Chapter 9: 

Hot Spot Analysis of Zones 

Ned Levine 

Ned Levine & Associates 

Houston, TX 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table of Contents 

Assigning Point Data to Zones 9.1 

Local Indicator of Spatial Association 9.3 

Anselin’s Local Moran 9.4 

Similarity of Dissimilarity 9.5 

 ID Field 9.6 

 Distance Weights 9.6 

Small distance adjustment 9.6 

 Output for Each Zone 9.7 

Simulation of Confidence Intervals for Anselin=s Local Moran 9.7 

Example 1: Local Moran Statistics for Baltimore Auto Thefts 9.8 

Example 2: Simulated Local Moran Confidence Intervals for Houston Burglaries 9.11 

Use of Anselin’s Local Moran 9.15 

Limitations of Anselin’s Local Moran 9.15 

Getis-Ord Local “G” 9.16

 ID Field 9.17 

 Search Distance 9.17 

 Getis-Ord Local AG@ Simulation of Confidence Intervals 9.17 

 Output for Each Zone 9.17 

Example: Testing Houston Burglaries with the Getis-Ord Local AG@ 9.18 

Uses of the Getis-Ord Local AG@ 9.20 

Limitations of the Getis-Ord Local AG@ 9.20 

Zonal Nearest Neighbor Hierarchical Clustering 9.21

 Weighting Variable 9.22 

 Clustering Criteria 9.22 

 Criterion 1: Threshold Distance 9.22 

 Fixed distance 9.22 

Random nearest neighbor distance 9.23 

Area must be defined correctly 9.25 

Criterion 2: Zones with the Highest Number of Attributes 9.25 

 First-order Clusters 9.26 

 Second-order Clusters 9.27 

 Simulating Confidence Intervals 9.27 

 Type of Graphical Output 9.27 

 Ellipse cluster output 9.28 

 Output size for ellipses 9.28 
 

 

 

 

 

 

 

 

 

 

Table of Contents (continued) 

 Convex hull cluster output 9.28 

 Tabular Output 9.29 

Example 1: Simulated Clustering of Zones 9.29 

Example 2: Clustering of Houston Burglaries by Traffic Analysis Zones 9.30 

Uses of Zonal Nearest Neighbor Hierarchical Clustering 9.44 

Limitations of Zonal Nearest Neighbor Hierarchical Clustering 9.44 

References 9.46 

Endnotes 9.47 

Attachments 9.48 

A. Using Local Moran’s “I” to Detect Spatial Outliers in Soil Organic 

Carbon Concentrations in Ireland 

By Chaosheng Zhang and David McGrath 9.49 
 

Chapter 9: 

Hot Spot Analysis of Zones 

In this chapter, we will discuss methods for identifying hot spots with zonal data. The 

user should be thoroughly familiar with the information presented in Chapter 5 on spatial 

autocorrelation indices because two of the same indices are used for the analysis of local 

variations in zones. 

We are going to look at four techniques for analyzing hot spots with zonal data or with 

individual level data that have attributes (count or interval variables that measure a characteristic 

associated with the X and Y coordinates). These are Anselin’s Local Moran, the Getis-Ord 

Local “G”, the Zonal Nearest Neighbor Hierarchical Clustering algorithm, and the Risk-adjusted 

Zonal Nearest Neighbor Hierarchical Clustering algorithm. Figure 9.1 shows the Hot Spot 

Analysis of Zones page. 

Assigning Point Data to Zones 

If a user has information on the location of individual events (e.g., robberies), then it is 

better to utilize that information with the hot spot techniques discussed in Chapters 7 and 8. The 

individual-level information will contain all the uniqueness of the events. 

However, sometimes it is not possible to analyze data at the individual level. The user 

may need to aggregate individual data points to spatial areas (zones) in order to compare the 

events to data that are only obtained for zones, such as census data, or to model environmental 

correlates of the data points or may find that individual data are not available (e.g., when a police 

department releases information by police beats but not individual streets). Zonal data can 

include crime counts by zone, socio-economic information (e.g., collected by the census or 

estimated by a Metropolitan Planning Organization), or some other data that are aggregated to 

the small areas. In other words, the zone becomes the unit of analysis instead of the individual 

data points. 

Since the zones are not events, they have to be spatially analyzed by assuming that all the 

data resides at a single point within the zone. This is usually the centroid (the geographical 

center of the zone) but sometimes the center of minimum distance (the point at which the sum of 

the distances to all other points is minimized) has been used, too, especially if the zone is very 

irregularly shaped. However, when individual data points are assigned to zones, information is 

lost. For example, the distance between zones is a singular value for all the points in those zones 

whereas there is much greater variability with the distances between individual events. Also, 

9.3
Figure 9.1: 

Hot Spot Analysis of Zones Screen 
 

 

 

topological information, such as the shape of the zone or the number of other zones that are 

adjacent, is lost. 

For the spatial autocorrelation indices, the interaction between zones is defined by 

distance. There are advantages and disadvantages. Contiguity (or adjacency) is a property of a 

zone, not a point. Thus, adjacency defines whether one zone is next to another zone whereas 

distance is the distance between single points that represent the zones (e.g., centroids). For 

example, if two zones are 0.25 miles apart, it is not known whether they are adjacent or not. In 

other words, in adopting a distance-based weight, information about adjacencies is lost. On the 

other hand, a distance-based weight is standardized. If two zones are adjacent, it is not known 

how far apart they are separated. Adjacencies can be misleading since they do not indicate the 

size of the adjacent zones whereas a specified distance is always constant. 

The zonal data also must include an attribute variable, a variable associated with the zone 

(e.g., number of robberies; median household income; percentage of households living below 

poverty level). The attribute can be a count or a continuous variable for a distributional property 

of the zone (e.g., median household income; percentage of households below poverty level) or 

even a binary variable (e.g.,1 v. 0).1 The indices discussed in this chapter are applied to the 

interaction between the attribute variable of the central zone and other zones, weighted by the 

distance between them. 

Individual level data can also have attributes. For example, Levine and Lee (2013) 

analyzed journey-to-crime distances for offenders in Manchester, England. In this case, the 

attribute variable was the distance traveled and the statistics discussed in this chapter are 

appropriate for analyzing that attribute data. Other examples of individual level data with 

attributes would be the age of the offender, the number of prior convictions, or the number of 

years of formal education. The key criterion is that the records must have an attribute which is 

either a count or an interval variable. 

Local Indicator of Spatial Association 

The basic concept behind a zone-specific measure of spatial autocorrelation is that of a 

local indicator of spatial association (LISA) and has been discussed by a number of researchers 

(Mantel, 1967; Getis, 1991; Anselin, 1995). For example, Anselin (1995) defines this as any 

statistic that satisfies two requirements: 

There is no fundamental difference between a count variable and a continuous interval or ratio variable 

since a real number can be converted into a count by multiplying by a power of 10 (e.g., 1.23 = 123 x 10-2). 

The statistics discussed in this chapter are applicable to either count or continuous data. 

9.5

1 
 

 

1. The LISA for each observation indicates the extent to which there is significant

spatial clustering of similar values around that observation; and

2. The sum of the LISAs for all observations is proportional to the global indicator of

spatial association:

(9.1) 

where Li is the local indicator of zone ݅, g(Yi ) is a function of the value of an 

intensity variable, Yi, at location ݅, h(Yji) is a weight function of the values of the 

intensity variable observed in the neighborhood ji of ݅, and f is a scaling constant 

to ensure that the sum of Li equals the global spatial autocorrelation index. 

