CrimeStat IV

Part IV: Spatial Modeling I 

 
10.2 

Chapter 10:

Kernel Density Interpolation

Ned Levine

Ned Levine & Associates 

Houston, TX 
Table of Contents 

Introduction 10.1 

Kernel Density Interpolation 10.1 

 Kernel Estimates as an Alternative to Histograms 10.3 

 Kernel Functions 10.7 

 Kernel Parameters 10.10 

 Shape and size of the bandwidth 10.11 

 Three-dimensional kernels 10.11 

CrimeStat Kernel Density Methods 10.13 

Single Kernel Density Interpolation 10.13

 File to be Interpolated 10.13 

 Method of Interpolation 10.16 

 Choice of Bandwidth 10.16 

 Fixed interval 10.16 

 Adaptive interval 10.17 

 Output Unit 10.17 

 Intensity or Weighting Variable 10.17 

 Density Calculation 10.18 

 Output File 10.18 

 Example 1: Kernel Density Estimate of Baltimore County Street Robberies 10.19 

Dual Kernel Density Interpolation 10.22

 File to be Interpolated 10.24 

 Method of Interpolation 10.24 

 Choice of Bandwidth 10.24 

 Fixed interval 10.25 

 Adaptive interval 10.25 

 Variable interval 10.25 

 Use kernel bandwidths that produce stable estimates 10.26 

 Output Unit 10.26 

 Intensity or Weighting Variable 10.26 

 Density Calculation 10.26 

 Output File 10.28 

 Example 2: Kernel Density Estimates of Vehicle Thefts 

 Relative to Population 10.28 

Example 3: Kernel Density Estimates and Risk-adjusted Clustering of 

 Robberies Relative to Population 10.31 

 Visually Presenting Kernel Estimates 10.34 
Table of Contents (continued) 

Advantages and Limitations of Kernel Density Interpolation 10.34

 Advantages of Kernel Density Interpolation 10.34 

 Limitations of Kernel Density Interpolation 10.34 

Conclusion 10.36 

References 10.37 

Endnotes 10.40 

Attachments 10.42 

A. Kernel Density Interpolation to Estimate Sampling Bias in 

 the Climatic Response of Sphagnum Spores in North America 

By Mike Sawada 10.43 

B. Describing Crime Spatial Patterns By Time of Day in Belo Horizonte 

By Renato Assunção, Cláudio Beato, & Bráulio Silva 10.44 

C. Using Kernel Density Smoothing and Linking to ArcGIS: 

 Examples from London, England 

By Spencer Chainey 10.45 

D. Infant Death Rate and Low Birth Weight in the I-5 Corridor of 

 Seattle and King County 

By Richard Hoskins 10.46 

E. The Risk of Violent Incidents Relative to Population Density in 

 Cologne Using the Dual Kernel Density Routine 

By Dietrich Oberwittler & Marc Wiesenhütter 10.47 

F. Kernel Density Interpolation of Police Confrontations in 

 Buenos Aires Province, Argentina: 1999 

By Gastón Pezzuchi 10.48 

G. Evolution of the Urbanization Process in the Brazilian Amazonia 

By Silvana Amaral, Antônio Miguel V. Monteiro, Gilberto Câmara, & 

José A. Quintanilha 10.49 

H. Using Small Area Estimation to Target Health Services in 

 Harris County, TX 

By Thomas F. Reynolds 10.50 

I. Identifying Voucher Holders and Crime Concentrations 

 Using Dual Kernel Density Estimation 

By Ron Wilson 10.51
10.1 

Chapter 10: 

Kernel Density Interpolation

Introduction

 In this chapter, we discuss tools aimed at interpolating incidents, using the kernel density 

approach. Kernel Density Interpolation (sometimes called Kernel Density Estimation) is a 

technique for generalizing incident locations to an entire area. Whereas the spatial distribution 

and hot spot statistics provide statistical summaries for the data incidents themselves, 

interpolation techniques generalize those data incidents to the entire region. In particular, they 

provide density estimates for all parts of a region (i.e., at any location). The density estimate is 

an intensity variable, a Z-value, that is estimated at a particular location. Consequently, it can 

be displayed by either surface maps or contour maps that show the intensity at all locations. 

 There are many interpolation techniques, such as Kriging, trend surfaces, local regression 

models (e.g., Loess, splines), and Dirichlet tessellations (Anselin, 1992; Cleveland, Grosse & 

Shyu, 1993; Venables & Ripley, 1997). Most of these require a variable that is being estimated 

as a function of location. However, kernel density estimation is an interpolation technique that 

is appropriate for individual point locations (Silverman, 1986; Härdle, 1991; Bailey & Gatrell, 

1995; Burt & Barber, 1996; Bowman & Azalini, 1997). 

Kernel Density Estimation

 Kernel density estimation involves placing a symmetrical surface over each point, 

evaluating the distance from the point to a reference location based on a mathematical function, 

and summing the value of all the surfaces for that reference location. This procedure is 

repeated for all reference locations. It is a technique that was developed in the late 1950s as an 

alternative method for estimating the density of a histogram (Rosenblatt, 1956; Whittle, 1958; 

Parzen, 1962). A histogram is a graphic representation of a frequency distribution. A 

continuous variable is divided into intervals of size, s (the interval or bin width), and the number 

of cases in each interval (bin) are counted and displayed as block diagrams. The histogram is 

assumed to represent a smooth, underlying distribution (a density function). However, in order to 

estimate a smooth density function from the histogram, traditionally researchers have linked 

adjacent variable intervals by connecting the midpoints of the intervals with a series of lines 

(Figure 10.1). 
CONSTRUCTING A DENSITY ESTIMATE FROM HISTO

Figure 10.1:

Method of Connecting Midpoints

50

ncy

40

Freque30

2

0

10

1

2

3

4

5

6

7

0

Variable Classification Interval (bin)
10.3 

Kernel Estimates as an Alternative to Histograms

 Unfortunately, doing this causes three statistical problems (Bowman & Azalini, 1997): 

1. Information is discarded because all cases within an interval are assigned to the 

midpoint. The wider the interval, the greater the information loss. 

2. The technique of connecting the midpoints leads to a discontinuous and not 

smooth density function even though the underlying density function is assumed 

to be smooth. To compensate for this, researchers will reduce the width of the 

interval. Thus, the density function becomes smoother with smaller interval 

widths, although still not very smooth. Further, there are limits to this technique 

as the sample size decreases when the bin width gets smaller, eventually 

becoming too small to produce reliable estimates. 

3. The technique is dependent on an arbitrarily defined interval size (bin width). 

By making the interval wider, the estimator becomes cruder and, conversely, by 

making the interval narrower, the estimator becomes finer. However, the 

underlying density distribution is assumed to be smooth and continuous and not 

dependent on the interval size of a histogram. 

 To handle this problem, Rosenblatt (1956), Whittle (1958) and Parzen (1962) developed 

the kernel density method in order to avoid the first two of these difficulties; the bin width issue 

still remains. What they did was to place a smooth kernel function over each point and sum the 

functions for each location on the scale. Figure 10.2 illustrates the process with five point 

locations. As seen, over each location, a symmetrical kernel function is placed; by symmetrical 

is meant that is falls off with distance from each point at an equal rate in both directions around 

each point. In this case, it is a normal distribution, but other types of symmetrical distribution 

have been used. The underlying density distribution is estimated by summing the individual 

kernel functions at all locations to produce a smooth cumulative density function. Notice that 

the functions are summed at every point along the scale and not just at the point locations. The 

advantages of this are that, first, each point contributes equally to the density surface and, 

second, the resulting density function is continuous at all points along the scale. 

 The third problem mentioned above, interval size, still remains since the width of the 

kernel function can be varied. In the kernel density literature, this is called bandwidth and refers 

essentially to the width of the kernel. Figure 10.3 shows a kernel with a narrow bandwidth 

placed over the same five points while Figure 10.4 shows a kernel with a wider bandwidth 

placed over the points. Clearly, the smoothness of the resulting density function is a result of 

the bandwidth size. 
Figure 10.2:

Kernel Density Estimate

0.7

Kernel Density Estimate

Summing of Normal Kernel Function for 5 Points

0.6

0.5

Kernel density estimate

0.4

0.3 Kernels over indivi

Density

0.2

0 2 4 6 8 10 12 14 16 18 

0.1

0.0

R l ti L ti

1 3 5 7 9 11 13 15 17 

Relative Location
Figure 10.3:

Kernel Density Estimate

1.00.9

Kernel Density Estimate

Narrower Bandwidth

0.8

0.7

Kernel density estimate

0.6

Density 0.5

0.4

0.3

0.2

0 2 4 6 8 10 12 14 16 18 

0.1

0.0

Rl i i

1 3 5 7 9 11 13 15 17 

Relative Location
Figure 10.4:

Kernel Density Estimate

0.5

Kernel Density Estimate

Wider Bandwidth

0.4

Kernel density estimate

0.3

Kernels over ind

Density

0.2

0 1

Kernels over ind

0.1

0.0

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

Relative Location
10.7 

Kernel Functions

 There are a number of different kernel functions that have been used in applications. 

