Gaussian Processes, Kriging and Co-Kriging

Gaussian Processes

Definition A Gaussian Process is a collection of random variables, any finite number of which have consistent joint Gaussian distributions. A Gaussian process f is fully specified by its mean function m(x) and covariance function k(x,x0), written as f∼(m,k)

Posterior Gaussian Process (Noise-free prediction)

Let f be the known function values of the training cases at X, and let f∗ be a set of function values corresponding to the test set inputs X∗, the joint distributions can be written as

[ff∗]∼([μμ∗],[ΣΣT∗Σ∗Σ∗∗])

Then the conditional distribution of f∗ given f is expressed as

f∗|f∼(μ∗+ΣT∗Σ−1(f−μ),Σ∗∗−ΣT∗Σ−1Σ∗)

or, we can written as

f|∼(m,k),

m=m(x)+Σ(X,x)TΣ−1(f−m)

k=k(x,x′)−Σ(X,x)TΣ−1Σ(X,x′)

Prediction using Noisy Observations

//to be done

Kriging

All kriging estimators are but variants of the basic linear regression estimator

To estimate f∗(x∗) given f(X), with X=x1,...,xn

1. Simple Kriging
assume m(x)=E(f(x))=m is constant

f∗(x∗)=m+∑in(x∗)λi(x∗)(f(xi)−m)

the estimator is unbiased, since E(f(xi)−m)=0, the goal is to determine weights λi, which minimize the variance of the estimator

σ2E(x∗)=Var(f∗(x∗)−f(x∗))s.t.E(f∗(x∗)−f(x∗))=0

// to be done

2. Ordinary Kriging
m(x)=E(f(x)) is not a global constant

f∗(x∗)=m(x∗)+∑in(x∗)λi(x∗)(f(xi)−m(x∗))

the unbiased condition is satisfied requiring the weights sum to 1, which leads to

f∗(x∗)=∑in(x∗)λi(x∗)f(xi)s.t.∑in(x∗)λi(x∗)=1

//to be done
3. Universal Kriging

4. Indicator Kriging

Co-Kriging