The function of the intensity variable can be a raw score, Yi, a Z-transformation of the 

intensity variable, such as: 

(9.2)

where ܻത is the mean of Y and SY is the standard deviation of variable Y, or some other function. 

In other words, a LISA is an indicator of the extent the value of an observation is affected 

by its neighboring observations. This requires two conditions. The first is that each observation 

has a value of an attribute variable that can be assigned to it (i.e., an intensity or weight value) in 

addition to its X and Y coordinates. For crime incidents, this means data must be aggregated 

into zones (e.g., number of incidents by census tracts, zip codes, or police reporting districts). 

Second, the neighborhood has to be defined. This could be either adjacent zones, all 

other zones negatively weighted by the distance from the observation zone, or all other zones 

negatively weighted by the distance from the observation zone up to some distance whereupon 

the weight is zero afterward (a bandwidth). Once these are defined, the LISA indicates the value 

of the observation zone in relation to its neighborhood. 

Anselin=s Local Moran 

Anselin=s Local Moran statistic was developed by Luc Anselin and is the oldest LISA 

statistic (Anselin, 1995). The procedure applies Moran=s “I” statistic to individual zones (see 

Chapter 5), allowing them to be identified as similar or different to their nearby pattern. 

9.6
The definition of “Ii” is from Getis and Ord (1996): 

 

 

 

The expected “Ii” is defined as: 

(9.3) 

where Zi is the intensity of observation i, ܼ̅ is the mean intensity over all observations, Zj is 

intensity for all other observations, j (where j =/ i), ܵ௓

ଶ is the variance over all observations, and 

Wij is a distance weight for the interaction between observations i and j. The first term in 

equation 9.3 refers only to observation ݅ while the second term is the sum of the weighted values 

for all other observations (but not including ݅ itself). 

(9.4)

where Wij is the distance weight for the interaction between observations ݅ and ݅. The variances 

of Ii are somewhat complicated (see endnote i for the formulas). 

Similarity or Dissimilarity 

Since the global Moran’s “I” statistic measures similarity in observations over a study 

area (see Chapter 5), the local Moran “Ii” also indicates the similarity of a zone relative to its 

neighbors. Thus, in neighborhoods where both the zone and its neighbors have high attribute 

values, the Local Moran will be positive indicating that the particular zone is similar (i.e., also 

>high=). Similarly, in neighborhoods where both the zone and its neighbors have >low= attribute

values, the Local Moran also will be positive indicating that the zone is similar to its neighbors

(i.e., also >low=). When the Local Moran statistic is positive, this is an indicator of similarity, not

absolute value of the intensity variable.

Conversely, if a zone has a high value of the intensity variable while its neighbors have 

low values or, alternatively, it has a low value while the neighbors have high values, then the 

Local Moran statistic will be negative. Dissimilarity is an indicator of either a hot spot or a cold 

spot, in other words zones that are different from their neighborhood. Hot spots would be seen if 

the number of incidents in a zone is much higher than in the nearby zones. Cold spots would be 

seen if the number of incidents in a zone is much lower than in the nearby zones. 

In other words, the Local Moran statistic indicates whether the zone is similar or 

dissimilar to its neighbors. 

9.7
in whichever distance units are specified (miles, kilometers, etc). 

First, there is a traditional distance decay function: 

 

 

 

ID Field 

The user should indicate a field for the ID of each zone. This ID will be saved with the 

output and can then be linked with the input file (Primary File) for mapping. 

 Distance Weights 

The weights, Wij, can be either an indicator of the adjacency of a zone to the observation 

zone (i.e., >1' if adjacent; 0 if not adjacent) or a distance-based weight which decreases with 

distance between zones i and j. Adjacency indices are useful for defining near neighborhoods; 

the adjacent zones have full weight while all other zones have no weight. Distance weights, on 

the other hand, are useful for defining spatial interaction; zones which are farther away can have 

an influence on an observation zone, although one that is much less. CrimeStat uses distance 

weights, in two forms. 

(9.5) 

where dij is the distance between the observation zone, i, and another zone, j. For example, a 

zone which is two miles away has half the weight of a zone that is one mile away. 

Small distance adjustment 

Second, there is an adjustment for small distances. The weight index becomes 

problematic with small distance between zones since the weight will approach infinity for dij -> 

0. To correct for this, the routine includes an adjustment for small distances so that the

maximum weight can be never be greater than 1.0 (see Chapter 5). The adjustment scales

distance to one mile, which is a typical distance for crime analysis. When the small distance

adjustment is turned on, the minimal distance is scaled automatically to be one mile. The

formula used is:

(9.6)

9.8
 

 

 

Output for Each Zone 

The output is for each zone includes: 

1. The sample size

2. The ID identifier

3. The X coordinate

4. The Y coordinate

5. The “Ii” value

6. The expected “Ii”.

If the variance box is checked, the program will also calculate the variance, standard 

error, and a Z-test of “Ii” for each zone. The default is for the variance not to be calculated. 

Simulation of Confidence Intervals for Anselin=s Local Moran 

There are two ways to estimate confidence intervals for Anselin’s Local Moran. First, 

the routine can calculate the variance and, for each zone, the standardized AIi@ score to produce a 

Z-test of the significance of the AIi@. Assuming the sample size is greater than 120, 95% percent

confidence intervals can be estimated by:

and 99% confidence intervals can be estimated by: 

(9.7) 

(9.8) 

One problem with this test is that AIi@ may not actually follow a normal standard 

distribution. That is, if AIi@ is calculated for all zones with random data, the distribution of the 

statistic may not be (and often will not be) normally distributed. This would be especially true if 

the variable of interest, Z, is skewed with some zones having very high values while the majority 

having low values, as is typically true with crime distributions. 

Second, the user can estimate confidence intervals (called credible intervals) using a 

Monte Carlo simulation. A permutation type simulation is run whereby the locations of the 

zones are kept and the original values of the intensity variable, Z, are maintained but randomly 

re-assigned to zones for each simulation run. This will maintain the structure of the attribute “Z” 

variable but will estimate the value of “Ii” for each under random assignment of this variable. 

Note that a simulation may take time to run especially if the data set 9.9

is large or if a large number of simulation runs are requested. 
 

 

If a permutation Monte Carlo simulation is run to estimate credible intervals, specify the 

number of simulations to be run (e.g., 1,000, 5,000, 10000). In addition to the AIi@ for each zone, 

the expected AIi@ and the variance (if requested), the output includes the results that were 

obtained by the simulation for: 

1. The minimum AIi@ value 

2. The maximum AIi@ value 

3. The 0.5 percentile of AIi@

4. The 2.5 percentile of AIi@

5. The 97.5 percentile of AIi@

6. The 99.5 percentile of AIi@

The two percentile pairs (2.5 and 97.5; 0.5 and 99.5) create approximate 95% and 99% 

credible intervals respectively. The minimum and maximum AIi@ values create an >envelope=

around each zone. It is important to run enough simulations to produce reliable estimates. 

The tabular results can be printed, saved to a text file or saved as a '.dbf' file with a 

LMoran<root name> prefix with the root name being provided by the user. For the latter, 

specify a file name in the ASave result to@ in the dialogue box. The >dbf= file can then be linked to 

the input >dbf= file by using the ID field as a matching variable. This would be done if the user 

wants to map the AIi@ variable, the Z-test, or those zones for which the AIi@ value is either higher 

than the 97.5 or 99.5 percentiles or lower than the 2.5 or 0.5 percentiles of the simulation results. 