Figure 10.5 illustrates five different kernel functions that are available in CrimeStat. 

 The first is the normal distribution and is the most commonly used (Kelsall & Diggle, 

1995a). It has the following functional form: 

݃ሺ݆ሻ ൌ ∑ ൥ܭܹ௜ܫ௜

ଵ

௛మଶగ

݁ି

೏೔ೕ

మ

మ೓మ൩ ே

௜ୀଵ (10.1) 

where g(xj) is the density of cell ݆, dij is the distance between cell	݆	and an incident location,	݅, h

is the standard deviation of the normal distribution (the bandwidth), K is a constant, Wi is a 

weight at the point location and Ii is an intensity at the point location. This function extends to 

infinity in all directions and, thus, will be applied to any location in the region. In CrimeStat, the 

constant K is initially set to 1 and then re-scaled to ensure that either the densities or probabilities 

sum to their appropriate values (i.e., N for densities and 1.00 for probabilities). 

 In other words, the density of cell	݆	is the sum over all incidents of a distance function 

where the function is the normal distribution. Each cell, in turn, is evaluated with this function 

and the result is a density estimate for every cell in the reference grid. 

 In CrimeStat, there are four alternative kernel functions that can be used, all of which 

have a circumscribed bandwidth (search area) unlike the normal distribution. The quartic

function is applied to a limited area around each incident point defined by the radius, h. It falls 

off gradually with distance until the bandwidth radius is reached. Its functional form is: 

 1. Outside the specified bandwidth, h: 

݃ሺ݆ሻ ൌ 0 (10.2) 

 2. Within the specified bandwidth, h: 

݃ሺ݆ሻ ൌ ∑ ൤ܭܹ௜ܫ௜

ଷ

௛మଶగ ሺ1 െ ௗ೔ೕ

మ

௛మ ሻଶ൨ ெೕ

௜ୀଵ (10.3) 

where g(j) is the density of cell	݆, dij is the distance between cell	݆	and an incident location,	݅, h is 

the radius of the search area (the bandwidth) , K is a constant, Wi is a weight at the point location, 

and Ii is an intensity at the point location. The summation is for the incidents that are within the 
Figure 10.5:

Five Types of Kernel Functions

Uniform

0.8

Quartic

Norma

0 4

Density

Triangular

0.40.0

0 2.5 5 7.5 10 12.5 

Distance
10.9 

bandwidth. Thus, each cell,	݆, has a different number of incidents that fall within the bandwidth 

search area, Mj. In CrimeStat, the constant K is initially set to 1 and then re-scaled to ensure 

that either the densities or probabilities sum to their appropriate values (i.e., N for densities and 

1.00 for probabilities). 

 The triangular (or conical) distribution falls off evenly with distance, in a linear 

relationship. It also has a circumscribed radius and is, therefore, applied to a limited area around 

each incident point, h. Compared to the quartic function, it decays more rapidly. Its 

functional form is: 

 1. Outside the specified bandwidth, h: 

݃ሺ݆ሻ ൌ 0 (10.4) 

 2. Within the specified bandwidth, h: 

݃ሺ݆ሻ ൌ ∑ ሾܹ௜ܫ௜ሺܭ െ ௄

௛

ெೕ

௜ୀଵ ሻ݀௜௝ሿ (10.5) 

where g(xj) is the density of cell	݆, dij is the distance between cell	݆	and an incident location,	݅, h

is the radius of the search area (the bandwidth) , K is a constant, Wi is a weight at the point 

location, and Ii is an intensity at the point location. The summation is for the incidents that are 

within the bandwidth. Thus, each cell,	݆, has a different number of incidents that fall within the 

bandwidth search area, Mj. In CrimeStat, the constant K is initially set to 0.25 and then 

re-scaled to ensure that either the densities or probabilities sum to their appropriate values (i.e., N 

for densities and 1.00 for probabilities). 

 The negative exponential (or peaked) distribution falls off very rapidly with distance up 

to the circumscribed radius. Its functional form is: 

1. Outside the specified bandwidth, h: 

݃ሺ݆ሻ ൌ 0 (10.6) 

 2. Within the specified bandwidth, h: 

݃ሺ݆ሻ ൌ ∑ ܹ௜ܫ௜ܭ݁஺ௗ೔ೕ ெೕ

௜ୀଵ (10.7) 

where g(xj) is the density of cell	݆, dij is the distance between cell	݆	and an incident location,	݅, h

is the radius of the search area (the bandwidth) , K is a constant, A is an exponent, Wi is a weight 

at the point location, and Ii is an intensity at the point location. The summation is for the 
10.10 

incidents that are within the bandwidth. Thus, each cell,	݆, has a different number of incidents 

that fall within the bandwidth search area, Mj. In CrimeStat, A is set to 3 while K is initially set 

to 1 and then re-scaled to ensure that either the densities or probabilities sum to their appropriate 

values (i.e., N for densities and 1.00 for probabilities). 

 Finally, the uniform distribution weights all points within the circle equally. Its 

functional form is: 

1. Outside the specified bandwidth, h: 

݃ሺ݆ሻ ൌ 0 (10.8) 

2. Within the specified bandwidth, h: 

݃ሺ݆ሻ ൌ ∑ ܹ௜ܫ௜ܭ ெೕ

௜ୀଵ (10.9) 

where g(xj) is the density of cell	݆, K is a constant, Wi is a weight at the point location, and Ii is an 

intensity at the point location. The summation is for the incidents that are within the bandwidth. 

Thus, each cell,	݆, has a different number of incidents that fall within the bandwidth search area, 

Mj. Initially, K is set to 0.1 but then re-scaled to ensure that either the densities or probabilities 

sum to their appropriate values (i.e., N for densities and 1.00 for probabilities). 

 Kernel Parameters

 The user can select these five different kernel functions to interpolate the data to the grid 

cells. They produce subtle differences in the shape of the interpolated surface or contour. The 

normal distribution weighs all points in the study area, though near points are weighted more 

highly than distant points. The other four techniques use a circumscribed circle around the grid 

cell. The uniform distribution weighs all points within the circle equally. The quartic function 

weighs near points more than far points, but the fall off is gradual. The triangular function 

weighs near points more than far points within the circle, but the fall off is more rapid. Finally, 

the negative exponential weighs near points much more highly than far points within the circle 

and the decay is very rapid. 

 The use of any of one of these depends on how much the user wants to weigh near points 

relative to far points. Using a kernel function which has a big difference in the weights of near 

versus far points (e.g., the negative exponential or the triangular) tends to produce finer 

variations within the surface than functions which weight more evenly (e.g., the normal 

distribution, the quartic, or the uniform); these latter ones tend to smooth the distribution more. 
10.11 

Shape and size of the bandwidth

 However, Silverman (1986) has argued that it does not make that much difference as long 

as the kernel is symmetrical. There are also edge effects that can occur and there have been 

different proposed solutions to this problem (Venables & Ripley, 1997). 

 There have also been variations on the size of the bandwidth with various formulas and 

criteria proposed (Silverman, 1986; Härdle, 1991; Venables & Ripley, 1997). Generally, 

bandwidth choices fall into either fixed or adaptive (variable) kernels (Kelsall & Diggle, 1995a; 

Bailey & Gatrell, 1995). CrimeStat follows this distinction, which will be explained below. 

 Another suggestion is to use the Moran correlogram, which was discussed in Chapter 5, 

to estimate the shape of the weighting function (Cliff & Haggett, 1988; Bailey & Gattrell, 1995). 

This would be appropriate for variables that have weights, such as population or employment. 

The Moran correlogram displays the degree of spatial autocorrelation as a function of distance. 

Whether the autocorrelation falls off quickly or more slowly can be used to select an 

approximate kernel function (e.g., a negative exponential function decays quickly whereas a 

quartic function decays very slowly). The bandwidth could also be selected by the distance at 

which the Moran correlogram levels off (i.e., approaches the global I value). This would lead 

to an estimate that minimizes spatial autocorrelation in the data set. It would be good for 

capturing major trends in the data, but would not be good for identifying local clusters (hot 

spots) since the bandwidth distance would incorporate most of a metropolitan area. 

 

Three-dimensional kernels

 The kernel function can be expanded to more than two dimensions (Härdle, 1991; Bailey 

& Gatrell, 1995; Burt & Barber, 1996; Bowman & Azalini, 1997). Figure 10.6 shows a 

three-dimensional normal distribution placed over each of five points with the resulting density 

surface being a sum of all five individual surfaces. Thus, the method is particularly appropriate 

for geographical data, such as crime incident locations. The method has also been developed to 

relate two or more variables together by applying a kernel estimate to each variable in turn and 

then dividing one by the other to produce a three-dimensional estimate of risk (Kelsall & Diggle, 

1995a; Bowman & Azalini, 1997). 

