � ( T 0 m | T 1 m , Q 0 T 1 m , Q 0 m ) ∼ N m

Observation models Common satellite derived data error model framework The observational&calibration errors are modelled as three error components: independent ( ǫ 0 ), spatially correlated ( ǫ 1 ), and systematic ( ǫ 2 ), with distributions determined by the uncertainty information from WP1 E.g., y i = T m ( s i , t i ) + ǫ 0 ( s i , t i ) + ǫ 1 ( s i , t i ) + ǫ 2 ( s i , t i ) Station homogenisation For station k at day t i J k � y k,i H k j ( t i ) e k,j m + ǫ k,i m = T m ( s k , t i ) + m , j =1 where H k j ( t ) are temporal step functions, e k,j m are latent bias variables, and ǫ k,i m are independent measurement and discretisation errors.

Observed data Observed daily T mean and T range for station FRW00034051 FRW00034051 20 10 Tmean 0 −10 1955 1960 1965 time (year) FRW00034051 20 15 Trange 10 5 0 1955 1960 1965 time (year)

Power tail quantile (POQ) model The quantile function (inverse cumulative distribution function) F − 1 θ ( p ) , p ∈ [0 , 1] , is defined through a quantile blend of generalised Pareto distributions: � 1 − (2 p ) − θ , θ � = 0 , f − 2 θ θ ( p ) = 1 2 log(2 p ) , θ = 0 , � (2(1 − p )) − θ − 1 θ � = 0 , , f + θ ( p ) = − f − 2 θ θ (1 − p ) = − 1 2 log(2(1 − p )) , θ = 0 . � � θ ( p ) = θ 0 + τ F − 1 (1 − γ ) f − θ 3 ( p ) + (1 + γ ) f + θ 4 ( p ) , 2 The parameters θ = ( θ 0 , θ 1 = log τ, θ 2 = logit[( γ + 1) / 2] , θ 3 , θ 4 ) control the median, spread/scale, skewness, and the left and right tail shape. This model is also known as the five parameter lambda model . A spatio-temporally dependent Gaussian field u ( s , t ) with expectation 0 and variance 1 can be transformed into a POQ field by u ( s , t ) = F − 1 � θ ( s ,t ) (Φ( u ( s , t )) , where the parameters can vary with space and time.