Example 1: Local Moran Statistics for Baltimore Auto Thefts 

Using data on 14,853 motor vehicle thefts for 1996 in both Baltimore County and 

Baltimore City, the number of incidents occurring in each of 1,349 census block groups was 

calculated (Figure 9.2). As seen, the pattern shows a higher concentration towards the center of 

the metropolitan area, as would be expected, but that the pattern is not completely uniform. 

There are many block groups within the City of Baltimore with very low counts of auto 

thefts and there are block groups within the County with very high counts. Using these data, 

CrimeStat calculated the Local Moran statistic with the variance box checked and the small 

distance adjustment used. The range of Ii values varied from -37.26 to +180.14 with a mean of 

5.20. The standardized Local Moran >Z= varied from -12.71 to 50.12 and with a mean of 1.61. 

Figure 9.3 maps the distribution. Because a negative Ii value indicates dissimilarity, these values 

have been drawn in red compared to blue for a positive Ii value. As seen, in both the City of 

Baltimore and the County of Baltimore, there are block groups with large negative Ii values, 

indicating that they differ from the surrounding block groups. 

9.10
Figure 9.2: 
Figure 9.3: 
 

For example, in the central part of Baltimore City, there is a small area of about eight 

block groups with low numbers of auto thefts, compared to the surrounding block groups. These 

form a >cold spot=. Consequently, they appear in dark tones in Figure 9.3 indicating that they 

have high Ii values (i.e., negative spatial autocorrelation). Similarly, there are several block 

groups on the western side of the County which have relatively high numbers of auto thefts 

compared to the surrounding block groups. They form a hot spot. Consequently, they also 

appear in dark tones in Figure 9.3 because this indicates positive spatial autocorrelation, having 

values that are similar to the surrounding blocks. In other words, similarity is shown in blue and 

dissimilarity in red. 

Example 2: Simulated Local Moran Confidence Intervals for Houston Burglaries 

To illustrate the simulated confidence intervals, we apply the Local Moran statistic to 

burglaries in the City of Houston shown in figure 9.4. The data were 26,480 burglaries that 

occurred in 2006. They were aggregated to 1,179 traffic analysis zones (TAZ). Anselin=s Local 

Moran statistic was calculated on each of the TAZ’s with 1,000 Monte Carlo simulations being 

calculated. Figure 9.5 shows a map of the calculated local AIi@ values. It can be seen that there 

are many more zones of positive spatial autocorrelation where the zones are similar to their 

neighbors. In most of these cases, the zone has few burglaries whereas it is surrounded by zones 

that also have few burglaries. A few zones have negative spatial autocorrelation. In most of the 

cases, the zones have many burglaries and are surrounded by zones with few burglaries. 

Confidence intervals were calculated in two ways. First, the theoretical variance was 

calculated and a Z-test computed. This is done in CrimeStat by checking the >theoretical 

variance= box. The test assumes that AIi@ is normally distributed, which may or may not be a 

valid assumption. Second, a Monte Carlo simulation was used to estimate the 99% confidence 

intervals (i.e., outside the 0.5 and 99.5 percentiles). 

Table 9.1 shows the results for four records. The four records illustrate different 

combinations. In the first record (TAZ 522), the AIi@ value is 0.000373, indicating positive 

spatial autocorrelation (i.e., nearby zones have similar values). Comparing it to the 95% credible 

intervals, it is larger than the 97.5th percentile. In addition, the Z-test, based on the theoretical 

variance, is positive. Thus, both the simulated confidence intervals and the theoretical 

confidence interval indicate that the AIi@ for this zone is significant. 

9.13
Figure 9.4: 
Figure 9.5: 
 

 

 

Table 9.1: 

Anselin=s Local Moran 95% Confidence & Credible Intervals 

4 Cases Estimated from Theoretical Variance and from Monte Carlo Simulation 

Simulated Theoretical 

TAZ X Y AIi@ Expected 0.5 % 97.5 % Z-test p 

522 3152030 13941900 0.000373 -0.000010 -0.000856 0.000216 2.29 0.05 

534 3200630 13955800 0.000345 -0.000007 -0.000516 0.000226 1.82 n.s. 

182 3126150 13842900 -0.040641 -0.000087 -0.014287 0.007292 -9.69 0.0001 

384 3156740 13879400 -0.000886 -0.000018 -0.001259 0.000593 -2.20 0.05 

In the second record (TAZ 534), the AIi@ value is 0.000345, also indicating positive spatial 

autocorrelation. However, the AIi@ value is greater than the 97.5th percentile, indicating that the 

simulation suggests the AIi@ is greater than what would be expected by chance. On the other 

hand, the Z-test, based on the theoretical distribution, is not significant. Thus, there is an 

inconsistency between simulation test and the Z-test. 

In the third record (TAZ 182), there is consistency between the simulated and theoretical 

significance tests. The AIi@ is negative (-0.040641), indicating negative spatial autocorrelation 

(i.e., the has different values than nearby zones). The simulation shows that the AIi@ is more 

negative than the simulated 5th percentile and the Z-test is also significantly negative. 

The fourth record (TAZ 384) shows a negative AIi@, indicating negative spatial 

autocorrelation (i.e., nearby zones have different values). But there is inconsistency in the test. 

The simulation shows that this AIi@ falls between the 5th and 97.5th percentiles, indicating nonsignificance, whereas the Z-test suggests the AIi@ is significant. 

In general, simulated confidence intervals will be similar to the theoretical ones. But, 

there can be discrepancies. The reason is that the sampling distribution of AIi@ may not be (and 

probably is not) normally distributed. Of the 1,179 traffic analysis zones, 661 showed significant 

AIi@ values according to the simulated 99% credible intervals (i.e., either equal to or smaller than 

the 0.5 percentile or equal to or greater than the 99.5 percentile) while 688 of the zones showed 

significant AIi” values according to the theoretical Z-test at the 99% level (i.e., having a Z-value 

equal to or less than -2.58 or equal to or greater than 2.58). It would behoove the user to 

estimate the number of zones that are significant according to both the simulated and theoretical 

confidence intervals before making a decision as to which criterion to use. 

Therefore, both the simulated confidence interval and the theoretical distribution should 

be used with caution. The best mapping solution may be to map only those zones that are highly 

9.16
significant with both tests showing substantial significance. Or, alternatively, map only those 

zones with the highest positive or highest negative AIi@ values. 

Uses of Anselin’s Local Moran 

Anselin’s Local Moran has a number of uses. First, it can identify zones that are different 

(dissimilar) from its neighbors. This can be a good first step in finding locations that either have 

higher crime numbers (a hot spot) or lower crime numbers (a cold spot) than the neighboring 

areas. This can focus police efforts on identifying the problems that cause the zone to be higher 

in the case of a hot spot or to identify factors that mitigate crime in the case of a cold spot. 

Second, another use of Anselin=s Local Moran statistic is to identify >outliers=, zones that 

are very different from their neighbors. In this case, zones with a high negative I value (e.g., 

with an “Ii” smaller than two standard deviations below the mean) are indicative of outliers. 

They either have a high number of incidents whereas their neighbors have a low number or, the 

opposite, a low number of incidents amidst zones with a high number of incidents. Identifying 

the outliers can focus on zones that are unique (and which should be studied) or, in multivariate 

analysis, on zones that need to be statistically treated differently in order to minimize a large 

modeling error (e.g., creating a dummy variable for the extreme outliers in a regression model). 

