Latent Gaussian Processes and Stochastic Partial Differential Equations

Transcription

1 Latent Gaussian Processes and Stochastic Partial Differential Equations Johan Lindström 1 1 Centre for Mathematical Sciences Lund University Lund Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 1/58

2 Interpolation Examples Kriging Big N Alternatives Spatial interpolation Given observations at some locations, Y(s i ), i = 1... n we want to make statements about the value at unobserved location(s), X(s). The general model formulation is Y(s i ) X, θ F(X(s i ), θ) X θ Spatial field(θ) often we assume X to be a Gaussian field. Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 2/58

3 Interpolation Examples Kriging Big N Alternatives Temperatures fall of Residuals Regression (~long+lat+elev) Reconstruction Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 3/58

4 Interpolation Examples Kriging Big N Alternatives Parana Rainfall Non-Gaussian data January Precip. January log(precip.) Gamma Y Normal log(y) Y(s i ) X, θ Γ (a, a exp( X(s i ))) E(Y(s i ) X) = exp(x(s i )) X θ N (μ(θ), Σ(θ)) Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 4/58

5 Interpolation Examples Kriging Big N Alternatives Larynx Cancer Count data Counts of Larynx Cancer Population (truncated) Y(s i ) X, θ Po (E i exp(x(s i ))) E(Y(s i ) X) = E i exp(x(s i )) X θ N (μ(θ), Σ(θ)) Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 5/58

6 Interpolation Examples Kriging Big N Alternatives Kriging (Krige, 1951; Matheron, 1969) In the simplest case we assume a Gaussian model for the observations, Y(s i ) N ( X(s i ), σ 2 ) ε giving a joint Gaussian density for the latent field and observations [ ([ ] [ ]) Y μy Σyy Σ yx N, X] μ x Σ, yx Σ xx with some parametric form for the covariance matrix and mean Y N (μ(θ), Σ(θ)). Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 6/58

7 Interpolation Examples Kriging Big N Alternatives Covariance functions We often need to estimate the N N covariance matrix Σ based on one sample (N observations) of the field. This implies estimating N(N + 1)/2 unknowns from N observations... The requirements (positive definite) on Σ implies restrictions on the estimation. The covariance matrix is often assumed to come from a parametric family of covariance functions. Reducing the problem to estimation of the covariance parameters. The resulting estimation problem does not have a closed form solution, leading to numerical optimisation. Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 7/58

8 Interpolation Examples Kriging Big N Alternatives Matérn: r(h) = Exponential: Matérn with ν = 1/2 Gaussian: Matérn with ν Cauchy: σ 2 Γ(ν) 2 ν 1 (κ h )ν K ν (κ h ), r(h) = σ 2 exp( κ h ) r(h) = σ 2 exp( 2 h 2 /ρ 2 ) r(h) = σ 2 (1 + ( h /ρ) 2) κ Spherical: r(h) = σ 2 { ( h /ρ) ( h /ρ) 3, h ρ 0, h > ρ Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 8/58

9 Interpolation Examples Kriging Big N Alternatives Examples of Covariance functions Spherical Gaussian Exponential Matern, k=1 Matern, k=3 Cauchy, k=1 Cauchy, k= distance Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 9/58

10 Interpolation Examples Kriging Big N Alternatives The Big N problem The log-likelihood becomes l (θ Y) = 1 2 log Σ(θ) 1 2 ( Σ(θ) 1 Y μ(θ)) ( ) Y μ(θ). Given (estimated) parameters, predictions at the unobserved locations are given by ( E X Y, θ ) = μ x + Σ xy Σ 1 yy (Y μ y ). The Big N problem Given N observations: The covariance matrix has O ( N 2) unique elements. Computations scale as O ( N 3) (due to Σ and Σ 1 ). Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 10/58

11 Interpolation Examples Kriging Big N Alternatives Getting around Big N Spectral representation (Whittle, 1954; Fuentes, 2007) Uses discrete Fourier transforms; limited to regular lattices. Covariance tapering (Furrer et al., 2006; Kaufman et al., 2008) Set small values in the covariance matrix to zeros. Likelihood approximation Various statistical or numerical approximation methods By sequential matrix approx. (Stein et al., 2004) Block composite likelihoods (Eidsvik et al., 2014) Krylov based numerical approx. (Stein et al., 2013; Aune et al., 2014) Low rank approximations Exact computations on a simplified model of reduced rank/size Predictive processes (Banerjee et al., 2008; Eidsvik et al., 2012) Fixed rank kriging (Cressie and Johannesson, 2008) Process convolution or kernel methods (Higdon, 2001) Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 11/58

