
This approach is aimed to model the overall distribution of properties throughout space. It will sketch the global trend of distribution as a simplified surface.
Trend surface modelling can be applied in the following geographic information context:
The principle of a trend surface model is a regression function that
estimates the property value Pi at any location, based on the Xi,Yi coordinates
of this location. The general function is:
A trend surface model is a particular case of a bivariate regression model with two independent variables, the coordinates X and Y and a dependent variable, the thematic variable P to be modelled. One can selected a linear regression function (first order) or, if the spatial distribution is more complex, a polynomial function (2nd, 3rd, …, or nth order). The modelled surface will correspond to respectively a flat oriented plane or a curved surface with an increasing number of curvatures.
In order to illustrate the principles of trend surface modelling, let’s take a phenomenon with a very obvious and observable spatial distribution: altitude. Let us suppose that we are starting our process of spatial distribution description with a sample of nn data point measurements, irregularly distributed throughout the study area to be described. One can identify 3 stages that are common to most modelling methods:
The following figure illustrates the real distribution of altitude values within a
study area as well as different trend surfaces modelling this
distribution.
Let us now illustrate the application of such a surface modelling process on a spatial sample of change index. Our objective is to summarise with the use of a trend surface the spatial distribution of change index values measured at different point locations in a study area. We have measured the population change of 55 localities in the district of the Sarine (in the Canton of Fribourg in Switzerland) during the period 19001986. The index change selected for the description of this evolution is the Normalised Difference (ND) as discussed in the section 2.1.1 of the Unit 2. Original X,Y coordinates were standardised in order to limit their unit of measurement and their influence on regression coefficient values.
Several regression functions were applied on this set of 55 points and Table summarises 3.2 their different parameters.
In order to test the significance of an individual trend surface model order, an F ratio should be derived from the coefficient of determination (%R^{2}) and the degrees of freedom (Df) associated with the fitted surface and its residuals (Davis 1986) (Unwin 1975). F ratio value is computed as follows:
The F ratio values computed for each regression function order ranging from 1 to 8 are listed in the last table. They are all greater than their respective critical F value at 95 % confidence level and therefore express a significant trend. However, as our sample is made of 55 observations and as the number of coefficients increases rapidly with high function orders, it sounds reasonable to consider trend surface models up to the quintic order (order 5). From a statistical point of view there is a technique to assess the significance of contribution for selecting a higher order a lower one already significant. It is based on an F ratio value that expresses the extra contribution of an n+1 order over an n order (Davis 1986) (Unwin 1975). It is an interesting test as it compares the improvement of the fit with the increase in complexity of the fitting model. This F ratio value is computed as follows:
The following illustrates the application of the test of increasing order significance for the 8 trend surface models applied to the 55 localities of the Sarine district.
The interpretation of significance of trend functions listed before (Table) leads to the conclusion that the 8th order is the most suitable for modelling the spatial distribution as it takes into account 98.1% of the point distribution. This conclusion is confirmed by the test of significance about the additional contribution of the order 8 against the previous order (Table). However, when comparing the interpolated surface (Figure) with the modelled trend surfaces illustrated in the next figure, one can observe several strong discrepancies and artefacts generated by trend models, particularly with higher orders:
Based on these general issues and keeping in mind that a relevant trend
surface model is aimed to summarise the overall spatial distribution of
properties, one should concludes that the most appropriate model for the
description of this distribution could be the 4th order function. It explains
around 60% of the overall variations and contains 15 coefficients, almost four
times less than the number of observations.