12.1 Introduction

Spatial statistics is a part of applied statistics and is concerned with modelling and analysis of spatial data. By spatial data we mean data where, in addition to the (primary) phenomenon of interest, the relative spatial locations of observations are also recorded because these may be important for the interpretation of data. This is of primary importance in earth‐related sciences such as geography, geology, hydrology, ecology, and environmental sciences, but also in other scientific disciplines concerned with spatial variations and patterns such as astrophysics, economics, agriculture, forestry, and epidemiology, and, at a microscopic scale, medical and health research. Spatial statistics uses nearly all methods described in the first eleven chapters of this book and also multivariate analysis and Bayesian methods, neither of which are discussed in this book. We therefore restrict ourselves in this chapter to a few basic principles and give hints for further reading for other important methods. As a consequence of this, the list of references is relatively long.

We restrict our attention to continuous characteristics and to Gaussian distributions and analyse examples with the program package R as we did in other chapters of this book. Those who prefer SAS and understand a bit of German are referred to the procedures in 6/61 of Rasch et al. (2008).

• 6/61/0000 Spatial Statistics – Introduction
• 6/61/1010 Estimation of the covariance function of a random variable with constant trend
• 6/61/1020 Estimation of the semi‐variogram of a random variable
• 6/61/1021 Estimation of the parameter of an exponential semi‐variogram model
• 6/61/1022 Estimation of the parameter of a spherical semi‐variogram model
• 6/61/1030 Definition of increments and of the generalised covariance function for non‐steady state random variables
• 6/61/1031 Estimation of the generalised covariance function for non‐steady state random variables
• 6/61/1100 Modelling spatial dependencies between two variables
• 6/61/2000 Spatial prediction – survey
• 6/61/2010 Prediction of stationary random variables using the covariance function
• 6/61/2020 Prediction of stationary random variables using the semi‐variogram
• 6/61/2030 Prediction of non‐steady state random variables
• 6/61/2040 Spatial prediction: Co‐kriging
• 6/61/2050 Prediction of probabilities
• 6/61/2051 Prediction of exceedance probabilities
• 6/61/2052 Hermitean prediction.

We restrict ourselves to one‐ and two‐dimensional regions D ⊂ R¹ and D ⊂ R², respectively. However, D ⊂ R³ (oil and mineral prospection, 3D imaging) is also possible. In some fields such as Bayesian data analysis, design and simulation one even requires spaces D of dimension >3; this pertains, in particular, to the design and analysis of computer experiments with a moderate to large number of input variables.

Points in D ⊂ R² are written as s^T = (x₁, x₂) and the coordinates x₁, x₂ are in geostatistics often the Gauss–Krüger coordinates on the earth based on the degrees (°) of the longitude meridional zone of the surface of the earth, see Krüger (1912). The surface is subdivided in meridional zones of a latitude of 3° running from the North Pole to the South Pole parallel to its central meridian. The degrees of the central meridian of each meridional zone counted from 0° eastwards are mapped to code numbers by dividing them by three as shown in Table 12.1.

The meridional zone is conformally mapped on a cylinder barrel with the axis in the equatorial plane and a radius equal to the curvature radius of the meridian. Its origin is the intersection of the central meridian and the equator. From the origin the coordinates of the points on the surface of the earth are defined like in a usual Cartesian coordinate system, positive to the east by the so‐called easting (x₁), and to the north by the so‐called grid north (x₂). The coordinates on the earth can be transformed to the Gauss–Krüger coordinates via https://www.koordinaten‐umrechner.de. As an example, we give the Gauss–Krüger coordinates of the (first author's) house in Feldrain 73 in Rostock, Germany.

Degrees minutes seconds E 12° 11′ 41; N 54° 06′ 18

Easting: 4507572; grid north 59922353.

First we must select a spatial model for the observations (variables) at the points in D ⊂ R²; their realisations are our observations.