12 Markov Precision Computations Gaussian Markov random fields (Rue and Held, 2005) Let the neighbours N i to a point s i be the points {s j, j N i } that are close to s i. Gaussian Markov random field (GMRF) A Gaussian random field x N (μ, Σ) that satisfies p ( x i {x j : j i} ) = p ( x i {x j : j N i } ) is a Gaussian Markov random field. The simplest example of a GMRF is the AR(1)-process x t = ax t 1 + ε t, ε t N(0, σ 2 ) and independent. x t 2 x t 1 x t x t+1 x t+2 N t Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 12/58

13 Markov Precision Computations Good neighbours Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 13/58

14 Markov Precision Computations Good neighbours Regional data Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 14/58

15 Markov Precision Computations Let me introduce: The precision matrix Q = Σ 1 Using the precision matrix the model becomes ( X N μ, Q 1) cf. X N (μ, Σ) With conditional expectation instead of E (X Y) = μ x Q 1 xx Q xy (Y μ y ). E (X Y) = μ x + Σ xy Σ 1 yy (Y μ y ). Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 15/58

16 Markov Precision Computations The conditional expectation for a single location is E ( x xj i, j i ) j i = μ i Q ij(x j μ j ) Q ii = μ i 1 Q ij (x j μ j ) + Q ij (x j μ j ) Q ii j Ni j / {N i,i} If Q ij = 0 for all j / N i then E ( x i x j, j i ) = E ( x i x j, j N i ). The precision matrix is sparse Elements in the precision matrix of a Gaussin Markov random field are non-zero only for neighbours and diagonal elements. j / {i, N i } Q ij = 0. Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 16/58

17 Markov Precision Computations Computational details If Q is a sparse matrix then (under mild conditions) the Cholesky factorisation Q = R R will also be sparse. ( Simulation of X N μ, Q 1). Conditional expectations X = μ + R 1 E, E N (0, I) ( E (X Y) = μ x R 1 xx Computing the determinant R xx ( )) Q xy (Y μ y ) 1 2 log Q = 1 R 2 log R = log R = log R ii i Never compute R 1, use back substitution for triangular systems instead. Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 17/58

18 Markov Precision Computations Computational costs For an AR(1) like-process the number of neighbours will be 2d for a grid in Z d. Matrix Elements Cholesky Σ O ( N 2) O ( N 3) 1D O (N) O (N) Q 2D O (N) O ( N 1.5) 3D O (N) O ( N 2) Reordering the precision matrix gives a sparser Cholesky factorisation. Some elements of Q 1 can be computed cheaply using the multifrontal approach (Campbell and Davis, 1995) Q 1 ij can be computed if R ij 0. Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 18/58

19 Markov Precision Computations Reordering Q chol(q) nz = nz = Q reordered chol(q) reordered nz = nz = Johan Lindstro m - johanl@maths.lth.se Gaussian Markov Random Fields 19/58

20 CAR Matérn SPDE Basis functions Lattice Introducing q For stationary fields the local q-pattern is used to define Q: q = Q,i = Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 20/58

21 CAR Matérn SPDE Basis functions Lattice CAR models (Whittle, 1954; Besag, 1974) In order to use GMRFs insead of full covariance models, we need to construct useful, sparse, Q-matrices. Conditional autoregressive models (CAR) A mean zero CAR(1) models is defined by x i {x j : j N i } N 1 1 κ 2 x j, + N i τ(κ 2 + N i ) j N i Remember x i {x j : j i} N 1 Q ij x j, Q 1 Q ii j i ii Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 21/58

22 CAR Matérn SPDE Basis functions Lattice Precision Matrix for CAR(1) The CAR(1) precision is constructed as κ 2 + N i if i = j, Q ij = τ 1 if j N i, 0 if j / N i. Often written as Q = τ ( κ 2 I + G ) Where G is a neighbourhood matrix. Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 22/58

23 CAR Matérn SPDE Basis functions Lattice CAR(1) Precision Example Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 23/58

24 CAR Matérn SPDE Basis functions Lattice Precision matrix for a non-gridded data Example Q = τ ( κ 2 I + G ) G = Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 24/58

