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Geostatistics for Large Data Sets Whitney Huang Motivation Methods Covariance tapering Low–rank approximation Likelihood approximation Gaussian Markov random ﬁeld approximation Geostatistical Modeling for Large Data Sets Whitney Huang Department of Statistics Purdue University October 28, 2014
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Whitney Huang
Geostatistical Modeling for Large Data Sets
Whitney Huang
October 28, 2014
Whitney Huang
Outline
Motivation
Geostatistics for Large Data Sets
Whitney Huang
Gaussian process (GP) geostatistics Model:
Y (s) = µ(s) + η(s) + ε(s), s ∈ S ⊂ Rd
where
µ(s) = XT (s)β, {η(s)}s∈S ∼ GP (0,C (·, ·))
C (s, s′) = σ2ρθ (s− s′), and ε(s) ∼ N(0, τ2) ∀s ∈ S Log-likelihood: Given data Y = (Y (s1), · · · ,Y (sn))T
ln(β,θ, σ2, τ2) ∝ −1 2 log Σ(θ, σ2) + τ2In
− 1
2 (Y− XTβ)T
[ Σ(θ, σ2) + τ2In
]−1 (Y− Xβ)
where Σ(θ, σ2)i ,j = σ2ρθ(si − sj), i , j = 1, · · · , n
Geostatistics for Large Data Sets
Whitney Huang
“Big n Problem” in geostatistics
I Modern environmental instrument has produced a wealth of space–time data ⇒ n is big
I Evaluation of the likelihood function involves factorizing large covariance matrices that generally requires
I O(n3) operations I O(n2) memory
I Modeling strategies are needed to deal with large spatial data set.
I parameter estimation ⇒ MLE, Bayesian I spatial interpolation ⇒ Kriging I multivariate spatial data, spatio-temporal data
Geostatistics for Large Data Sets
Whitney Huang
“Big n Problem” in geostatistics
I Modern environmental instrument has produced a wealth of space–time data ⇒ n is big
I Evaluation of the likelihood function involves factorizing large covariance matrices that generally requires
I O(n3) operations I O(n2) memory
I Modeling strategies are needed to deal with large spatial data set.
I parameter estimation ⇒ MLE, Bayesian I spatial interpolation ⇒ Kriging I multivariate spatial data, spatio-temporal data
Geostatistics for Large Data Sets
Whitney Huang
“Big n Problem” in geostatistics
I Modern environmental instrument has produced a wealth of space–time data ⇒ n is big
I Evaluation of the likelihood function involves factorizing large covariance matrices that generally requires
I O(n3) operations I O(n2) memory
I Modeling strategies are needed to deal with large spatial data set.
I parameter estimation ⇒ MLE, Bayesian I spatial interpolation ⇒ Kriging I multivariate spatial data, spatio-temporal data
Geostatistics for Large Data Sets
Whitney Huang
Modeling strategies in the literature
I Covariance tapering (Furrer et al. 06, Kaufman et al. 08, Du et al. 09)
I Low–rank approximation (Cressie & Johannesson 08, Banerjee et al. 08)
I Likelihood approximation (Vecchia 88, Stein 04)
I Gaussian Markov random field approximation (Rue & Tjelmeland 02, Rue & Held 05, Lindgren et al. 11)
Geostatistics for Large Data Sets
Whitney Huang
Modeling strategies in the literature
I Covariance tapering (Furrer et al. 06, Kaufman et al. 08, Du et al. 09)
I Low–rank approximation (Cressie & Johannesson 08, Banerjee et al. 08)
I Likelihood approximation (Vecchia 88, Stein 04)
I Gaussian Markov random field approximation (Rue & Tjelmeland 02, Rue & Held 05, Lindgren et al. 11)
Geostatistics for Large Data Sets
Whitney Huang
Modeling strategies in the literature
I Covariance tapering (Furrer et al. 06, Kaufman et al. 08, Du et al. 09)
I Low–rank approximation (Cressie & Johannesson 08, Banerjee et al. 08)
I Likelihood approximation (Vecchia 88, Stein 04)
I Gaussian Markov random field approximation (Rue & Tjelmeland 02, Rue & Held 05, Lindgren et al. 11)
Geostatistics for Large Data Sets
Whitney Huang
Modeling strategies in the literature
I Covariance tapering (Furrer et al. 06, Kaufman et al. 08, Du et al. 09)
I Low–rank approximation (Cressie & Johannesson 08, Banerjee et al. 08)
I Likelihood approximation (Vecchia 88, Stein 04)
I Gaussian Markov random field approximation (Rue & Tjelmeland 02, Rue & Held 05, Lindgren et al. 11)
Geostatistics for Large Data Sets
Whitney Huang
Outline
Motivation
Geostatistics for Large Data Sets
Whitney Huang
Methods Covariance tapering Low–rank approximation Likelihood approximation Gaussian Markov random field approximation
Covariance tapering (Furrer et al. 06) We replace the C (h) by
Ctap(h; γ) = ρtap(h; γ) C (h)
where ρtap(h; γ) is an isotropic correlation function with compact support (ρtap(h) = 0 if h ≥ γ) and denotes the Schur product
Geostatistics for Large Data Sets
Whitney Huang
Covariance tapering cont’d
Geostatistics for Large Data Sets
Whitney Huang
Low–rank approximation
Y = η + ε, ε ∼ MVN(0,Σε)
η = Hα+ ξ, ξ ∼ MVN(0,Σξ)
α ∼ MVN(0,Σα)
where α = (α1, · · · , αp)T such that p n and H is mapping from the latent process, α, to the true spatial process of interest, η. Σε and Σξ and diagonal.
