Measuring Geographic Distributions with GeoPandas: Standard Distance

Glen Bambrick

October 10, 2024
8:35 am
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Introduction

The Standard Distance, also know as the Standard Distance Deviation, is the average distance all features vary from the Mean Center and measures the compactness of a distribution. The Standard Distance is a value representing the distance in units from the Mean Center and is usually plotted on a map as a circle for a visual indication of dispersion, the Standard Distance is the radius.

The Standard Distance works best in the absence of a strong directional trend. According to Andy Mitchell, if a directional trend is present you are better off using the Standard Deviational Ellipse.

You can use the Standard Distance to compare territories between species, which has the wider/broader territory, or to compare changes over time such as the dispersion of burglaries for each calendar month.

In a Normal Distribution you would expect around 68% of all points to fall within the Standard Distance.

Sources:
The Esri Guide to GIS Analysis, Volume 2: Spatial Measurements and Statistics.
An Introduction to Statistical Problem Solving in Geography

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The Formula

The mean x-coordinate is subtracted from the x-coordinate value for each point and the difference is squared. The sum of all the squared values for x minus the x-mean is divided by the number of points. This is also performed for y-coordinates. The resulting values for x and y are summed and then we take the square root of this value to return the value to original distance units.

First we get the mean X and Y…

…and then the Standard Distance.

For Point features the X and Y coordinates of each feature is used, for Polygons the centroid of each feature represents the X and Y coordinate to use, and for Linear features the mid-point of each line is used for the X and Y coordinate.

Using GeoPandas to Calculate the Standard Distance

The code below represents previous endeavours from when we used GeoPandas and Shapely to find the mean center for a dataset. We require the mean center for the standard distance calculations. In our example we will use a Shapefile, but you can use any input and output filetypes that you have available with your GeoPandas setup.

The code is heavily commented for ease of understanding the workflow. For a Point, we now we need get the mean value for all x values and y values. For a Polyline, we need to first get the midpoint of each line, and then get the mean value for all x values and y values for the midpoints. For a Polygon, we need to the the centroid value for each polygon, and then get the mean value for all x values and y values for the centroids.

				
					import geopandas as gpd
from shapely.geometry import Point
import math

## input shapefile path
in_shp = r"path\to\input\shapefile\input.shp"

## the output shapefile path for the standard distance polygon
out_shp = r"path\to\output\shapefile\output.shp"

## read in the shapefile to a GeoDataFrame
gdf = gpd.read_file(in_shp)

## get the geometry type from the first record
geom_type = gdf.geom_type[0]

## get the EPSG code
crs = gdf.crs

## required for the formula
num_features = len(gdf)

## for Point geometry
if geom_type == "Point":
    ## get all x and y values
    gdf["x"] = gdf.geometry.x
    gdf["y"] = gdf.geometry.y

## for LineString geometry
elif geom_type == "LineString":
    ## get all x and y values for the midpoints
    gdf["midpoint"] = gdf.geometry.interpolate(0.5, normalized=True)
    gdf["x"] = gdf["midpoint"].x
    gdf["y"] = gdf["midpoint"].y

## for Polygon geometry
elif geom_type == "Polygon":
    ## get all x and y values for the centroids
    gdf["centroid"] = gdf.geometry.centroid
    gdf["x"] = gdf["centroid"].x
    gdf["y"] = gdf["centroid"].y

## get the mean of the x and y values
mean_x = gdf['x'].mean()
mean_y = gdf['y'].mean()

## calculate the sum of the squared differences for x and y
sum_of_sq_diff_x = sum([math.pow(x - mean_x, 2) for x in gdf["x"]])
sum_of_sq_diff_y = sum([math.pow(y - mean_y, 2) for y in gdf["y"]])

## get the sum of the results
sum_of_results = (sum_of_sq_diff_x/num_features) + (sum_of_sq_diff_y/num_features)

## get the square root of the sum of the results to find the standard distance
standard_distance = math.sqrt(sum_of_results)

## create a point geometry representing the mean center
mean_center = Point(mean_x, mean_y)

## buffer the mean center by the standard distance
standard_distance_geom = mean_center.buffer(standard_distance)

## create a GeoDataFrame with the polygon geometry
gdf_standard_distance = gpd.GeoDataFrame(geometry=[standard_distance_geom], crs=crs)

## add fields for the CenterX, CenterY, and StdDist
gdf_standard_distance["CenterX"] = mean_x
gdf_standard_distance["CenterY"] = mean_y
gdf_standard_distance["StdDist"] = standard_distance

## write the standard distance polygon to the output shapefile
gdf_standard_distance.to_file(out_shp, driver="ESRI Shapefile")

Standard Distance in Action

I downloaded crime data from DATA.POLICE.CO.UK for all crimes in the West Midlands which were a series of CSV files. I further processed the data for burglaries only and you can downland the shapefile for the example below here.

Running the script produces a Shapefile that contains the Standard Distance polygon for all burglaries in 2023

Below is a comparison between our GeoPandas tool and the Standard Distance tool output from ArcGIS Pro. Spot on!

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