25 CAR Matérn SPDE Basis functions Lattice CAR(1) models For observations on a regular grid (Z 2 ) the resulting local q-pattern (with pixel x i marked in red) is q = τ κ With special treatment of the edges 1 0 q edge = τ 3 + κ [ ] 2 + κ 2 1 q corner = τ 1 0 Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 25/58

26 CAR Matérn SPDE Basis functions Lattice Precision matrix for a CAR(1) The q-pattern can be divided into a component containing the parameter, κ 2, and a 2 nd order finite difference operator q = τ κ 2 ( Q = τ κ 2 I + G }{{}}{{} (I) Δ (G) ) How do we change the parameters if we change the spatial resolution? (Besag and Kooperberg, 1995; Besag and Mondal, 2005). Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 26/58

27 CAR Matérn SPDE Basis functions Lattice How to create Q? The Matérn covariance family (Matérn, 1960) The covariance between two points at distance h is r( h ) = σ 2 2 ν 1 Γ(ν) (κ h )ν K ν (κ h ), h R d Fields with Matérn covariances are solutions to a Stochastic Partial Differential Equation (SPDE) (Whittle, 1954, 1963), ( κ 2 Δ ) α/2 x(s) = W(s). Here W(s) is white noise, Δ = i 2, and α = ν + d/2. s 2 i Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 27/58

28 CAR Matérn SPDE Basis functions Lattice The link between the SPDE and the Matérn covariances is proved by (Whittle, 1954, 1963) using a spectral argument on Z d. Spectral Density For a stochastic process in time, x(t) with stationary covariance function, r(t) the spectral density and covariance function form a Fourier transform pair f(ω) = 1 2π r(t) = r(t)e iωt dt = Fr f(ω)e iωt dω = F 1 f Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 28/58

29 CAR Matérn SPDE Basis functions Lattice Linear Filters Applying a linear filter with impulse response, h(u), to a Gaussian process results in a transformed Gaussian process y(t) = with spectral density h(t u)x(u) du = where H(ω) is the transfer function f y (ω) = H(ω) 2 f x (ω) H(ω) = h(u)e iωu du h(u)x(t u) du, Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 29/58

30 CAR Matérn SPDE Basis functions Lattice Spectral Density For a stochastic field, x(s), in R d with stationary covariance function, r(h) the spectral density is f(ω) = 1 r(h)e iω h dh, (2π) d R d r(h) = f(ω)e iω h dω R d Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 30/58

31 CAR Matérn SPDE Basis functions Lattice Spectral Density Isotropy For an isotropic stationary covariance function in R 2 a change to polar coordinates gives the spectral density as f(ω) = 1 (2π) 2 = 1 2π 0 0 r(h) 2π 0 r(h) h J 0 (ωh) dh, e iωh cos θ dθ h dh where ω = ω, h = h, and J 0 is a Bessel function of the first kind. Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 31/58

32 CAR Matérn SPDE Basis functions Lattice Spectral Density Isotropy For a stochastic field, x(s), in R d with isotropic stationary covariance function, r(h) the spectral density is f(ω) = 1 (2π) d/2 0 r(h) h d 1 J (d 2)/2(ωh) (ωh) (d 2)/2 dh, r(h) = (2π) d/2 0 f(ω) ω d 1 J (d 2)/2(ωh) (ωh) (d 2)/2 where J k is a Bessel function of the first kind. dω, Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 32/58

33 CAR Matérn SPDE Basis functions Lattice Spectral Density Matérn The Matérn covariance family The covariance between two points at distance h is r M (h) = σ 2 Γ(ν) 2 ν 1 (κ h )ν K ν (κ h ) has spectral density (in R 2 ) f M (ω) = σ 2 2πΓ(ν) 2 ν 1 = νσ2 κ 2ν π 0 1 (κ 2 + ω 2 ) ν+1 (κh) ν K ν (κh) h J 0 (wh) dh Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 33/58

34 CAR Matérn SPDE Basis functions Lattice Matérn covariances Using the spectral representation, fields with Matérn covariances are solutions to a Stochastic Partial Differential Equation (SPDE), ( κ 2 Δ ) α/2 x(s) = 1 τ W(s). Here W(s) is spatial white noise. Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 34/58

35 CAR Matérn SPDE Basis functions Lattice Matérn covariances from an SPDE If we consider the SPDE as a linear filter we can write ( κ 2 Δ ) α/2 x(s) = 1 τ W(s). with transfer function ( (κ H(ω) = F 2 Δ ) ) α/2 ( κ 2 + ω 2) α/2 since F (f ) = iωf(f) F (f ) = ω 2 F(f) Proportional to since we re ignoring 2π-constants in the Fourier-transform Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 35/58