In short, the Local Moran statistic can be a useful tool for identifying zones that are 

dissimilar from their neighborhood. To use the Local Moran statistic, however, requires that the 

data be summarized into zones in order to produce the necessary intensity value. Given that 

most crime incident databases will list individual events without intensity or weight values 

assigned, this will entail additional work by a law enforcement agency. 

Limitations of Anselin’s Local Moran 

There are several limitations to the method. First, because it is an index of similarity, a 

positive “Ii” value does not necessarily indicate a hot spot. The positive “Ii” value could be due 

to zones with low values of the intensity variable surrounded by other zones that also have low 

values. Thus, in terms of using the method to identify hot spots of zones can lead to ambiguous 

results. It is best seen as a first step in identifying hot spot zones. 

Second, there are concerns about the statistical criterion used to identify a zone as being 

similar or dissimilar to its neighbors. One has to be suspect about a technique that finds 

significance in more than half the cases. It would probably be more conservative to use 99% 

confidence intervals for identifying zones that show positive or negative spatial autocorrelation 

rather than using 95% confidence intervals or, better yet, choosing only those zones that have 

9.17
 

 

 

 

 

 

 

 

 

 

very negative or very positive AIi@ values. Unfortunately, this characteristic of Anselin=s local 

Moran is also true of the local Getis-Ord statistic, which is discussed below. The significance 

tests, whether simulated or theoretical, are not strict enough and, thereby, increase the likelihood 

of a Type I (false positive) error. A user must be very careful in interpreting AIi@ values for 

individual zones and would be better served choosing only the very highest or lowest. 

For a detailed discussion of problems in conducting tests on local spatial autocorrelation 

statistics, such as the local Moran or Getis-Ord Local “G” (to be discussedbelow), see Waller 

and Gottway (2004; p. 238). 

Getis-Ord Local AG@ 

The Getis-Ord Local G statistic applies the Getis-Ord "G" statistic to individual zones to 

assess whether particular zoness are spatially related to the nearby zones (see Chapter 5). Unlike 

the global Getis-Ord AG@ but like Anselin’s Local Moran, the Getis-Ord Local AG@ is applied to 

each individual zone. The formulation presented here is taken from Wong and Lee (2005). The 

AG@ value is calculated with respect to a specified search distance (defined by the user), namely: 

(9.9)

 (9.10)

 (9.11) 

(9.12)

where wj is the weight of zone Aj@ from zone Ai@, Wi is the sum of weights for zone Ai@, and n is 

the number of cases. 

The standard error of G(d) is the square root of the variance of G. Consequently, a Z-test 

can be constructed by:

 (9.13) 

(9.14)

9.18
 

 

A good example of using the Getis-Ord local AG@ statistic in crime mapping is found in 

Chainey and Racliffe (2005, pp. 164-172). 

ID Field 

The user should indicate a field for the ID of each zone. This ID will be saved with the 

output and can then be linked with the input file (Primary File) for mapping. 

Search Distance 

The user must specify a search distance for the test and indicate the distance units (miles, 

nautical miles, feet, kilometers, meters, 

Getis-Ord Local AG@ Simulation of Confidence Intervals 

Since the Getis-Ord AG@ statistic may not be normally distributed, the significance test is 

frequently inaccurate. Instead, a permutation type Monte Carlo simulation can be run whereby 

the original values of the intensity variable, Z, for the zones are maintained but are randomly reassigned to zones for each simulation run. This will maintain the distribution of the variable Z 

but will estimate the value of G for each zone under random assignment of this variable. Specify 

the number of simulations to be run (e.g., 100, 1000, 10000). 

Output for Each Zone 

The output is for each zone includes: 

1. The sample size

2. The ID

3. The X coordinate

4. The Y coordinate

5. The AG@

6. The expected AG@

7. The difference between AG@ and the expected AG@

8. The standard deviation of AG@

9. A Z-test of "G" under the assumption of normality for the zone

and if a simulation is run: 

10. The 0.5 percentile of AG@ for the zone

9.19
11. The 2.5 percentile of AG@ for the zone

12. The 97.5 percentile of AG@ for the zone

13. The 99.5 percentile of AG@ for the zone

The two pairs of percentiles (5 and 95; 2.5 and 97.5; 0.5 and 99.5) create approximate 

95% and 99% credible intervals respectively around each zone. The minimum and maximum AG@

values create an >envelope= around each zone. However, unless a large number of simulations 

are run, the actual AG@ value may fall outside the envelope for any zone. The tabular results can 

be printed, saved to a text file or saved as a '.dbf' file. For the latter, specify a file name in the 

ASave result to@ in the dialogue box. The file is saved with a LGetis-Ord<root name> prefix 

with the root name being provided by the user. 

The >dbf= output file can be linked to the Primary File by using the ID field as a matching 

variable. This would be done if the user wants to map the AG@ variable, the expected AG@, the Ztest, or those zones for which the AG@ value is either higher than the 97.5 or 99.5 percentiles or 

lower than the 2.5 or 0.5 percentiles of the simulation results respectively (95% or 99% 

confidence intervals). 

Example: Testing Houston Burglaries with the Getis-Ord Local AG@ 

Using the same data set on the Houston burglaries as above, the Getis-Ord Local AG@ was 

run with a search radius of 2 miles. The output file was then linked to the input file using the ID 

field to allow the mapping of the local AG@ values. Figure 9.6 illustrates the Z-test of the GetisOrd Local AG@ for different zones. The map displays the significance of the Z-test (the 

difference between the AG@ and the expected AG” relative to the standard error of “G”). Zones 

with a Z-test of +1.96 or higher are shown in blue (hot spots). Zones with Z-tests of -1.96 or 

smaller are shown in red (cold spots) while zones with a Z-test between -1.96 and +1.96 are 

shown in yellow (no pattern). 

As seen, there are some very distinct patterns of zones with high positive spatial 

autocorrelation and low positive spatial autocorrelation. Examining the original map of 

burglaries by TAZ (Figure 9.4), it can be seen that where there are many burglaries, the zones 

tend to show high positive spatial autocorrelation (hot spots) in Figure 9.6. Conversely, where 

there are few burglaries, the zones show either low positive spatial autocorrelation (‘cold spots’) 

or, more commonly, no pattern in Figure 9.6. In particular, the greater downtown Houston area, 

and area southwest of downtown that includes the Texas Medical Center and a commercial area 

west of downtown around the IH 610 ‘loop’ show areas of significant ‘cold spots’. These are 

areas dominated by commercial or office buildings and generally have relatively few burglaries. 

9.20
Figure 9.6: 
 

Uses of the Getis-Ord Local AG@ 

The Getis-Ord Local AG@ is very good at identifying hot spots and also good at 

identifying cold spots. As mentioned, Anselin=s Local Moran can only identify positive or 

negative spatial autocorrelation, that is, whether the zones are similar or dissimilar. Those zones 

with positive spatial autocorrelation could occur because zones with high values are nearby other 

zones with high values or they could occur because zones with low values are nearby other zones 

with low values. The Getis-Ord Local AG@ can distinguish those two types. 

Limitations of the Getis-Ord Local AG@ 

The biggest limitation with the Getis-Ord Local AG@, which also applies to the global 

Getis-Ord and Getis-Ord Correlogram routines (see Chapter 5), is that it cannot detect negative 

spatial autocorrelation where a zone is surrounded by neighbors that are different (either having a 

high value surrounded by zones with low values or having a low value and being surrounded by 

zones with high values). In actual use, both the Anselin=s Local Moran and the Getis-Ord Local 

AG@ should be used to produce a full interpretation of the rsults. 

