LETANÍA DE LA DOLOROSA

Mixed or hybrid spatial prediction models comprise of a combination of the techniques listed previously. For example, a mixed geostatistical model employs both correlation with auxiliary predictors and spatial autocorrelation simultaneously. There are two main sub-groups of mixed geostatistical models: (a) co-kriging-based and (b) regression- kriging-based techniques (Goovaerts, 1997), but the list could certainly be extended.

Note also that, in the case of environmental correlation by linear regression, we as- sume some basic (additive) model, although the relationship can be much more complex. To account for this, a linear regression model can be extended to a diversity of statisti- cal models ranging from regression trees, General Additive Models, neural networks and similar. Consequently, the mixed models are more generic than pure kriging-based or regression-based techniques and can be used to represent both discrete and continuous changes in the space, both deterministic and stochastic processes.

1.3 Statistical spatial prediction models 25

One can also combine deterministic, statistical and expert-based estimation models. For example, one can use a deterministic model to estimate a value of the variable, then use actual measurements to fit a calibration model, analyse the residuals for spatial correlation and eventually combine the statistical fitting and deterministic modelling (Hengl et al., 2007b). Most often, expert-based models are supplemented with the actual measurements, which are then used to refine the rules, e.g. using the neural networks (Kanevski et al., 1997).

Important sources:

F Rossiter D.G., 2005. Geostatistics, lecture notes, ITC, Enschede, Netherlands. F Nielsen, D. and Wendroth, O., 2003. Spatial and Temporal Statistics — Sampling

Field Soils and Their Vegetation. Catena-Verlag, Reiskirchen, 614 pp.

F Webster, R. and Oliver, M.A., 2001. Geostatistics for Environmental Scientists. Statistics in Practice. John Wiley & Sons, Chichester, 265 pp.

F Goovaerts, P., 1997. Geostatistics for Natural Resources Evaluation (Applied Geostatistics). Oxford University Press, New York, 496 pp.

F Isaaks, E.H. and Srivastava, R.M. 1989. An Introduction to Applied Geostatistics. Oxford University Press, New York, 542 pp.

F http://www.wiley.co.uk/eoenv/ — The Encyclopedia of Environmetrics. F http://geoenvia.org — A research association that promotes use of geostatisti-

2

Regression-kriging

As we saw in the previous chapter, there seems to be many possibilities to map environ- mental variables using geostatistics. In reality, we always try to go for the most flexible, most comprehensive and the most robust technique (preferably implemented in a soft- ware with an user-friendly GUI). In fact, many (geo)statisticians believe that there is only one Best Linear Unbiased Prediction (BLUP) model for spatial data (Gotway and Stroup, 1997; Stein, 1999; Christensen, 2001). As we will see further on in this chapter, one such generic mapping technique is the regression-kriging. All other techniques men- tioned previously — ordinary kriging, environmental correlation, averaging of values per polygons or inverse distance interpolation — can be seen as its special cases.

2.1 The Best Linear Unbiased Predictor of spatial data

Matheron (1969) proposed that a value of a target variable at some location can be modelled as a sum of the deterministic and stochastic components:

Z(s) = m(s) + ε0(s) + ε00 (2.1.1)

Residuals Actual distribution of the target variable

(if we would sample it very densely)

Regression function (trend) Field samples geographical space (s) Target variable ) ) ) ( ( ( ˆ ˆ s s s z ) (s z m e Final estimate by regression-kriging

Fig. 2.1: A schematic example of regression-kriging: fitting a vertical cross-section with assumed distribution of an environmental variable in horizontal space.

which he termed universal model of spatial variation. We have seen in the previous sections (§1.3.1and§1.3.2) that both deterministic and stochastic components of spatial variation can be modelled separately. By combining the two approaches, we obtain:

ˆ z(s0) = ˆm(s0) + ˆe(s0) = p X k=0 ˆ βk· qk(s0) + n X i=1 λi· e(si) (2.1.2)

where ˆm(s0) is the fitted deterministic part, ˆe(s0) is the interpolated residual, ˆβk are estimated deterministic model coefficients ( ˆβ0 is the estimated intercept), λi are kriging weights determined by the spatial dependence structure of the residual and where e(si) is the residual at location si. The regression coefficients ˆβk can be estimated from the sample by some fitting method, e.g. ordinary least squares (OLS) or, optimally, using Generalized Least Squares (Cressie, 1993, p.166):

ˆ

βGLS= qT· C−1· q
−1

· qT· C−1· z (2.1.3)

where ˆβGLS is the vector of estimated regression coefficients, C is the covariance matrix of the residuals, q is a matrix of predictors at the sampling locations and z is the vector of measured values of the target variable. The GLS estimation of regression coefficients is, in fact, a special case of the geographically weighted regression (compare with Eq.1.3.20). In the case, the weights are determined objectively to account for the spatial auto-correlation between the residuals.

