Ripley K – Fisher et al.. Ripley K - Issues Assumes the process is homogeneous (stationary random...
-
Upload
jane-joseph -
Category
Documents
-
view
218 -
download
0
Transcript of Ripley K – Fisher et al.. Ripley K - Issues Assumes the process is homogeneous (stationary random...
![Page 1: Ripley K – Fisher et al.. Ripley K - Issues Assumes the process is homogeneous (stationary random field). Ripley K was is very sensitive to study area.](https://reader033.fdocuments.in/reader033/viewer/2022051214/56649f225503460f94c3a806/html5/thumbnails/1.jpg)
Ripley K – Fisher et al.
![Page 2: Ripley K – Fisher et al.. Ripley K - Issues Assumes the process is homogeneous (stationary random field). Ripley K was is very sensitive to study area.](https://reader033.fdocuments.in/reader033/viewer/2022051214/56649f225503460f94c3a806/html5/thumbnails/2.jpg)
![Page 3: Ripley K – Fisher et al.. Ripley K - Issues Assumes the process is homogeneous (stationary random field). Ripley K was is very sensitive to study area.](https://reader033.fdocuments.in/reader033/viewer/2022051214/56649f225503460f94c3a806/html5/thumbnails/3.jpg)
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
![Page 4: Ripley K – Fisher et al.. Ripley K - Issues Assumes the process is homogeneous (stationary random field). Ripley K was is very sensitive to study area.](https://reader033.fdocuments.in/reader033/viewer/2022051214/56649f225503460f94c3a806/html5/thumbnails/4.jpg)
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
![Page 5: Ripley K – Fisher et al.. Ripley K - Issues Assumes the process is homogeneous (stationary random field). Ripley K was is very sensitive to study area.](https://reader033.fdocuments.in/reader033/viewer/2022051214/56649f225503460f94c3a806/html5/thumbnails/5.jpg)
Pottery Survey Points“Random”
![Page 6: Ripley K – Fisher et al.. Ripley K - Issues Assumes the process is homogeneous (stationary random field). Ripley K was is very sensitive to study area.](https://reader033.fdocuments.in/reader033/viewer/2022051214/56649f225503460f94c3a806/html5/thumbnails/6.jpg)
NND Test
![Page 7: Ripley K – Fisher et al.. Ripley K - Issues Assumes the process is homogeneous (stationary random field). Ripley K was is very sensitive to study area.](https://reader033.fdocuments.in/reader033/viewer/2022051214/56649f225503460f94c3a806/html5/thumbnails/7.jpg)
Minimum Enclosing RectangleRipley Correction Formula
ExpectedKObservedKConfidence Env.
K FunctionClustered
Dispersed
Distance700600500400300200100
L(d)
800
750
700
650
600
550
500
450
400
350
300
250
200
150
100
50
![Page 8: Ripley K – Fisher et al.. Ripley K - Issues Assumes the process is homogeneous (stationary random field). Ripley K was is very sensitive to study area.](https://reader033.fdocuments.in/reader033/viewer/2022051214/56649f225503460f94c3a806/html5/thumbnails/8.jpg)
Used Survey Shapefile
ExpectedKObservedKConfidence Env.
K FunctionClustered
Dispersed
Distance50045040035030025020015010050
L(d)
550
500
450
400
350
300
250
200
150
100
50
![Page 9: Ripley K – Fisher et al.. Ripley K - Issues Assumes the process is homogeneous (stationary random field). Ripley K was is very sensitive to study area.](https://reader033.fdocuments.in/reader033/viewer/2022051214/56649f225503460f94c3a806/html5/thumbnails/9.jpg)
Reduce Analysis AreaUsed Shapefile
ExpectedKObservedKConfidence Env.
K FunctionClustered
Dispersed
Distance700600500400300200100
L(d)
750
700
650
600
550
500
450
400
350
300
250
200
150
100
50
![Page 10: Ripley K – Fisher et al.. Ripley K - Issues Assumes the process is homogeneous (stationary random field). Ripley K was is very sensitive to study area.](https://reader033.fdocuments.in/reader033/viewer/2022051214/56649f225503460f94c3a806/html5/thumbnails/10.jpg)
Weighted Points
![Page 11: Ripley K – Fisher et al.. Ripley K - Issues Assumes the process is homogeneous (stationary random field). Ripley K was is very sensitive to study area.](https://reader033.fdocuments.in/reader033/viewer/2022051214/56649f225503460f94c3a806/html5/thumbnails/11.jpg)
WeightedReduce Analysis Area - Shapefile
ExpectedKObservedKConfidence Env.
