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Fig. 7.2 Concept of trend illustrated by means of two-dimensional graphs (After Davis
1973
).
Value (vertical scale) is plotted against distance (horizontal scale). (a) Collection of original data
points and smooth curve on which they lie; (b) Straight-line trend fitted to the observations; (c)
Quadratic trend; (d) Cubic trend.
Shadings
represent positive and negative residuals from the
trends (Source: Agterberg
1984
, Fig. 2)
Some problems of trend surface analysis can be illustrated by the fitting of curves
to data collected along a line. An example is shown in Fig.
7.2
(after Davis
1973
). The
observations fall on a smooth curve which, however, is not a low-degree polynomial.
The magnitudes of the residuals decrease when the degree of the polynomial is
increased. This artificial example illustrates two important points:
1. A good fit is not necessarily the object of trend analysis. Instead of this, the aim
usually is to divide the data into a regional trend component and local residuals
which can be linked to separate spatial processes of regional and local signifi-
cance, respectively. Examples of interpretation of trends and residuals will be
given later in this chapter.
2. Each curve-fitting of Fig.
7.2
yields residuals that form a continuous curve.
Adjoining residuals along a line therefore are not statistically independent.
Instead of this, the residuals are spatially correlated.
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