The general model is for observations y(s) at

12.1

with side conditions for weak stationarity (second order stationarity):

12.2

12.3

Formula (12.3) leads for h = 0 to var(y(s))= C(0).

If we further assume that C(h) = C(∥h∥) with the Euclidian norm ∥h∥ of h, then C is called isotropic, otherwise anisotropic.

The positions of observation sites s ∈ D can be fixed in advance (as, e.g. the position of wind power stations in an area) or may be random.

As examples where observation points occur randomly, we mention meteor strikes in a special area. To this situation also belongs the possibly oldest mapping of clusters of cholera cases in the London epidemic of 1854 (Snow 1855). Randomly means in this connection that we assume that a point is equally probable to occur at any location and that the position of a point is not affected by any other point.

Further, the observation points may not be fixed but determined by the scientist (monitoring). In this case, besides problems of analysis, design problems also exist by selecting the optimal observation points in an area. For this, readers are referred to Müller (2007).

Comprehensive treatments of the whole field of spatial statistics are given in Ripley (1988), Cressie (1993), and Gaetan and Guyon (2010).

Basically, there are four classes of problems which spatial statistics is concerned with: point pattern analysis, geostatistical data analysis, areal/lattice data analysis and spatial interaction analysis. These sub‐problems are treated in overview papers such as: Pilz (2010), Mase (2010), Kazianka and Pilz (2010a), Diggle (2010), and Spöck and Pilz (2010).We discuss mainly geostatistical data analysis with some hints to areal data analysis.

For a good overview on software for different problem areas of spatial data analysis with R we recommend the book by Bivand et al. (2013), for the important issue of simulation of spatial models we refer to Lantuéjoul (2002) and Gaetan and Guyon (2010). An overview of methodology and software for interfacing spatial data analysis and geographic information system (GIS) for visualising spatial data is given in Pilz (2009).

Due to the fact that in some fields of application of spatial statistics special methods have been applied, and further because the theory and applications are still in development, we cannot give a closed presentation of the field. We describe basic methods used in geostatistics using Euclidean distances and then give examples. For point pattern analysis, areal/lattice data analysis, and spatial interaction analysis the reader is referred to the books mentioned above.

12.2 Geostatistics

In geostatistics D is a continuous subspace of R² or R³ and the random variable (field) is observed at n > 2 fixed sites s₁, s₂, … , s_n ∈ D. Typical examples include rainfall data, data on soil characteristics (porosity, humidity, etc.), oil and mineral exploration data, air quality, and groundwater data. In this chapter only one characteristic observation variable is measured per observation point on a line or plane. Multivariate geostatistics dealing with observation vectors per observation point is described in detail in Wackernagel (2010).

The concept of stationarity is key in the analysis of spatial and/or temporal variation: roughly speaking, stationarity means that the statistical characteristics (e.g. mean and variance) of the random variable of interest do not change over the considered area. However, testing for stationarity is not possible. For spatial prediction the performance of a stationary and a non‐stationary model could be compared through assessment of the accuracy of predictions.

In this chapter we assume that the random vectors y^T = [y(s₁), y(s₂), … , y(s_n) ] follow an n‐dimensional normal (Gaussian) distribution for any collection of spatial locations {s₁, s₂, … , s_n} ⊂ D and any n ≥ 1. In the literature the collection of random variables {y(s) : s ∈ D} is then usually termed a Gaussian random field (GRF) over D. For other types of random fields and a detailed explanation of their mathematical structure and most important properties we refer to Cressie and Wikle (2011), where also extensions to so‐called spatio‐temporal random fields are considered. Often non‐normal random variables may be transformed by a so‐called Box–Cox transformation as a generalisation of the normal case by including an additional parameter λ. This transformation is given by:

12.4

A GRF is completely determined by its expectation (trend function)

and covariance function C(s_i − s_j) = cov[y(s_i), y(s_j)] : i, j = 1, … , n.