 Significance testing of density estimates is more complicated. Current techniques tend 

to focus on simulating surfaces under spatially random assumptions (Bowman & Azalini, 1997; 

Kelsall & Diggle, 1995b). Because of the still experimental nature of the testing, CrimeStat

does not include any testing of density estimates in this version. 
Figure 10.6:

Kernel Density Surface Kernel Density Surface

Summing of Normal Kernel Surface for 5 Points

+

+

+

=

+
10.13 

CrimeStat Kernel Density Methods

 CrimeStat has two kernel density interpolation routines. The first applies to a single 

variable while the second to the relationship between two variables. Both routines have a 

number of options. In addition, kernel density interpolation is used in several other CrimeStat

routines including journey-to-crime modeling, Bayesian journey-to-crime modeling, and 

Head-Bang interpolation. Those latter techniques will be discussed in Chapters 11, 13, and 14. 

 Figure 10.7 shows the Interpolation I screen in CrimeStat and the two routines that are 

available. Users indicate their choices by clicking on the tab and menu items. For either 

technique, it is necessary to have a reference file, which is usually a grid placed over the study 

region (see chapter 3). The reference file represents the region to which the kernel estimate will 

be generalized. Figure 10.8 illustrates a reference grid over the Baltimore region with 100 

columns and 90 rows. 

Single Kernel Density Interpolation

 The single kernel density routine in CrimeStat is applied to a distribution of point 

locations, such as crime incidents. It can be used with either a primary file or a secondary file; 

the primary file is the default. For example, the primary file can be the location of motor 

vehicle thefts. The points can also have a weighting or an associated intensity variable (or 

both). For example, the points could represent the location of police stations while the weights 

(or intensities) represent the number of calls for service. Again, the user must be careful in 

having both weighting and intensity variables as the routine will use both variables in calculating 

densities, which could lead to double weighting. 

 It is necessary to define the appropriate file on the Primary or Secondary file pages. Also, 

it is necessary to define a reference file, either an existing file or one generated by CrimeStat (see 

Chapter 3). There are other parameters that must be defined. 

File to be Interpolated

 First, the user must indicate whether the Primary file or the Secondary file (if used) is to 

be interpolated. 
Figure 10.7:

Interpolation I Screen
Figure 10.8:

Upper

Lower-left coordinate
10.16 

Method of Interpolation

 Second, the user must indicate the method of interpolation. Five types of kernel density 

estimators are used: 

1. Normal distribution (bell; default) 

2. Uniform (flat) distribution 

3. Quartic (spherical) distribution 

4. Triangular (conical) distribution 

5. Negative exponential (peaked) distribution 

 In our experience, there are advantages to each. The normal distribution produces an 

estimate over the entire region whereas the other four produce estimates only for the 

circumscribed bandwidth radius. If the distribution of points is sparse towards the outer parts of 

the region, then the four circumscribed functions will not produce estimates for those areas, 

whereas the normal will. Conversely, the normal distribution can cause some edge effects to 

occur (e.g., spikes at the edge of the reference grid), particularly if there are many points near 

one of the boundaries of the study area. The four circumscribed functions will produce less of a 

problem at the edges, although they still can produce some spikes. Within the four 

circumscribed functions, the uniform and quartic tend to smooth the data more whereas the 

triangular and negative exponential tend to emphasize >peaks= and >valleys=. The differences 

between these different kernel functions are small, however. The user should probably start 

with the default normal function and adjust accordingly to how the surface or contour looks. 

Choice of Bandwidth

 Third, the user must indicate how bandwidths are to be defined. There are two types of 

bandwidth for the single kernel density routine, fixed interval or adaptive interval. 

Fixed interval

 With a fixed bandwidth, the user must specify the interval to be used and the units of 

measurement (square miles, square nautical miles, square feet, square kilometers, or square 

meters). Depending on the type of kernel estimate used, this interval has a slightly different 

meaning. For the normal kernel function, the bandwidth is the standard deviation of the normal 

distribution. On the other hand, for the uniform, quartic, triangular, or negative exponential 

kernels, the bandwidth is the radius of the search area to be interpolated. 

 There are few guidelines for choosing a particular bandwidth other than by visual 

inspection (Venables & Ripley, 1997). Some have argued that the bandwidth be no larger than 
10.17 

the finest resolution that is desired and others have argued for a variation on random nearest 

neighbor distances (see Spencer Chainey’s article in the attachments section of this chapter). 

Others have argued for particular sizes (Silverman, 1986; Härdle, 1991; Kadafar, 1996; Farewell, 

1999; Talbot, Kulldorff, Forand, & Haley, 2000; see endnote ࢏(. There does not seem to be 

consensus on this issue. Consequently, CrimeStat leaves the definition up to the user. 

 Typically, a narrower bandwidth interval will lead to a finer mesh density estimate with 

lots of ‘peaks and valleys’. A larger bandwidth interval, on the other hand, will lead to a 

smoother distribution and, therefore, less variability between areas. While smaller bandwidths 

show greater differentiation among areas (e.g., between >hot spot= and >low spot= zones), one has 

to keep in mind the statistical precision of the estimate. If the sample size is not very large, 

then a smaller bandwidth will lead to more imprecision in the estimates, and the ‘peaks and 

valleys’ may show nothing more than random variation. On the other hand, if the sample size 

is large, then a finer density estimate can be produced. In general, it is a good idea to 

experiment with different fixed intervals to see which results make the most sense. 

Adaptive interval

 

 An adaptive bandwidth adjusts the bandwidth interval so that a minimum number of 

points are found. This has the advantage of providing constant precision of the estimate over 

the entire region. Thus, in areas that have a high concentration of points, the bandwidth is 

narrow whereas in areas where the concentration of points is sparser, the bandwidth will be 

larger. This is the default bandwidth choice in CrimeStat since we believe that consistency in 

statistical precision is paramount. The degree of precision is generally dependent on the sample 

size of the bandwidth interval. The default is a minimum of 100 points within the bandwidth 

radius. The user can make the estimate more fine grained by choosing a smaller number of 

points (e.g., 25) or more smooth by choosing a larger number of points (e.g., 200). Again, 

experimentation is necessary to see which results make the most sense. 

Output Unit

 Fourth, the user must indicate the measurement units for the density estimate in points 

per square miles, square nautical miles, square feet, square kilometers, or square meters. The 

default is points per square mile. 

Intensity or Weighting Variable

 If an intensity or weighting variable is to be used (and has been defined on the Primary or 

Secondary file page), the appropriate box must be checked. Be careful about using both 

intensity and weighting variables to avoid >double weighting=. 
10.18 

 Density Calculation

 Finally, the user must indicate the type of output for the density estimates. There are 

three types of calculation that can be conducted with the kernel density routine. The 

calculations are applied to each reference cell: 

1. The kernel estimates can be calculated as absolute density estimates using 

equations 10.1-10.9, depending on what type of kernel function is used. The 

estimates at each reference cell are re-scaled so that the sum of the densities over 

all reference grids equals the total number of incidents. That is, the estimate is 

the number of incidents/points that occurred in each grid cell. This is the default 

choice. 

2. The kernel estimates can be calculated as relative density estimates. These 

divide the absolute densities by the area of the grid cell. It has the advantage of 

interpreting the density in terms that are familiar. Thus, instead of a density 

estimate represented by points per grid cell, the relative density will convert this 

to points per square mile or points per square kilometer. 

3. The densities can be converted into probabilities by dividing the density at any 

one cell by the total number of incidents. 

 Since the three types of calculation are directly interrelated, the output surface will not 

differ in its variability. The choice would depend on whether the calculations are used to 

estimate absolute densities, relative densities, or probabilities. For comparisons between 

different types of crime or between the same type of crime and different time periods, usually 

absolute densities are the unit of choice (i.e., incidents per grid cell). However, to express the 

output as a probability, that is, the likelihood that an incident would occur at any one location, 

then outputting the results as probabilities would make more sense. For display purposes, 

however, it makes no difference as both look the same. 

Output File

 The results can be displayed in an output table or can be output into two formats: 1) 

Raster grid formats for display in a surface mapping program- Surfer for Windows >.dat= format 

(Golden Software, 2008) or ArcGIS Spatial Analyst >asc= format (ESRI, 2012); or 2) Polygon 

grids in ArcGIS >.shp=, MapInfo >.mif= or various Ascii formats. However, all but Surfer for 

Windows require that the reference grid be created by CrimeStat.

1

1 CrimeStat will output the geographical boundaries of the reference grid and will assign a third-variable 
10.19 

 Example 1: Kernel Density Estimate of Baltimore County Street Robberies

 An example can illustrate the use of the single kernel density routine. Figure 10.9 

shows a Surfer for Windows output of 1180 street robberies for 1996 in Baltimore County. The 

reference grid was generated by CrimeStat and had 100 columns and 90 rows. Thus, the routine 

calculated the distance between each of the 10,800 reference cells and each of the 1180 robbery 

incident locations, evaluated the kernel function for each measured distance, and summed the 

results for each reference cell. The normal distribution kernel function was selected for the 

kernel estimator and an adaptive bandwidth with a minimum sample size of 100 was chosen as 

the parameters. 

 There are three views in the figure: 1) a map view showing the location of the incidents; 

2) a surface view showing a three-dimensional interpolation of robbery density; and 3) a contour 

view showing contours of high robbery density. The surface and contour views provide 

different perspectives. The surface shows the peaks very clearly and the relative density of the 

peaks. As can be seen, the peak for robberies on the eastern part of the County is much higher 

than the two peaks in the central and western parts of the County. The contour view can show 

where these peaks are located; it is difficult to identify location clearly from a three-dimensional 

surface map. Highways and streets could be overlaid on top of the contour view to identify 

more precisely where these peaks are located. 

