DGFCMS MOVIMIENTO

This section will introduce the topic of interpolation and then the interpolation tech- nique of kriging.

Interpolation is the method of obtaining (estimating) values within the range of a set of known points that have been obtained by, for example, sampling or experimentation. The intuitive way of calculating these values is to take a weighted average of the N

surrounding known values:

ˆ X(s0) = N X i=1 λiX(si) (5.1) where λi ∝ ||s0−si|| (5.2)

and ˆX(s0) is the value being estimated at s0, X(si) are the known values at si (i ∈

1...N). This interpolation method, although intuitive (its usage dates back to Ancient Babylonian astronomers and Greek mathematicians) does not provide results with a measure of certainty for interpolated values. To combat this disadvantage kriging was developed.

Kriging

Kriging interpolation has been used extensively in geographical and geostatistical ap- plications. The mathematics behind it were developed by Andrey Kolmogorov (1903- 1987) [97] and Herman Wold (1908-1992) [181] in the 1930s and by Norbert Wiener (1894-1964) in 1949 [178]. The word ‘kriging’ was coined by Pierre Carlier (French: krigeage) but brought into Anglo-Saxon mining terminology in 1963 by the French mathematician Georges Matheron (1930-2000) [123]. The technique is named after the South African mining engineer Danie G. Krige (1919-2013) who wrote a Master’s thesis on using kriging to evaluate concentrations of gold and other metals in blocks of rock from samples collected from drill holes [99, 38].

Kriging is based on the assumption that data being interpolated has a continuous nature and that nearby points have a higher degree of spatial correlation than points found further away. Interpolation is hence performed by giving larger weights to closer observed points. An advantage of kriging interpolation over standard linear interpola- tion methods includes the fact that it provides an estimation of the uncertainty along with interpolation results [5].

There has recently been a resurgence of kriging and today it may be better known as Gaussian Process Regression (GPR) in the spatial statistics field. Kriging is in fact a subcase of GPR with slightly differing terminology. For example, the terms ‘sill’ and ‘range’ (see Figure 5.2) would be referred to as the ‘variance paramater’ and ‘scale parameter’ in GPR literature [32, 80].

Ordinary kriging

Ordinary kriging is by far the most popular form of kriging in practice so it will be explained here first. It uses the standard interpolation technique with weights (Eq. 5.1) but to ensure an unbiased estimate, the weights are made to sum to 1:

N

X

j=i

The expected error is assumed to beE[ ˆX(s0)−X(s0)] = 0 and the estimation variance is given by: var[ ˆX(s0)] = E[{Xˆ(s0)−X(s0)}2] = 2 N X i=1 λiγ(si, s0)− N X i=1 N X j=1 λjλiγ(si, sj) (5.4)

whereγ(si, sj) is the semivariance of X between the data pointssi and sj.

The goal in kriging is to find the weights λ that minimise the variance in 5.4. To impose the constraint in 5.3 a Lagrange multiplier (ψ) is added such that the auxiliary functionf(λi, ψ) containing the variance we wish to minimise is defined as:

f(λi, ψ) = arg minvar[ ˆX(s0)−X(s0)]−2ψ

( _N X i=i λi−1 ) (5.5)

With the two partial derivatives of the auxiliary function set to 0 fori= 1,2, ..., N this gives a set of N + 1 equations withN + 1 unknowns:

N

X

i=i

λiγ(si, sj) +ψ(s0) =γ(sj, s0),∀j (5.6)

This is the kriging system for points that provides the values for the weights λ in Equation 5.1. The variance of the estimated results can be obtained by:

σ2(s0) = N X i=i λiγ(si, s0) +ψ(s0) (5.7) Variogram

To solve Equation 5.5, a variogram functionγ, which is directly linked to the covariance of the stochastic processes, is chosen to model the spatio-temporal dependencies. In practice, this variogram is chosen such that it best describes a computed empirical variogram. Depending on the empirical variogram, a different model can be chosen. One of the most common models is the exponential model, which can be described as:

γ(x) = c 1−exp(−x r ) (5.8)

where c is the sill and r the range as shown in Figure 5.2. The sill and range will ultimately dictate the size of the weights λ. A nugget can also be added to the model to more closely control the covariance close to the origin.

Partial

Sill

Nugget

Range

Sill

ɣ

Figure 5.2: Exponential variogram model.

Some other common variogram models include the bounded linear, circular, spher- ical and pentaspherical.

Simple and universal kriging

As was mentioned earlier, the most common form of kriging is ordinary kriging, which assumes that the mean is unknown. Two other well known forms of kriging are simple and universal kriging. Simple kriging is used when the mean of a random variable is known (from past experience, for example). This additional knowledge can be used in calculations. With a known mean,µ, Equation 5.1 becomes

ˆ X(s0) = N X i=1 λiX(si) +{1− N X i=1 λi}µ (5.9)

Universal kriging is used when the spatial processes are comprised of stochastic and deterministic components. It assumes that we have a model telling us how the mean evolves in the spatio-temporal domain. The general formula then becomes

ˆ X(s0) = K X k=1 N X i=1 akλifkX(si) (5.10)

where f is a set of polynomials representing the non-stationary components with unknown coefficients ak.