Another limitation is that the significance tests are too weak, allowing too many zones to 

show significance. In the data shown in Figure 9.6, 63% of the zones (740) were statistically 

significant by the Z-test! A simulation of credible intervals also showed a very high proportion 

having G values greater or less than the 95% credible intervals. Thus, there is a substantial Type 

I error with this statistic (false positives), a similarity it shares with Anselin=s Local Moran. 

Reducing the search radius will reduce the number of zones with significant Z-scores. 

For example, with a 1 mile search radius, only 44% of the zones were statistically significant by 

the Z-test. But, given the size of the zones, there is a limit to how small a search radius can be 

made. With the Houston block groups, for example, the average area of a block group is 0.48 

square miles. If a typical block group size is viewed as a square having that area, then each side 

would be about 0.7 miles in length. Choosing a search radius smaller than 0.7 would end up 

with many zones not having neighbors selected, especially farther away from the city center 

where zones are generally much larger in size. This would lead to an unrealistic estimate of the 

amount of spatial autocorrelation. In other words, there is a trade-off between the precision of 

the search radius and the accuracy of the “G” estimate. In this case, a search radius of two miles 

is a realistic search radius for this geographical distribution. 

Waller and Gottway (2004, p. 238) point out that there are four problems with the testing 

of LISA statistics since the measures are interrelated: First, the distributional properties remain 

largely unknown. Second, multiple tests lead to overly rejecting the null hypothesis, which we 

9.22
 

have demonstrated above. Third, the LISA’s of neighboring zones are often highly correlated 

due to using the same data and, fourth, many of the tests are based on small samples sizes since 

the number of events in any one zone may be limited. A random simulation can overcome the 

first problem by using the empirical distribution as a basis for calculating credible intervals, but 

it cannot overcome the next three. 

In short, a user should be very careful in interpreting zones with significant AG@ values 

and would probably be better served by choosing only those zones with the highest or lowest AG@

values. 

Zonal Nearest Neighbor Hierarchical Clustering 

The zonal nearest neighbor hierarchical spatial clustering routine applies the nearest 

neighbor hierarchical clustering algorithm (Nnh; see Chapter 7 for the background and details) to 

zonal data. The point-based Nnh is a constant-distance clustering routine that groups points 

together on the basis of spatial proximity. A threshold distance is defined and the minimum 

number of points that are required for each cluster specified. The output can be displayed with 

ellipses or convex hulls. 

On the other hand, in the zonal Nnh (Znnh), the algorithm is adjusted to allow weighting 

of each zone usually applied to a single point within the zone (e.g., a centroid). Thus, if the 

‘point’ is a centroid of a zone, then the weighting is an attribute assigned to that centroid (e.g., 

population, employment, median household income). Clusters are groups of adjacent zones that 

have much higher weights than non-clustered zones. 

The routine requires a primary file (e.g., robberies) that is weighted with the weight or 

intensity variable (see Primary File). On the Znnh routine, the user defines a weighting variable, 

a threshold distance, the minimum number of values of the weighting variable that are required 

for each cluster, and the type of output size, either standard deviational ellipses or convex hulls. 

The routine identifies first-order clusters that represent groups of zones that are closer 

together than the threshold distance, that have the highest weights, and in which there is at least 

the minimum number of zones specified by the user (the minimum is 3 zones). Clustering is 

hierarchical in that the first-order clusters are treated as separate ‘points’ to be clustered into 

second-order clusters, and the second-order clusters are treated as separate ‘points’ to be 

clustered into third-order clusters, and so on. Higher-order clusters will be identified only if the 

distances between their centers are closer than the new threshold distance. 

9.23
For example, if the attribute to be grouped is the number of crimes in a zone, then the 

routine identifies adjacent zones that have high concentrations of crimes. The user can modify 

the number of clusters identified and the relative size of them by changing the search radius or 

the minimum number of attributes that must be grouped together. The results can be output as 

either standard deviational ellipses or convex hulls. 

Weighting Variable 

Each zone must be weighted by an attribute variable. This is the weight or intensity 

variable defined on the Primary File page. The user specifies whether the weight or the intensity 

variable is to be used for the attribute. The default is Intensity. 

Clustering Criteria 

Two criteria are used to group zones together. 

Criterion 1: Threshold Distance 

The first criterion in identifying clusters is whether zones are closer than a specified 

threshold distance. There are two alternatives in selecting the threshold distance: 1) a fixed 

distance (the default is 2 miles); or 2) a random nearest neighbor distance. 

Fixed distance 

Unlike the Nnh routine for clustering points (Chapter 7), the default alternative for 

selecting a threshold distance in the Znnh is to choose a fixed distance (in miles, nautical miles, 

feet, kilometers, or meters). The user checks the AFixed distance@ box and selects a threshold 

distance. The default value is 2 miles but the user can change this. 

The main advantage of this method is that, first, the search radius can be specified exactly 

and, second, unlike points, zones do not overlap and are spatially dispersed. The distance 

between adjacent zones may be substantial especially for large zones at the periphery of an urban 

area. Thus, to capture adjacent zones that have high values of the attribute variable requires 

choosing a search radius that is large. 

The main disadvantage of this method is that the choice of a threshold is subjective. 

There is no reason why any particular search radius should be chosen. Further, the larger the 

distance that is selected, the greater the likelihood that clusters will be found by chance. This 

can be tested using a Monte Carlo simulation (see below). 

9.24
 

 

 (9.17)

ൌ ݈ܽݒݎ݁ݐ݅݊ 	 	݂݊݅݀݁݊ܿ݁݋ܿ	݂݋ ݐ݈݅݉݅ 	 	ݎ݁ݓ݋ܮ

 (9.18)

 

Random nearest neighbor distance 

The alternative is to use the expected random nearest neighbor distance for first-order 

nearest neighbors. The user specifies a one-tailed confidence interval around the random 

expected nearest neighbor distance. The t-value corresponding to this probability level, t, is 

selected from the Student=s t-distribution under the assumption that the degrees of freedom are at 

least 120.2

 This selection is controlled by a slide bar under the routine (see Figure 9.1). From 

Chapter 6, the mean random distance is defined as: 

(9.15)

where A is the area of the region and N is the number of zones and the standard error of the mean 

random distance is: 

(9.16)

where A is the area of the region and N is the number of zones. The confidence interval around 

that distance is defined as: 

This is the next highest degree of freedom in the Student=s t-table below infinity. 

9.25

2 

where t is the t-value associated with a probability level in the Student=s t-distribution. 

 

The approximate lower limit of this confidence interval is: 
	 	 	 	

 

ൌ ݈ܽݒݎ݁ݐ݅݊ ݂݊݅݀݁݊ܿ݁݋ܿ	݂݋ ݐ݈݅݉݅ ݎ݁݌݌ܷ

 

 

 

and the upper limit of this confidence interval is: 

(9.19) 

The confidence interval defines a probability for the distance between any pair of zones. 

For example, for a specific one-tailed probability, p, fewer than p% of the zones would have 

nearest neighbor distances smaller than this selected limit if the distribution was spatially 

random. If the data were spatially random and if the mean random distance is selected as the 

threshold criteria (the default position on the slide bar), approximately 50% of the pairs will be 

closer than this distance. For randomly distributed data, if a p#.05 level is taken for t (two steps 

to the left of the default or the fifth in from the left), then only about 5% of the pairs would be 

closer than the threshold distance. Similarly, if a p#.75 level is taken for t (one step to the right 

of the default or the fifth in from the right), then about 75% of the pairs would be closer than the 

threshold distance. 

