4. Lesson: Spatial Statistics¶

4.1. Follow Along: Create a Test Dataset¶

In order to get a point dataset to work with, we’ll create a random set of points.

To do so, you’ll need a polygon dataset defining the extents of the area you want to create the points in.

We’ll use the area covered by streets.

• Create a new empty map.
• Add the roads_33S layer from the exercise_data/projected_data directory, as well as the srtm_41_19.tif raster (elevation data) found in exercise_data/raster/SRTM/.
• Use the Convex hull(s) tool (available under Vector ‣ Geoprocessing Tools) to generate an area enclosing all the roads:

• Save the output under exercise_data/spatial_statistics/ as roads_hull.shp.
• Add it to the TOC (Layers list) when prompted.

4.1.1. Creating random points¶

• Create random points in this area using the tool at Vector ‣ Research Tools ‣ Random points:

• Save the output under exercise_data/spatial_statistics/ as random_points.shp.
• Add it to the TOC (Layers list) when prompted:

4.1.2. Sampling the data¶

• To create a sample dataset from the raster, you’ll need to use the Point sampling tool plugin.
• Refer ahead to the module on plugins if necessary.
• Search for the phrase point sampling in the Plugin –> Manage and Install Plugins... and you will find the plugin.
• As soon as it has been activated with the Plugin Manager, you will find the tool under Plugins ‣ Analyses ‣ Point sampling tool:

• Select random_points as the layer containing sampling points, and the SRTM raster as the band to get values from.
• Make sure that “Add created layer to the TOC” is checked.
• Save the output under exercise_data/spatial_statistics/ as random_samples.shp.

Now you can check the sampled data from the raster file in the attributes table of the random_samples layer, they will be in a column named srtm_41_19.

A possible sample layer is shown here:

The sample points are classified by their value such that darker points are at a lower altitude.

You’ll be using this sample layer for the rest of the statistical exercises.

4.2. Follow Along: Basic Statistics¶

Now get the basic statistics for this layer.

• Click on the Vector ‣ Analysis Tools ‣ Basic statistics menu entry.
• In the dialog that appears, specify the random_samples layer as the source.
• Make sure that the Target field is set to srtm_41_19 which is the field you will calculate statistics for.
• Click OK. You’ll get results like this:

• Close the plugin dialog when done.

To understand the statistics above, refer to this definition list:

Mean

The mean (average) value is simply the sum of the values divided by the amount of values.

StdDev

The standard deviation. Gives an indication of how closely the values are clustered around the mean. The smaller the standard deviation, the closer values tend to be to the mean.

Sum

All the values added together.

Min

The minimum value.

Max

The maximum value.

N

The amount of samples/values.

CV

The spatial covariance of the dataset.

Number of unique values

The number of values that are unique across this dataset. If there are 90 unique values in a dataset with N=100, then the 10 remaining values are the same as one or more of each other.

Range

The difference between the minimum and maximum values.

Median

If you arrange all the values from least to greatest, the middle value (or the average of the two middle values, if N is an even number) is the median of the values.

4.3. Follow Along: Compute a Distance Matrix¶

• Create a new point layer in the same projection as the other datasets (WGS 84 / UTM 33S).
• Enter edit mode and digitize three point somewhere among the other points.
• Alternatively, use the same random point generation method as before, but specify only three points.
• Save your new layer as distance_points.shp.

To generate a distance matrix using these points:

• Open the tool Vector ‣ Analysis Tools ‣ Distance matrix.
• Select the distance_points layer as the input layer, and the random_samples layer as the target layer.
• Set it up like this:

• Save the result as distance_matrix.csv.
• Click OK to generate the distance matrix.
• Open it in a spreadsheet program to see the results. Here is an example:

4.4. Follow Along: Nearest Neighbor Analysis¶

To do a nearest neighbor analysis:

• Click on the menu item Vector ‣ Analysis Tools ‣ Nearest neighbor analysis.
• In the dialog that appears, select the random_samples layer and click OK.
• The results will appear in the dialog’s text window, for example:

4.5. Follow Along: Mean Coordinates¶

To get the mean coordinates of a dataset:

• Click on the Vector ‣ Analysis Tools ‣ Mean coordinate(s) menu item.
• In the dialog that appears, specify random_samples as the input layer, but leave the optional choices unchanged.
• Specify the output layer as mean_coords.shp.
• Click OK.
• Add the layer to the Layers list when prompted.

Let’s compare this to the central coordinate of the polygon that was used to create the random sample.

• Click on the Vector ‣ Geometry Tools ‣ Polygon centroids menu item.
• In the dialog that appears, select roads_hull as the input layer.
• Save the result as center_point.
• Add it to the Layers list when prompted.

As you can see from the example below, the mean coordinates and the center of the study area (in orange) don’t necessarily coincide:

4.6. Follow Along: Image Histograms¶

The histogram of a dataset shows the distribution of its values. The simplest way to demonstrate this in QGIS is via the image histogram, available in the Layer Properties dialog of any image layer.

