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Lecture 5

# Lecture 5 - Nearest-Neighbor Methods Defining"connectivity...

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Point data can be used in various ways to measure the degree to which the point pattern exhibits spatial autocorrelation . But first, care must be taken in describing the nature of connectivity or the degree to which any two points are likely to be defined as “connected” the extent to which they might be dependent based on distance or separation . Defining “connectivity” between points Nearest-Neighbor Methods

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Point data may also be converted to polygons to better understand the spatial distribution of potential point influence as defined by the implied areal units which divide up a study area based on an observed point pattern. Conversion: Point data Areal Unit data
In short, the point pattern itself may also be used to create areal units (polygons) that divide up a study area . Instead of overlaying a grid or a series of cells, polygons may be drawn/derived to reflect the spacing of points and the extent to which individual points are separated from their neighbors… in effect to create polygons where all locations within a given polygon are closer to a corresponding point than to any other points in the study region (while allowing points to share polygon boundaries).

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Common technique: convert point data polygon data An alternative to area/grid- based methods or quadrat counts… is to evaluate the pattern using “ Voronoi-Thiessen polygons (aka Voronoi tessellation ), by dividing the study region or plane up into non-overlapping polygons (or space into polyhedra) .
The construction of a net of Thiessen polygons may proceed in two ways (a) the inter-point line method ; and (b) the intersecting circles method . Voronoi-Thiessen polygons

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Thiessen polygon (shown in gray) constructed from k=5 surrounding or neighboring points.
Classic application of Thiessen polygons to John Snow’s 1854 study of the spatial incidence of Cholera (cases shown in red) in the city of London; an outbreak said to be linked to tainted drinking water from public wells (blue dots). In this case well locations were used to create the polygons.

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Inter-point line method : 1. Lines are drawn from each point to each adjacent point; 2. Each of these inter- point lines is ‘bisected” to yield the midpoint of the line; and 3. From each mid-point, a boundary line is drawn at right-angles to the original inter-point line to create a series of convex polygons, such that the area within each polygon is nearer to the enclosed point than it is to any other point. Such polygons have been widely used in the central place theory literature to define the sphere of influence of urban centers under the assumption that each center dominates the area that lies nearest to it (geometrically speaking).
An illustration of the inter-point line method: Thiessen polygons bisectors network based on adjacency A B C Intersection of bisector extension lines now forms corner of polygon

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Kopec’s intersecting circles method for delineation of Voronoi-Thiessen polygons: 1. Consider two adjacent points which are at a distance d apart.
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