Geostatistics for Large Data Sets
Whitney Huang
Low–rank approximation cont’d
To carry out the spatial interpolation (i.e. kriging) of η(s0)|{Y (si )}ni=1 one need to compute(
HΣαH T + V
where V = Σε + Σξ.
Sherman–Morrison–Woodbury formula (A + BCD)−1 = A−1 − A−1B
( C−1 + DA−1B
In the case of low–rank model, we have( HΣαH
T + V )−1
)−1 HTV−1
Whitney Huang
Fixed Rank Kriging (Cressie & Johannesson 08)
Y = Xβ + ZW∗ + ε
Let W ∗ = {w(s∗i )pi=1} be be latent variables at p n known knots {s∗i }
p i=1 and Z (·) be a known basis function
The fixed rank kriging is equivalent to the following low rank model
Y (s) = X(s)β +
p∑ j=1
Geostatistics for Large Data Sets
Whitney Huang
Gaussian Predictive Process (Banerjee et al. 08)
Use a model
to approximate the original spatial process
Y (s) = X(s)Tβ + η(s) + ε(s)
Knots: {s∗1, · · · , s∗p} where p n
⇒ α = {α(s∗i )}pi=1, H(θ) = [ Cov(si , s∗j ;θ)
]T [Σα]−1
Whitney Huang
Likelihood approximation (Vecchia 88)
Partition the observation vector Y into sub–vector Y1, · · · ,Yb and let Y(j) = (YT
1 , · · · ,YT j )T
p(Yj |Y(j−1);β,θ)
Approximate the exact likelihood by replacing Y(j−1) by a sub–vector S(j−1) of Y(j−1)
Geostatistics for Large Data Sets
Whitney Huang
Markov Random Fields
Whitney Huang
Gaussian Markov Random Fields (GMRF)
Definition Let the neighbors to a point i be the points Ni that are “close" to i . A Gaussian random field X ∼ N(µ,Σ = Q−1)
that satisfies
p(Xi |Xj , j 6= i) = p(Xi |Xj : j ∈ Nj)
is a Gaussian Markov random field (GMRF) with Qij = 0 iff Xi ⊥ Xj |X−ij
Geostatistics for Large Data Sets
Whitney Huang
Remarks: GP vs. GMRF in geostatistical modeling
I +: GP model is widely used in modeling continuously indexed spatial data in which the covariance function characterizes the process properties
I –: Inference involves factorizing covariance matrices
I +: GMRF model is computationally efficient due to the sparse precision matrix
I –: Only for discretely indexed spatial data
Main idea of GMRF approach:
GP inference
Whitney Huang
Remarks: GP vs. GMRF in geostatistical modeling
I +: GP model is widely used in modeling continuously indexed spatial data in which the covariance function characterizes the process properties
I –: Inference involves factorizing covariance matrices
I +: GMRF model is computationally efficient due to the sparse precision matrix
I –: Only for discretely indexed spatial data
Main idea of GMRF approach:
GP inference
Whitney Huang
Remarks: GP vs. GMRF in geostatistical modeling
I +: GP model is widely used in modeling continuously indexed spatial data in which the covariance function characterizes the process properties
I –: Inference involves factorizing covariance matrices
I +: GMRF model is computationally efficient due to the sparse precision matrix
I –: Only for discretely indexed spatial data
Main idea of GMRF approach:
GP inference
Whitney Huang
Remarks: GP vs. GMRF in geostatistical modeling
I +: GP model is widely used in modeling continuously indexed spatial data in which the covariance function characterizes the process properties
I –: Inference involves factorizing covariance matrices
I +: GMRF model is computationally efficient due to the sparse precision matrix
I –: Only for discretely indexed spatial data
Main idea of GMRF approach:
GP inference
Whitney Huang
Remarks: GP vs. GMRF in geostatistical modeling
I +: GP model is widely used in modeling continuously indexed spatial data in which the covariance function characterizes the process properties
I –: Inference involves factorizing covariance matrices
I +: GMRF model is computationally efficient due to the sparse precision matrix
I –: Only for discretely indexed spatial data
Main idea of GMRF approach:
GP inference