36 CAR Matérn SPDE Basis functions Lattice Matérn covariances from an SPDE If the SPDE ( κ 2 Δ ) α/2 x(s) = 1 τ W(s). is driven by spatial white noise, W(s), with f W (ω) 1 τ and r W (h) = 1 τ δ 0(h) then the resulting spectral density for x(s) is Matérn. H(ω) 2 f x (ω) = f W (ω) f x (ω) = H(ω) 2 f W (ω) 1 τ 1 (κ 2 + ω 2 ) α Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 36/58

37 CAR Matérn SPDE Basis functions Lattice Does the Matérn covariance produce Markov fields? The Matérn covariance has spectral density 1 f(ω) (κ 2 + ω 2 cf. ) α ( κ 2 Δ ) α/2 x(s) = W(s). Spectral density for Markov fields According to Rozanov (1977) a stationary field is Markov if and only if the spectral density is a reciprocal of a polynomial. For the SPDE this implies α Z (or ν Z for R 2 ). Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 37/58

38 CAR Matérn SPDE Basis functions Lattice Basic idea Construct a discrete approximation of the continious field using basis functions, {ψ k }, and weights, {w k }, x(s) = k ψ k (s)w k Find the distribution of w k by solving ( κ 2 Δ ) α/2 x(s) = W(s) Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 38/58

39 CAR Matérn SPDE Basis functions Lattice Solving the SPDE A stochastic weak solution to the SPDE is given by [ φ k, ( κ 2 Δ ) ] α/2 D x = [ φ k, W ] k=1,...,n for each set of test functions {φ k } Replacing x with k ψ kw k gives k=1,...,n [ φ i, ( κ 2 Δ ) α/2 ψj ] Study the case α = 2 and φ i = ψ i (Galerkin) i,j w D = [ φ k, W ] k κ 2 [ ] ψ i, ψ j + [ ] ψ i, Δψ j w = D [ ψ k, W ] }{{}}{{}}{{} C G N(0,C) Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 39/58

40 CAR Matérn SPDE Basis functions Lattice Solution to the SPDE A weak solution to the SPDE ( κ 2 Δ ) x(s) = W(s). is given by x(s) = k ψ k (s)w k where ( κ 2 C + G ) w N (0, C) The precision of the weights, w, is V (w) 1 = Q 2 = ( κ 2 C + G ) C 1 ( κ 2 C + G ) Q 1 =κ 2 C + G Q 2 = ( κ 2 C + G ) C 1 ( κ 2 C + G ) Q α = ( κ 2 C + G ) C 1 Q α 2 C 1( κ 2 C + G ), α = 3, 4, 5,... Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 40/58

41 CAR Matérn SPDE Basis functions Lattice What s a good basis? Several possible basis functions exist: Harmonic functions gives a spectral representation (Thon et al., 2012; Sigrist et al., 2012). Eigenfunctions of the covariance matrix give Karhunen-Loéve. Piecewise linear basis gives (almost) a GMRF (Lindgren et al., 2011). Waveletts also gives a GMRF (Bolin and Lindgren, 2013). Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 41/58

42 CAR Matérn SPDE Basis functions Lattice But it s not a Markov field! Yet Using a piecewise linear basis only neighbouring basis functions overlap, so both G ij = ψ i, Δψ j and C ij = ψ i, ψ j are sparse. However, C 1 is not sparse. GMRF approximation To obtain sparse precision matrices we replace the C-matrix with a diagonal matrix C with elements C ii = ψ i, 1 = ψ i (s) ds The resulting approximation error is small (Bolin and Lindgren, 2013; Simpson et al., 2010) Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 42/58

43 CAR Matérn SPDE Basis functions Lattice Technicalities Green s first identity (integration by parts) gives G ij = ψ i, Δψ j Ω = ψ i, ψ j Ω ψ i, n ψ j Ω Strictly we should account for boundary conditions by solving ( κ 2 Δ ) α/2 x(s) = W(s) s Ω x(s) = condition see A.4 in Lindgren et al. (2011) for details. Given Neumann boundaries ψi, n ψ j Ω = 0 s Ω for the piecewise linear basis. Boundary effects implies that the solution is only approximately equal to the stationary field, this can often be handled by extending the domain. Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 43/58

