For point sets in the plane a simple and often useful boundary is the convex hull (Figure 4‑26). This is taken as the convex polygon of smallest area that completely encloses the set (it may also be used for line and polygon sets).

Figure 4‑26 Convex hull of sample point set

 

If the data you are working with consist solely of a single point set then the convex hull of this set encloses all the information you have available, even though you may know that the data extend beyond these limits.

The convex hull may also be useful in the analysis of optimum location and networking problems. For example the centre of gravity of a set of points in the plane will always lie inside or on the convex hull of the point set, and the sequence of points which comprises the convex hull form part of the solution to certain network routing problems (systematic minimal tours of a set of locations). However, it is important to note that the convex hull of a set of objects (points, lines etc.) is affected markedly by extreme values, so outliers and/or errors can have a substantial impact on the shape, area, perimeter length and centroid of a convex hull. For example, one or two extreme values can substantially increase the area enclosed by the convex hull. Typically, however, convex hulls under-estimate area for many applications and hence will over-estimate densities if computed using an area measure based upon this hull.

By definition the convex hull includes a number of points as part of its boundary. For some problems this may be satisfactory whilst for others it may not be suitable. For example, when analysing the distribution of points in a study region it may be preferable to minimise the potential distortions to sample statistics by avoiding points that lie on or close to the convex hull. Creating an inner buffer of the convex hull and analysing the subset of points lying within the core of the region, with or without consideration of points lying in the buffer zone, may be preferable. Alternatively the convex hull may be uniformly increased (buffered) in size, for example by a given percentage of area or by a specified distance such as half the mean distance to the nearest-neighbours of enclosed points.