5 Spatial Weights

This section is about how we can turn geography into numbers that statistics can understand. At this point, we dive right into the more methodological part of the course, so you can expect the conceptual sections to be more challenging. At the same time, the coding side of each block will start looking more and more familiar because we are starting to repeat concepts. We will introduce less new building blocks and instead rely more on what we have seen, just adding small bits here and there.

Slides can be downloaded “here”

Space, formally

How do you express geographical relations between objects (e.g. areas, points) in a way that can be used in statistical analysis? This is exactly the core of what we get into in here. There are several ways but one of the most widespread approaches is what is termed spatial weights matrices. Their role is to:

Take all richness of geographical relationships and express it in a language that is well understood by statistics and computers
Relate to concepts of spatial ‘smoothing’ and interpolating data, facilitating visualisation and exploratory analysis
Check how the characteristics or outcomes of one spatial object might be correlated with those of its neighbours: e.g. education, criminality, etc.

Furthermore, they are a core element in several spatial analysis techniques:

Spatial autocorrelation
Spatial clustering/geo-demographics
Spatial regression

We define spatial weights matrices as structured sets of numbers that formalise geographical relationships between the objects in a dataset. Essentially, a spatial weights matrix of a given geography is a positive definite matrix of dimensions \(N \times N\), where \(N\) is the total number of observed objects:

\(W = \begin{pmatrix} 0 & w_{12} & \dots & w_{1N}\\ w_{21} & \ddots & w_{ij} & \vdots \\ \vdots & w_{ji} & 0 & \vdots \\ w_{N1} & \dots & \dots & 0 \end{pmatrix}\)

where each cell 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 neighbor and neighborhood. 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 spatial connection. In terms of \(W\), \(i\) and \(j\) are neighbours if \(w_{ij} > 0\). Following this logic, the neighbourhood of \(i\) will be the set of observations in the system with which it has certain connection, or those observations with a weight greater than zero.

Types of Weights

There are specific types of spatial weights that we can define for our particular analyses.

Contiguity-based weights

The neighbourhood of an observation is defined by those observations which share boundaries, e.g. rook, bishop or queen neighbourhood (can be based on a point, an edge, etc.).

Distance-based weights

The neighbourhood of an observation is defined by those spatial units which are at a certain distance from the central one, with the weights typically decreasing as distance increases, e.g. inverse distance (1/distance or threshold), \(k\) nearest-neighbours or KNN (fixed number of closest neighbors).

Block-based weights

The neighbourhood of an observation is defined by those spatial units which share a certain attribute with the central one.

The Spatial Lag

We wrap up the the set of concepts in this block with one of the applications that makes spatial weights matrices so important: the spatial lag.

Given a set of observations for a specific variable in a geography and a spatial weights matrix \(W\) for that geography, the spatial lag is defined as the product of \(W\) and the observations.

The spatial lag can be interpreted as a measure that captures the behaviour of the variable in the neighbourhood of each observation \(i\). If \(W\) is standardised, the spatial lag is the average value of the variable in the neighbourhood.

Space, formally

Types of Weights

The Spatial Lag

Further readings