Contrary to traditional statistics, in a geostatistical setting we usually observe only one realisation of y at a finite number of locations . Therefore, the distribution underlying the random field cannot be inferred without imposing further assumptions. The simplest assumption is that of (strict) stationarity, which means that the normal distributions do not change when all positions are translated by the same (lag) vector h and this implies that (12.2) and (12.3) are valid.

Often (in areal/lattice data analysis) measurements are not related to points but to areas A_i, i = 1, … n . In such cases in place of distances between points measures of spatial proximity w_ij between two areas A_i and A_j are used and represented in a square n × n matrix W = (w_ij).

According to Bailey and Gatrell (1995) some possible criteria for determining proximities might be:

• w_ij = 1 if A_j shares a common boundary with A_i and w_ij = 0 else.
• w_ij = 1 if the centroid of A_j is one of the k nearest centroids to that of A_i and w_ij = 0 else.
• w_ij =  if the inter‐centroid distance d_ij < δ (δ > 0, γ < 0); and w_ij = 0 else.
• w_ij = , where l_ij is the length of the common boundary between A_iand A_j and l_i is the perimeter of A_i.

All diagonal elements w_ii are set to 0. Note that the spatial proximity matrix W must not necessarily be symmetric.

For more proximity measures we refer to Bailey and Gatrell (1995) and any other publications on areal spatial analysis like Anselin and Griffith (1988).

12.2.1 Semi‐variogram Function

From now on, we focus on geostatistics and assume second order stationarity, i.e. (12.2) and (12.3) hold, and additionally, we assume isotropy. In geostatistics it is common to use the so‐called semi‐variogram function

12.5

Observing that under the assumption of stationarity it holds that

12.6

we can simply use the empirical moment estimate to estimate the semi‐variogram according to

12.7

where denotes the number of sampling locations separated from each other by the .

We call the estimated semi‐variogram function or the sample semi‐variogram function. Since a vector h is uniquely determined by its length and its direction, it is customary to form lag distance classes along given directions, usually this is done for the four main directions 0°, 45°, 90°, and 135°.

Then

This is called the variogram function. Dividing by two leads to the semi‐variogram function

12.9

as a special case of (12.5).

For ‘classical’ estimation methods for the trend function and variogram parameters see Mase (2010), for Bayesian approaches we refer to Banerjee et al. (2014), Kazianka and Pilz (2010a) and Pilz et al. (2012). For non‐stationary variogram modelling we refer to the review provided by Sampson et al. (2001) and Schabenberger and Gotway (2005).

Theoretically, at zero separation distance (lag = 0), the semi‐variogram value γ(0) is zero, because . However, at an infinitesimally small separation distance, the semi‐variogram function often exhibits a so‐called nugget effect, which is some value greater than zero. For example, if the semi‐variogram model intercepts the x₂‐axis at 3, then the nugget effect is 3. Furthermore, often is an asymptotic value of the semi‐variogram model called sill. The nugget effect can be attributed to measurement errors or spatial sources of variation at distances smaller than the sampling interval (or both). Measurement error occurs because of the error inherent in measurement devices. Natural phenomena can vary spatially over a range of scales. Variation at microscales smaller than the sampling distances will appear as part of the nugget effect. Before collecting data, it is important to gain some understanding of the scales of spatial variation.

