1. Inference for Geostatistical Data: Kriging for Spatial Interpolation
GRAD6104/8104 INES 8090
Spatial Statistic- Spring 2017
Inference for Geostatistical Data: Kriging for Spatial Interpolation

2. Kriging for Spatial Interpolation
Ordinary Kriging
Universal Kriging
Block Kriging
Co-Kriging
Indicator-Kriging
Image source

3. Ordinary Kriging Danie G. Drige (1919-2013), South African
Mining Engineer
Pioneer of geostatistics
Professor at the University of Witwatersrand
Definition (Krige, 1978)
“The name given … to the multiple regression procedure for arriving at the best linear unbiased predictor or best linear weighted moving average predictor of the ore grade of an ore block (of any size) by assigning an optimal set of weights to all the available and relevant data inside and outside the ore block.”

4. Ordinary Kriging Assumptions The mean m(s) is constant
The semivariogram γ(h) is known
One-dimension example
One-dimension example
Image Source:

5. Ordinary Kriging Goal of Kriging
To predict the values of Z(s0) at S0, using the observed responses Z(s1), …, Z(sn) (i.e., spatial interpolation)

6. Ordinary Kriging Properties of OK predictor
A linear combination of the data values
Unbiased, i.e., it satisfies
Among all functions of the data that satisfy the first two properties, it minimizes the variance of prediction error

7. Ordinary Kriging Minimization of variance of prediction error
Subject to

8. Ordinary Kriging Problem solution
Using Lagrange multipliers to solve the constrained minimization problem

9. Ordinary Kriging where
Let the optimal coefficients λ1, …, λn of the OK predictor are the first n elements of the vector λo that satisfies that following system of linear equations, known as the (ordinary) kriging equations:
where
m is a Lagrange multiplier
is symmetric

10. Ordinary Kriging
Kriging variance, the minimized variance is thus:

11. Ordinary Kriging
Example
γ(||h||)=1-exp(-||h||/2

12. Ordinary Kriging
Example
γ(||h||)=1-exp(-||h||/2

13. Ordinary Kriging Influence of Spatial Dependence on Ordinary Kriging
Consider the same spatial configuration in the toy example, but with each of the following four semivariograms

14. Ordinary Kriging Influence of Spatial Dependence on Ordinary Kriging
Kriging weights and variances for these models
More weights are given to near responses when spatial autocorrelation are stronger
Adding nugget effect tends to equalize the weights

15. Ordinary Kriging Influence of Spatial Dependence on Ordinary Kriging
Kriging weights and variances for these models
More weights are given to near responses when spatial autocorrelation are stronger
Adding nugget effect tends to equalize the weights
More weights are given to near responses when spatial autocorrelation are stronger
Adding nugget effect tends to equalize the weights

16. Ordinary Kriging Remarks
The uncertainty in the Kriging predictor: using a confidence interval 100(1-α)%
Assume the random field is Gaussian, and all the Z(si) are normally distributed
zα/2 is the upper α/2 percentage point of the standard normal distribution

17. Ordinary Kriging Remarks
Theoretically, the OK predictor is a linear combination of all the observations
In practice, only the observations within a moving window or kriging neighborhood are used
The nugget/sill ratio, the range, and the sampling configuration are important factors in the choice of window size

18. Ordinary Kriging Remarks
Ordinary kriging is derived under an assumption of intrinsic stationarity (the mean is constant).
The variation of kriging that can handle trends is called universal kriging

19. Ordinary Kriging Remarks
Ordinary kriging is derived under an assumption that the semivariogram is known
In practice, the semivariogram is unknown and must be estimated
The estimated kriging variance tends to underestimate the prediction error variance of the estimated OK predictor (it does not account for the estimation error incurred in estimating θ)

20. Ordinary Kriging Sampling Design
The kriging variance at any given site s0 does not depend on the observed responses
Thus, it is suitable for addressing sampling design questions
Where to allocate more observations to maximize reduction in the kriging variance

21. Ordinary Kriging Sampling Design
The kriging variance at any given site s0 does not depend on the observed responses b c a d

22. Ordinary Kriging Sampling Design
The kriging variance at any given site s0 does not depend on the observed responses b c a d
So, the best additional site is ?

23. Extensions of Ordinary Kriging
Universal Kriging
Block Kriging
Indicator Kriging
Co-Kriging
Simple Kriging

24. Universal Kriging The mean function is not constant Suppose that Where
fj(.) are functions of spatial location
ε(0) is intrinsically stationary

25. Universal Kriging Minimize the variance of prediction error
Subject to the unbiasedness constraint
i.e., corresponding to a set of p+1 constraints

26. Universal Kriging Minimize the variance of prediction error
Subject to the unbiasedness constraint
i.e., a set of p+1 constraints
P+1 Lagrange multipliers
=> p+1 Lagrange multipliers

27. Universal Kriging
Similar to OK, we could write UK using the following forms:
Where
ΓU is a symmetric (n+p+1) x (n+p+1) matrix
The universal kriging variance is thus:
P+1 Lagrange multipliers

28. Extensions of Ordinary Kriging
Universal Kriging
Block Kriging
Using averaged values in blocks instead of exact values of points
Indicator Kriging
indicator of an additional variable
Co-Kriging
for multiple variables
Simple Kriging
The mean is constant and known

29. Kriging vs. Inverse Distance Weighting
IDW
The predicted (or estimated) value at any unknown location is a weighted linear combination of the observed data, with weights inversely related to the distance from the observed data location to the place at which we conduct prediction

30. Kriging vs. Inverse Distance Weighting
IDW
Example

31. Kriging vs. Inverse Distance Weighting
IDW yields unbiased predictors, assuming that the random field is intrinsically stationary
IDW is inferior to OK since IDW is linear and unbiased under intrinsic stationary, and OK yields the best linear unbiased predictor. Empirical work suggests that OK variance is often 10-30% less than that from IDW
IDW does not account for how strong (or weak) the spatial autocorrelation is
Prediction error variances are not usually reported with IDW interpolations

32. Reading Assignment Chapter 5 by Schabenberger and Gotway (2005)
Spatial Prediction and Kriging