Whitney Huang
Remarks: GP vs. GMRF in geostatistical modeling
I +: GP model is widely used in modeling continuously indexed spatial data in which the covariance function characterizes the process properties
I –: Inference involves factorizing covariance matrices
I +: GMRF model is computationally efficient due to the sparse precision matrix
I –: Only for discretely indexed spatial data
Main idea of GMRF approach:
GP inference
Whitney Huang
Methods Covariance tapering Low–rank approximation Likelihood approximation Gaussian Markov random field approximation
GP/Stochastic Partial Differential Equation (SPDE) connection (Whittle 1954, 1963)
Gaussian process Y (s) with Matern covariance function is a stationary solution to the linear fractional stochastic partial differential equation:(
α2 − )κ
2 , ν > 0
I = ∑
I d is the dimension of the spatial domain
Geostatistics for Large Data Sets
Whitney Huang
Methods Covariance tapering Low–rank approximation Likelihood approximation Gaussian Markov random field approximation
An explicit link between GP and GMRF via SPDE (Lindgren et al. 11)
I Establish the link between GP with Matérn covariance function (with ν + d
2 are integers) and GMRF
I (Bayesian) inference can be done by using Integrated nested Laplace approximation (INLA) approach
I The extensions to nonstationary models, models on manifolds, multivariate models, spatio-temporal models are relatively easy
Geostatistics for Large Data Sets
Whitney Huang
Methods Covariance tapering Low–rank approximation Likelihood approximation Gaussian Markov random field approximation
An explicit link between GP and GMRF via SPDE (Lindgren et al. 11)
I Establish the link between GP with Matérn covariance function (with ν + d
2 are integers) and GMRF
I (Bayesian) inference can be done by using Integrated nested Laplace approximation (INLA) approach
I The extensions to nonstationary models, models on manifolds, multivariate models, spatio-temporal models are relatively easy
Geostatistics for Large Data Sets
Whitney Huang
Methods Covariance tapering Low–rank approximation Likelihood approximation Gaussian Markov random field approximation
An explicit link between GP and GMRF via SPDE (Lindgren et al. 11)
I Establish the link between GP with Matérn covariance function (with ν + d
2 are integers) and GMRF
I (Bayesian) inference can be done by using Integrated nested Laplace approximation (INLA) approach
I The extensions to nonstationary models, models on manifolds, multivariate models, spatio-temporal models are relatively easy
Geostatistics for Large Data Sets
Whitney Huang
Extensions I non-stationary model on a sphere(
α2(s) + )κ
I non-separable anisotropic space-time model( ∂
∂t + (α2 + m · ∇ −∇ ·H∇
Whitney Huang
H. Rue, and L. Held Gaussian Markov Random Fields: Theory and Applications. Chapman & Hall/CRC, 2005.
S. Banerjee, A. E. Gelfand, A. O. Finley, and H. Sang Gaussian Predictive Process Models for Large Spatial Data Sets JRSSB, 70:825–848, 2008.
N. A. C. Cressie, and G. Johannesson Fixed Rank Kriging for Very Large Spatial Data Sets JRSSB, 70:209–226, 2008.
Geostatistics for Large Data Sets
Whitney Huang
J. Du, H. Zhang, and V. S. Mandrekar Fixed–Domain Asymptotic Properties of Tapered Maximum Likelihood Estimators The Annals of Statistics, 37:3330–3361, 2009.
R. Furrer, M. G. Genton, and D. W. Nychka Covariance Tapering for Interpolation of Large Spatial Datasets Journal of Computational and Graphical Statistics, 15:502–523, 2006.
C. G. Kaufman, M. J. Schervish, and D. W. Nychka Covariance Tapering for Likelihood–Based Estimation in Large Spatial Data Sets Journal of the American Statistical Association, 103:1545–1555, 2008.
Geostatistics for Large Data Sets
Whitney Huang