Parametric regression analysis assumes the following:

• The error is a random variable with a mean of zero, conditional on the explanatory variables.
• The independent variables are measured with no error.
• The predictors are linearly independent; it is not possible to express any predictor as a linear combination of the others.
• The error terms are uncorrelated, that is, the variance–covariance matrix of the error terms is diagonal and each nonzero element is the variance of the error.
• The variance of the error is constant across observations (homoscedasticity).

This method is a simple way to model large-scale trend and can include additional explanatory variables to improve predictions. This method cannot be used to model complex surfaces unless additional explanatory variables are included that explain most of the variation.

In the context of optimization, see how to use the results of the geospatial methods to address specific optimization questions.

In addition to mapping, splines and kernel smoothers can also be used to help detect outliers, if the value at a known sampling point is significantly different from the value of the interpolated value at that point.

Using splines and kernel methods to visualize general trends in data can be extremely useful as an extension of exploratory spatial data analysis. These methods do not require the estimation of a statistical model of spatial correlation, and can therefore be used early in the data analysis before more complex and computationally intensive methods, such as kriging, are used. Smoothing methods can help in the development of a CSM, capture general trends in data, and detect outlying data points. Another major strength is the flexibility to incorporate break lines and barriers into the interpolation.

Smoothing methods, however, should not be regarded as a replacement for more formal statistical methods. In particular, the manual choice of smoothing parameter leads to some subjectivity in the resulting maps. Although the maps resulting from splines and kernel methods may be similar to those from advanced methods such as kriging, in general smoothing methods do not provide uncertainty estimates.

In the context of optimization, see how to use the results of the geospatial methods to address specific optimization questions.

Same as parametric regression.

This method is flexible and can model almost any surface. The method can include explanatory variables other than the spatial coordinates to improve predictions. Few parameters must be estimated. Results can be very similar to kriging with less effort, but the uncertainty estimates may not be as accurate.

In the context of optimization, see how to use the results of the geospatial methods to address specific optimization questions.

A basic exploratory data analysis is performed using a histogram (Figure 62). The histogram shows the zinc concentrations are highly skewed. Any interpolation method will work better if it transforms the concentrations to make the data more symmetric. A logarithmic (log) transformation is often used to transform concentrations, and so that approach is used here.

Figure 62. Histograms of zinc concentrations.

The map shows that the zinc concentrations are spatially correlated. This conclusion can be confirmed by looking at the h-scattergram in Figure 63, which shows on a scatter plot all pairs of data points having a specified range of separation distances. The correlation coefficients are also shown on the plots. For example, the correlation between pairs of points less than 80 m apart is 0.683. Up to a separation distance of approximately 500 m, there is still significant spatial correlation between zinc concentrations. This correlation suggests that the spatial coordinates will make reasonable predictor variables in the regression. The variogram shown in Figure 64 is another way to examine spatial correlation. In this case, the variogram confirms that the range of spatial correlation is approximately 500 m. The horizontal dashed line is the variance of the log of zinc concentrations (log zinc). The semivariance reaches the variance at approximately 500 m.

Figure 63. H-scattergrams.

Figure 64. Variogram.

The concentrations of zinc are higher near the river. Figure 65 shows that the relationship between log zinc concentrations and distance to the river is nonlinear. By taking the square root of the distance, however, the relationship becomes approximately linear. Figure 66 shows the log zinc concentrations as a function of the square root of the distance.

Figure 65. Relationship between log zinc concentration and distance to the river.

Figure 66. Log zinc concentrations as a function of the square root of the distance.

The other available covariates (or predictor variables) for this site are soil type and flood frequency. Three soil types have been identified throughout the area of interest (Figure 67). As shown in Figure 68, the boxplot, type 1 soil is associated with higher zinc concentrations than the other types. An alternative to the box plot is a density plot. The density plot in Figure 69 shows that type 3 soil is associated with a wide range of log zinc concentrations, so this soil type may not contribute much to prediction accuracy. Flood frequency is shown in Figure 70, and the flood frequency class box plot in Figure 71. Flood frequency has also been categorized into three classes, with class 1 (closest to the river) representing the locations that flood most frequently.