Table 9.2: 

Approximate Probability Values Associated with Threshold Scale Bar

Scale Bar

Position Probability Description 

1 0.00001 Far left point of slide bar 

2 0.0001 Second from left 

3 0.001 Third from left 

4 0.01 Fourth from left 

5 0.05 Fifth from left 

6 0.1 Sixth from left 

7 0.5 Sixth from right (default value) 

8 0.75 Fifth from right 

9 0.9 Fourth from righ 

10 0.95 Third from righ 

11 0.99 Second from righ 

12 0.999 Far right point of slide bar 

9.26
 

 

(9.18) ൌ ݏ݊݋݅ݐܾ݉݅݊ܽ݋ܥ

In other words, the threshold distance is a probability level for selecting any two zones (a 

pair) on the basis of a chance distribution. The slide bar has 12 levels and is associated with a 

probability level for a t-distribution from a sample of 120 or larger. From the left, the p-values 

are approximately (see Table 9.2 above): 

Taking a broader conception of this, if there is a spatially random distribution, then for all 

distances between pairs of zones, of which there are 

fewer than p% will be shorter than this threshold distance. 

Area must be defined correctly 

Note that it is very important that area be defined correctly for this routine to work. If the 

user defines the area on the measurement parameters page (see Chapter 3), the Znnh routine uses 

that value to calculate the threshold distance. If the user does not define the area on the 

measurement parameters page, the routine calculates the area from the minimum and maximum 

X/Y values (the bounding rectangle), which will usually be a larger area. In either case, the 

routine will be able to calculate a threshold distance and run the routine. 

However, if the area units are defined incorrectly on the measurement parameters page, 

then the routine will certainly calculate the threshold distance wrongly. For example, if data are 

in feet but the area on the measurement parameters page are defined in square miles, most likely 

the routine will not find any zones that are farther apart the threshold distance since that distance 

is defined in miles. In other words, it is essential that the area units be consistent with the data 

for the routine to properly work. 

Criterion 2: Zones with the Highest Number of Attributes 

The second criterion involves the weighting of each zone. With zonal data, each zone 

has an attribute value, defined either by the intensity variable or weight variable on the Primary 

File page. Clusters are defined by those zones that are within the threshold distance but which 

have the highest combined value of the attribute variable. The algorithm looks for a ‘center’ of 

three of more zones for which the total value of the attribute variable is highest. Like the Nnh 

routine, the process is iterative, first finding an approximate center and then re-calculating it with 

respect to the total value of the attribute variable for those zones within the threshold distance of 

the center. Eventually, the process stabilizes and the routine quits. 

9.27
 

 

 

 

 

 

 

Table 9.3 presents a simple example. Suppose there are two zones (A and B) within the 

second matrix and each has three other zones closer than the threshold distance (C, D, E for Zone 

A and F, G, H for Zone B). In this example, Zone A would be chosen as the initial center for the 

first cluster because the sum of the weights (for itself and for the three other zones that are within 

the threshold distance) add to 85 whereas the sum of the weights for the other points for Zone B 

only add to 65 even though Zone B had a higher weight for itself than Zone A. 

Table 9.3: 

Example of Weighting Pairs of Zones by Attributes 

Zone A Zone B 

Other Zones 

A (itself) 

C

Weighting 

10

 20

Other Zones 

B (itself) 

F

Weighting

20

 10

 D 30 G 15

 E 25 H 20

--- ---

TOTAL: 85 65 

The routine then removes the zones selected for the first cluster (A, C, D, and E). It then 

attempts to find a second cluster. In this example, there is only one other (B, F, G, and H), which 

is then removed from the matrix. If there were more zones, the routine would look for additional 

centers of clusters. 

Having completed an initial identification of cluster centers, the routine then calculates 

the center of minimum distance (CMD) for the selected points and then calculates those zones 

that are within the threshold distance of the CMD. It repeats the process for a second cluster. 

After a second round of clustering, the routine repeats the process for a third cluster. The 

iterations continue until no zones change clusters and the calculated center of minimum distance 

changes very little. 

First-order Clusters 

Using these criteria, CrimeStat constructs a first-order clustering of the zones. For each 

first-order cluster, the center of minimum distance is output as the cluster center, which can be 

saved as a >.dbf= file. 

9.28
Second and Higher-order Clusters 

The first-order clusters are then tested for second-order clustering. The procedure is 

similar to first-order clustering except that the cluster centers (the center of minimum distance 

for each) are now treated as >points= which themselves are clustered (see endnote ݅݅). The process 

is repeated until no further clustering can be conducted. Either all sub-clusters converge into a 

single cluster, the threshold distance criterion fails, or there are fewer than four seeds in the 

higher-order cluster. 

Note that this process is similar to that of the Nnh routine discussed in Chapter 7 except 

the selection of clusters is function of the total value of the attribute variable and not just the 

distance between zones. 

Simulating Confidence Intervals 

A Monte Carlo simulation can be run to estimate the approximate confidence intervals 

around first-order Znnh clusters. Second- and higher-order clusters are not simulated since their 

structure depends on first-order clusters. The user specifies the number of simulation runs and 

the Znnh clustering is calculated for randomly assigned data. The random output is sorted and 

percentiles are calculated. The output includes the number of first-order clusters, the area, the 

number of points, the number of zones, and the density. 

Confidence intervals can be estimated from these percentiles. The two most commonly 

used ones are the 95% (defined by the 2.5 and 97.5 percentiles) and the 99% (defined by the 0.5 

and 99.5 percentiles). The simulated data that is used can be viewed by checking the 'Dump 

simulation data' box on the Options tab. 

Type of Graphical Output 

The type of graphical output is specified, either standard deviational ellipses or convex 

hulls around the zones identified in each cluster. If the output is to be ellipses, then the output 

size for the clusters can be adjusted by the second slide bar. These are the number of standard 

deviations defined by the ellipse, from one standard deviation (the default value) to three 

standard deviations. Typically, one standard deviation will cover about 50-60% of the zones 

(and a higher percentage of the total of the weighting variable) whereas three standard deviations 

will cover more than 99% of the zones. On the other hand, if the output is to be convex hulls, the 

routine outputs a convex hull for each identified cluster. 

9.29
 

 

Ellipse cluster output 

The results can be output graphically as an ellipse to ArcGIS ‘shp’, MapInfo ‘mif’, 

various ASCII formats, or Google Earth ‘kml’ (if the coordinate system is spherical) files. A file 

name should be provided. For MapInfo ‘mif’ format, the user has to define up to nine parameters 

including the name of the projection and the projection number. If the MapInfo system file 

MAPINFOW.PRJ is placed in the same directory as CrimeStat, then a list of common 

projections with their appropriate parameters is available to be selected.

 First and higher-order ellipses will be output as separate objects. The prefix will be 

‘Znnh1’ for the first-order ellipses, ‘Znnh2’ for the second-order ellipses, and ‘Znnh3’ for the 

third-order ellipses. Higher-order ellipses will only index the number. 