K FunctionClustered
Dispersed
Distance700600500400300200100
L(d)
750
700
650
600
550
500
450
400
350
300
250
200
150
100
50
![Page 12: Ripley K – Fisher et al.. Ripley K - Issues Assumes the process is homogeneous (stationary random field). Ripley K was is very sensitive to study area.](https://reader033.fdocuments.in/reader033/viewer/2022051214/56649f225503460f94c3a806/html5/thumbnails/12.jpg)
High-Low Clustering (Getis-Ord)
![Page 13: Ripley K – Fisher et al.. Ripley K - Issues Assumes the process is homogeneous (stationary random field). Ripley K was is very sensitive to study area.](https://reader033.fdocuments.in/reader033/viewer/2022051214/56649f225503460f94c3a806/html5/thumbnails/13.jpg)
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.
![Page 14: Ripley K – Fisher et al.. Ripley K - Issues Assumes the process is homogeneous (stationary random field). Ripley K was is very sensitive to study area.](https://reader033.fdocuments.in/reader033/viewer/2022051214/56649f225503460f94c3a806/html5/thumbnails/14.jpg)
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.
![Page 15: Ripley K – Fisher et al.. Ripley K - Issues Assumes the process is homogeneous (stationary random field). Ripley K was is very sensitive to study area.](https://reader033.fdocuments.in/reader033/viewer/2022051214/56649f225503460f94c3a806/html5/thumbnails/15.jpg)
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.
![Page 16: Ripley K – Fisher et al.. Ripley K - Issues Assumes the process is homogeneous (stationary random field). Ripley K was is very sensitive to study area.](https://reader033.fdocuments.in/reader033/viewer/2022051214/56649f225503460f94c3a806/html5/thumbnails/16.jpg)
Point to Area Transformations
Data Rasterization
Zone of InfluenceVoronmoi Polygons
Zone/Vornomoi Irregular Polygons
![Page 17: Ripley K – Fisher et al.. Ripley K - Issues Assumes the process is homogeneous (stationary random field). Ripley K was is very sensitive to study area.](https://reader033.fdocuments.in/reader033/viewer/2022051214/56649f225503460f94c3a806/html5/thumbnails/17.jpg)
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.
![Page 18: Ripley K – Fisher et al.. Ripley K - Issues Assumes the process is homogeneous (stationary random field). Ripley K was is very sensitive to study area.](https://reader033.fdocuments.in/reader033/viewer/2022051214/56649f225503460f94c3a806/html5/thumbnails/18.jpg)
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.
![Page 19: Ripley K – Fisher et al.. Ripley K - Issues Assumes the process is homogeneous (stationary random field). Ripley K was is very sensitive to study area.](https://reader033.fdocuments.in/reader033/viewer/2022051214/56649f225503460f94c3a806/html5/thumbnails/19.jpg)
![Page 20: Ripley K – Fisher et al.. Ripley K - Issues Assumes the process is homogeneous (stationary random field). Ripley K was is very sensitive to study area.](https://reader033.fdocuments.in/reader033/viewer/2022051214/56649f225503460f94c3a806/html5/thumbnails/20.jpg)
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.
![Page 21: Ripley K – Fisher et al.. Ripley K - Issues Assumes the process is homogeneous (stationary random field). Ripley K was is very sensitive to study area.](https://reader033.fdocuments.in/reader033/viewer/2022051214/56649f225503460f94c3a806/html5/thumbnails/21.jpg)
Minimum Convex Polygon
Hawth's Analysis Tools is an extension for ESRI's ArcGIS
http://spatialecology.com
![Page 22: Ripley K – Fisher et al.. Ripley K - Issues Assumes the process is homogeneous (stationary random field). Ripley K was is very sensitive to study area.](https://reader033.fdocuments.in/reader033/viewer/2022051214/56649f225503460f94c3a806/html5/thumbnails/22.jpg)
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.
