Spatial Distribution Centrographic Measures

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Spatial DistributionCentrographic Measures
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Centrographic Measures

• Descriptive statistics of spatial point patterns.
• Similar to classical measures but demonstrate the
  spatial location and distribution.
• Work only on the locations and not the underlying
  data values.
• Measure central tendency and dispersion.
• Are first-order measures of distribution
  properties.

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Centrographic Spatial Statistics

• Central Tendency
• Mean Center
• Geometric Mean
• Harmonic Mean
• Weighted Mean Center
• Median Center
• Center of Minimum Distance
• Dispersion
• Standard Distance Deviation
• Standard Deviational Ellipse
• Convex Hull

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Distribution of Burglary
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Distribution of Theft
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Measures ofCentral Tendency
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Mean Center Standard Distance Deviation
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Mean Center
Average for X coordinates.
Average for Y coordinates.
ith X coordinate in a collection of points.
ith Y coordinate in a collection of points.
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Geometric Mean
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Geometric Mean
The equations can be evaluated by
and then calculated with
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Harmonic Mean
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Weighted Mean Center
weight on ith coordinate in a collection of
points.
Average for X coordinates.
Average for Y coordinates.
ith X coordinate in a collection of points.
ith Y coordinate in a collection of points.
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Center of Minimum Distance
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Center of Minimum Distance
CMD
is minimized.
the Center of Minimum Distance (CMD).
distance between a single point i and C.
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Median Center
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Measures ofDispersion
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Standard Deviation of X Y
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Standard Distance Deviation
distance between ith observation and the mean
center (MC).
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Standard Deviational Ellipse
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Standard Deviational Ellipse (SDE)
two dimensional bivariate distribution.
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Rotation of SDE
angle at which the minor axis (Y) is rotated.
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SD Calculation for X Y
Major Axis (X)
Minor Axis (Y)
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SDE Length and Area
Length of Major (X) and Minor (Y) Axes
Area of the Ellipse
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Advantages with SDEs

• Advantages
• Shows distribution in two dimensions of the
  spread of observations.
• Shows directionality on the major and minor axes
  of the incident distributions.
• Can compare between centers, dispersions and
  shapes of SDEs, both of time slices or other
  distributions.
• Can compare with other spatial, social,
  demographic, economic, etc. processes to
  eliminate variables that might explain the same
  thing.
• Can conform to distributions along streets unlike
  circles.

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Tests for Measures

• Test for equality of (two means) - use
  t-test independent samples.
• Test for equality of (more than two
  means) use one-way analysis of variance (ANOVA)
  with post hoc multiple comparisons.
• Test for equality of (two variances)
  largest variance in the numerator and smallest in
  denominator. Test
• Test for equality of (more than two
  variances). Test

Langworthy, Robert H. and Jefferis, Eric S.
(2000) The Utility of Standard Deviational
Ellipses for Evaluating Hot Spots in Goldsmith,
Victor, et. Al. (2000) Analyzing Crime
Pattersns pp 104.
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Disadvantages with SDEs

• Disadvantages
• Only describes the symmetry of the locations and
  not the underlying data values.
• Clustered groups of points can affect global
  statistics of mean centers and variances.
• Shape and direction of ellipse can be affected by
  shape of the jurisdiction.
• Other spatial, social, demographic, economic,
  etc. processes are assumed to have Complete
  Spatial Randomness (CSR).

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Variable Distributions inside SDE
For Burglary
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Variable Distributions inside SDE
For Theft
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Directional Mean Variance
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Measures ofSpatial Autocorrelation
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Global Spatial Autocorrelation

• Measures spatial dependence in a point data set.
  Tests whether spatial processes are
  non-stationary.
• Three available in CrimeStat
• Morans I
• Gearys C
• Moran Correlogram
• These are only global measures and cannot
  indicate where these spatial process are changing.

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Spatial Autocorrelation
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Global Morans I
inverse distance between locations i and j i
¹ j
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Global Gearys C
inverse distance between locations i and j i
¹ j
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Adjustment for Small Distances
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Global Spatial Autocorrelation
For Burglary
Adjustment for Small Distances
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Global Spatial Autocorrelation
For Theft
Adjustment for Small Distances
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Comparing Morans I
observed Morans I for crime type.
observed Morans I for population.
expected standard deviation of I.
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Comparing Morans I
Burglary to Population
Note square both values in numerator, add them
together and take square root for denominator.
This will be the SE of the difference of the two
variances.
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Comparing Morans I
Theft to Population
is replaced with the I for theft, which
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Comparing Morans I
Burglary to Theft
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Moran Correlogram Output
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For Burglary
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Monte Carlo Simulation

• These results were just the first run at the
  global level. For consistency, these need to be
  run multiple times and can be viewed in the graph
  function.

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Monte Carlo Simulation

• At a more local level the fluctuations in the
  envelope are much more pronounced.

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Global measures make blanket statements about an
entire area. That is, spatial homogeneity is
assumed.
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Exercise

• Calculate global spatial measures of central
  tendency and dispersion. Then do this on
  subsets.
• Take a look at the 3 different means.
• Take a look at the SDEs and how they differ.
• Select out block groups and graph several
  variables such as crime counts, demographic, SES,
  etc to see how their distribution looks.
• Break data down into time slices for at least 6
  months.
• Compare MC, HM and GM for significant difference.
• Compare the SDE mean centers, standard deviation
  of the X Y coordinates, and the ratio of major
  axis to the minor axis.

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Exercise (cont.)

• Calculate the spatial autocorrelations measures.
• Take a look at the Morans I and Gearys C for a
  crime type(s) and compare them with the I and C
  for the population or another crime type.
• Use the Morans Correlogram to look at global and
  local subsets of your data. Use these to help
  determine what distances might be used to
  understand spatial dependences in later analysis.

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To Do

• Change numerators in I comparison to SE of
  variances between two values in numerator.
• List output naming conventions.
• Morans I example from GeoDa as a comparison to
  point based.
• Put in example of value distributions of SDEs at
  the substation level.
• Highlight output that can be tested statistically
  on slides.
• Insert Convex Hull to show can be used for
  determining study area.
• Change CS graphs to Excel graphs.
• Graphics of
• SDE construction.
• CMD process.
• Morans I and Gearys C process.