Once the deterministic part of variation has been estimated, the residual can be interpolated with kriging and added to the estimated trend (Fig.2.1). The estimation of the residuals is an iterative process: first the deterministic part of variation is estimated using ordinary least squares (OLS), then the covariance function of the residuals is used to obtain the GLS coefficients. Next, these are used to re-compute the residuals, from which an updated covariance function is computed, and so on. Although this is by many geostatisticians recommended as the proper procedure, Kitanidis (1994) showed that use of the covariance function derived from the OLS residuals (i.e. a single iteration) is often satisfactory, because it is not different enough from the function derived after several iterations; i.e. it does not affect much the final predictions. Minasny and McBratney (2007) recently reported similar results — it is much more important to use more useful and higher quality data then to use more sophisticated statistical methods.

In matrix notation, regression-kriging is commonly written as (Christensen, 2001, p.277):

ˆ

zRK(s0) = qT0 · ˆβGLS+ λT0 · (z − q · ˆβGLS) (2.1.4) where ˆz(s0) is the predicted value at location s0, q0 is the vector of p + 1 predictors and λ0 is the vector of n kriging weights used to interpolate the residuals. The model in Eq.(2.1.4) is considered to be the Best Linear Predictor of spatial data. It has a prediction variance that reflects the position of new locations (extrapolation) in both geographical and feature space:

ˆ σ_RK2 (s0) = (C0+ C1) − cT0 · C−1· c0 + q0− qT· C−1· c0
T · qT· C−1· q
−1 · q0− qT· C−1· c0
(2.1.5)

2.1 The Best Linear Unbiased Predictor of spatial data 29 Field measurements horizontal space (s) Sampled variable (z) Covariate (q) (a) (b) (c) R2® 1.00 R2 = 0.50 R2® 0.00

Fig. 2.2: Whether we will use pure regression model, pure kriging or hybrid regression-kriging is basically determined by R-square: (a) if R-square is very high, then the residuals will be infinitively small; (c) if R-square is insignificant, then we will probably finish with using ordinary kriging; (b) in most cases, we will use a combination of regression and kriging.

where C0+ C1 is the sill variation and c0 is the vector of covariances of residuals at the unvisited location.

Obviously, if the residuals show no spatial auto-correlation (pure nugget effect), the regression-kriging (Eq.2.1.4) converges to pure multiple linear regression (Eq.1.3.14) because the covariance matrix (C) becomes identity matrix:

C =    C0+ C1 · · · 0 .. . C0+ C1 0 0 0 C0+ C1   = (C0+ C1) · I (2.1.6)

so the kriging weights (Eq.1.3.4) at any location predict the mean residual i.e. 0 value. Similarly, the regression-kriging variance (Eq.2.1.5) reduces to the multiple linear re- gression variance (Eq.1.3.16):

σ2_RK(s0) = (C0+ C1) − 0 + qT0 ·  qT· 1 (C0+ C1) · q −1 · q0 σ_RK2 (s0) = (C0+ C1) + (C0+ C1) · qT0 · qT· q
−1 · q₀

and since (C0+ C1) = C(0) = MSE , the RK variance reduces to the MLR variance: ˆ

σ_RK2 (s0) = ˆσOLS2 (s0) = MSE · h

1 + qT₀ · qT_{· q}
−1

· q₀i (2.1.7)

Likewise, if the target variable shows no correlation with the auxiliary predictors, the regression-kriging model reduces to ordinary kriging model because the deterministic part equals the (global) mean value (Fig. 2.2c, Eq.1.3.25).

The formulas above show that, depending on the strength of the correlation, the RK might turn to pure kriging — if predictors are uncorrelated with the target variable —

or pure regression — if there is significant correlation and the residuals show pure nugget variogram (Fig. 2.2). Hence, pure kriging and pure regression should be considered as only special cases of regression-kriging (Hengl et al., 2004a, 2007b).