# Standard Deviation Analysis

This approach involves determining the mean center of the series and drawing rectangles or ellipses around the mean center showing the areas that represent one or two standard deviations away from the mean (Boba, 2009). They are drawn to demonstrate clusters of points that would not be expected from random chance (Paynich and Hill, 2010).

Like choropleth mapping, a limitation to depicting hot spots using ellipses is that hot spots are rarely depicted accurately with bounded polygons (Eck et al., 2005; Paynich and Hill, 2010). However they can be useful for making comparisons of hot spots across time (Paynich and Hill, 2010). For example, an analyst can be tasked with evaluating the effects of a police operation on a hot spot. As part of his or her evaluation, the analyst creates standard deviation ellipses with incident data before and after a police operation. The analyst can then determine whether there was a reduction or displacement in crime.

# Grid Cell Mapping Analysis

This method is sometimes described as density analysis and compensates for the limitations noted above with choropleth mapping and standard deviation analysis. This approach uses surface estimation techniques and illustrates the surface of a geographical area (Ratcliffe, 2004) with rasters (Gorr and Kurland, 2012). This process first involves an analyst calibrating two parameters, specifically cell size and search radius, so that the results are meaningful and useful. The method involves placing a grid or fishnet on top of a map (creating a matrix of cells). Then a mathematical function visits the center of each cell and performs a calculation on that cell, as well as within a predetermined search radius or bandwidth (Harries, 1999; Eck et al., 2005; Paynich and Hill, 2010).

In simple density analysis, when the mathematical function is applied to each cell, the number of incidents within a given radius are added together and then divided by the area of the radius; this value is then assigned to the cell (Harries, 1999; Mitchell, 1999; Eck et al., 2005; Boba, 2009; Gwinn et al., 2008). Therefore “a cell’s score does not represent the number of incidents in that cell but the number of incidents

‘near’ that cell divided by the area ‘around’ that cell, approximating the concentration of activity” (Boba, 2009, p. 271). In other words, the cell’s density value is an estimate and is influenced by incidents found within the search radius placed on top of the grid cell (Boba, 2009).

*Kernel Density Interpolation*

Another approach to density analysis is kernel density interpolation or smoothing techniques. Instead of simply adding up all the points within a radius as with simple density analysis, a bell-shaped function or kernel is applied over every cell (Gorr and Kurland, 2012). In other words, greater weight is given to incidents closer to the center of the radius (Eck et al., 2005; Gwinn et al., 2008; Paynich and Hill, 2010). Dual kernel density interpolation is similar but involves producing “a risk value associated with crime density” (Paynich and Hill, 2010, p. 378) and allows for comparative density analysis. In other words, comparisons can be made between two different crime types or crimes at two different time periods (Paynich and Hill, 2010). It is apparent that density analysis does not depict physical boundaries and is consequently a “much more realistic image of the shape of the hot spot distribution” (Paynich and Hill, 2010, p. 378). It is also advantageous over point maps because overlapping points or stacked points are added together and represented with a single color (Harries, 1999). However, analysts must still consider three parameters when constructing density maps. First, the analyst must determine a threshold for what defines a hot spot. Values are assigned to the output raster cells, and it is at the discretion of the analyst to determine the numerical value at which a location is considered a hot spot. In other words, those areas of greater density above this threshold are then considered hot spots (Gorr and Kurland, 2012). Second, changes to either the search radius or grid cell size can yield different maps (Harries, 1999; Boba, 2009). A smaller search radii will reveal greater local variation or more specificity (Eck et al., 2005), while a larger selection will show long rolling hills for the surface (Gorr and Kurland, 2012). Likewise, the choice of cell size or spatial resolution (Gwinn et al., 2008) will affect the smoothness of the surface, with smaller cell sizes showing finer resolution. A third consideration is the size of the study area. Choosing different study areas can have an effect on the appearance of the computed density surface. As an example, an analyst could choose as his or her study area an artificially drawn square on a GIS map to represent a city’s boundaries. Alternatively, if an analyst uses a more accurate GIS shape file containing the official boundaries of a city, kernel density calculations would then be more accurate, but they would produce a different type of map.

Currently, there are no hard rules for how analysts decide on setting these parameters in a GIS. Analysts first study crime points and visually determine the boundaries of hot spots. The analyst then calibrates parameter values in the GIS for density analysis until they resemble the analyst’s expert judgment as to where the boundaries are (Gorr and Kurland, 2012). One limitation of density analysis, and another reason why density maps need to be calibrated manually, is that they do not consider “natural or manmade barriers that may affect directionality of data density” (Gwinn et al., 2008, p. 303). In other words, the radius or cell grid that is placed on top of a map does not conform to the presence of nature or man-made barriers, such as a body of water, a freeway, or a wall.

A potential solution to this problem is to incorporate raster masking in GIS. The analyst builds a mask around areas that are not appropriate for inclusion in the density analysis. For example, a large body of water is unlikely to have many crimes occurring in the center. A mask for this body of water would exclude that area in calculating the density surface. The result is a density surface based on a more realistic risk of crime. A final consideration of analysts when producing density maps is their audience. If audience members are not familiar with density analysis, it is recommended that a legend with labels such as “low density” or “medium density” be used.

Although high-crime areas can easily be identified based on the past experiences of police officers or based on the characteristics of those areas, GIS allows police departments to more accurately pinpoint hot spots to confirm trouble areas, identify the specific nature of the activity occurring within the hot spot, and then develop strategies to respond.