This is page 46 Printer: Opaque this 3 Gaussian models for geostatistical data Gaussian stochastic processes are widely used in practice as models for geostatis- tical data. These models rarely have any physical ustification. Rather# they are used as convenient empirical models which can capture a wide range of spatial $ehaviour according to the specification of their correlation structure. Histori- cally# one very good reason for concentrating on Gaussian models was that they are uniquely tractable as models for dependent data. With the increasing use of computationally intensive methods, and in particular of simulation-based methods of inference, the analytic tractability of Gaussian models is $ecom- ing a less compelling reason to use them. Nevertheless# it is still convenient to work within a standard model class in routine applications. The scope of the Gaussian model class can $e extended $y using a transformation of the origi- nal response variable# and with this extra *e)i$ility the model often provides a good empirical !t to data. +lso, within the speci!c context of geostatistics# the Gaussian assumption is the model-based counterpart of some widely used geostatistical prediction methods, including simple# ordinary and universal krig- ing (Journel and Hui $regts# ./012 Chil4es and 5elfiner# 1999). &e shall use the Gaussian model initially as a model in its own right for geostatistical data with a continuously varying response, and later as an important component of a hierarchically speci!ed generalised linear model for geostatistical data with a discrete response variable# as previously discussed in 7ection 1.4. 3.1 Covariance functions and the variogram + Gaussian spatial process# S,x6 :x 8" 2 # is a stochastic process with the property that for any collection{ of∈ locations} x , . , x with eachx 8"2# 1 n i ∈ 3.1. Covariance functions and the variogram 47 the joint distribution ofS9 S,x 16,...,S,xn6 is multivariate Gaussian. +ny process of this (ind is completely{ specified $y its} mean function#µ,x6 9 :;S,x6<# and its covariance function#γ,x, x �6 9 Cov S,x6, S,x �6 . { } In any such process# consider an arbitrary set of locationsx 1, . , xn# define S9 S,x 16,...,S,xn6 # writeµ S for then-element vector with ele mentsµ,x i6 and{G for then n matri)} with elementsG 9γ,x , x ). Then#S follows a × ij i j multivariate Gaussian distribution with mean vectorµ S and covariance matri) G. &e write this asS =>',µ S,G). n ∼ Now# letT9 i=1 aiS,xi). ThenT is univariate Gaussian with meanµ T 9 n aiµ,xi6 and variance i=1 � � n n 2 σT 9 aiajGij 9a �Ga, i=1 j=1 � � wherea9,a , . , a ). 8t must therefore $e the case thata �Ga 0. This 1 n ≥ condition# which must hold for all choices ofn# ,x 1, . , xn6 and ,a1, . , an6 constrainsGto$ea positive definite matrix# and the correspondingγ, 6 to $e a positive definite function. 3onversely# any positive definite function· γ, 6 is a legitimate covariance function for a spatial Gaussian process. · + spatial Gaussian process is stationary ifµ,x6 9µ# a constant for allx# and γ,x, x�6 9γ,u6# whereu9x x � i.e.# the covariance depends only on the vec- − tor difference $etweenx andx �. Additionally# a stationary process is isotropic ifγ,u69γ, u 6# where denotes :uclidean dist ance i.e.# the covariance $e- tween values|| || ofS,x6 at any|| · || two locations d epends only on the distance $etween them. Note that the variance of a stationary process is a constant#σ 2 9γ(0). &e then define the correlation function to$eρ,u6 9γ,u6/σ 2. The correlation function is symmetric inu i.e.#ρ, u6 9ρ,u). This follows from the fact that for anyu# Corr S,x6, S,x u6 9 Corr− S,x u6,S,u6 9 Corr S,x6,S,xAu6 # the second equality{ followin− g} from the{ stationarity− }ofS,x6. Hence#{ ρ,u6 9ρ,} u). − Brom now on, we will useu to mean either the vect orx x � or the scalar x x � according to context. &e will also use the term stationary− as a shorthand || − for|| stationary and isotropic. + process for whichS,x6 µ,x6 is stationary is called covariance stationary. Processes of this kind are very− widely used in practice as models for geostatistical data. In Chapter C# we introduced the empirical variogram as a tool for exploratory data analysis. &e now consider the theoretical variogram as an alternative characterisation of the second-order dependence in a spatial stochastic process. The variogram of a spatial stochastic processS,x6 is the function . V,x, x �6 9 >ar S,x6 S,x �6 . (3..6 C { − } 1 Note thatV,x, x �6 9 2 ;>ar S,x6 A >ar S,x �6 2Cov S,x6,S,x �6 <. In the stationary case# this simpli!es{ to}V,u6 9{σ 2 }−. ρ,u6 {which# incidentally#} explains why the factor of one-half is convention{ally− included} in the definition of the variogram. The variogram is also well defined as a function ofu for a limited class of non-stationary processes2 a one-dimensional example is a simple random walk# for whichV,u6 9 αu. Processes which are n on-stationary but for which 48 3. Gaussian models for geostatistical data V,u6 is well-defined are ca lled intrinsic random functions ,=atheron# 1973). &e discuss these in more detail in Section 3.9. In the stationary case the variogram is theoretically equivalent to the covari- ance function# but it has a num$er of advantages as a tool for data analysis, especially when the data locations form an irregular design. &e discuss the data analytic role of the variogram in Chapter D. Conditions for the theoretical validity of a specified class of variograms are usually discussed in terms of the corresponding family of covariance functions. Gneiting, SasvEari and 7chlather ,C??.6 present analogous results in terms of variograms. 3.2 Regularisation In 7ection 1.2.. we discussed briefly how the support of a geostatistical mea- surement could affect our choice of a model for the data. When the support for each measured value e)tends over an area, rather than $eing con!ned to a single point# the modelled signalS,x6 should strictly $e rep resented as S,x6 9 w,r6S ∗,x r6dr, (3 .C6 − � whereS ∗, 6 is an underlying, unobserved signal process andw, 6 is a weighting function. In· this case, the form ofw, 6 constrains the allowa ·$le form for the · covariance function ofS, ). 7peci!cally# ifγ, 6 andγ ∗, 6 are the covariance · · · functions ofS, 6 andS ∗, 6# respectively# it follows from (3.C6 that · · γ,u69 w,r6w,s6γ ∗,uAr s6drds. (3.3) − � � Now make a change of variable in (3.3) froms tot9r s# and de!ne − W,t6 9 w,r6w,t r6dr. − � Then (3.3) $ecomes γ,u69 W,t6γ ∗,uAt6dt. ,3 .4) � Typical weighting functionsw,r6 would $e radially sym metric, non-negative val- ued and non-increasing functions of r 2 this holds for the effect of the gamma camera integration in :xample 1.3, || where|| w,r6 is not known explicit ly $ut is smoothly decreasing in r # and for the soil core data of Example 1.4, where w, 6 is the indicator corres || ||ponding to the circular cross section of each core. In general,· the effect of weighting functions of this kind is to ma(eS,x6 vary more smoothly thanS ∗,x6# with a similar effect onγ,u6 $y comparison withγ ∗,u). An analogous result holds for the relationship $etween the variograms ofS, 6 · andS ∗, 6. Using the relationship thatV,u6 9γ,?6 γ,u6 it follows from (3.46 that · − V,u69 W,t6 V ∗,tAu6 V ∗,t6 dt. (3.D6 { − } � 3.3. Continuity and differentiability of stochastic processes 49 If the form of the weighting functionw, 6 is known# it would $ e possi$le to incorporate it into our model for the data.· This would mean specifying a model for the covariance function ofS �, 6 and evaluating (3.4) to derive the corresponding covariance function ofS, ).· Note that this woul d enable data with different supports to $e com$ined· naturally# for example soil core data using different sizes of core. + more pragmatic strategy# and the only availa$le one ifw, 6 is unknown# is to spec ify directly an appropriately smooth model for the covariance· function ofS, ). The question of regularisation· can also arise in connection with prediction# rather than model formulation. The simplest geostatistical prediction problem is to map the spatial signalS,x6# $ut in some applica tions a more relevant target for prediction might $e a map of a regularised signal, T,x6 9 S,u6du, � where the integral is over a disc with centrex i.e.#T,x6 is a spatial average ov er the disc. &e return to questions of this kind in Chapter 6. 3.3 Continuity and differentiability of stochastic processes The specification of the covariance structure of a spatial processS,x6 directly affects the smoothness of the surfaces which the process generates. +ccepted mathematical descriptors of the smoothness of a surface are its continuity and di@erentiability. %owever# for stochastically generated surfacesS,x6 we need to distinguish two kinds of continuity or differentiability. In what follows# we shall consider a one-dimensional spacex# essentially for notatio nal convenience. &e first consider mean-square properties, de!ned as follows. + stochastic processS,x6 is mean-square continuou s if :; S,xAh6 S,x6 2< ? ash 0. { − } → → Also,S,x6 is mean-square differenti able# with mean-square derivativeS �,x6# if S,xAh6 S,x6 2 : − S �,x6 ? h − → �� � � ash 0.