Spatial Distribution Hot Spot Analysis I

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Spatial DistributionHot Spot Analysis I
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Approaching Hot Spots Analysis

• Determination of approach should be general or
  focused.
• Definition of a Hot Spot must be specified
  using theory or empirical evidence will give an
  indication of scale, time, similarity and
  distances to select.
• Selection of intensity and/or weight variables
  beyond the X Y locations.
• Number of clusters must be specified in that
  there are to be either a fixed or variable set of
  clusters.
• Visual display should be based on thematic
  mapping principals.

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Nearest Neighbor Methods
Event Point
x Sample Point
Z
W
X
Y
x
X2
W Event to Nearest Event
X Sample Point To Nearest Event
X2 Sample Point To 2nd Nearest Event
Y Event-Nearest-Sample Point to Nearest Event
Z Event-Nearest-Sample Point to
Nearest-Event-In-Half Plane-Not-Containing-Sample
Point
Cressie, Noel A.C. (1993) Statistics for
Spatial Data Analysis Revised Edition John
Wiley and Sons, Inc., New York, NY pp 602-603
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Hot Spot I Measures

• Point Techniques
• Mode Fuzzy Mode
• Hierarchical Techniques
• Nearest Neighbor Hierarchical Clustering
• Risk Adjusted Nearest Neighbor Hierarchical
  Clustering

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Mode Fuzzy Mode
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Mode Fuzzy Mode

• Mode
• For locations that have multiple incidents.
• Calculates frequency of incidents occurring at a
  single location and are ranked from highest to
  lowest.
• Is more precise but less flexible.
• Fuzzy Mode
• For individual incident locations.
• A fixed distance search radius counts the number
  of incidents near a single incident location.
• Incidents are counted multiple times as
  neighboring incidents are visited.
• Is less precise but more flexible.

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Mode Fuzzy Mode Input
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Mode Fuzzy Mode Output
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Fuzzy Mode Statistics Burglary
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Fuzzy Mode Statistics Theft
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Fuzzy Mode Statistics TABC
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Nearest Neighbor Hierarchical Clustering
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NN Hierarchical Clustering

• Two or more incidents are grouped on the basis of
  criteria, such as the nearest neighbor form
  first-order.
• Those pairs are then grouped on, again, being the
  nearest neighbor form second-order.
• Those grouped pairs are also grouped based on the
  nearest group neighbor form third-order.
• Those groups converge in to the final group
  forming the fourth-order.
• OR
• Grouping criteria fails and the points are not
  included in the hierarchy.

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NN Hierarchical Clustering Technique
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Clustering Criteria 1

• Threshold Distance
• Random Nearest Neighbor Distance (default). This
  is based on a one-tailed confidence interval
  around the random expected NN distance. A
  t-value is computed under the assumption that the
  degrees of freedom are at least 120 (next degree
  in the t-distribution is .) This creates a
  distance probability between two points.
• Fixed Distance. The search distance is specified
  exactly. This allows for comparisons against
  crime types. The flexibility for exploration of
  various distances is greater as theory or
  empirical findings can be tested. Distances from
  the Moran Correlogram and Ripleys K can be
  examined.

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Mean Random Distance
Random Nearest Neighbor Distance Index.
Total area of the region.
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Confidence Interval for Mean Random Distance
This confidence interval defines the probability
for the distance between any pair of points.
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Confidence Interval for Mean Random Distance
0.999
0.00001
0.99
0.95
0.0001
0.9
0.001

0.75
0.01
0.5
0.05
0.1
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Clustering Criteria 2

• Minimum Number of Points
• Can be specified exactly. This allows for
  comparisons against crime types. Again, the
  number selected could come from theory or
  empirical findings as they can be tested.
• Used in combination with search distance to
  minimize the possibility of over identifying
  clusters based on a minimum distance only, that
  is, it reduces the chance of finding numerous
  small clusters. This is particularly needed if a
  small scale such as city is being searched.
• More points reduces the number of clusters found
  and vice versa for less points.

