#### Table of Contents

### Introduction

The Central Feature is the point that is the shortest distance to all other points in the dataset and thus identifies the most centrally located feature. The *Weighted Central Feature* considers weights when calculating the central feature, where points with higher weights have a larger influence on the result.

Sources:

The Esri Guide to GIS Analysis, Volume 2: Spatial Measurements and Statistics.

An Introduction to Statistical Problem Solving in Geography

### The Formula

For each feature, calculate the distance to all other features and multiply each distance by the assigned weight. Get the sum of all the (distance x weigth). The feature with the minimal sum of weighted distances with the Weighted Central Feature.

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 Weighted Central Feature

The code below uses GeoPandas and Shapely to find the weighted central feature for a dataset and create an output file. 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 and Polygon, we use the centroid. You could use the Point geometry itself for a Point shapefile, but in order to get the “total_distance” calculation on one line of code it was easier to assign the Point geometry to a column called “point” for each geometry type. For a Polyline, we use the midpoint of each line.

We calculate the distance from each point to all other points, multiplying each distance by a weight, and getting the sum of all distance * weight, and then find the point with the smallest summed distance * weight, this represents the weighted central feature. Although, there could be multiple features with the same smallest shortest-distance so we account for this.

Lastly, we export the central feature(s) from the original input Shapefile to a new Shapefile.

` ````
```import geopandas as gpd
## input shapefile path
in_shp = r"path\to\input\shapefile\input.shp"
## the output shapefile path for the weighted central feature(s)
out_shp = r"path\to\output\shapefile\output.shp"
## the field that contains the numerical weight
weight_fld = "FIELD_NAME"
## 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]
## for Point and Polygon geometry get the centroid
if geom_type in ("Point", "Polygon"):
## get the centroid of each feature as a Point geometry
gdf["point"] = gdf.geometry.centroid
## for LineString geometry get the midpoint
elif geom_type == "LineString":
## get the midpoint of each line as a Point geometry
gdf["point"] = gdf.geometry.interpolate(0.5, normalized=True)
## calculate the weighted sum of distances for all points
gdf["weighted_sum_distance"] = gdf["point"].apply(lambda geom: (gdf["point"].distance(geom) * gdf[weight_fld]).sum())
## get the value of the minimum weighted sum distance
min_distance = gdf["weighted_sum_distance"].min()
## there could be multiple weighted central features with the same smallest
## cumulative distance
weighted_central_feature = gdf[gdf["weighted_sum_distance"] == min_distance]
## sanitize the weighted central feature(s) gdf and make ready for output
weighted_central_feature = weighted_central_feature.drop(columns=["point", "weighted_sum_distance"])
## write the weighted central feature(s) to the output shapefile
weighted_central_feature.to_file(out_shp, driver="ESRI Shapefile")

### Weighted Central Feature in Action

Data for Primary School location was downloaded from the Department of Education (Ireland) and processed to contain Primary Schools in County Kildare in a projected coordinate system – Irish Transverse Mercator (EPSG:2157). You can download the Shapefile containing the data used below here.

Running the script produces a Shapefile that contains the Weighted Central Feature from the original Primary Schools Shapefile.

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

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### Also in this series...

- Mean Center
- Central Feature
- Median Center
- Standard Distance
- Weighted Mean Center
- Mean Center by Case
- Standard Distance by Case