各位大侠，在arcmap地统计学模块下面有这样一个功能：Voronoi图。曾经有人对这一功能做的解释是：用来发现离群值。Voronoi图的生成方法：每个多边形内有一个样点，多变形内任一点到该点的距离都小于其他多边形到该点的距离，生成多边形后。某个样点的相邻样点便会与该样点的多边形有相邻边。至于多边形值的计算有多种方法，可以用生成多边形的样点值作为多边形的值（Simple方法），也可以以相邻样点的平均值为多边形的值（Mean方法），具体计算方法可以在Type下拉菜单中选择。我想请教各位：第一，这里的离群值的具体涵义是什么？可以等同于特异值吗？第二，Type下拉菜单中的各种计算方法是什么意思？第三，进行计算之后如何确定哪些为离群值？

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Voronoi maps

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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, given as red. Every location within the red polygon is closer to the bright green sample point than 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.

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. After this is repeated for all polygons and their neighbors, a color ramp shows the relative values of the local means, which helps visualize regions of high and low values.

The Voronoi Map tool provides a number of methods for assigning or calculating values to polygons.

• Simple: The value assigned to a cell is the value recorded at the sample point within that cell.

• Mean: The value assigned to a cell is the mean value that is calculated from the cell and its neighbors.

• Mode: All cells are placed into five class intervals. The value assigned to a cell is the mode (most frequently occurring class) of the cell and its neighbors.

• Cluster: All cells are placed into five class intervals. If the class interval of a cell is different from each of its neighbors, the cell is colored gray to distinguish it from its neighbors.

• Entropy: All cells are placed into five classes based on a natural grouping of data values (smart quantiles). The value assigned to a cell is the entropy that is calculated from the cell and its neighbors, that is,

  Entropy = - Σ (p_i * Log p_i ),

  where p_i is the proportion of cells that are assigned to each class. For example, consider a cell surrounded by four neighbors (a total of five cells). The values are placed into the corresponding classes:

Class	Frequency	Pi
1	3	3/5
2	0	0
3	1	1/5
4	0	0
5	1	1/5

The entropy assigned to the cell 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 cell values are all located in the same class. Then,

E_min = -[1 * log₂ (1)] = 0

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

E_max = -[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

• Median: The value assigned to a cell is the median value calculated from the frequency distribution of the cell and its neighbors.

• Standard Deviation: The value assigned to a cell is the standard deviation that is calculated from the cell and its neighbors.

• Interquartile Range: The 1st and 3rd quartiles are calculated from the frequency distribution of a cell and its neighbors. The value assigned to the cell is the interquartile range calculated by subtracting the value of the 1st quartile from the value of the 3rd quartile. The different Voronoi statistics are used for different purposes.
  The statistics can be grouped into the following general functional categories:

Functional category	Voronoi statistics
Local Smoothing	Mean Mode Median
Local Variation	Standard deviation Inter-quartile range Entropy
Local Outliers	Cluster
Local Influence	Simple

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