Models of the spatial location of individuals
This chapter introduces the exploratory tool known as the “ЛГ-fu action” to characterize a univariate homogeneous point pattern. It discusses the behaviour of the ЛГ-function in the benchmarking case of complete spatial randomness and in the two major violations related to clustering and inhibition. It also introduces the “Monte Carlo procedure” to test the significance of the departure from the complete spatial randomness benchmark.
Ripley’s K-function
The methods used to identify the spatial location patterns of economic agents described in Chapter 6, both the quadrat-based methods and the Clark-Evans test, rely on zoning and hence require the arbitrary selection of a partitioning scheme. Therefore, they do not allow for the inspection of spatial location patterns at varying spatial scales simultaneously. In light of this, Ripley (1976; 1977) introduced the idea that a single number (as given by an index or a test statistic) cannot summarize the observed pattern at different spatial scales, and, by extending the work of Bartlett (1964), he proposed a functional statistic able to overcome the problem. The importance of this statistic, the ЛГ-fu notion, lies in its dual role as a summary descriptive measure of the spatial characteristics of the observed pattern and as a tool that can be used for parameter estimation and goodness-of-fit testing of the point pattern generating process. It is, indeed, not an exaggeration to say that Ripley’s ^-function is the most common basic tool used to analyze the spatial distribution of dimensionless events on a continuous space. In particular, it has proved a powerful tool for testing for the presence of spatial dependence or interaction within a micro-geographical pattern. As a result, since its introduction, the АГ-function has been widely applied in various fields such as ecology, epidemiology and, more recently, economics (Arbia, 1989; Arbia and Espa, 1996; Barff, 1987; Feser and Sweeney, 2000; Sweeney and Feser, 1998; Marcon and Pucch, 2003).
Formally, the ЛГ-function is an alternative description, with respect to the second-order intensity function Я2 (x, r) introduced in Chapter 6, of the second-order properties of a spatial point process that can be adopted under the assumption of stationarity and isotropy (Diggle, 2003). A stationary and isotropic spatial point process is such that Я2 (x,y) = Я2 (
where X denotes the intensity (that is constant because of the stationarity assumption), corresponding to the mean number of events per unitary area. Therefore, XK (d) indicates the expected number of further points up to a distance d of a typical point (Ripley, 1977). In the empirical economic analyses where the data-generating point process is stationary and isotropic (that is when the territory is essentially homogeneous), the ЛГ-function quantifies properly the mean (global) level of spatial interactions between the economic agents (such as firms or consumers) up to each distance d.
Under the further tenable assumption that the point process is orderly, which essentially implies that each location cannot host more than one point, Ripley (1976) has shown that the link between К (d) and Я2 (d) is:
The link between the two functions lies in the fact that both describe the distribution of the distances between pairs of points in a point pattern, where К (d) is related to the cumulative distribution function and Я2 (d) to the probability density function (Diggle, 2003). In the context of practical applications, however, it is more convenient to work with the ЛГ-function rather than with the second-order intensity function, as the former can be more straightforwardly estimated from the observed dataset (Diggle, 2003). It is because of this important practical advantage that the ЛГ-function has become the most popular statistic used to detect the presence of spatial dependence, positive or negative, in a spatial point pattern.