![Page 23: Ripley K – Fisher et al.. Ripley K - Issues Assumes the process is homogeneous (stationary random field). Ripley K was is very sensitive to study area.](https://reader033.fdocuments.in/reader033/viewer/2022051214/56649f225503460f94c3a806/html5/thumbnails/23.jpg)
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:
nD(s) = (si / 2) hi <=
i=1where:
D(s) = density (intensity) at point s (grid cell center)si = observation point i (equals 1 or a quantity) = radius of circle neighborhoodhi = Euclidean distance between point and cell center.n = number of observations points within the
neighborhood
![Page 24: Ripley K – Fisher et al.. Ripley K - Issues Assumes the process is homogeneous (stationary random field). Ripley K was is very sensitive to study area.](https://reader033.fdocuments.in/reader033/viewer/2022051214/56649f225503460f94c3a806/html5/thumbnails/24.jpg)
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
![Page 25: Ripley K – Fisher et al.. Ripley K - Issues Assumes the process is homogeneous (stationary random field). Ripley K was is very sensitive to study area.](https://reader033.fdocuments.in/reader033/viewer/2022051214/56649f225503460f94c3a806/html5/thumbnails/25.jpg)
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.
![Page 26: Ripley K – Fisher et al.. Ripley K - Issues Assumes the process is homogeneous (stationary random field). Ripley K was is very sensitive to study area.](https://reader033.fdocuments.in/reader033/viewer/2022051214/56649f225503460f94c3a806/html5/thumbnails/26.jpg)
![Page 27: Ripley K – Fisher et al.. Ripley K - Issues Assumes the process is homogeneous (stationary random field). Ripley K was is very sensitive to study area.](https://reader033.fdocuments.in/reader033/viewer/2022051214/56649f225503460f94c3a806/html5/thumbnails/27.jpg)
Kernel Density Function
• There are several commonly used kernel functions:– Gaussian
– Quadratic (in ESRI)
– Uniform– Triangle
![Page 28: Ripley K – Fisher et al.. Ripley K - Issues Assumes the process is homogeneous (stationary random field). Ripley K was is very sensitive to study area.](https://reader033.fdocuments.in/reader033/viewer/2022051214/56649f225503460f94c3a806/html5/thumbnails/28.jpg)
Kernel Density Mapping• ArcGIS uses a quadratic kernel function where: n
D (s) = si (3/2) [1 – (hi2/2)]2 hi <=
i=1
D(s) = 0 otherwise
where: = radius of circle neighborhoodhi = distance between the point s and the observation point sin = number of observation points D(s) = density (intensity) at point s (grid cell center)si = observation point i (equals 1 or a quantity)
![Page 29: Ripley K – Fisher et al.. Ripley K - Issues Assumes the process is homogeneous (stationary random field). Ripley K was is very sensitive to study area.](https://reader033.fdocuments.in/reader033/viewer/2022051214/56649f225503460f94c3a806/html5/thumbnails/29.jpg)
Kernel Weights – Quadratic Function
Kernel
0
0.2
0.4
0.6
0.8
1
1.2
Distance
Ke
rne
l We
igh
t
Kernel
![Page 30: Ripley K – Fisher et al.. Ripley K - Issues Assumes the process is homogeneous (stationary random field). Ripley K was is very sensitive to study area.](https://reader033.fdocuments.in/reader033/viewer/2022051214/56649f225503460f94c3a806/html5/thumbnails/30.jpg)
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.
![Page 31: Ripley K – Fisher et al.. Ripley K - Issues Assumes the process is homogeneous (stationary random field). Ripley K was is very sensitive to study area.](https://reader033.fdocuments.in/reader033/viewer/2022051214/56649f225503460f94c3a806/html5/thumbnails/31.jpg)
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.
