Ripley K

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Ripley K Fisher et al.
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Ripley K - Issues

• Assumes the process is homogeneous (stationary
  random field).
• Ripley K was is very sensitive to study area
  size.
• Riley K is influenced by study area shape, the
  expected L(d) assumes a simple geometry.
• Ripley K has strong basis near the edge/boundary.
  You should use a boundary correction method, and
  if your study area is not a simple shape you
  should use study are polygon.
• Weighting points

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Boundary Correction Methods

• Boundary Correction Methods
• RIPLEY EDGE CORRECTION FORMULA
• SIMULATE OUTER BOUNDARY VALUES
• REDUCE ANALYSIS AREA
• Study Area Method
• MINIMUM ENCLOSING RECTANGLE
• USER PROVIDED STUDY AREA

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Pottery Survey PointsRandom
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NND Test
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Minimum Enclosing RectangleRipley Correction
Formula
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Used Survey Shapefile
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Reduce Analysis AreaUsed Shapefile
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Weighted Points
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WeightedReduce Analysis Area - Shapefile
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High-Low Clustering (Getis-Ord)
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Point Transformations

• In many situations we collect point
  measurements of an entity we wish to study, but
  prefer/need to have an area (i.e. polygon) or
  field (e.g. raster) representation to relate the
  measurements to other information or have
  information at locations not measured to make
  decisions.

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Four Basic Types of Methods

• Point to Area Transformations (Deterministic)
• Delineate areas and assign the point
  measurement(s) to the area. An Area is related to
  one or more points
• Points are usually weighted.
• Density Mapping point to field (Deterministic)
• A field element (e.g. a raster cell) is assigned
  a value based on sampling the surrounding
  neighborhood and computing the density of
  observations around the element. Density is the
  quantity per area.
• Points can weighted or un-weighted.

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Four Basic Types of Methods

• Interpolation Methods point to field
  (Deterministic)
• A field element is assigned a value based on a
  mathematical transformation that predicts what
  the value should be at the field element location
  based on known point observations.
• Points must be weighted.
• Local Interpolation Methods
• Uses a sub-sample of point observations to
  develop the mathematical equation and make the
  prediction.
• Global Interpolation Methods
• Uses all the point observations to develop the
  mathematical equation. Regression and trend
  analysis are examples.
• Stochastic Modeling field generation
  (Stochastic)
• Use point observations to understand the
  statistical properties of an entity and to
  develop a model that generates a field of values
  that retain the statistical properties of the
  entity.

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Point to Area Transformations
Data
Rasterization
Voronmoi Polygons
Zone of Influence
Irregular Polygons
Zone/Vornomoi
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Point to Area Transformation Methods

• Voronmoi (Thiessen) Polygons
• The most commonly used.
• Based on the concept of Nearest Neighbor.
• Creates (usually) unequal size areas around each
  point. The areas are assigned the value of the
  origin point.
• Depending on the distribution of points the range
  in sizes can be relatively large, but the entire
  analysis area is covered.

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Thiessen Polygons

• The Thiessen polygons are constructed as follows
  All points are triangulated into a triangulated
  irregular network (TIN) that meets the Delaunay
  criterion. The perpendicular bisectors for each
  triangle edge are generated, forming the edges of
  the Thiessen polygons. The location at which the
  bisectors intersect determine the locations of
  the Thiessen polygon vertices.

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Point to Area Transformation Methods

• Irregular Shaped Polygons
• Based on modeling or analysis, sometimes using
  additional data.
• Watershed Delineation
• Need outlet point and surface representation
  (e.g. DEM).
• Model flow across surface to outlet point.
• All areas that flow to outlet point are in
  watershed.
• Minimum Convex Polygons (MCP)
• Minimum area around all selected points.
• Can create MCP that contain a percentage of the
  points.

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Minimum Convex Polygon
Hawth's Analysis Tools is an extension for ESRI's
ArcGIS http//spatialecology.com
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Density Mapping

• Also referred to as Intensity.
• Point to field ? raster data structure
• Calculates the density of points in a
  neighborhood around each output grid cell.
  Neighbor is usually larger then the cell.
• Points can be weighted (using a numeric
  attribute) or un-weighted (all points 1).
• Units of density are quantity per unit area.
• Good for when the density of points is small
  relative to the desired cell size.
• Two methods Simple and Kernel.

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Simple Density Mapping

• The density is calculated using the number of
  points that fall within the neighborhood of each
  output grid cell, divided by the area of the
  neighborhood. For a circle neighborhood the
  equation is
• n
• D(s) ? (si / ? ?2) hi lt ?
• i1
• where
• D(s) density (intensity) at point s (grid
  cell center)
• si observation point i (equals 1 or a
  quantity)
• ? radius of circle neighborhood
• hi Euclidean distance between point and cell
  center.
• n number of observations points within the
  
• neighborhood

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Simple Density Mapping

• Rough surfaces, all points have the same weight
  within search radius regardless of distance.
• No assumptions regarding the kernal method type.
  