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Selecting Parameters

• Threshold Distance
• Theory or empirical evidence are best for
  starting the exploration. For example, it has
  been found that convenience store crime occurs
  within ¾ of a mile from limited access highway
  intersections.
• Use distance found from Ripleys K or Moran
  Correlogram.
• Minimum Number of Points
• Theory or empirical evidence to does not really
  exist as number of incidents do not have known
  patterns. However, results from NNI would be
  useful.
• Assign incident counts to aerial units and
  calculate descriptive statistics such a mean,
  median, standard deviation and percentiles.

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Descriptive Statistics Burglary
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Descriptive Statistics Theft
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NNH Clustering Input
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NNH Clustering Output
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Simulation of NNH Clustering

• Routine is not just clustering pairs or fixed
  orders of points, but clustering as many points
  as possible within both the threshold and minimum
  number of points.
• Given that the threshold and minimum number of
  points can vary the probability distribution is
  not, or rather can not be known.
• Monte Carlo simulation of randomness, therefore,
  needs to be employed. Produces approximate
  confidence intervals for the first-order clusters
  but not higher-orders.

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NNH Output with Simulation
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Advantages of NNH Clustering

• Can identify small geographical environments with
  concentrated incidents.
• Can be applied to the full data set instead of
  having to carve up the study area into sub
  juridictions.
• Links between, and among other types of, clusters
  can be made.
• Demonstrates at which level various neighborhood,
  policing, community, policy making, etc
  strategies can be focused.

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Limitations of NNH Clustering

• No intensity of weight variable can be assigned.
  Therefore, this technique is based solely on
  locations.
• Size of grouping area is dependent on the sample
  size when the confidence interval around the mean
  random distance is used for the threshold.
• There is a certain arbitrariness to the selection
  of minimum number of points. There is no theory
  to serve as a guide and empirical evidence likely
  varies from place to place, but descriptive
  statistics help.
• There is not theory in regards to clusters
  themselves and must be interpreted in regards to
  environment.

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Risk-Adjusted Nearest Neighbor Hierarchical
Clustering
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Risk Adjusted Process

• Primary and Secondary files are required
• Primary observation/incident point locations.
• Secondary aerial unit with baseline or other
  location.
• Grid is defined from the minimum bounding
  rectangle (MBR) specified in the Reference file
  tab. A standard number of 50 columns is specified
  making up the number of cells.
• Area is defined from the Measurement Parameters
  tab. If no area is defined area, then of the
  grid is used. Thus it is based on the MBR of the
  Secondary file.

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Risk Adjusted Process

• Kernel Density parameters must be specified.
• Type of Density Estimation/Interpolation
• Type of Bandwidth
• Minimum Number of Points
• The Secondary file is interpolated to the grid
  using the density estimation parameters. It uses
  the absolute densities, which are the number of
  points per unit of analysis assigned to that grid
  cell and is rescaled to add up to the same number
  of points as in the Primary file (incidents).
  This provides a distribution that has been
  standardized from which to compare.

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Risk Adjusted Process

• The incidents in the Primary file are assigned to
  the cell in the grid that it falls in
  (point-in-polygon). A unique threshold distance,
  based on the confidence interval set on the
  slider bar, is also assigned to that cell.
• Once the pairs of points are selected the
  Risk-Adjusted NNH proceeds in the same fashion as
  the NNH.

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Risk-Adjusted NNH Input
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Exercise

• Calculate Mode and Fuzzy Mode and exclude values
  within 2 standard deviations and/or
  Percentiles.
• Aggregate crime counts to block groups and derive
  descriptive statistics of mean, median, mode,
  standard deviation and 75, 80 and 90th
  percentiles.
• Use descriptive to set minimum number of points
  and thresholds for at least two crime types.
  Compare the various order ellipses for linkages
  and similar/different patterns.

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Exercise

• Make several thematic maps showing population
  social, economic or other distributions in
  comparison to ellipses. Map out any other data
  layers that might have relevance to the NNH
  outpt.
• Run simulations on a subset and adjust minimum
  number of points and thresholds and compare
  output with previous output.

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To Do

• Definition of Hot Spot and the 3 examples of
  what might constitute one.
• Cautions with Fuzzy Mode.
• Graphics of
• Some Process.