Figure 67. Soil type map.

Figure 68. Soil type box plots.

Figure 69. Soil type probability density plot showing the frequency of zinc concentrations for each soil type.

Figure 70. Flood frequency class 1 (closest to the river) representing the locations that flood most frequently.

Figure 71. Flood frequency class box plots.

A simple parametric linear regression model predicts zinc concentrations from the three predictor variables available: soil type, flood frequency, and distance to the river. The model diagnostic output from the R package mgcv is shown below. The fit is reasonably good, with an R-squared value of 66%, meaning that the model can predict approximately 66% of the variation in the data. The generalized cross validation (GCV) score is 0.163. Comparing this score to the GCV score for an alternative model determines which model is better. Figure 72 includes the diagnostic plots which show that the assumptions of linear regression are approximately satisfied.

Diagnostic Output from the R Package mgcv

##
## Family: Gamma
## Link function: log
##
## Formula:
## zinc ~ soil + sqrt(dist) + ffreq
##
## Parametric coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 7.06916 0.07571 93.376 < 2e-16 ***
## soil2 -0.37666 0.09493 -3.968 0.000112 ***
## soil3 -0.23286 0.14926 -1.560 0.120854
## sqrt(dist) -1.75893 0.20808 -8.453 2.39e-14 ***
## ffreq2 -0.46702 0.08726 -5.352 3.22e-07 ***
## ffreq3 -0.46302 0.10744 -4.310 2.96e-05 ***
## —
## Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ‘ 1
##
##
## R-sq.(adj) = 0.658 Deviance explained = 71.7%
## GCV = 0.16301 Scale est. = 0.17469 n = 155

Figure 72. Diagnostic plots.

Nonparametric regression captures the remaining spatial structure by using a thin-plate regression spline, one of many kinds of splines and kernel methods that could be used. Once the spline is added to the existing model, the R-squared rises to 82%, which is considered good. The GCV score is also significantly lower (0.115). Figure 73 and Figure 74 include the predicted zinc concentrations and associated error.

Diagnostic Output from the R package

##
## Family: Gamma
## Link function: log
##
## Formula:
## zinc ~ s(x, y, bs = “ts”, k = 100) + s(sqrt(dist)) + soil + ffreq
##
## Parametric coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 6.26383 0.05608 111.699 < 2e-16 ***
## soil2 -0.11421 0.10748 -1.063 0.2900
## soil3 -0.30312 0.15727 -1.927 0.0561.
## ffreq2 -0.62153 0.07155 -8.687 1.43e-14 ***
## ffreq3 -0.63759 0.10922 -5.837 4.05e-08 ***
## —
## Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ‘ 1
##
## Approximate significance of smooth terms:
## edf Ref.df F p-value
## s(x,y) 17.261 99.000 1.035 2.69e-14 ***
## s(sqrt(dist)) 3.721 4.572 12.387 2.48e-09 ***
## —
## Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ‘ 1
##
## R-sq.(adj) = 0.818 Deviance explained = 87.4%
## GCV = 0.11476 Scale est. = 0.082056 n = 155

Figure 73. Predicted zinc concentrations.

Figure 74. Prediction standard error.

If the residuals of the nonparametric model are evaluated using an h-scatterplot (see Figure 75), it becomes clear that they have no remaining spatial correlation. Thus, the model has combined the information from the predictor variables and the spatial correlation.

The model output (predictions and prediction standard error) can now be used for optimization. For example, the output could be used to determine where additional sampling would be most beneficial to reducing uncertainty about the extent of elevated concentrations. The map of predictions shows that the highest concentrations are located along the river shoreline. The map of prediction standard errors shows that within the area of higher concentrations the highest levels of uncertainty are located in the southwest corner of the map at the bend in the river. These areas with the highest uncertainty (highest standard error) should be targeted for additional sampling. Conversely, if resampling of the entire area was planned, the prediction standard errors could be used to indicate areas of low uncertainty where less dense sampling might be appropriate.

Figure 75. Residuals of the nonparametric model.