• Called Point Density in ArcMap.
• Can use different neighborhood shapes
• Circle (most common)
• Rectangle
• Wedge
• Annulus

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Kernel Density Mapping

• A kernel function is used to fit a smoothly
  tapered surface to each point, and the density is
  calculated from these surfaces where they overlap
  the center of the output grid cell. This gives a
  smoother output grid, while maintaining the same
  general values for density. A circular
  neighborhood is always used with the KERNEL
  option.

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Kernel Density Function

• There are several commonly used kernel functions
• Gaussian
• Quadratic (in ESRI)
• Uniform
• Triangle

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Kernel Density Mapping

• ArcGIS uses a quadratic kernel function where
• n
• D (s) ? si (3/??2) 1 (hi2/?2)2 hi lt ?
• i1
• D(s) 0 otherwise
• where ? radius of circle neighborhood
• hi distance between the point s and the
  observation point si
• n number of observation points
• D(s) density (intensity) at point s (grid
  cell center)
• si observation point i (equals 1 or a
  quantity)

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Kernel Weights Quadratic Function
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Kernel Density Mapping

• Assume ? 10m si 1 with 5 points
• If hi 0 D(si) si (3/??2) 1.0000 0.0095
• If hi 2 D(si) si (3/??2) 0.9216 0.0088
• If hi 5 D(si) si (3/??2) 0.5625 0.0054
• If hi 7 D(si) si (3/??2) 0.2601 0.0025
• If hi 9 D(si) si (3/??2) 0.0361 0.0003
• D(S) 0.0265 units per square meter
• If all points hi 9 D(S) 0.0015 units per sq.
  m.
• D(S)simple 5 / ??2 5 /314.159 m2
• 0.0159 units per sq. m.

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Kernel Density Mapping

• Factors that influence surface characteristics
• Method Simple would have a rougher looking
  surface, but typically less variance.
• Neighborhood Size the greater the number of
  points used to compute density the less variance
  in the surface.
• Cell Size the larger the cell the rougher,
  greater potential relative change per cell.
• You can made a surface to smooth and loss the
  natural variance of the surface, areas with high
  and low density.
• You should experiment, what creates the best
  surface for you. Some use the search distance
  where variance starts to become stable.

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Density Mapping
Kernel, Radius 10 CellSize 5x5m Mean
0.0094 S.D. 0.0073
Kernel, Radius 20 CellSize 5x5m Mean
0.0088 S.D. 0.0038
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Density Mapping
Kernel, Radius 20 CellSize 2x2m Mean
0.0088 S.D. 0.0038
Kernel, Radius 20 CellSize 5x5m Mean
0.0088 S.D. 0.0038
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Density Mapping
Kernel, Radius 20 CellSize 2x2m Mean
0.0088 S.D. 0.0038
Simple, Radius 20 CellSize 2x2m Mean
0.0083 S.D. 0.0031
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Density Mapping
Patchy
Trend
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Density surface using bivariate normal density
kernel
This figure is a display of the location points
(shown in yellow) within the selected 50, 75, and
90 probability polygons.
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• Here is a kernel density map of the cholera
  deaths (kernel size 1.0
• cellsize 0.0025) with density contours
  overlaid.   The density of
• cholera deaths derived from this map is 36.8 at
  the Broad Street pump,
• versus 2.4 at Carnaby Street, 1.9 at Rupert
  Street, 0.8 at Marlborough
• Mews, 0.2 at Bridle Street, 0.1 at Newman Street
  and zero at all other
• pumps.  A simple density analysis with no
  smoothing yielded a similar
• map with discrete edge segments.

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Interpolation

• Global Interpolation Methods
• Trend Analysis (Global Polynomials)
• Regression (spatial and non-spatial)

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Interpolation

• Trend Analysis
• Surface is approximated by a polynomial
• Value (z) at any point (x,y) on the surface is
  given by an equation in powers of x and y.
• Linear equation (1 degree) describes a tilted
  plane surface
• z a bx cy
• Quadratic equation (2 degree describes a simple
  hill or valley
• z a bx cy dx2 exy fy2

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Interpolation

• Trend Analysis
• In general, any cross-section of a surface of
  degree n can have at most n-1 alternating maxima
  and minima.
• Assumes the general trend of the surface is
  independent of random errors found at each sample
  point.
• Good at addressing non-stationary cases.