Output size for ellipses 

The cluster output size can be adjusted by the lower slide bar. This specifies the number 

of ellipse standard deviations to be calculated for each cluster: one standard deviation (1X - the 

default value), one and a half standard deviations (1.5X), or two standard deviations (2X). The 

default value is one standard deviation. Typically, one standard deviation will cover more than 

half the zones in a cluster whereas two standard deviations will cover more than 99% of the 

zones in a cluster, though the exact percentage will depend on the distribution. Slide the bar to 

select the number of standard deviations for the ellipses. The output file is saved as 

Znnh<number><file name> with the file name being provided by the user. The number is the 

order of the clustering (i.e., 1, 2...). 

Restrictions on the number of clusters can be placed by defining a minimum number of 

zones that are required. The default is 10 and the minimum is 3. If there are too few zones 

allowed, then there will be many very small clusters. By increasing the number of required 

zones, the number of clusters will be reduced. 

Convex hull cluster output 

The clusters can also be output as convex hulls to ArcGIS ‘shp’, MapInfo ‘mif’, various 

ASCII formats, or Google Earth ‘kml’ (if the coordinate system is spherical) files. Specify a file 

name. For MapInfo ‘mif’ format, the user has to define up to nine parameters including the 

name of the projection and the projection number. If the MapInfo system file MAPINFOW.PRJ 

is placed in the same directory as CrimeStat, then a list of common projections with their 

appropriate parameters is available to be selected. 

9.30
The name will be output with a ‘CZnnh1’ prefix for the first-order clusters, a ‘CZnnh2’ 

prefix for the second-order clusters, and a ‘Cznnh3’ prefix for the third-order clusters. Higherorder clusters will index only the number. 

Note that unlike the Nnh clustering algorithm for points, discussed in Chapter 7, the zonal 

Nnh generally has much larger search areas. Consequently the convex hulls will be much larger 

than the ellipses, even the 2x ellipse (the opposite is true with the Nnh). 

Tabular Output 

The routine outputs six results for each cluster that is calculated: 

1. The hierarchical order and the cluster number

2. The mean center of the cluster (Mean X and Mean Y)

3. The standard deviational ellipse of the cluster (the rotation and the lengths of the

X and Y axes)

4. The number of zones in the cluster

5. The area of the cluster

6. The density of the cluster (the total weight of the zones divided by area)

and if a simulation is run: 

7. The minimum for the spatially random Znnh simulations:

8. The maximum for the spatially random Znnh simulations

9. The 0.5 percentile for the spatially random Znnh simulations

10. The 1 percentile for the spatially random Znnh simulations

11. The 2.5 percentile for the spatially random Znnh simulations

12. The 5 percentile for the spatially random Znnh simulations

13. The 10 percentile for the spatially random Znnh simulations

14. The 90 percentile for the spatially random Znnh simulations

15. The 95 percentile for the spatially random Znnh simulations

16. The 97.5 percentile for the spatially random Znnh simulations

17. The 99 percentile for the spatially random Znnh simulations

18. The 99.5 percentile for the spatially random Znnh simulations

Example 1: Simulated Clustering of Zones 

To illustrate the Znnh routine, a dispersed cluster structure for an arbitrary variable with 

five main groupings was created with 1,179 City of Houston Traffic Analysis Zones (TAZ). The 

9.31
five clusters can be labeled as central, southwest, northwest, northeast and southeast. Figure 9.7 

illustrates the pattern that was created. 

Four separate search areas were selected with a minimum of 25 ‘events’ being required of 

the attribute variable: 

1. 2 miles

2. 5 miles

3. 8 miles

4. 12 miles

Figures 9.8-9.12 illustrate the results of the clustering using these search distances with 

the standard deviational ellipse. Figure 9.11 also shows the convex hull of the search radius. 

Notice that a search radius of 2 miles produces small clusters and did not cover the clusters in the 

northeast, the southeast and most of the southwest. The reason is that TAZs for those areas are 

quite large with many being larger than 2 miles. 

A 5 mile search radius covered the five clusters though the clusters are still small. The 8 

mile search radius appeared to fit the data better while the 12 mile search radius produced too 

large ellipses with one large one for the central area. Note that Figure 9.11 shows the convex 

hulls of the 8 mile search radius and which covers most of the TAZs of the City of Houston. 

Example 2: Clustering of Houston Burglaries by Traffic Analysis Zones 

The second example examines burglaries in the City of Houston in 2006. In that year, 

24,935 burglaries were recorded. The data from which these came were assigned to blocks. Each 

of the burglaries was geocoded to the mid-block and then aggregated into 1,179 TAZs. Figure 

9.4 above illustrates the pattern of burglaries in Houston. 

The Znnh routine was run with four different search radii and with a minimum of 25 

burglaries being required for each cluster: 

1. 0.5 miles

2. 2 miles

3. 5 miles

4. 8 miles

9.32
Figure 9.7: 
Figure 9.8: 
Figure 9.9: 
Figure 9.10: 
Figure 9.11: 
Figure 9.12: 
Figures 9-13 shows the results of the 0.5 mile search radius. Four clusters were identified, 

but they were very small and covered only the downtown Houston area. The reason is that with 

a half mile radius, only very small TAZ’s can be captured within the radius and these are 

typically in the central downtown area. Further, they do not capture many burglaries, only 139 

of the 24,935. However, they do a better job of capturing high density burglary TAZ’s, defined 

as burglaries per square mile (Figure 9.14) 

Figure 9.15 through 9.17 show the results of using 2, 5 and 8 mile search radii. The 2 

mile search radius produced 10 small clusters; the 5 mile search radius produced 9 medium-sized 

clusters, and the 8 mile search radius identified 5 moderately large clusters. Clearly the cluster 

structure produced by the 2 mile search radius was also too small to fit the citywide pattern 

whereas either the 5 mile search radius or the 8 mile search radius seemed to best fit the overall 

data. Depending on whether the user wants smaller or larger clusters would determine which of 

these is selected. 

Keep in mind that there is a danger is using large search radii since the likelihood of 

obtaining clusters by chance increases. To illustrate this, two Monte Carlo simulations of 1000 

runs was made with both the 0.5 and the 8 mile search radius. Table 9.4 compares the actual 

clusters with the simulated clusters. 

With the 0.5 mile search radius, no clusters were identified in the Monte Carlo 

simulation. This indicates that the clusters identified in Figure 9.13 are most likely real. On the 

other hand, with the 8 mile search radius and randomly distributed data, the expected number of 

clusters would be expected to vary between 5 and 8 clusters 95% of the time. This is calculated 

as the credible interval defined by the 2.5th and 97.5th percentiles. Thus, the five clusters 

obtained by the Znnh are not significantly greater than or smaller than what would be expected 

by chance. Similarly, the area of the ellipses, the number of attribute points captured and the 

number of zones are not significantly different than what would be expected by chance. 

In short, the distribution that was obtained was not fundamentally different from a chance 

distribution. This is primarily the result of selecting a very large search radius. A user has to 

balance the choice between a small search radius which would capture clusters that are 

statistically much less likely to be due to chance but which cover only a small proportion of the 

study area with a larger search radius to capture the overall pattern but which increases the 

likelihood of identifying clusters by chance. In other words, there is a precision versus utility 

choice with a zonal clustering algorithm such as the Znnh. 

