Examining local variation

Voronoi maps are constructed from a series of polygons formed around the location of a sample point.

Voronoi polygons are created so that every location within a polygon is closer to the sample point in that polygon than any other sample point. After the polygons are created, neighbors of a sample point are defined as any other sample point whose polygon shares a border with the chosen sample point. For example, in the following figure, the bright green sample point is enclosed by a polygon, which has been highlighted in red. Every location within the red polygon is closer to the bright green sample point than to any other sample point (given as small dark blue dots). The blue polygons all share a border with the red polygon, so the sample points within the blue polygons are neighbors of the bright green sample point.

Voronoi map

Using this definition of neighbors, a variety of local statistics can be computed. For example, a local mean is computed by taking the average of the sample points in the red and blue polygons. This average is then assigned to the red polygon. This process is repeated for all polygons and their neighbors, and the results are shown using a color ramp to help visualize regions of high and low local values.

The Voronoi Map tool provides the following methods to assign or calculate values for the polygons.

Class

Frequency

P i

1

3

3/5

2

0

0

3

1

1/5

4

0

0

5

1

1/5

Entropy class/frequency

The entropy assigned to the polygon will be

E = -[0.6*log2 (0.6) + 0.2* log2 (0.2) + 0.2* log2 (0.2)] = 1.371

Minimum entropy occurs when the polygon values are all located in the same class. Then,

Emin = -[1 * log2 (1)] = 0

Maximum entropy occurs when each polygon value is located in a different class interval. Then,

Emax = -[0.2 * log2 (0.2) + 0.2 * log2 (0.2) + 0.2 * log2 (0.2) + 0.2 * log2 (0.2) + 0.2 * log2 (0.2)] = 2.322

The Voronoi statistics can be used for different purposes and can be grouped into the following general functional categories:

Functional category

Voronoi statistics

Local Smoothing

Mean, Mode, Median

Local Variation

Standard deviation, Interquartile range, Entropy

Local Outliers

Cluster

Local Influence

Simple

General functional categories

7/10/2012