Spatial weights in PySAL

In this session, you will be learning the ins and outs of one of the key pieces in spatial analysis: spatial weights matrices. These are structured sets of numbers that formalise geographical relationships between the observations in a dataset. Essentially, a spatial weights matrix of a given geography is a positive definite matrix of dimensions \(N\) by \(N\), where \(N\) is the total number of observations:

\[ W = \left(\begin{array}{cccc} 0 & w_{12} & \dots & w_{1N} \\ w_{21} & \ddots & w_{ij} & \vdots \\ \vdots & w_{ji} & 0 & \vdots \\ w_{N1} & \dots & \dots & 0 \end{array} \right) \]

where each cell \(w_{ij}\) contains a value that represents the degree of spatial contact or interaction between observations \(i\) and \(j\). A fundamental concept in this context is that of neighbour and neighbourhood. By convention, elements in the diagonal (\(w_{ii}\)) are set to zero. A neighbour of a given observation \(i\) is another observation with which \(i\) has some degree of connection. In terms of \(W\), \(i\) and \(j\) are thus neighbors if \(w_{ij} > 0\). Following this logic, the neighbourhood of \(i\) will be the set of observations in the system with which it has a certain connection or those observations with a weight greater than zero.

There are several ways to create such matrices and many more to transform them so they contain an accurate representation that aligns with the way you understand spatial interactions between the elements of a system. This session will introduce the most commonly used ones and show how to compute them with PySAL.

1import matplotlib.pyplot as plt
import contextily
import geopandas as gpd
import numpy as np
import pandas as pd
import seaborn as sns

from libpysal import graph

1

A common standard is to do all the imports on top of your notebook.

Data

For this tutorial, you will use a dataset of Basic Settlement Units (ZSJ) in Prague for 2021. The table is available as part of this course, so it can be accessed remotely through the web. If you want to see how the table was created, a notebook is available here.

To make things easier, you will read data from a file posted online so, for now, you do not need to download any dataset:

prague = gpd.read_file(
    "https://martinfleischmann.net/sds/spatial_graphs/data/zsj_prague_2021.gpkg"
)
1prague = prague.set_index("NAZ_ZSJ")
prague.explore()

1

Use the name of each observation as an index. It will help you link them to the weights matrix.

Make this Notebook Trusted to load map: File -> Trust Notebook