 Figure 10.10 shows an ArcGIS map of robbery density with the robbery incident 

locations overlaid on top of the density contours. Here, we can see quite clearly that there are 

three strong concentrations of incidents, one on the west side, one on the northern border 

between Baltimore City and Baltimore County, and one on the east side which blends with a 

smaller peak in the southeast corner of the County. 

 From one perspective, the kernel estimate is a better >hot spot= identifier than the cluster 

analysis routines discussed in Chapters 7 and 8. Cluster routines group incidents into clusters 

and distinguish between incidents which belong to the cluster and those which do not belong. 

Depending on which mathematical algorithms are used, different clustering routines will return 

differing allocations of incidents to clusters. The kernel estimate, on the other hand, is a 

continuous surface; the densities are calculated at all locations; thus, the user can visually inspect 

(called Z) as the density estimate. ArcGIS >.shp= files can be read directly into the program. For 

MapInfo, on the other hand, the output is in MapInfo Interchange Format (a >.mif= and a >.mid= file). The 

files must be imported to a MapInfo >.tab= file. For both output formats, the values of Z can be shown as a 

thematic map but the ranges must be adjusted to illustrate the high density locations (i.e., the default values 

in most GIS programs will not display the densities very well). On the other hand, the default interval 

values for Surfer for Windows and ArcGIS Spatial Analyst provide a reasonably good visualization. 
Figure 10.9:
Figure 10.10:
10.22 

the variability in density and decide what to call a >hot spot= without having to define arbitrarily 

where to cut-off the >hot spot= zone. 

 Going back to the Surfer for Windows output, Figure 10.11 shows the effects of varying 

the bandwidth parameters. There are three fixed bandwidth intervals (0.5, 1, and 2 miles 

respectively) and there are two adaptive bandwidth intervals (a minimum of 25 and 100 points 

respectively). As can be seen, the fineness of the interpolation is affected by the bandwidth 

choice. For the three fixed intervals, an interval of 0.5 miles produced a finer mesh 

interpolation than an interval of 2 miles, which tended to >oversmooth= the distribution. 

Perhaps, the intermediate interval of 1 mile gives the best balance between fineness and 

generality? For the two adaptive intervals, the minimum sample size of 25 gave very specific 

peak locations whereas the adaptive interval with a minimum sample size of 100 gave a 

smoother distribution. 

 Which of these should be used as the best choice would depend on how much confidence 

the analyst has in the results. A key question is whether the >peaks= are real or merely 

by-products of small sample sizes. The best choice would be to produce an interpolation that 

fits the experience of the department and officers who travel an area. Again, experimentation and 

discussions with beat officers will be necessary to establish which bandwidth choice should be 

used in future interpolations. 

 

 Note in all five of the interpolations, there is some bias at the edges with the City of 

Baltimore (the three-sided area in the central southern part of the map). Since the primary file 

only included incidents for the County, the interpolation nevertheless has estimated some 

likelihood at the edges; these are edge biases and need to be ignored or removed with an ASCII 

editor.2 Further, the wider the interval chosen, the more bias was produced at the edge. 

Dual Kernel Density Interpolation

 The dual kernel density routine in CrimeStat is applied to two distributions. For 

example, the primary file could be the location of auto thefts while the secondary file could be 

the centroids of census tracts with the population of the census tract being an intensity variable. 

The dual routine must be used with both a primary file and a secondary file. Also, it is 

2

 All the CrimeStat outputs except for ArcGIS >shp= files are in ASCII. There are usually >edge effects= and 

values interpreted outside the actual geographical area. These can be removed with an ASCII editor by 

substituting >0' for the values at the edges or outside the study region. For >shp= files, the values at the edges 

can be edited within the ArcGIS program. Another alternative is to >cut out= the cells that are beyond the 

study area. Care must be taken, however, to not edit an output file too much otherwise it will bear little 

relationship to the calculated kernel estimate. 
Figure 10.11:

Interpolation of Baltimore County Vehicle Thefts: Interpolation of Baltimore County Vehicle Thefts: 1

Different Smoothing Parameters
10.24 

necessary to define a reference file, either an existing file or one generated by CrimeStat (see 

Chapter 3). Several parameters need to be defined. 

File to be Interpolated

 The user must indicate the order of the interpolation. The routine uses the language first

file and second file in making the comparison (e.g., dividing the first file by the second; adding 

the first file to the second). The user must indicate which is the first file - the Primary or the 

Secondary. The default is that the Primary file is the first file. 

Method of Interpolation

 The user must indicate the type of kernel estimator. As with the single kernel density 

routine, five types of kernel density estimators are used 

1. Normal distribution (bell; default) 

2. Uniform (flat) distribution 

3. Quartic (spherical) distribution 

4. Triangular (conical) distribution 

5. Negative exponential (peaked) distribution 

 In our experience, there are advantages to each. The normal distribution produces an 

estimate over the entire region whereas the other four produce estimates only for the 

circumscribed bandwidth radius. If the distribution of points is sparse towards the outer parts of 

the region, then the four circumscribed functions will not produce estimates for those areas, 

whereas the normal will. Conversely, the normal distribution can cause some edge effects to 

occur (e.g., spikes at the edge of the reference grid), particularly if there are many points near 

one of the boundaries of the study area. The four circumscribed functions will produce less of a 

problem at the edges, although they still can produce some spikes. Within the four 

circumscribed functions, the uniform and quartic tend to smooth the data more whereas the 

triangular and negative exponential tend to emphasize >peaks= and >valleys=. The differences 

between these different kernel functions are small, however. The user should probably start 

with the default normal function and adjust accordingly to how the surface or contour looks. 

Choice of Bandwidth

The user must define the bandwidth parameter. There are three types of bandwidths for 

the single kernel density routine - fixed interval, adaptive interval, or variable interval. 
10.25 

 Fixed interval

With a fixed bandwidth, the user must specify the interval to be used and the units of 

measurement (square miles, square nautical miles, square feet, square kilometers, or square 

meters). Depending on the type of kernel estimate used, this interval has a slightly different 

meaning. For the normal kernel function, the bandwidth is the standard deviation of the normal 

distribution. For the uniform, quartic, triangular, or negative exponential kernels, the 

bandwidth is the radius of the search area to be interpolated. Since there are two files being 

compared, the fixed interval is applied both to the first file and the second file. 

Adaptive interval

 An adaptive bandwidth adjusts the bandwidth interval so that a minimum number of 

points (sample size) is found. This sample size is applied to both the first file and the second 

file. It has the advantage of providing constant precision for the kernel estimate over the entire 

region. Thus, in areas that have a high concentration of points, the bandwidth is narrow while 

in areas where the concentration of points is sparser, the bandwidth will be wider. This is the 

default bandwidth choice in CrimeStat since consistency in statistical precision is important. 

The degree of precision is generally dependent on the sample size of the bandwidth interval. 

The default is a minimum of 100 points. The user can make the estimate finer by choosing a 

smaller number of points (e.g., 25) or smoother by choosing a larger number (e.g., 200). 

 Variable interval

 With a variable interval, each file (the first and the second) have different intervals. For 

both, the units of measurements must be specified (square miles, square nautical miles, square 

feet, square kilometers, or square meters). 

 There is a good reason why a user might want to use variable intervals. In comparing 

two kernel estimates, the most common comparison is to divide one by the other. However, if 

the density estimate for a particular cell for the denominator variable approaches zero, then the 

ratio will blow up and become a very large number. Visually, this will be seen as spikes in the 

distribution, the result, usually, of too few cases. In this case, the user might decide to smooth 

the denominator more than numerator to reduce these spikes. For example, the interval for the 

first file (the numerator) could be 0.5 miles whereas the interval for the second file (the 

denominator) could be 1 mile. Experimentation will be necessary to see whether this is 

warranted. But, in our experience, excessively large densities happen when either there are too 

few cases or there is an irregular boundary to the region with a number of incidents grouped at 

one of the edges. 
10.26 

 Use kernel bandwidths that produce stable estimates

 Note that with a dual kernel calculation, particularly the ratio of one variable to another, 

it is important not to choose too small a bandwidth. This could have the effect of creating spikes 

at the edges of the study area or in low population density areas. For example, in low 

population density areas, there will probably be fewer events than in more built-up area. For 

the denominator of a ratio estimate, an extremely low value could cause the ratio to be 

exaggerated (a >spike=) relative to neighboring grid cells. Using a larger bandwidth will produce 

a more stable average. 

Output Unit

 The user must indicate the measurement units for the density estimate in points per 

square miles, square nautical miles, square feet, square kilometers, or square meters. 

Intensity or Weighting Variable

 If an intensity or weighting variable is to be used (and has been defined on the Primary or 

Secondary file page), the appropriate box must be checked. Be careful about using both 

intensity and weighting variables to avoid >double weighting=. 

