Plate 9.5: Gas Welders Working along Madzindadzi Road
2.9 Fundamentals of Spatial Statistics
47 gives the fundamentals of spatial statistics.
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analysing data from agriculture trials gave an implicit application of continued spatial variation (Fisher, 1966). Geostatistical methods were first used in South Africa to forecast the availability of minerals over a defined spatial region between 1955 and 1965 (Matheron, 1963).
In the arena of forestry, it has been noted that Matern developed a parametric family of correlation functions to represent the position of trees in a densely populated forest in the 60s (Diggle, 2010).
The development of methods for studying continued spatial variation was followed by two methodological breakthroughs, that is, discrete spatial variation and spatial point process. This breakthrough gave rise to second and third branches of spatial statistics. It is further noted that
“Besag (1974) suggested models and related methods of inference for analysing spatially discrete, or “lattice,” data, while Ripley (1977) set out a methodical approach for analysing spatial point process data” (Diggle, 2010, p. 9). This thesis focuses on the third branch of spatial statistics, spatial point processes.
The fundamental feature of spatial point pattern is its concern in handling measures of analysis whose realisation consists of a finite or countable infinite set of points in the plane (Diggle, 2010). Two classical examples of situations that can be examined using spatial point pattern have been given (Diggle, 2010). First, in studying the locations of petty crimes that occurred in each neighbourhood per given period, key research questions would include the incidence of crimes, spatial variation in intensity, and evidence for concentration of crimes. Second, defects in the crystal surface of a silicon wafer can be inspected and their locations recorded.
In such a scenario, a study can aim to determine the occurrence of defects, spatial trends in intensity, and spacing between defects. Interpreting these examples, spatial point pattern is worried about the intensity of an observation, spacing and spatial dependents between observed
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points, the overall design of observed realities, and whether the perceived pattern is explained by existence of covariates.
2.9.2 Data Needs for Spatial Point Pattern
Spatial point processes require data that satisfy four essential needs (Baddeley, 2010). These four pre−requisites set the basis of the sampling design in spatial point pattern analysis. Firstly, the measures of analysis should be observed inside a clearly defined sampling window where the boundaries known. Where they are unknown, they must be accurately estimated using defensible methods (Ripley and Rosson, 1977; Moore, 1984). Secondly, actual position of points under study must be accurately established or estimated with substantial accuracy. This, therefore, calls for the implementation data capturing methods that give the accurate spatial location of individual points. Thirdly, use of accurate data capturing methods is of highest significance since one point should be linked with a spatial location, that is, no two points should lie on one location. Lastly, all points in the sampling window must be surveyed.
Reduced to sampling language, a census as opposed to a sample survey must be implemented to enable accurate mapping of a phenomenon in question.
A qualifying example of disease mapping was exploited to suggest that observing the four conditions outlined in the foregoing paragraph, there exist two more additive conditions (Waller and Carlin, 2010). The sampling window must be small enough to allow the production of maps that present a good geographical resolution. On the other, population in individual windows must be huge enough to maintain statistical precision, that is, low variance. Satisfying the two additive conditions is not a simple task. It is claimed that it is hard to have a small geographical area with a very large, satisfying population (Waller and Carlin 2010). There are further details on small expanse estimate (Ghosh and Rao, 1994; Ghosh et al. 1998; Rao, 2003).
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The goals of analysing spatial point pattern have been categorised into three; intensity of observed measures of analysis, interaction between points, and conjoint focus on intensity and contact (Baddeley, 2010). On one hand, intensity aims to determine the “rate of occurrence, abundance, or incidence of the events recorded in the point pattern” (Baddeley, 2010, p. 342).
This is essential in quantifying the number of measures of analysis per unit area, and subsequently maps them. In the current study, the goal is to enable relevant authorities to collect tax from the informal trade sector, so intensity would provide revenue authorities with number of operators per unit area, and whether this number constant or spatially varying. On the other, inter−point interaction aims to compile data related to interdependence of points in space and the assumed positive correlation between them. Perhaps, reviewing how the spatial point data needs were satisfied in other studies clarifies the application of spatial point pattern.
2.9.3 Application of Spatial Point Pattern
This section presents one instrumental case that used the spatial point pattern to analyse measures of analysis. In that case, a spatial statistical approach was exploited to identify and map deprived areas where access to grocery foods is restricted due to impoverishment in the sampling window (Lee and Lim, 2009). This was accomplished by creating census block clusters in the city of Buffalo, New York. In each census block group, standardised local statistics was mapped. It has been espoused that mapping only provides a rough visual impression, without providing useful understanding into the actual needs of the study population (Lee and Lim, 2009). In this thesis, mapping will be complemented by ‘hard’ proof of the spatial variation of manufacturers’ anticipated behavioural reaction to the suggested tax measure.
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