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.

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 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

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

Pottery Survey Points“Random”

NND Test

Minimum Enclosing RectangleRipley Correction Formula

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WeightedReduce Analysis Area - Shapefile

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High-Low Clustering (Getis-Ord)

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.

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.

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.

Point to Area Transformations

Data Rasterization

Zone of InfluenceVoronmoi Polygons

Zone/Vornomoi Irregular Polygons

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.

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.

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.

Minimum Convex Polygon

Hawth's Analysis Tools is an extension for ESRI's ArcGIS

http://spatialecology.com

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.

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

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

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.

Kernel Density Function

• There are several commonly used kernel functions:– Gaussian

– Quadratic (in ESRI)

– Uniform– Triangle

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)

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.

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.

Density Mapping

Kernel, Radius = 10 CellSize = 5x5m Mean = 0.0094S.D. = 0.0073

Kernel, Radius = 20 CellSize = 5x5m Mean = 0.0088S.D. = 0.0038

Density Mapping

Kernel, Radius = 20 CellSize = 2x2m Mean = 0.0088S.D. = 0.0038

Kernel, Radius = 20 CellSize = 5x5m Mean = 0.0088S.D. = 0.0038

Density Mapping

Kernel, Radius = 20 CellSize = 2x2m Mean = 0.0088S.D. = 0.0038

Simple, Radius = 20 CellSize = 2x2m Mean = 0.0083S.D. = 0.0031

Density MappingChange in S.D. (#/ha)

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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.

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.

Interpolation

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

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

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.