Density Calculation 

The user must indicate the type of density output. There are six types of density 

calculations that can be conducted with the dual kernel density routine. The calculations are 

applied to each reference cell: 

1. There is the ratio of densities. This is the first file divided by the second file. 

This is the default choice. For example, if the first file is the location of auto thefts 

incidents and the second file is the location of census tract centroids with the 

population assigned as an intensity variable, then ratio of densities would divide 

the kernel estimate for auto thefts by the kernel estimate for population and would 

be an estimate of auto thefts risk. 

2. There is also the log ratio of densities. This is the natural logarithm of the 

density ratio, that is 

Log ratio of densities = Ln [ g(xj) / g(yj) ] (10.10) 
10.27 

where g(xj) is the density estimate for the first file and g(yj) is the density estimate 

for the second file. For a variable that has a spatially skewed distribution such 

that most reference cells have very low density estimates, but a few have very 

high density estimates, converting the ratio into a log function will tend to mute 

the spikes that occur. This measure has been used in studies of risk (Kelsall & 

Diggle, 1995b). 

3. There is the absolute difference in densities. This is the first file minus the 

second file. This can be a useful output for examining differential effects. For 

example, by using the centroids of census block groups (see example 2 below) 

with the population of the census block group assigned as an intensity variable, 

the difference in population between two different census years can be estimated. 

Since the spatial arrangements of the block groups changes slightly from one 

census to the next (the U. S. Census Bureau suggests that census units be drawn 

so that there are approximately equal populations in each unit), estimating the 

difference in kernel densities between two census can show where the changes 

have occurred irrespective of the particular census units. 

4. There is the relative difference in densities. Like the relative density in the 

single-kernel routine (discussed above), the relative difference in densities first 

standardizes the densities of each file by dividing by the grid cell area in familiar 

units (square miles or square kilometers) and then subtracts the secondary file 

relative density from the primary file relative density. This can be useful in 

calculating changes between two time periods, for example in calculating a 

change in relative density between two censuses or a change in the crime density 

between two time periods. 

5. There is the sum of the densities. This is the density estimate for the first file 

plus the density estimate for the second file. A possible use of the sum operation 

is to combine two different density surfaces, for example the density of robberies 

plus the density of assaults; 

6. Finally, there is the relative sum of densities. The relative sum of densities first 

standardizes the densities of each file by dividing by the grid cell area in familiar 

units (square miles or square kilometers) and then adds the secondary file relative 

density to the primary file relative density. This can be useful for identifying the 

total effects of two distributions. For example, the total impact of robberies and 

burglaries on an area can be estimated by taking the relative density of robberies 

and adding it to the relative density of burglaries. The result is the combined 
10.28 

relative density of robberies and burglaries per unit area (e.g., robberies and 

burglaries per square mile). 

 Output File

 Finally, the user must specify the file formats for the output. The results can be output in 

three forms. First, the results are displayed in an output table. Second, the results can be output 

into two raster grid formats for display in a surface mapping program: Surfer for Windows

format as a >.dat= file (Golden Software, 2008) and ArcGIS Spatial Analyst format as an ‘asc= file 

(ESRI, 2012). Third, the results can be output as polygon grids into ArcGIS >.shp=, MapInfo

>.mif= and various Ascii formats (see footnote 1). All but Surfer for Windows require that the 

reference grid be created by CrimeStat. 

 Example 2: Kernel Density Estimates of Vehicle Thefts Relative to Population 

 As an example of the use of the dual kernel density routine, the dual routine is applied in 

both the City of Baltimore and the County of Baltimore to 14,853 motor vehicle theft locations 

for 1996 relative to the 1990 population of census block groups. Again, a reference grid of 100 

columns by 108 rows was generated by CrimeStat. 

 Figure 10.12 shows the resulting density estimate as a Surfer for Windows output; again, 

there is a map view, a surface view, and a contour view. The normal kernel function was used 

and an adaptive bandwidth of 100 points was selected. As seen, there is a very high 

concentration of auto theft incidents within the central part of the metropolitan area. The 

contour view suggests five or six peak areas that are close to each other. 

 Much of this concentration, however, is produced by high population density in the 

metropolitan center. Figure 10.13, for example, shows the kernel estimate for 1349 census 

block groups for both the City of Baltimore and the County of Baltimore with the 1990 

population assigned as the intensity variable. Again, the normal kernel function was used with 

an adaptive bandwidth of 100 points being selected. The map shows three views: 1) a surface 

view; 2) a contour view; and 3) a ground level view looking directly north. The distribution of 

population is, of course, also highly concentrated in the metropolitan center with two peaks, 

quite close to each other with several smaller peaks. 

 When these two kernel estimates are compared using the dual kernel density routine, a 

more complicated picture emerges (Figure 10.14). This routine has conducted three operations: 

1) it calculated the distance between each of the 10,800 reference cells and the 14,853 auto theft 

locations, evaluated the kernel function for each measured distance, and summed the results for 

each reference cell; 2) it calculated the distance between each of the 10,800 reference cells and 
Baltimore Metropolitan Vehicle Thefts: 1996

Figure 10.12:

Surface View Contou

39 70

N

Baltimore Metropolitan Vehicle Thefts: 1996

Three Views of Kernel Density Interpolation

39.6039.70

N

39.4039.50

-76.80 -76.70 -76.60 -76.50 

39.2039.30

Ground Level View

76.80 76.70 76.60 76.50 
Baltimore Metropolitan Population: 1990

Figure 10.13:

Surface View Contour View

39.70

Baltimore Metropolitan Population: 1990

Three Views of Kernel Density Interpolation

39.6039.70

N

39.4039.5039.2039.30

Ground Level View -76.90 -76.80 -76.70 -76.60 -76.50 
10.31 

the 1349 census block groups with population as an intensity variable, evaluated the kernel 

function for each intensity-weighted distance, and summed the results for each reference cell; 

and 3) divided the kernel density estimate for auto thefts by the kernel density estimate for 

population for each reference cell location. 

 While the concentration of motor vehicle thefts relative to population (>motor vehicle 

theft risk@) is still high in the metropolitan center, there are bands of high risk that spread 

outward, particularly along major arterials. There are now many >hot spot= areas that have a 

high distribution of motor vehicle thefts relative to the residential population. We could, of 

course, refine this analysis further by taking, for example, employment as a baseline variable 

rather than population; employment is a better indicator for the daytime population distribution 

whereas the residential population is a better indicator for nighttime population distribution 

(Levine, Kim, & Nitz, 1995a; 1995b). 

Example 3: Kernel Density Estimates and Risk-adjusted Clustering of Robberies 

Relative to Population 

 The final example shows how the dual kernel interpolation compares with the 

risk-adjusted nearest neighbor clustering, discussed in chapter 6. Figure 10.15 shows 15 

first-order and two second-order risk-adjusted clusters overlaid on the dual kernel estimate of 

1996 robberies relative to 1990 population.3 As seen, there is a correspondence between the 

identified risk-adjusted clusters and the dual kernel interpolation of the ratio of robberies to 

population. For a broad regional perspective, the interpolation produces an adequate model of 

where there is a high robbery risk. At the neighborhood level, however, the risk-adjusted 

clusters are more specific and would be preferable for use by police in identifying high-risk 

locations. 

 The advantage of a dual kernel density interpolation routine is that two variables can be 

related together. By interpolating one variable to a reference grid and then interpolating a second 

variable to the same reference grid, the two variables have been interpolated to the same 

geographical units. The two interpolations can then be related, by dividing, subtracting, or 

summing. As has been mentioned throughout this manual, one of the problems with techniques 

that depend on the concentration of incidents is that they ignore the underlying population 

at-risk. With the dual routine, however, we can start to examine the risk and not just the 

concentration. 

3 The risk-adjusted hierarchical clustering (Rnnh) method defined the largest search radius but a minimum of 

25 points being required to be clustered. The kernel estimate for both the Rnnh and the dual-kernel 

routines used the normal distribution function with an adaptive bandwidth of 25 points. 
Baltimore Metropolitan Vehicle Theft Risk: 1990

-1

Figure 10.14:

Surface View Contour View

39.70

Baltimore Metropolitan Vehicle Theft Risk: 1990

-1

Ratio of Interpolation of 1996 Auto Thefts to 1990 Populatio

N

39.5039.6039.7039.3039.40

-76.80 -76.70 -76.60 -76.50 

39.20

Ground Level View
Figure 10.15:
10.34 

 Visually Presenting Kernel Estimates

 Whether the single- or dual-kernel estimate is used, the result is a grid interpretation of 

the data. By scaling these values in a GIS program, a visualization of the data is obtained. 

Areas with higher densities can be shown in darker tones and those with lower densities can be 

shown in lighter tones; some people do the opposite with the high density areas being lighter. 

 To make the visualization even more realistic, one could use a GIS program to cut out 

those grid cells that are outside the study area or are on water bodies. Before doing this, however, 

be sure to re-scale the estimated AZ@ values so that they will sum to the total of the original grid. 

For example, if the original sample size was 1000, then the grid cells will sum to 1000 if the 

absolute density option is chosen. If, say, 20% of these cells are then removed to improve the 

visualization, then the grid cell Z values have to be re-scaled so that their sum will continue to be 

1000. A simple way to do this is to, first, add up the Z values for the remaining cells and, 

second, multiply each grid cell Z by the ratio of the original sum to the reduced sum. 

