Geographic Data Science - Lecture V

Space, formally

Dani Arribas-Bel

Today

  • The need to represent space formally
  • Spatial weights matrices
    • What
    • Why
    • Types
  • The spatial lag
  • The Moran Plot

Space, formally

For a statistical method to be explicitly spatial, it needs to contain some representation of the geography, or spatial context

One of the most common ways is through Spatial Weights Matrices

  • (Geo)Visualization: translating numbers into a (visual) language that the human brain "speaks better"
  • Spatial Weights Matrices: translating geography into a (numerical) language that a computer "speaks better".

Core element in several spatial analysis techniques:

  • Spatial autocorrelation
  • Spatial clustering / geodemographics
  • Spatial regression

W as a formal representation of space

W

N x N positive matrix that contains spatial relations between all the observations in the sample

wii = 0 by convention

...What is a neighbor???

Types of W

A neighbor is "somebody" who is:

  • Next door  →  Contiguity-based Ws
  • Close  →  Distance-based Ws
  • In the same "place" as us  →  Block weights
  • ...

Contiguity-based weights

Sharing boundaries to any extent

  • Rook
  • Queen
  • ...

Distance-based weights

Weight is (inversely) proportional to distance between observations

  • Inverse distance (threshold)
  • KNN (fixed number of neighbors)
  • ...

Block weights

Weights are assigned based on discretionary rules loosely related to geography

For example:

  • LSOAs into MSOAs
  • Post-codes within city boundaries
  • Counties within states
  • ...

Other types of weights

  • Combinations of the above
  • Kernel
  • Statistically-derived
  • ...

See Anselin & Rey (2014) for an in-detail discussion.

How much of a neighbor?

No neighbors receive zero weight: wij = 0

Neighbors, it depends, wij can be:

  • One wij = 1  →  Binary
  • Some proportion (0 < wij < 1, continuous) which can be a function of:

    • Distance
    • Strength of interaction (e.g. commuting flows, trade, etc.)
    • ...

Choice of W

Should be based on and reflect the underlying channels of interaction for the question at hand.

Examples:

  • Processes propagated by inmediate contact (e.g. disease contagion)  →  Contiguity weights
  • Accessibility  →  Distance weights
  • Effects of county differences in laws  →  Block weights

Do your own (contiguity) weights time!

Standardization

In some applications (e.g. spatial autocorrelation) it is common to standardize W

The most widely used standardization is row-based: divide every element by the sum of the row:

where $w_{i\cdotp}$ is the sum of a row.

The spatial lag

The spatial lag

Product of a spatial weights matrix W and a given variably Y


Ysl = WY

ysl − i = ∑jwijyj

  • Measure that captures the behaviour of a variable in the neighborhood of a given observation i.
  • If W is standardized, the spatial lag is the average value of the variable in the neighborhood

  • Common way to introduce space formally in a statistical framework
  • Heavily used in both ESDA and spatial regression to delineate neighborhoods. Examples:

    • Moran's I
    • LISAs
    • Spatial models (lag, error...)

Moran Plot

Moran Plot

  • Graphical device that displays a variable on the horizontal axis against its spatial lag on the vertical one
  • Usually, variables are standardized ($\dfrac{y - mean(y)}{std(y)}$), which divides the space into quadrants
  • Tool to start exploring spatial autocorrelation

Moran Plot

Recapitulation

  • Spatial Weights matrices: matrix encapsulation of space
  • Different types for different cases
  • Useful in many contexts, like the spatial lag and Moran plot, but also many other things!

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Geographic Data Science'16 - Lecture 5 by Dani Arribas-Bel is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.