Spatial Modeling Kernel Density Estimation

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Spatial ModelingKernel Density Estimation
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Histogram Density Interpolation

• Continuous variable is divided up into bins of a
  specified interval with counts falling into a
  bin.
• Histogram is assumed to represent a smoothed
  distribution, that is, a density function.
• Estimating a smooth density function is done by
  linking the center points of the interval with a
  line.
• Causes three statistical problems
• Information is discarded by center point
  assignment.
• Creates a discontinuous density function.
• Dependent on arbitrarily specified bin sizes.

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Kernel Density Interpolation

• Overcomes the first two statistical problems. Bin
  size, or bandwidth, is still a problem.
• Involves placing a symmetrical surface over each
  central reference point, which is the centroids
  on a grid overlaid on the study area.
• Evaluates and sums the distance from all points
  to the central reference point
• The functions of all the symmetrical surfaces are
  summed over each other to produce an estimate at
  that central reference point.
• Was developed as an alternative method for
  estimating density of a frequency histogram.

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Uniform
Quartic
Normal
Negative Exponential
Triangular
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Normal Density Functions

• Functional Form
• Function extends in all directions over entire
  study area.
• All points are factored in.

Weight at point location i.
Intensity at point location i.
Bandwidth at reference point i (Standard
Deviation).
Distance weight between incident i and
reference point j.
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Quartic Function

• Within radius
• Outside radius

Weight at point location i.
Intensity at point location i.
Bandwidth at reference point i (Standard
Deviation).
Distance weight between incident i and
reference point j.
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Triangular Function

• Within radius
• Outside radius

Constant is set to 0.25.
Bandwidth at reference point i (Standard
Deviation).
Distance weight between incident i and
reference point j.
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Negative Exponential Function

• Within radius
• Outside radius

Constant is set to 1.
Exponent is set to 3.
Distance weight between incident i and
reference point j.
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Uniform Function

• Within radius
• Outside radius

Constant, is set to 0.1.
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Size Shape of Bandwidth

• Spatial effects/autocorrelation are to be
  captured.
• Too large of a bandwidth hides local clustering
  trends by producing a large, combined, hot
  spot.
• Too small of a bandwidth may produce too many
  peaks and valleys possibly indicating false hot
  spots or cold spots.
• Use results to determine size and shape from
• Previous Results, such as Moran Correlogram,
  Nearest Neighbor Index or Ripleys K.
• Theoretical Guidelines.
• Knowledge of the Environment.

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Number of Points in Bandwidth

• Spatial process/autocorrelation are to be
  captured.
• Too many produces a lot of hot spots with
  several possibly being false (random variation).
• Too few produces an even surface and nothing
  indistinguishable as a hot spot.
• Use results to determine minimum number of points
  from
• Previous Results, such as Nearest Neighbor Index,
  Ripleys K or count distributions from aerial
  units.
• Theoretical Guidelines.
• Knowledge of the Environment.

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Kernel Interpolation Input
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Prue NNI Base Comparison
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Prue K Statistic Base Comparison
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Type of Bandwidth (Single)

• Fixed
• A non changing distance interval must be
  specified in units of measurement.
• Should be based of some distance from theoretical
  guidelines or empirical evidence.
• Assumes stationary spatial processes.
• Variable (for Dual Interpolation)
• Adaptive
• An adjusted distance based on capturing the
  minimum number of points.
• Improves statistical precision by being narrower
  in areas with a higher concentration of incidents
  and wider in areas with more dispersed incidents.

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Density Calculations (Single)

• Absolute Estimates of each cell are re-scaled so
  that the sum of the densities over all cells is
  equal to the total number of observations
  estimates are points per grid cell. Used for
  comparisons between crime types or same crime
  type and different time period.
• Relative Absolute densities of each cell are
  divided by the area of the cell estimates are
  points per square unit of measurement. Used for
  expression in units across the study area.
• Probabilities Absolute density is divided by the
  total number of observations within the grid
  estimates that are the likelihood of an incident
  occurring within a cell.

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Single Kernel Density Estimation
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Single Kernel Interpolation Input
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Single Kernel Interpolation Output
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Dual Kernel Density Estimation
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Dual Kernel Interpolation Input
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Type of Bandwidth (Dual)

• Fixed (same as for Single Interpolation)
• Variable
• Each file (Primary Secondary) has different
  intervals.
• Divides cell value from primary file with value
  from same cell in secondary file. However, as a
  value in a cell from the secondary file
  approaches zero the quotient will be become
  exponentially larger providing an overestimation
  for that cell.
• Adaptive (same as for Single Interpolation)

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Density Calculations (Dual)

• Ratio The primary file cell values are divided
  by the secondary file cell values to produce a
  risk ratio.
• Log Ratio Natural logarithm of the density ratio
  for those a set of grid cells that have a very
  skewed distribution of density values. This will
  mute the over-estimations, or spikes.
• Absolute Difference The primary file cell values
  are subtracted from the secondary file cell
  values producing a differentials. Also used for
  comparing grids that are created with and without
  a weight or intensity variable to produce more
  precise estimates when clustering occurs in a
  spatial process.

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Density Calculations (Dual)

• Relative Difference Standardizes the values from
  the primary and secondary cells and subtracts
  secondary cell relative density from the primary
  relative density. Used for comparing the
  relative density change between two periods of
  the same crime type.
• Sum Adds the values from primary and secondary
  cells. Used for combining two density surfaces
  to show additive effect of two different crime
  types.
• Relative Sum Standardizes the values from the
  primary and secondary cells and adds the
  secondary cell relative density from the primary
  relative density. Used for identifying the total
  effect of two different crime types.

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Dual Kernel Interpolation Output
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Population at RiskBurglary
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Population at RiskBurglary
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Exercise

• Calculate several single densities for different
  crime types using different bandwidths, minimum
  number of points and kernel type. Each of these
  should be based on what was found in previous
  analysis from descriptives and/or from NNI,
  Ripleys K or Morans Correlogram.
• Examine the differences and identify the
  parameters that make sense for further analysis.
• Repeat this process for any base data that will
  be used as a baseline.
• Calculate several dual densities for different
  crime types with the input parameters from the
  single density analysis.