Advantages and Limitations of Kernel Density Interpolation

 There are advantages and limitations to the kernel density interpolation method for hot 

spot analysis. 

Advantages of Kernel Density Interpolation

 The main advantage of kernel density interpolation is its ability to visualize a broad, 

regional view of events. Whereas each of the hot spot analysis techniques discussed in 

Chapters 7 and 8 (and 9 for the Zonal Nnh) drew boundaries around the hot spots, kernel density 

interpolation provides density estimates through the study area. One can see all the high density 

and low density areas simultaneously. For example, this can provide a police department with 

an overview of the high crime areas and can form the basis of patrol deployment. Essentially, for 

a city-wide or region-wide view, there is no better technique (Chainey, Thompson & Uhlig, 

2008). 

Limitations of Kernel Density Interpolation

 At the same time, there are limitations to the approach for hot spot analysis. There are 

three statistical problems. First, the method does depend on overgeneralizing data. By 

interpolating N data points to M grid cells where M is almost always much greater than N, means 

that the data are being shared across many grid cells. This can lead to overgeneralization of 

results. For example, 10,000 cases seems like a large data set (which it normally is), but when 

it is generalized to 10,000 grids (100 columns x 100 rows), this leaves an average of 1 data point 
10.35 

per grid cell. It is well known that sampling error is very high with small samples and almost 

infinitely high with a sample size of 1. Yet the method pools the data so that every grid cell is 

represented by all (for the normal distribution kernel) or most (for the other kernels) data points. 

This leads to additional spatial autocorrelation among the estimates since each grid cell shares 

the same data points with adjacent grid cells. The practical effect of this is that a hot spot can 

appear to be larger than it truly is. Too many users are taking kernel density interpolations as 

evidence of hot spots even with very small samples. These hot spots will turn out to be nothing 

more than random variation. Again, if used carefully, the method can provide an overview of 

crime density in a study area. But, one has to be very, very careful in using the method to 

define specific hot spots. 

 Second, like other hot spot analysis methods, kernel density interpolation is effected by 

the choice of kernel used and the selected bandwidth. The normal distribution kernel, for 

example, will smooth the data and eliminate small nodules (‘peaks and valleys’) whereas the 

quartic and exponential kernels will emphasis the small nodules. Whether the more granular 

variation in density estimates is valid or not depends on the sample size. With a small sample 

size, small hot spots may be nothing more than random variation and may not be real. Unless 

the sample size is very large (meaning 10,000 or more cases), we recommend using the normal 

distribution kernel to avoid finding false hot spots. In addition, the selected bandwidth 

determines the smoothness of the visualization. Again, if the sample size is large, a smaller 

bandwidth is appropriate whereas a larger bandwidth is more desirable for smaller samples. 

One has to consider the precision of the estimates, which is a function of the sample size (i.e., 

larger is better). 

 Third, because the technique smoothes data, it is often inappropriate for small area 

analysis. It will lead to generalization of data points into adjacent areas from where the events 

occur and can lead to false conclusions (Levine, 2008). For example, motor vehicle crashes 

typically occur on freeways, highways, major arterial roads, and minor arterial roads. Few 

occur on residential (neighborhood) streets, typically less than 15%. Levine (2009) found that 

only 11% of motor vehicle crashes in the Houston metropolitan area occurred on local roads 

even though these roads accounted for 61% of the total road mileage in the region. The 

likelihood of a crash occurring on any particular local road is extremely small. However, since 

nearly one half of the crashes occurred at intersections, the method would generalize crashes at 

two intersecting arterial roads into the adjacent neighborhood streets when, in reality, very few 

crashes will occur on those streets. 

 Similarly, Levine (2008) showed how vehicle thefts that were concentrated in parking 

lots in a commercial area of Houston were generalized by kernel density interpolation into the 

local residential neighborhoods. In other words, the method produces spatial distortion 
10.36 

especially for small area (large-scale), neighborhood-level analysis. Very often, hot spots are 

very limited spatially, sometimes into an area less than half a block wide. 

 Taking another example, in a 1986 study of dangerous bus stops in Los Angeles, the most 

serious one was identified on one corner of a Hollywood (CA) intersection (Levine, Wachs & 

Shirazi, 1986). The hot spot involved a drug trade that occurred at a food stand on the corner 

and was supplied by drug dealers who used a bar near the intersection for cover. The crimes 

occurred only on that corner of the intersection; the other three corners had no crime events. 

 In other words, the amount of smoothing involved in kernel density interpolation will 

distort spatial relationships for very small hot spots and will make it appear as if there is a risk in 

nearby blocks when there might not be such a risk. The use of one of the cluster routines 

discussed in chapters 7 and 8 would be more appropriate for small area analysis. 

Conclusion

 Kernel density estimation is one of the most utilized spatial statistical techniques. There 

is currently research on the use of this technique in both the statistical theory and in developing 

applications. For crime analysis, the technique represents a powerful way of conducting both 

regional hot spot analysis as well as being able to link the hot spots to an underlying 

population-at-risk. It can be used both for police deployment by targeting areas of high 

concentration of incidents as well as for prevention by targeting areas with high risk. It can also 

be used as a research tool for analyzing two or more distributions. Caution has to be used in 

adapting the method for small area (large scale) neighborhood type of analysis. Other 

techniques are more appropriate for that level. 
10.37 

References

Anselin, L. (1992). SpaceStat: A Program for the Statistical Analysis of Spatial Data. Santa 

Barbara, CA: National Center for Geographic Information and Analysis, University of 

California. 

Bailey, T. C. & Gatrell, A. C. (1995). Interactive Spatial Data Analysis. Longman Scientific & 

Technical: Burnt Mill, Essex, England. 

Bowman, A. W. & Azzalini, A. (1997). Applied Smoothing Techniques for Data Analysis: The 

Kernel Approach with S-Plus Illustrations. Oxford Science Publications, Oxford University 

Press: Oxford, England. 

Burt, J. E. & Barber, G. M. (1996). Elementary Statistics for Geographers (second edition). 

The Guilford Press: New York. 

Chainey, S., Thompson, L., & Uhlig, S. (2008). The utility of hotspot mapping for predicting 

spatial patterns of crime. Security Journal, 21, 4-28. 

Cleveland, W. S., Grosse, E., & Shyu, W. M. (1993). Local regression models. In John M. 

Chambers & Trevor J. Hastie, Statistical Models in S. Chapman & Hall: London. 

Cliff, A. D. & Haggett, P. (1988). Atlas of Disease Distributions. Blackwell Reference: 

Oxford. 

 

ESRI (2012). ArcGIS 10.0. Environmental Systems Research Institute: Redlands, CA. 

http://www.esri.com/software/arcgis/index.html. 

Farewell, D.l (1999). Specifying the bandwidth function for the kernel density estimator. 

http://www.iph.cam.ac.uk/bugs/documentation/coda03/node44.html.

Golden Software. 2008. Surfer® for Windows (Ver. 10). Golden Software, Inc.: Golden, CO. 

Härdle, W. (1991). Smoothing Techniques with Implementation in S. Springer-Verlag: New 

York. 

Kafadar, K. (1996). Smoothing geographical data, particularly rates of disease. Statistics in 

Medicine 15(23), 2539-2560. 

 
10.38 

References (continued) 

Kelsall, J. E. & Diggle, P.J. (1995a). Kernel estimation of relative risk, Bernoulli, 1, 3-16. 

Kelsall, J. E. & Diggle, P.J. (1995b). Non-parametric estimation of spatial variation in relative 

risk. Statistical Medicine, 14, 2335-2342. 

Levine, N. (2009). “A motor vehicle safety planning support system: The Houston 

experience”. In S. Geertman and J. Stillwell, Planning Support Systems: Best Practice and 

New Methods. Springer. 93-111. 

Levine, N. (2008). “The ‘hottest’ part of a crime hotspot: Comments on “The utility of hotspot 

mapping for predicting spatial patterns of crime” by Spencer Chainey, Lisa Tompson, and 

Sebastian Uhlig”. Security Journal, 21, 295-302. 

Levine, N., Kim, K. E. & Nitz, L. H. (1995a). Spatial analysis of Honolulu motor vehicle 

crashes: I. Spatial patterns. Accident Analysis & Prevention, 27(5), 663-674. 

Levine, N., Kim, K. E.& Nitz, L. H. (1995b). Spatial analysis of Honolulu motor vehicle 

crashes: II. Generators of crashes. Accident Analysis & Prevention, 27(5), 675-685. 

Levine, N., Wachs, M. & Shirazi, E. (1986). Crime at Bus Stops: A Study of Environmental 

Factors. Journal of Architectural and Planning Research. 3 (4), 339-361. 

Parzen, E. (1962). On the estimation of a probability density and mode. Annuals of 

Mathematical Statistics, 33, 1065-1076. 

Rosenblatt, M. (1956). Remarks on some non-parametric estimates of a density function. 

Annuals of Mathematical Statistics, 27, 832-837. 

