Title:

The detection of change in spatial processes with environmental applications

Ever since Halley (1686) superimposed onto a map of land forms, the direction of trade winds and monsoons between and near the tropics and attempted to assign them a physical cause, homosapiens has attempted to develop procedures which quantify the level of change in a spatial process, or assess the relationship between associated spatially measured variables. Most spatial data, whether it be originally point, linear or areal in nature, can be converted by a suitable procedure into a continuous form and plotted as an isarithmic map i.e. points of equal height are joined. Once in that form it may be regarded as a statistical surface in which height varies over area in much the same way as the terrain varies on topographic maps. Particularly in environmental statistics, the underlying shape of the surface is unknown, and hence the use of nonparametric techniques is wholly appropriate. For most applications, the location of data points is beyond the control of the mapmaker hence the analyst must cope with irregularly spaced data points. A variety of possible techniques for describing a surface are given in chapter two, with attention focusing on the methodology surrounding kernel density estimation. Once a surface has been produced to describe a set of data, a decision concerning the number of contours and how they should be selected has to be taken. When comparing two sets of data, it is imperative that the contours selected are chosen using the same criteria. A data based procedure is developed in chapter three which ensures comparability of the surfaces and hence spurious conclusions are not reached as a result of inconsistencies between surfaces. Contained within this chapter is a discussion of issues which relate to other aspects of how a contour should be drawn to minimise the potential for inaccuracies in the surface fitting methodology. Chapter four focuses on a whole wealth of techniques which are currently available for comparing surfaces. These range from the simplest method of overlaying two maps and visually comparing them to more involved techniques which require intensive numerical computation. It is the formalisation of the former of these techniques which forms the basis of the methodology developed in the following two chapters to discern whether change/association has materialised between variables. One means of quantifying change between two surfaces, represented as a contoured surface, is in terms of the transformation which would be required for the two surfaces to be matched. Mathematically, transformations are described in terms of rotation, translation and scalar change. Chapter five provides a geometrical interpretation of the three transformations in terms of area, perimeter, orientation and the centre of gravity of the contour of interest and their associated properties. Although grid resolution is fundamentally a secondary level of smoothing, this aspect of surface fitting has generally been ignored. However to ensure consistency across surfaces, it is necessary to decide firstly, whether data sets of different sizes should be depicted using different mesh resolutions and secondly, how fine a resolution provides optimal results, both in terms of execution time and inherent surface variability. This aspect is examined with particular reference to the geometric descriptors used to quantify change. The question of random noise contained within a measurement process has been ignored in the analysis to this point. However in practice, some form of noise will always be contained within a process. Quantifying the level of noise attributable to a process can prove difficult since the scientist may be over optimistic in his evaluation of the noise level. In developing a suitable set of test statistics, four situations were examined, firstly when no noise was present and then for three levels of noise, the upper bounds of which were 5,15 and 25%. Based on these statistics, a series of hypothesis tests were developed to look at the question of change for individual contour levels i.e. local analysis, or alternatively for a whole surface by combining the statistics and effectively performing a multivariate test.