• In your Layers list, right-click on the SRTM DEM layer.
• Select Properties.
• Choose the tab Histogram. You may need to click on the Compute Histogram button to generate the graphic. You will see a graph describing the frequency of values in the image.
• You can export it as an image:

• Select the Metadata tab, you can see more detailed information inside the Properties box.

The mean value is 332.8, and the maximum value is 1699! But those values don’t show up on the histogram. Why not? It’s because there are so few of them, compared to the abundance of pixels with values below the mean. That’s also why the histogram extends so far to the right, even though there is no visible red line marking the frequency of values higher than about 250.

Therefore, keep in mind that a histogram shows you the distribution of values, and not all values are necessarily visible on the graph.

• (You may now close Layer Properties.)

4.7. Follow Along: Spatial Interpolation¶

Let’s say you have a collection of sample points from which you would like to extrapolate data. For example, you might have access to the random_samples dataset we created earlier, and would like to have some idea of what the terrain looks like.

To start, launch the Grid (Interpolation) tool by clicking on the Raster ‣ Analysis ‣ Grid (Interpolation) menu item.

• In the Input file field, select random_samples.
• Check the Z Field box, and select the field srtm_41_19.
• Set the Output file location to exercise_data/spatial_statistics/interpolation.tif.
• Check the Algorithm box and select Inverse distance to a power.
• Set the Power to 5.0 and the Smoothing to 2.0. Leave the other values as-is.
• Check the Load into canvas when finished box and click OK.
• When it’s done, click OK on the dialog that says Process completed, click OK on the dialog showing feedback information (if it has appeared), and click Close on the Grid (Interpolation) dialog.

Here’s a comparison of the original dataset (left) to the one constructed from our sample points (right). Yours may look different due to the random nature of the location of the sample points.

As you can see, 100 sample points aren’t really enough to get a detailed impression of the terrain. It gives a very general idea, but it can be misleading as well. For example, in the image above, it is not clear that there is a high, unbroken mountain running from east to west; rather, the image seems to show a valley, with high peaks to the west. Just using visual inspection, we can see that the sample dataset is not representative of the terrain.

4.8. Try Yourself¶

• Use the processes shown above to create a new set of 1000 random points.
• Use these points to sample the original DEM.
• Use the Grid (Interpolation) tool on this new dataset as above.
• Set the output filename to interpolation_1000.tif, with Power and Smoothing set to 5.0 and 2.0, respectively.

The results (depending on the positioning of your random points) will look more or less like this:

The border shows the roads_hull layer (which represents the boundary of the random sample points) to explain the sudden lack of detail beyond its edges. This is a much better representation of the terrain, due to the much greater density of sample points.

Here is an example of what it looks like with 10 000 sample points:

4.9. Follow Along: Additional Spatial Analysis Tools¶

Originally a separate project and then accessible as a plugin, the SEXTANTE software has been added to QGIS as a core function from version 2.0. You can find it as a new QGIS menu with its new name Processing from where you can access a rich toolbox of spatial analysis tools allows you to access various plugin tools from within a single interface.

• Activate this set of tools by enabling the Processing ‣ Toolbox

menu entry. The toolbox looks like this:

You will probably see it docked in QGIS to the right of the map. Note that the tools listed here are links to the actual tools. Some of them are SEXTANTE’s own algorithms and others are links to tools that are accessed from external applications such as GRASS, SAGA or the Orfeo Toolbox. This external applications are installed with QGIS so you are already able to make use of them. In case you need to change the configuration of the Processing tools or, for example, you need to update to a new version of one of the external applications, you can access its setting from Processing ‣ Options and configurations.

4.10. Follow Along: Spatial Point Pattern Analysis¶

For a simple indication of the spatial distribution of points in the random_samples dataset, we can make use of SAGA’s Spatial Point Pattern Analysis tool via the Processing Toolbox you just opened.

• In the Processing Toolbox, search for this tool Spatial Point Pattern Analysis.
• Double-click on it to open its dialog.

4.10.6. Using SAGA¶

• Open the SAGA dialog.
• SAGA produces three outputs, and so will require three output paths.
• Save these three outputs under exercise_data/spatial_statistics/, using whatever file names you find appropriate.

The output will look like this (the symbology was changed for this example):

The red dot is the mean center; the large circle is the standard distance, which gives an indication of how closely the points are distributed around the mean center; and the rectangle is the bounding box, describing the smallest possible rectangle which will still enclose all the points.

4.11. Follow Along: Minimum Distance Analysis¶

Often, the output of an algorithm will not be a shapefile, but rather a table summarizing the statistical properties of a dataset. One of these is the Minimum Distance Analysis tool.

• Find this tool in the Processing Toolbox as :guilabel:`Minimum

Distance Analysis`.

It does not require any other input besides specifying the vector point dataset to be analyzed.

• Choose the random_points dataset.
• Click OK. On completion, a DBF table will appear in the Layers list.
• Select it, then open its attribute table. Although the figures may vary, your results will be in this format:

4.12. In Conclusion¶

QGIS allows many possibilities for analyzing the spatial statistical properties of datasets.

4.13. What’s Next?¶

Now that we’ve covered vector analysis, why not see what can be done with rasters? That’s what we’ll do in the next module!