{"id":5579,"date":"2018-07-03T02:20:59","date_gmt":"2018-07-03T00:20:59","guid":{"rendered":"http:\/\/www.sigterritoires.fr\/?p=5579"},"modified":"2018-10-17T13:01:30","modified_gmt":"2018-10-17T11:01:30","slug":"exploratory-data-analysis-for-geostatistics-voronoi-diagrams","status":"publish","type":"post","link":"https:\/\/www.sigterritoires.fr\/index.php\/en\/exploratory-data-analysis-for-geostatistics-voronoi-diagrams\/","title":{"rendered":"Exploratory data analysis for geostatistics: Voronoi diagrams"},"content":{"rendered":"

Following the article\u00a0\u00a0\u00a0Introduction to exploratory data analysis for geostatistics<\/a>\u00a0\u00a0\u00a0we will discuss all the available tools to carry out the exploratory analysis of spatialized data.\u00a0We have discussed\u00a0\u00a0\u00a0the histograms<\/a>\u00a0,\u00a0\u00a0\u00a0\u00a0the QQ-Plots<\/a>\u00a0, and now we will address the Voronoi maps.<\/p>\n

We must introduce a notion not yet introduced in the previous articles that concerns the extent or influence of a phenomenon.\u00a0In geostatistics we can consider two types of extension for a phenomenon: GLOBAL or LOCAL extent.<\/p>\n

Global and local phenomena.<\/strong><\/p>\n

We refer to a global phenomenon when we use as reference all the available data.\u00a0We refer to a local phenomenon when we use as reference a sampling point and its neighbouring points.<\/p>\n

A simple example is when we refer to outliers.\u00a0If we look for global outliers, we will look for values \u200b\u200bthat are outside the logical range of our data.\u00a0For example, if we have a batch of seawater temperature data, with values \u200b\u200bthat are between 2 \u00b0 C and 16 \u00b0 C, a value of -3 \u00b0 C or a value of 35 \u00b0 C, will appear as\u00a0\u00a0\u00a0global aberrant<\/em><\/strong>\u00a0values.\u00a0Suppose that our measurements are spread over a whole year and that we have in winter the following series of values \u200b\u200b2 \u00b0 C, 2.5 \u00b0 C, 11 \u00b0 C, 2.2 \u00b0 C, 2.5 \u00b0 C.\u00a0The value 11 \u00b0 C is not aberrant from a global point of view, because in the course of the year it can appear quite often.\u00a0But since it is found in the middle of much lower and regular temperatures we can deduce that it is a\u00a0local aberrant<\/em><\/strong>\u00a0value.<\/p>\n

The histograms and the QQ-plot that we have already discussed are global tools.\u00a0They allow us to work and understand phenomena that affect all our data.\u00a0With the Voronoi maps we will tackle tools that will allow us to\u00a0\u00a0\u00a0visualize and understand local phenomena, ie they will only concern a part of our data.<\/p>\n

The Voronoi maps <\/strong><\/p>\n

The Voronoi maps are built from a series of polygons formed around the location of each sampling point.<\/p>\n

The Voronoi polygons are created so that each location in a polygon is closer to the sampling point present in that polygon rather than to any other sampling point.<\/p>\n

\"\"<\/p>\n

For example, in this figure, the yellow dot is surrounded by a polygon, displayed in red.\u00a0Each location in the red polygon is closer to the yellow sampling point than to any other sampling point (dark blue dots).<\/p>\n

After creating the polygons, the\u00a0\u00a0\u00a0neighbours<\/em><\/strong>\u00a0of a sampling point are defined as any other sampling point whose polygon shares a border with the chosen sampling point.\u00a0The blue polygons all share a border with the red polygon, so the sampling points in the blue polygons are\u00a0\u00a0\u00a0neighbours<\/em><\/strong>\u00a0of the clear green sampling point.<\/p>\n

Using this neighbourhood definition, an array of local statistics can be calculated.\u00a0For example, a local average will be calculated by using the average of the sampling points in the central polygon and the neighbouring polygons (red and blue polygons).\u00a0This average will then be assigned to the red polygon.\u00a0After repeating this task for all the polygons and their neighbours, a colour scale will show the relative values \u200b\u200bof the local averages, which allows to visualize regions of high and low values.<\/p>\n

\"\"<\/p>\n

At the top right, the colour scale indicates the values \u200b\u200bof the calculated averages.\u00a0We can see that the top and right corner have the lowest values \u200b\u200bof the set and the bottom and left corner have the highest values.<\/p>\n

The different Geostatistical Analyst Voronoi maps <\/strong><\/p>\n

The tool Geostatistical Analyst Voronoi map provides a number of methods for assigning or calculating values \u200b\u200bto the polygons.<\/p>\n

Let’s first look at the list of possibilities and how they are calculated.\u00a0Then we will then discuss their use.<\/p>\n

MAPS TYPES <\/strong><\/p>\n

Simple<\/strong>\u00a0: The value assigned to each polygon a cell is the value of the sampling point of that polygon.<\/p>\n

Average<\/strong>\u00a0: The value assigned to a polygon is the average calculated from this polygon and its neighbours.<\/p>\n

Mode<\/strong>\u00a0: All polygons are classified in\u00a0\u00a0\u00a0five class intervals.\u00a0The value assigned to a polygon is the most current value (mode) between the polygon and its neighbours.<\/p>\n

Cluster<\/strong>\u00a0: All polygons are classified into five colour class intervals.\u00a0If the class range of the polygon is different from all its neighbours, the cell is grey (to distinguish it from its neighbours).<\/p>\n

Entropy<\/strong>\u00a0: All polygons are classified into five classes using the\u00a0\u00a0\u00a0smart quartiles<\/em><\/strong> method, a variation of the quartile method.\u00a0Entropy is calculated using the following formula<\/p>\n

\"\"<\/p>\n

Where pi is the proportion of polygons, among the central polygon and the neighbouring polygons, for each of the five classes, and Log is the logarithm base 2.<\/p>\n

Since this is not simple, let’s look at an example.\u00a0We have a polygon with 5 neighbouring \u00a0polygons.\u00a0We apply the smart quartiles method and we obtain 3 class 1 polygons, 1 class 3 polygon and 2 class 5 polygons.<\/p>\n

\"\"<\/p>\n

 <\/p>\n

We will have an entropy<\/p>\n

– [0.6 *\u00a0\u00a0\u00a0-0.736966\u00a0\u00a0\u00a0+ 0.2 * -2.321928\u00a0\u00a0\u00a0+ 0.4 *\u00a0\u00a0\u00a0-1.321928] = 1.4353<\/p>\n

In all cases we will have values \u200b\u200branging from 0 to 2.322.<\/p>\n

If all polygons (central polygon and neighbours) have the same class, the entropy is zero (1 * log2 (1)).<\/p>\n

If we find the five classes, each one will have a proportion 0.2 and the resulting entropy will be 2.322.<\/p>\n

Median<\/strong>: The value assigned to a cell is the median value calculated from the frequency distribution of the cell and its neighbours.<\/p>\n

Standard Deviation<\/strong>: The value assigned to a cell is the standard deviation calculated from the cell and its neighbours.<\/p>\n

Interquartile Deviation<\/strong>: The first and third quartiles are calculated using the frequency distribution of a polygon\u00a0and its neighbours.
\nThe value assigned to the cell is calculated by subtracting the first quartile value from the third quartile value:<\/p>\n