44 CAR Matérn SPDE Basis functions Lattice Regular lattice in R 2 Solving the SPDE gives a GMRF with precision elements Order α = 1 (ν = 0): κ } 1 {{ } Δ Order α = 2 (ν = 1): κ κ }{{} 1 Δ }{{} Δ 2 Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 44/58

45 Global Data General SPDEs Global Temperature Data January 2003 July 2003 Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 45/58

46 Global Data General SPDEs Global Temperature Data Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 46/58

47 Global Data General SPDEs GMRFs based on SPDEs (Lindgren et al., 2011) GMRF representations of SPDEs can be constructed for to oscillating, anisotropic, non-stationary, non-separable spatio-temporal, and multivariate fields on manifolds. (κ 2 Δ)(τ x(s)) = W(s), s R d Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 47/58

48 Global Data General SPDEs GMRFs based on SPDEs (Lindgren et al., 2011) GMRF representations of SPDEs can be constructed for to oscillating, anisotropic, non-stationary, non-separable spatio-temporal, and multivariate fields on manifolds. (κ 2 Δ)(τ x(s)) = W(s), s Ω Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 47/58

49 Global Data General SPDEs GMRFs based on SPDEs (Lindgren et al., 2011) GMRF representations of SPDEs can be constructed for to oscillating, anisotropic, non-stationary, non-separable spatio-temporal, and multivariate fields on manifolds. (κ 2 e iπθ Δ)(τ x(s)) = W(s), s Ω Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 47/58

50 Global Data General SPDEs GMRFs based on SPDEs (Lindgren et al., 2011) GMRF representations of SPDEs can be constructed for to oscillating, anisotropic, non-stationary, non-separable spatio-temporal, and multivariate fields on manifolds. (κ 2 s + m s M s )(τ s x(s)) = W(s), s Ω Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 47/58

51 Global Data General SPDEs GMRFs based on SPDEs (Lindgren et al., 2011) GMRF representations of SPDEs can be constructed for to oscillating, anisotropic, non-stationary, non-separable spatio-temporal, and multivariate fields on manifolds. ( t + κ2 s,t + m s,t M s,t ) (τ s,t x(s, t)) = E(s, t), (s, t) Ω R Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 47/58

52 Observations E(Y X) Hierarchical Observations The A-matrix The field is created as a weighted sum of basis functions. x(s) = N ψ k (s) w k, k=1 The locations of the basis functions do not need to match observation locations. Observations y(s) = x(s) + ε = k ψ k (s)w k + ε We introduce a sparse matrix A i = [ ψ 1 (s i ) ψ N (s i ) ] linking the field to the observation. Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 48/58

53 Observations E(Y X) Hierarchical Observations The A-matrix We can also allow for area observations (change of support) Area observations y(m) = M = k x(s) ds + ε w k ψ k (s) ds + ε We introduce a sparse matrix M A i = [ M ψ 1(s) ds linking the field to the observation. M ψ N(s) ds ] Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 49/58

54 Observations E(Y X) Hierarchical Conditional expectation Given a matrix A with rows A i = [ ψ 1 (s i ) ψ N (s i ) ] we can write the observation equation on matrix form as ( ) Y w N Aw, Q 1 ε w N (μ, Q 1) Kriging with GMRF E (w y) = μ + Q 1 w y A Q ε (y Aμ) V (w y) = Q 1 w y = (Q + A Q ε A ) 1 Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 50/58

55 Observations E(Y X) Hierarchical Bayesian hierarchical modelling using GMRF Data model, p (y x, θ): Describing how observations arise assuming a known latent field x. Latent model, p (x θ): Describing how the latent field behaves. ( ) X = Aw + Bβ w N 0, Q 1 (θ) Parameters, p (θ): Prior knowledge of the parameters. Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 51/58

56 Observations E(Y X) Hierarchical Inference Given a Bayesian hierarchical model we are interested in Posteriors for the parameters p (θ y) p (y θ) p (θ) Posteriors for the latent field p (x y) p (x y, θ) p (θ y) dθ Next lecture... Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 52/58

