McCullagh Page 1 09/08/2006 Detecting Hotspots in Time and Space Michael J McCullagh Senior Research Fellow, University of Nottingham 1. Introduction The numerical analysis of spatial point distributions has interested academics for many years. A very early work that resulted in the famous Moran’s I statistic was that by Moran (1948). An early important collection of work on general spatial analysis is that edited by Berry and Marble (1968). The development of spatial point analysis theory is illustrated by such later books by Cliff and Ord (1972, 1981), more recent papers by Ord and Getis (1995), Anselin (1995) and Gatrell et al (1996), the overview by Getis and Ord (1996), leading to large number of recent studies such as Cho (2003) using the techniques that have been developed over the last fifty years. The intention of applying the analytical tools is to outline areas on a map that exhibit significant difference from their surroundings, either on the basis of areal data such as administrative counties, or on the basis of point data such as the location of crimes, disease, or intentional fires. This paper will concentrate on the analytical process in relation to point data distributions. Point pattern events on maps change through time, so an important aspect stressed by many authors (see Ma et al, 2006; Kulldorff, 2001; Diggle et al, 1991) is to try to make the linkage between the spatial map and temporal changes. This has proved quite difficult. The importance of attempting to recognise significant events in time is demonstrated by a simple example (Figure 1, from Ratcliffe et al, 2001), where the graph shows the number of burglaries committed in part of Nottingham City over a period of rather more than a year. Figure 1: Looking for Hotspots in Time: Linear Weighting Adaptive Thresholding The course of events can be represented by a weighted fifteen week running mean, and the standard deviation limits calculated and drawn either side of the dataset. Every so often a value will lie outside the ±1SD range indicating possibly significant changes in the burglary events in the district, either up or down. These ISG06 Shah AlamMcCullagh Page 2 09/08/2006 represent a simple series of hotspots (or coldspots) through the time period and are of great importance in terms of the police targeting their scarce resources at the time and in the areas that need them most and also being able to relax their vigilance in areas that are locally improving. Figure 2: Timeline Hotspot Recognition - Police beats overlaid with one kilometre square grids cells. (a) shows standard mapping practice. The weighted threshold result is shown in (b) where the number of upper threshold crossings during the year is mapped. Figure 2a, also from Ratcliffe et al, 2001, shows the traditional mapping strategy of displaying the total number of crimes in each small cell area for the year. This gives an overall picture of crime, but does not show where significant increases or decreases are occurring. Figure 2b shows areas that are worsening rapidly as those with high counts of threshold crossing during the year. It is apparent that targeting of resources to prevent further deterioration is needed in different cells from those where the highest continuous level of crime is found. This simple temporal example demonstrates the need for a mixed spatial and temporal approach to hotspot formation. In most cases, as will be seen later in this paper, emphasis is usually placed on the spatial hotspot with only simplistic attempts to tie in temporal changes because of the complexities involved. 2. What is a Hotspot? The significantly different areas detected by the applied statistics (usually termed LISA: Local Indicators of Spatial Autocorrelation) are traditionally termed hotspots, hotpoints or occasionally hotbeds (Ratcliffe et al, 1999). The statistics used to define the hotbed coverage may result in circular, elliptical or amoeboid shapes depending on the method or application used. There is a wide choice of approaches available. A few are shown below in Figure 3 (Ratcliffe et al, 2001). In all cases the point data set for residential burglary is identical; only the hotspot calculation method has changed. In Figure 3a the STAC (2005) ellipse method, one of the tools in Crimestat (Levine, 2004, 2006) has been used. The location and size of possible hotspots is calculated using n hotspot set by the user. The problem with STAC is that neither crime nor disease nor intentional fires follow an ellipsoid pattern, so the shape of the hotspot is ill-defined although the statistical significance of the ellipse is assured. Figure 3b uses a standard interpolation program, Vertical Mapper (2006) in MapInfo, to develop a crime surface from which hotspots may be inferred. Note that unlike STAC or Crimestat no boundary is set to the hotspot coverage. Figure 3c provides and isodensity surface based on the number of burglary locations within a given radius for a fixed rectangular grid. There is, thankfully, considerable correspondence between the isodensity map and the Vertical Mapper surface, but again no significant boundary hotspot edge limit. Figure 3d solves this problem by applying Getis’s G* LISA statistic (Getis ISG06 Shah AlamMcCullagh Page 3 09/08/2006 and Ord, 1992, 1995) to the isodensity surface in Figure 3c using the Hot Spot Detective add-on to MapInfo. In this case boundaries are stated at given Figure 3: Four different ways of determining hotspot location. Some show “significant” boundaries, and some do not. See text for detail. significance levels. It is clear from 3d that the clear is a very clear limit to the hotspot areas where the burglary numbers fall off rapidly. Note that the hotspot shape is not really ellipsoid or circular, but more usually – for burglary at least – follows the residential housing pattern. 3. How are Spatial Hotspots Detected? The resumé above has tried to show what hotspots are, but has not discussed the wide range of measures used to calculate them This section of the paper will outline some of the more common approaches in greater detail. A number of books cover this area quite well; in particular sections of Chainey and Ratcliffe (2005) with the accent on crime mapping, and Cromley and McLafferty (2002) in the field of public health. The original research thrust considered the development of LISA statistics for geographical areas, such as local municipal