Scott, D. W. (1992). Multivariate Density Estimation: Theory, Practice, and Visualization. 

John Wiley & Sons: New York. 

Silverman, B. W. (1986). Density Estimation for Statistics and Data Analysis. Chapman & 

Hall: London. 

Talbot T. O., Kulldorff, M., Forand S. P., & Haley V. B. (2000). Evaluation of spatial filters to 

create smoothed maps of health data. Statistics in Medicine, 19, 2399-2408. 
10.39 

References (continued) 

Venables, W.N. & Ripley, B. D. (1997). Modern Applied Statistics with S-Plus (second edition). 

Springer-Verlag: New York. 

Whittle, P. (1958). On the smoothing of probability density functions. Journal of the Royal 

Statistical Society, Series B, 55, 549-557. 

 
10.40 

Endnotes 

i. There are differences in opinion about how wide a particular fixed bandwidth should be determined. The 

smoothing is done for a distribution of values, Z. If there are only unique points (and, hence, there is no Z 

value at a point), the distances between points can be substituted for Z. Thus, MeanD is the mean 

distance, sd(D) is the standard deviation of distance, and iqr(D) is the inter-quartile range of distances 

between points. These would be substituted for MeanZ, sd(Z), and iqr(Z) respectively 

Silverman (1986; 45-47; Härdle, 1991; Farewell, 1999) proposed a bandwidth, h, of: 

݄ ൌ 1.06 ∗ min	ሼݏ݀ሺܼሻ ௜௤௥ሺ௓ሻ

ଵ.ଷସ ሽܰିభ

ఱ

where min is the minimum of the next two terms, sd(Z) is the standard deviation of the variable, Z, being 

interpolated, iqr(Z) is the inter-quartile range of Z, and N is the sample size. 

Bowman and Azzalini (1997; 31) defined a slightly different optimal bandwidth for a normal kernel: 

݄ൌሼ ସ

ଷேሽ

భ

ఱ ∗ ܵܦሺܼሻ

To avoid being influenced by outlier, they suggested using the median absolute deviation estimator for 

sd(Z): 

ܦܣܯሺܼሻ ൌ ݉݁݀݅ܽ݊ሼ௓ሺ௜ሻିெ௘ௗ௜௔௡௓

଴.଺଻ସହ ሽ

Scott (1992) suggested an upper bound on the normal kernel of 

h = 1.144 * sd(Z) * N-1/5

Bailey and Gatrell (1995, 85-87) offered a rough choice for the bandwidth of 

 ݄ ൌ 0.68ܰିభ

ఱ

but suggested that the user could experiment with different bandwidths to explore the surface. 

On the other hand, the concept of an adaptive bandwidth is based more on sampling theory (Bailey & 

Gatrell, 1995). By increasing the bandwidth until a fixed number of points are counted ensures that the 

level of precision is constant throughout the region. As with all sampling, the standard error of the 

estimate is a function of the sample size; a larger sample leads to smaller error. In general, if there was 

independent sampling, the 95% confidence interval of a bandwidth for a normal kernel could be 

approximated by: 

ହ. ଴1.96 േܼ̅ ൌ ݈ܽݒݎ݁ݐ݅݊	݂݊݅݀݁݊ܿ݁݋ܿ	95%

ேሺ௛ሻ

భ

మ

ݏ݀ሺܼሻ

where N(h) is the adaptive sample size (the number of points counted within the bandwidth for the adaptive 

kernel). This assumes that a point has an equal likelihood of falling within the bandwidth of one cell 

compared to an adjacent cell (i.e., it sits on the boundary of the bandwidth circle). The adaptive bandwidth 

criterion requires that the bandwidth be increased until it captures the specified number of points. 
10.41 

Endnotes (continued) 

On average, if there are N points in a region of area, A, and if the adaptive sample size is N(p), then the 

average area required to capture N(p) points is: 

ܣሺ݌ሻ ൌ ஺ேሺ௣ሻ

ே

and the average bandwidth, Mean(h), is: 

ܯ݁ܽ݊ሺ݄ሻ ൌ ට஺ሺ௣ሻ

గ ൌ ට஺ேሺ௣ሻேగ

Each of these provide different criteria for the bandwidth size with the adaptive being the most 

conservative. For example, for a standardized distribution with 1000 data points, a standardized mean of Z 

of 0 and a standardized standard deviation of 1, the Silverman criteria would produce a bandwidth of 

0.2663; the Bowman and Azzalini criteria would produce a bandwidth of 0.2661; the Scott criteria would 

produce a bandwidth of 0.2874 and the Bailey and Gatrell criteria would produce a bandwidth of 0.1708. 

For the adaptive interval, if the required adaptive sample size is 25, then the average bandwidth would be 

approximately 0.3162 (this assumes that the area is a circle with a radius of 2 standardized standard 

deviations). 

 
10.42 

Attachments

 
Kernel Density Interpolation to Estimate Sampling Bias in the Climatic 

Response of Sphagnum Spores in North America 

Mike Sawada 

 Laboratory for Applied Geomatics and GIS Science 

 University of Ottawa, Department of Geography, Canada 

Sphagnum moss, the dominant species of bogs, thrives under certain ranges of 

temperature and precipitation. Sphagnum releases spores for reproduction and these are 

transported, often long distances, by wind and water. Thus, the presence of a spore in the fossil 

record may not indicate nearby Sphagnum plants. However, spores should be most numerous 

near Sphagnum plants. Over time, these spores and pollen from other plants accumulate in lake 

and bog sediments and leave a fossil record of vegetation history. 

We wanted to use the amount of fossil Sphagnum spores in different parts of North 

America to infer past climates. To do so, we had to first show that Sphagnum spores are most 

abundant in climates where Sphagnum plants thrive and secondly, that this center of abundance 

is not biased sampling because of under sampling in parts of climate space. First, we developed 

a Sphagnum spore response surface showing the relative abundance of spores along the axes of 

temperature and precipitation (Fig. A). 

CrimeStat was used in the second stage to develop a kernel density surface using a 

quartic kernel for 3007 sample sites within climate space (Fig. B). These were smoothed and 

visualized in Surfer. The surface showed that the intensity of points is higher in regions 

surrounding the response maximum. This gave us confidence that the Sphagnum response was 

real since other parts of climate space are well sampled but unlikely to produce high spore 

proportions. This fact allowed climate inferences to be made within the fossil record for past 

time periods using the amount of Sphagnum spores present. 

Figures modified from Gajewski, Viau, Sawada et al. 2001. Global Biogeochemical Cycles,
Describing Crime Spatial Patterns By Time of Day in Belo Horizonte 

Renato Assunção, Cláudio Beato, Bráulio Silva 

CRISP, Universidade Federal de Minas Gerais, Brazil 

We used the kernel density estimate to visualize time trends for crime occurrences 

on a typical weekday. We found markedly different spatial distributions depending on 

the time, with the amount of crime varying and the hot spots, identified by the ellipses, 

appearing in different places. 

 

The analysis used 1114 weekday robberies from 1995 to 2000 in downtown Belo 

Horizonte. Breaking the data into hours, we used the normal kernel, a fixed bandwidth of 

450 meters and outputted densities option (points per square unit of area). Note that the 

latter option could be useful if one is interested only in the hot spot locations, and not in 

the distribution during the day. To make the ellipses, we used the nearest neighbor 

hierarchical spatial clustering technique with a minimum of 35 incidents. We output the 

results to MapInfo, keeping the same scale for all maps. Four of them are shown below. 

9:00 AM 1:00 PM

7:00 PM 11:00 PM
Using Kernel Density Smoothing and Linking to ArcGIS: 

Examples from London, England 

Spencer Chainey 

Jill Dando Institute of Crime Science 

University College 

London, England 

CrimeStat offers an effective method for creating kernel density surfaces. The 

example below uses residential burglary incidents in the London Borough of Croydon, 

England for the period June 1999 – May 2000 (N=3104). The single kernel routine was 

used to produce a kernel density surface representing the distribution of residential 

burglary. 

The kernel function used was the quartic, which is favoured by most crime 

mappers as it applies added weight to crimes closer to the centre of the bandwidth. 

Rather than choosing an arbitary interval it is useful to use the mean nearest neighbour 

distance for different orders of K, which can be calculated by CrimeStat as part of a 

nearest neighbour analysis. For the Croydon data, an interval of 269 metres was chosen, 

which relates to a mean nearest neighbour distance at a K-order of 13. The output units 

were densities in square kilometres and was output to ArcGIS. 

Kernel density estimation is a particularly useful method as it helps to precisely 

identify the location, spatial extent and intensity of crime hotspots. It is also visually 

attractive, so helping to invoke further enquiry and the reasoning behind why crime and 

disorder is concentrated. The density surface that is created can reflect the distribution of 

incidents against the natural geography of the area of interest, including representing the 

natural boundaries, such as reservoirs and lakes, or an alignment that follows a particular 

street in which there is a high concentration of offending. The method is also less 

subjective if clear guidelines are followed for the setting of parameters. 
Infant Death Rate and Low Birth Weight 

in the I-5 Corridor of Seattle and King County 

Richard Hoskins 

Washington State Department of Health 

Olympia, Washington 

 Although the infant death rate (< 1 year old) has been steadily declining in 

Washington, the incidence of low birth weight (< 2500 gms) is increasing. This is a 

significant public health problem, resulting in suffering and high medical cost. If we 

know where the rates are high at a neighborhood level we can develop more efficient and 

effective programs. The goal is to determine regions where rates are clustered and to 

characterize those regions with respect to SES variables from the US Census. 