57 Observations E(Y X) Hierarchical Questions? Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 53/58

58 Bibliography I Aune, E., Simpson, D. P., and Eidsvik, J. (2014), Parameter estimation in high dimensional Gaussian distributions, Stat. Comput., 24, Banerjee, S., Gelfand, A. E., Finley, A. O., and Sang, H. (2008), Gaussian predictive process models for large spatial data sets, J. R. Stat. Soc. B, 70, Besag, J. (1974), Spatial Interaction and the Statistical Analysis of Lattice Systems (with discussion), J. R. Stat. Soc. B, 36, Besag, J. and Kooperberg, C. (1995), On Conditional and Intrinsic Autoregression, Biometrika, 82, Besag, J. and Mondal, D. (2005), First-order intrinsic autoregressions and the de Wijs process, Biometrika, 92, Bolin, D. and Lindgren, F. (2013), A comparison between Markov approximations and other methods for large spatial data sets, Comput. Stat. Data Anal., 61, Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 54/58

59 Bibliography II Campbell, Y. E. and Davis, T. A. (1995), Computing the sparse inverse subset: an inverse multifrontal approach, Tech. rep., University of Florida, rep Cressie, N. and Johannesson, G. (2008), Fixed rank kriging for very large spatial data sets, J. R. Stat. Soc. B, 70, Eidsvik, J., Finley, A. O., Banerjee, S., and Rue, H. (2012), Approximate Bayesian inference for large spatial datasets using predictive process models, Comput. Stat. Data Anal., 56, Eidsvik, J., Shaby, B. A., Reich, B. J., Wheeler, M., and Niemi, J. (2014), Estimation and Prediction in Spatial Models With Block Composite Likelihoods, J. Comput. Graphical Stat., 23, Fuentes, M. (2007), Approximate Likelihood for Large Irregularly Spaced Spatial Data, J. Am. Stat. Assoc., 102, Furrer, R., Genton, M. G., and Nychka, D. (2006), Covariance Tapering for Interpolation of Large Spatial Datasets, J. Comput. Graphical Stat., 15, Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 55/58

60 Bibliography III Higdon, D. (2001), Space and space time modeling using process convolutions, Tech. rep., Institute of Statistics and Decision Sciences, Duke University, Durham, NC, USA. Kaufman, C. G., Schervish, M. J., and Nychka, D. W. (2008), Covariance Tapering for Likelihood-Based Estimation in Large Spatial Data Sets, J. Am. Stat. Assoc., 103, Krige, D. G. (1951), A statistical approach to some basic mine valuation problems on the Witwatersrand, J. Chem. Metall. Min. Soc. S. Afr., 52, Lindgren, F., Rue, H., and Lindström, J. (2011), An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach, J. R. Stat. Soc. B, 73, Matérn, B. (1960), Spatial variation. Stochastic models and their application to some problems in forest surveys and other sampling investigations. Ph.D. thesis, Stockholm University, Stockholm, Sweden. Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 56/58

61 Bibliography IV Matheron, G. (1969), Le Krigeage Universel: Recherche d Opérateurs Optimaux en Présence d une Dérive, vol. 1, École nationale supérieure des mines de Paris. Rozanov, Y. A. (1977), Markov random fields and stochastic partial differential equations, Math. USSR Sb., 32, Rue, H. and Held, L. (2005), Gaussian Markov Random Fields; Theory and Applications, vol. 104 of Monographs on Statistics and Applied Probability, Chapman & Hall/CRC. Sigrist, F., Künsch, H. R., and Stahel, W. A. (2012), Stochastic partial differential equation based modeling of large space-time data sets, Tech. Rep. arxiv: v5, arxiv preprint. Simpson, D., Lindgren, F., and Rue, H. (2010), In order to make spatial statistics computationally feasible, we need to forget about the covariance function, Environmetrics, 23, Stein, M. L., Chen, J., and Anitescu, M. (2013), Stochastic approximation of score functions for Gaussian processes, Ann. Stat., 7, Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 57/58

62 Bibliography V Stein, M. L., Chi, Z., and Welty, L. J. (2004), Approximating likelihoods for large spatial data sets, J. R. Stat. Soc. B, 66, Thon, K., Rue, H., Skrøvseth, S. O., and Godtliebsen, F. (2012), Bayesian multiscale analysis of images modeled as Gaussian Markov random fields, Comput. Stat. Data Anal., 56, Whittle, P. (1954), On Stationary Processes in the Plane, Biometrika, 41, (1963), Stochastic processes in several dimensions, Bull. Int. Stat. Inst., 40, Johan Lindström - johanl@maths.lth.se Gaussian Markov Random Fields 58/58