12.2.3 Kriging

In geostatistics kriging is a method of interpolation to predict values of the variable of interest in D at positions where no measurement has been made. It is based on a master thesis of Krige (1951) and the mathematical foundation by Matheron (1963). Observed values are modelled by a GRF. Under suitable assumptions, kriging gives the best linear unbiased prediction (BLUP) of values not observed in the corresponding area. The basic idea of kriging is to predict the value of a function at a given point by computing a weighted average or linear combination of the observed values of the function in the neighbourhood of the point. The methods used are a kind of regression analysis in two dimensions. Basically, we distinguish between ordinary and universal kriging: ordinary kriging assumes a constant trend function, E(y(s)) = const for all s in D, whereas universal kriging assumes a non‐constant trend. Placing the problem in a stochastic framework permits precision‐defining optimality for estimations of unknown parameters from the random variables for which the measurements are realisations. A criterion imposed is that the estimator be unbiased, or that in an average sense the difference between the predicted value and the actual value is zero. Another optimality criterion is that the prediction variance be minimised. This variance (kriging variance) is defined to be the expectation of the average squared difference between predicted and actual values. The kriging estimator minimises this variance. This minimisation is performed algebraically and results in a set of equations known as the kriging equations – in ordinary kriging we call them ordinary kriging equations (OK system).

For ordinary kriging, we must make the following assumptions already made above:

• y_i; i = 1, … , n are normally distributed.
• Second order stationarity, meaning, in particular, that the y_i; i = 1, … , n all have the same constant mean and variance.

We restrict ourselves to observation points in D ⊂ R² and the univariate case with one character measured at each of n points in D. In the case that in (12.1) μ(s) is known we speak about simple kriging. This is in many practical situations an unrealistic assumption. Therefore, we describe the so‐called ordinary kriging where μ(s) is assumed to be unknown and constant.

The kriging prediction at an unobserved location s₀ ∈ D is then given by a linear combination

12.16

where the weights in λ^T = (λ₁, … , λ_n) are determined as solutions of the OK system of the observations .

Here G = (γ(‖s_i − s_j‖))_{i, j = 1, … , n} is the (n × n)‐semi‐variogram matrix of the observations,

is the vector of semi‐variogram values, and λ₀ is the Lagrange multiplier, which is necessary due to the minimisation under the equality constraint , which, in turn, results from the unbiasedness condition E() = μ. The kriging variance then becomes σ²(s₀) = var() = λ₀ + λ^Tγ₀. We illustrate this with the following toy example where n = 4.

12.3 Special Problems and Outlook

We now briefly outline more recent developments which go beyond the traditional kriging and trans‐Gaussian kriging considered in Section 12.2.3.

12.3.2 Copula Based Geostatistical Prediction

The GLGM framework does not allow distributions outside the exponential family. In particular we cannot use it for heavy‐tailed/extreme value distributions, which have a much slower decay of probability in the tails than the normal distribution. A prominent example of such a distribution is the generalised extreme value distribution with distribution function

with location, scale, and shape parameter μ, σ, and τ, respectively. Observations from non‐Gaussian, skewed or heavy‐tailed distributions are dealt with in the library (intamap). A general framework of handling such data are copulas which are distribution functions on the unit cube [0, 1]ⁿ with uniformly distributed margins, introduced by Sklar (1959). Copulas are invariant under strictly increasing transformations of the marginals; thus, frequently applied data transformations (e.g. square root and log transformations) do not change the copula. The relation between two locations separated by the lag‐vector h is characterised by the bivariate distribution with distribution function

12.23

The copula C_h thus becomes a function of the separating vector h. Spatial copulas have been introduced by Bardossy (2006) and Kazianka and Pilz (2010b). Spatial copulas describe the spatial dependence over the whole range of quantiles for a given separating vector h, not only the mean dependence, as the variogram does. The main difference to GLGMs is that these only model the means μ_i = E(y(s_i)), i = 1, … , n through some link function , where Z(·) is a stationary GRF with mean zero and given covariance function. The spatial copula, however, allows us to build a complete multivariate distribution of the random variables whose realisations are the observed values [y(s₁), … , y(s_n)]. Recently, so‐called vine copula have been developed which extend the concept of the bivariate copula C_h in (12.23) to higher dimensions – see e.g. Gräler (2014). The corresponding R packages copula, spcopula and VineCopula allow flexible spatial data modelling. A Matlab toolbox for copula‐based spatial analysis is given in Kazianka (2013).

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