Geographic Data Science - Lecture V Space, formally Dani - - PowerPoint PPT Presentation

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Geographic Data Science - Lecture V Space, formally Dani - - PowerPoint PPT Presentation

Dec 17, 2023 113 likes •552 views

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

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Geographic Data Science - Lecture V

Space, formally

Dani Arribas-Bel

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Today

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

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Space, formally

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

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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

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(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".

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Core element in several spatial analysis techniques: Spatial autocorrelation Spatial clustering / geodemographics Spatial regression

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W as a formal representation

  • f space
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W

N x N positive matrix

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W

N x N positive matrix that contains spatial relations

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W

N x N positive matrix that contains spatial relations between all the

  • bservations in the sample
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W

N x N positive matrix that contains spatial relations between all the

  • bservations in the sample

wii = 0 by convention

= { } wij x > 0 if i and j are neighbors

  • therwise
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W

N x N positive matrix that contains spatial relations between all the

  • bservations in the sample

wii = 0 by convention ...What is a neighbor???

= { } wij x > 0 if i and j are neighbors

  • therwise
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Types of W

A neighbor is "somebody" who is: Next door Close In the same "place" as us ...

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Types of W

A neighbor is "somebody" who is: Next door → Contiguity-based Ws Close In the same "place" as us ...

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Types of W

A neighbor is "somebody" who is: Next door → Contiguity-based Ws Close → Distance-based Ws In the same "place" as us ...

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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 ...

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Contiguity-based weights

Sharing boundaries to any extent Rook Queen ...

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Distance-based weights

Weight is (inversely) proportional to distance between observations Inverse distance (threshold) KNN (fixed number of neighbors) ...

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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 ...

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Other types of weights

Combinations of the above Kernel Statistically-derived ... See for an in-detail discussion. Anselin & Rey (2014)

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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.) ...

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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

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Do your own (contiguity) weights time!

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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 is the sum of a row.

= wij ¯ wij wi⋅ wi⋅

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The spatial lag

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The spatial lag

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

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The spatial lag

Product of a spatial weights matrix W and a given variably Y Ysl = WY ysl − i = ∑jwijyj

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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

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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...)

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Moran Plot

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Moran Plot

Graphical device that displays a variable on the horizontal axis against its spatial lag on the vertical one Usually, variables are standardized ( ), which divides the space into quadrants Tool to start exploring spatial autocorrelation

y − mean(y) std(y)

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Moran Plot

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Moran Plot

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Recapitulation

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