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The validation was conducted for the model including no spatial trend function (Eq. 8.3), the model with a spatial trend function of low dimension (Eq. 8.4) and the model with a trend function of optimal dimension concerning AIC (Eq. 8.51).

The results of the investigation were presented as the trend function.

Additionally the dimension of the spatial trend function was decreased slightly (edf = 93.38).

By increasing the dimension of the 2-dimensional spatial trend function, the model was improved significantly.

An optimum value was reached at approximately 120 degrees of freedom (edf) for the spatial trend function (Eq. 8.5).

The test is based on the separation between an empirical version of the relative trend function and its greatest convex minorant.

The methodology employed allowed the simultaneous estimation of non-linear model effects of causal covariates and, to account for spatial autocorrelation, of a 2-dimensional spatial trend function.

The 2-dimensional trend function indicated a strong spatial effect, and thus explained by far the highest proportion of variation in larva density.

The inventory plots are located exactly via coordinates, and hence a 2-dimensional spatial trend function is fitted instead of spatially correlated random effects for distinct areas.

Again model improvement was validated by a stepwise integration of predictors, and by increasing the basis dimension of the 2-dimensional spatial trend function (Tables 3 and 4).

In this investigation the modeling approach for determining the effect of spatial heterogeneity on overdispersion was much more flexible due to the complex spatial trend function (Table 2).

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