9.39
Figure 9.13: 
Figure 9.14: 
Figure 9.15: 
Figure 9.16: 
Figure 9.17: 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 9.4: 

Zonal Nearest Neighbor Hierarchical Clustering of Houston Burglaries 

(N= 24,935) 

0.5 mile search radius, Minimum points per cluster=25 

Area of Number Number 

Cluster Ellipse (sq mi) of Points of Zones Density 

1 0.005 56 35 11,633.1 

2 0.310 68 155 219.3 

3 0.081 29 36 359.4 

4 0.374 99 34 264.7 

No clusters found in simulation 

8 mile search radius, Minimum points per cluster=25 

Area of Number Number 

Cluster Ellipse (sq mi) of Points of Zones Density 

1 171.758 12,749 623 74.227 

2 99.028 3,253 91 32.849 

3 130.048 5,070 288 38.986 

4 65.936 2,418 86 36.672 

5 31.450 681 26 21.653 

Area of Number 

Percentile Clusters Ellipse (sq mi) of Zones Density 

0.5 4 10.34 25 0.707 

1.0 5 11.04 25 0.730 

2.5 5 12.29 25 0.777 

5.0 5 13.74 25 0.862 

10.0 5 15.74 26 0.938 

90.0 7 239.21 467 2.092 

95.0 8 241.59 471 2.183 

97.5 8 243.05 477 2.353 

99.0 8 244.67 481 2.631 

99.5 8 245.06 483 2.798 

9.45
Uses of Zonal Nearest Neighbor Hierarchical Clustering 

This brings up one of the dilemmas in using a zonal clustering technique. On the one 

hand, since zones do not overlap, the dispersion is much more spread out than with individual 

events. As seen in Chapter 7, the regular nearest neighbor hierarchical clustering routine (Nnh) 

produced quite small clusters. With zonal data, however, all the events are assigned to a single 

point within the zone which either creates a cluster associated with a single or else a dispersion 

between adjacent zones that have a higher concentration. Since the identification of a single 

zone is not very useful, the Znnh routine requires a minimum of three adjacent zones to be 

included in a cluster. 

Still, the Znnh can be useful for describing overall cluster patterns in a study area even 

with the increased uncertainty associated with large search radii. As Figures 9.16 and 9.17 

illustrate, meaningful areas of higher concentration can be identified even though the identified 

clusters cannot be empirically distinguished from a chance distribution. This gives the user 

flexibility in defining groupings of zones which can then be used for various purposes (e.g., 

assigning patrols or defining contingency areas). 

Limitations of Zonal Nearest Neighbor Hierarchical Clustering 

On the other hand, the Znnh routine does have some limitations. The first was shown 

above, namely that to ensure that clusters are substantially different from that expected by 

chance, only small search radii can be chosen. However, given that most zones are associated 

with population density with the smallest zones being in the downtown center but increasing in 

size with distance from the center, the use of a small search radius becomes less useful. 

Second, choosing a larger search radius can produce a cluster structure that appears to fit 

the data better but cannot be empirically distinguished from a chance distribution. Since there is 

not a single criterion that can be used to select among these, there is a certain amount of 

arbitrariness in the selection of a search radius or in the minimum number of events/attribute 

values specified. A user will have to experiment with different combinations to find the cluster 

structure that best fits the data. In this sense, the Znnh routine is more similar to the K-means 

clustering routine discussed in Chapter 8 than the Nnh routine in Chapter 7. 

The best solution, of course, is to use the location of individual events and cluster them 

with either either Nnh, STAC or K-means routines discussed in Chapters 7 and 8. The Znnh 

routine should only be used if the data are organized by zones and cannot be disaggregated. In 

this case, the user must be aware of the limitations of the Znnh method and of the trade-off 

between precision (certainty) and utility. 

9.46
A third limitation is that the cluster structure will almost certainly be different than had 

the individual events been clustered using the point-based Nearest Neighbor Hierarchical 

Clustering routine (Nnh). The requirement that zones do not overlap and that all events are 

assigned to the centroid of the zone ensures that the Znnh clusters will almost always be larger in 

size than the point-based Nnh clusters. In short, assigning events to zones and then clustering the 

zones will produce a larger and less focused cluster structure than the events themselves. The 

Znnh is only useful when it is not possible to disaggregate events to individual locations. 

9.47
 

References 

Anselin, L. (1995). Local indicators of spatial association - LISA. Geographical Analysis. 27, 

No. 2 (April), 93-115. 

Chainey, S. & Ratcliffe, J. (2005). GIS and Crime Mapping, John Wiley & Sons, 

Inc.:Chichester, Sussex, England. 

Getis, A. (1991). Spatial interaction and spatial auto-correlation: a cross-product approach. 

Environment and Planning A, 23, 1269-1277. 

Getis, A. & Ord, J. K. (1996). Local spatial statistics: an overview. In Longley, Paul & Batty, 

Michael (eds), Spatial Analysis: Modelling in a GIS Environment. GeoInformation International: 

Cambridge, England, 261-277. 

Mantel, N. (1967). The detection of disease clustering and a generalized regression approach. 

Cancer Research, 27, 209-220. 

Waller, L. A. & Gotway, C. A. (2004). Applied Spatial Statistics for Public Health Data. John 

Wiley & Sons: Hoboken, NJ. 

Wong, D. W. S. & Lee, J. (2005). Statistical Analysis of Geographic Information with ArcView 

GIS and ArcGIS. J. Wiley & Sons, Inc.: New York. 

9.48
where k =/ i and h =/ i. This term is twice the sum of the cross-products of all

weights for i with themselves, using k and h to avoid the use of identical

subscripts. Since each pair of observations, i and j, has its own specific weight, a

cross-product of weights are two weights multiplied by each other (where i =/ j)

and the sum of these cross-products is twice the sum of all possible interactions

irrespective of order (i.e., Wij = Wji). Because the weight of an observation with

itself is zero (i.e., Wii = 0), all terms can be included in the summation.

C. Third, define the variance, standard deviation, and an approximate (pseudo)

standardized score of Ii:

 

Endnotes 

i. The variance of the Local Moran is defined in three steps:

A. First, define b2.

This is the fourth moment around the mean divided by the squared second 

moment around the mean. 

9.49

B. Second, define
 

Attachments 

9.50
Using Local Moran’s “I” to Detect Spatial Outliers in Soil Organic 

Carbon Concentrations in Ireland 

Chaosheng Zhang1

Lecturer in GIS 

David McGrath2 

Research Officer 

1 Department of Geography, National University of Ireland, Galway, Ireland 

2 Teagasc, Johnstown Castle Research Centre, Wexford, Ireland 

One objective in the study of soil organic carbon concentrations is to produce 

a reliable spatial distribution map. A geostatistical variogram analysis was applied 

to study the spatial structure of soils in Ireland for the purpose of carrying out a 

spatial interpolation with the Kriging method. The variogram looks at similarities

in organic carbon concentrations as a function of distance. In the analysis, a

relatively poor variogram was observed, and one of the main reasons was the 

existence of spatial outliers. Spatial outliers make the variogram curve erratic and 

hard to interpret, and impair the quality of the spatial distribution map. 

CrimeStat was used to identify the spatial outliers. The parameter of the 

standardized Anselin’s Local Moran’s “I (z)” was used. When z < -1.96, the sample 

was defined as a spatial outlier. Out of 678 soil samples, a total of 39 samples were 

detected as spatial outliers, and excluded in the spatial structure calculation. As a 

consequence, the variogram curve was significantly improved. This improvement 

made the final spatial distribution map more reliable and trustable. 

Spatial outliers are clearly different from the majority of samples nearby. 

Compared with the samples nearby, high value spatial outliers are found in the 

southeastern part, and low value spatial outliers are located in the western and 

northern parts of the country.