![Page 32: Ripley K – Fisher et al.. Ripley K - Issues Assumes the process is homogeneous (stationary random field). Ripley K was is very sensitive to study area.](https://reader033.fdocuments.in/reader033/viewer/2022051214/56649f225503460f94c3a806/html5/thumbnails/32.jpg)
Density Mapping
Kernel, Radius = 10 CellSize = 5x5m Mean = 0.0094S.D. = 0.0073
Kernel, Radius = 20 CellSize = 5x5m Mean = 0.0088S.D. = 0.0038
![Page 33: Ripley K – Fisher et al.. Ripley K - Issues Assumes the process is homogeneous (stationary random field). Ripley K was is very sensitive to study area.](https://reader033.fdocuments.in/reader033/viewer/2022051214/56649f225503460f94c3a806/html5/thumbnails/33.jpg)
Density Mapping
Kernel, Radius = 20 CellSize = 2x2m Mean = 0.0088S.D. = 0.0038
Kernel, Radius = 20 CellSize = 5x5m Mean = 0.0088S.D. = 0.0038
![Page 34: Ripley K – Fisher et al.. Ripley K - Issues Assumes the process is homogeneous (stationary random field). Ripley K was is very sensitive to study area.](https://reader033.fdocuments.in/reader033/viewer/2022051214/56649f225503460f94c3a806/html5/thumbnails/34.jpg)
Density Mapping
Kernel, Radius = 20 CellSize = 2x2m Mean = 0.0088S.D. = 0.0038
Simple, Radius = 20 CellSize = 2x2m Mean = 0.0083S.D. = 0.0031
![Page 35: Ripley K – Fisher et al.. Ripley K - Issues Assumes the process is homogeneous (stationary random field). Ripley K was is very sensitive to study area.](https://reader033.fdocuments.in/reader033/viewer/2022051214/56649f225503460f94c3a806/html5/thumbnails/35.jpg)
Density MappingChange in S.D. (#/ha)
0
50
100
150
200
250
4 6 8 10 12 14 16 18 20 22 24 26 28 30 40 50
Search Distance
Sta
nd
ard
De
via
tio
n
S.D.
20m
40m
10m15m
Patchy
Trend
![Page 36: Ripley K – Fisher et al.. Ripley K - Issues Assumes the process is homogeneous (stationary random field). Ripley K was is very sensitive to study area.](https://reader033.fdocuments.in/reader033/viewer/2022051214/56649f225503460f94c3a806/html5/thumbnails/36.jpg)
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.
![Page 37: Ripley K – Fisher et al.. Ripley K - Issues Assumes the process is homogeneous (stationary random field). Ripley K was is very sensitive to study area.](https://reader033.fdocuments.in/reader033/viewer/2022051214/56649f225503460f94c3a806/html5/thumbnails/37.jpg)
Here is a kernel density map of the cholera deaths (kernel size = 1.0; cellsize = 0.0025) with density contours overlaid. The density ofcholera 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 MarlboroughMews, 0.2 at Bridle Street, 0.1 at Newman Street and zero at all otherpumps. A simple density analysis with no smoothing yielded a similarmap with discrete edge segments.
![Page 38: Ripley K – Fisher et al.. Ripley K - Issues Assumes the process is homogeneous (stationary random field). Ripley K was is very sensitive to study area.](https://reader033.fdocuments.in/reader033/viewer/2022051214/56649f225503460f94c3a806/html5/thumbnails/38.jpg)
Interpolation
• Global Interpolation Methods– Trend Analysis (Global Polynomials)– Regression (spatial and non-spatial)
![Page 39: Ripley K – Fisher et al.. Ripley K - Issues Assumes the process is homogeneous (stationary random field). Ripley K was is very sensitive to study area.](https://reader033.fdocuments.in/reader033/viewer/2022051214/56649f225503460f94c3a806/html5/thumbnails/39.jpg)
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
![Page 40: Ripley K – Fisher et al.. Ripley K - Issues Assumes the process is homogeneous (stationary random field). Ripley K was is very sensitive to study area.](https://reader033.fdocuments.in/reader033/viewer/2022051214/56649f225503460f94c3a806/html5/thumbnails/40.jpg)
![Page 41: Ripley K – Fisher et al.. Ripley K - Issues Assumes the process is homogeneous (stationary random field). Ripley K was is very sensitive to study area.](https://reader033.fdocuments.in/reader033/viewer/2022051214/56649f225503460f94c3a806/html5/thumbnails/41.jpg)
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.