 Birth and infant death data were geocoded to the street level. In order to detect 

clusters of high infant death and low birth weight, several CrimeStat tools were used. We 

find that using several tools at once helps detect regions where something untoward is 

going on and also helps develops guesses about where other problems might be expected 

develop. 

 

 The result of a kernel density interpolation using a normal estimator is shown 

above along with an empirical Bayes rate and standardized mortality ratio (SMR) 

calculated in SAS and mapped in Maptitude (www.caliper.com). Starting with over 2,500 

infant deaths, about 25,000 low weight births (out of over 500,000 live births) occurred in 

the Seattle I-5 corridor region in King County from 1989-2002. The kernel density 

method was used to detect high rate regions. A clearly articulated region and ridge 

appears on the grid of the kernel density map and the 3D and prism maps. 

 

I-5 corridor in Kernel density Top: 3-D map: empirical Bayes rate 

King County interpolation Bottom: Prism map: SMR 
The Risk of Violent Incidents Relative to Population Density in Cologne Using 

the Dual Kernel Density Routine 

Dietrich Oberwittler and Marc Wiesenhütter 

Max Planck Institute for Foreign and International Criminal Law 

Freiburg, Germany 

When estimating the density of street crimes within a metropolitan area by interpolating 

crime incidents, the result is usually a very high concentration in the city center. However, there 

is also a very high concentration of people either living or pursuing their daily routine activities 

in these areas. The question emerges how likely is a criminal event when taking into account the 

number of people spending their time in these areas. The CrimeStat dual kernel density routine 

is able to estimate a ratio density surface of crime relative to the 'population at risk'. 

 

In this example, data on ‘calls to the police’ for assault and battery from April 1999 to 

March 2000 (N=6363 calls) and population from Cologne were used. Exact information on the 

number of people spending their time in the city does not exist. Therefore, 1997 counts of passengers entering and leaving the public transport system at each of 550 stations and bus stops in 

the city was used as a proxy variable. The number of persons at each station or bus stop was assigned to adjacent census tracts and added to the resident population resulting in a crude measure 

of the 'population at risk'. 

In the dual kernel routine, the density estimate of crime incidents is compared to the density estimate of the population at risk, defined by the centroids of census tracts with the number 

of persons as an intensity variable. We chose the normal method of interpolation and adaptive 

intervals with a minimum of five points. The adaptive bandwidth adjusts for the fact that there 

are fewer incidents and census tracts at the edges of the city, resulting in a relatively smoother 

density surface for the ratio. The results were output to ArcView. 

The effect of adjusting the crime distribution for the underlying 'population at risk' becomes quite visible. Whereas the concentration of crime is highest in the city center (left map), 

the crime risk (right map) is in fact much higher in several more distant areas that are known for 

high concentrations of socially disadvantaged persons. Given the imperfect nature of the population data these results should be interpreted as a broad view on the distribution of crime risk that, 

nevertheless, has important policy implications. 

 Single kernel density of crime incidences 

(assault & battery, Cologne 1999/2000) 

Dual kernel density of crime incidences 

relative to population at risk 
Kernel Density Interpolation of 

Police Confrontations in 

Buenos Aires Province, Argentina: 1999

Gastón Pezzuchi 

Crime Analyst 

Buenos Aires Province Police Force 

Buenos Aires, Argentina

One of our first tryouts with the CrimeStat software involved the calculation of 

both single and dual kernel density interpolations using data on 1999 confrontations with 

the police within Buenos Aires Province, an area that covers 29 counties around the 

Federal Capital. The confrontations include mostly gun fights with the police but also 

other attacks (e.g., knives, rocks, sticks). In the last three years, there has been an 

increase in confrontations with the police. The single interpolation shows a density 

surface that gives a good picture of the ongoing level of violence while the dual 

interpolations shows a risk surface using the personnel deployment data; the latter are 

confrontations relative to the number of police deployed. Typically, police are allocated 

to areas according to crime rates.\ 

Events = Police shootings (aprox. 800)

Buenos Aires City

(No-Data)

Buenos Aires City

(No-Data)

Example: Kernel Density Estimation

(CrimeStat)

Single Interpolation - Density of Events Dual Interpolation - Ratio of densities (Risk)

N N

Density

Low

Medium

High

No Data

Frontera

Risk

Low

Medium

High

No Data

Frontera

Both images are quite different, suggesting varying policing strategies. For 

example, though there are two well-defined hot spot areas in the Province (one in the 

north, the other in the south), the high levels of risk detected in the southern areas came 

as a complete surprise. The northern area has a higher crime rate than the southern area, 

hence a high police deployment. However, the level of confrontation is approximately 

equal between the two areas. 
Evolution of the Urbanization Process in the Brazilian Amazonia 

Silvana Amaral, Antônio Miguel V. Monteiro, Gilberto Câmara, José A. Quintanilha 

INPE, Instituto Nacional de Pesquisas Espaciais, Brazil 

The Brazilian Amazon rain forest is the world’s largest contiguous area of tropical 

rain forest in the world. During the last three decades, the region has experienced the 

largest urban growth rates in Brazil, a process that has reorganized the network of human 

settlements in the region. We used the CrimeStat single and dual kernel density routines 

to visualize trends in urbanization from 1996 to 2000 in Amazonia. Two variables were 

used to measure urbanization: 1) the concentration of urban nuclei (city density); and 2) 

the ratio of urban to total population. 

The concentration of cities was spatially associated with federal roads in the 

eastern and southern portions, and along the Amazonas River in the middle of the region. 

Additionally, the surfaces of urban population show that city density is not always 

associated with large urban populations. From 1996 to 2000 city density increased in the 

western Amazonia (Pará state) at a greater rate than the growth of the urban population. 

In the southeastern part of the region (Rondônia state), there were many urban centers. 

But the ratio of urban to total population was small, indicating that they are 

predominately agricultural regions. 

Urban Pop/Total Pop-1996 

City density - 2000 

City density - 1996 

Urban Pop/Total Pop-2000 
Using Small Area Estimation to Target Health Services 

in Harris County, TX

Thomas F. Reynolds, MS 

University of Texas-Houston School of Public Health 

 In Texas, the City of Houston and Harris County organized a Public Health Task 

force to make recommendations concerning the provision of health services for those 

without health insurance. Task force members wanted to know approximately how many 

area citizens did not have health insurance. 

 Data from the two most recent Current Population Survey Annual Social and 

Economic Supplements (CPS-ASEC, 2003-04) were used to derive a synthetic estimate 

using a stratified model. Estimates were calculated at census tract and block group 

levels. Selected political divisions were clipped from base maps for political officials and 

legislators. 

 Percentages are indicative of risk. On the other hand, numbers are essential for 

targeting physical resources. There is seldom a perfect correspondence between high 

percentages and large numbers. For example, an area with a concentration of multifamily housing may have a relatively small percentage, but a large number, of uninsured. 

Percentage maps of the uninsured (figure 1) are generally clustered and informative; 

however, due to large variations in population numbers at both levels of census 

geography, maps of the population densities of uninsured proved most valuable to 

officials (figure 2). 

CrimeStat was used to develop the density maps. The single kernel density 

routine was used to estimate the density of block group values using the centroid to 

represent the values and the number of uninsured as an intensity value. The Moran 

Correlogram was used to select the type of kernel for the single-kernel interpolation (a 

uniform distribution) and an optimal bandwidth. 

Fig. 1: Percent Uninsured Fig. 2: Population Density of Uninsured 
Identifying Voucher Holder and Crime Concentrations using Dual 

Kernel Density Estimation 

Ron Wilson, U.S. Department of Housing and Urban Development 

Public housing authorities and law enforcement are cooperating to reduce neighborhood 

crime where Housing Choice Voucher Program (HCVP) participants concentrate. When 

police departments and housing authorities can identify geographically combined 

voucher holder and crime concentrations, more specific strategies can be employed to 

reduce those crimes and prevent victimization. Aggravated assaults are common in and 

around neighborhoods where HCVP households concentrate. I used the relative sum of 

densities option of the Dual Kernel Density Estimation routine in CrimeStat IV to identify 

where HCVP households and aggravated assaults were concentrated in 2010. 

Several areas of voucher holder and aggravated assault concentrations are revealed with 

gradations in density, some in census tracts with high poverty. The Dallas Police 

Department might deploy varying community policing approaches in these areas based 

on concentration grade to reduce assault opportunities while building relationships with 

neighborhood residents. The Dallas Public Housing Authority may help voucher holders 

find safer neighborhoods to relocate outside the concentrated areas, in particular to areas 

with low poverty. These findings may also help Dallas city officials craft separate placebased polices that work to eliminate the root causes of aggravated assaults in these areas.