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Chapter 4 - Fundamentals of spatial processes

Lecture notes, part 2, Theory

Odd Kolbjørnsen and Geir Storvik February 6, 2017 STK4150 - Intro 1

Last time, spatial process

Spatial correlation Stationary Isotropic/Anisotropic Permutation test for independence Variogram Link to covariance Nugget effect Sill Spatial prediction Interpolation Minimum Mean Squared Prediction Error (Kriging) Prediction in a Gaussian model Kriging Simple Kriging (known mean) Bayesian Kriging (prior on mean) Universal Kriging (unknown mean) Ordenary Kriging (Special case of UK) STK4150 - Intro 2

General setup for Kriging type problems

What is the process vs data relation? Set up the prior for the process model. Do the relevant computations to derive joint distribution/second order moments Recall: ( X 1 X 2

)

∼MVN ((

μ 1 μ 2

E (X 1 |X 2 ) =μ 1 + Σ 12 Σ− 221 (X 2 − μ 2 ) var(X 1 |X 2 ) =Σ 11 − Σ 12 Σ− 221 Σ 21 STK4150 - Intro 4

A standard problem

Z =aY + ε, ε ∼ N(0, σ^2 ) Y ∼N(μ, τ 2 ) ( Y Z

)

∼MVN ((

μ aμ

τ 2 aτ 2 aτ 2 a^2 τ 2 + σ^2

E (Y |Z ) =μ + aτ 2 a^2 τ 2 + σ^2 (Z − aμ) = μ

σ^2 a^2 τ 2 + σ^2

)

Z

a

a^2 τ 2 a^2 τ 2 + σ^2

var(Y |Z ) =τ 2 − τ 2

a^2 τ 2 a^2 τ 2 + σ^2

STK4150 - Intro 5

Kriging example

STK4150 - Intro 7

Kriging example

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Change of support, Area support on Data

Zi =

Ai Y (s)ds + εi , εi ∼ N(0, σ^2 ), iid G T^ =[g 1 T , ..., g (^) nT ], with gi = I (s ∈ Ai ) E(Zi ) =

Ai μY (s) ds Cov(Y (s 0 ), Zi ) =

Ai CY (s 0 , s) ds Cov(Zi , Zj ) =

Ai

Aj CY (s 1 , s 2 ) ds 1 ds 2 + σ^2 I (i = j) STK4150 - Intro 10

Change of support,Prediction of a a region

V =

∫

A Y (s)ds

E (V |Z ) =

∫

A E (Y (s)|Z )ds Var (V |Z ) =

A

A Cov(Y (s 1 ), Y (s 2 )|Z ) ds 1 ds 2 STK4150 - Intro 11

Spectral expansion

Eigen-representation of moving average kernel:

k(s, u) =

∑^ ∞

i=

λi φi (s)φi (u)

⇒ Spectral representation random field:

Y (s) =

∑^ ∞

i=

yi φi (s), with yi ∼ N(0, λ^2 i ), independent

Spectral representation korrelation function (Karhunen-Lo´eve expansion):

CY (s, u) =

∑^ ∞

i=

λ^2 i φi (s)φi (u) (Σ = V Λ^2 V T^ )

Common to use Fourier transform, Fast Fourier Transform (FFT)

Estimation

Likelihood: L(θ) = p(Z ; θ) =

Y

p(Z |Y θ) dY

Gaussian process: can be derived analytically Optimization can still be problematic Many routines in R available

Boreality - fitted covariance function

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Hierarchical and Bayesian model

Hierarchical model Variable Densities Notation in book Data model: Z p(Z|Y, θ) [Z|Y, θ] Process model: Y p(Y|θ) [Y|θ] (Gaussian in 4.1) Parameter: θ

Hierarchical and Bayesian model

Hierarchical model Variable Densities Notation in book Data model: Z p(Z|Y, θ) [Z|Y, θ] Process model: Y p(Y|θ) [Y|θ] (Gaussian in 4.1) Parameter: θ Simultaneous model: p(y, z|θ) = p(z|y, θ)p(y|θ) Marginal model: L(θ) = p(z|θ) =

y p(z,^ y|θ)dy Inference: ̂θ = argmaxθL(θ) Bayesian approach: Include model on θ Variable Densities Notation in book Data model: Z p(Z|Y, θ) [Z|Y, θ] Process model: Y p(Y|θ) [Y|θ] (Gaussian in 4.1) Parameter model: θ p(θ) [θ] Simultaneous model: p(y, z, θ) Marginal model: p(z) =

θ

y p(z,^ y|θ)dydθ Inference: ̂θ =

θ θp(θ|z)dθ

Non-Gaussian data

Counts, binary data: Gaussian assumption inappropriate Can still have Gaussian assumption on latent process, but non-Gaussian data-distribution Y (s) =x(s)T^ β + ε(s), {ε(s)} Gaussian process Z (si )|Y (si ), θ 1 ∼ind.f(Y (si ), θ 1 ) Best linear predictor still possible, but is it reasonable? Conditional expectation E [Y (s 0 )|Z] still optimal under square loss Not easy to compute anymore Exponential-family model (EFM) f (z) = exp{(zη − b(η))/a(θ 1 ) + c(z, θ 1 )} η =xT^ β Include Binomial, Poisson, Gaussian, Gamma STK4150 - Intro 20