Spatial Data, Analysis, and Regression - II

W as a formal representation of space

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

$w_{ij} = \left\{ \begin{array}{rl} x_{ij}>0 &\mbox{ if $i$ and $j$ are neighbors} \\ 0 &\mbox{ otherwise} \end{array} \right\}$

Defined ex-ante (sometimes criticized by its ad-hoc nature)
Very often, x_ij may follows standard criteria:
- Contiguity (queen, rook, bishop)
- Some function of distance
- Nearest neighbors (knn)
- ...
In some contexts, additional requirements:
- Exogeneity
- Close match to theoretical framework
- Empirically motivated reflection of actual interactions

Example: rook contiguity

↓

→ GeoDaSpace demo with weights

y_sl − i = ∑ _jw_ijy_j

Y_sl = WY

Measure that captures the behaviour of a variable in the neighborhood of a given observation i.
Similar to the time lag, but not completely (I am my neighbor's neighbor)
Typically standardized to reflect some sort of average, although not always, depends on purpose (e.g. market potential).

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...)

[Exploratory] Focus on discovery and assumption-free investigation
[Spatial] Analysis of patterns related to the spatial distribution of observations

Thematic map in which values of a variable are encoded using a color gradient of some sort
Preliminar and very exploratory but super useful as a check
Many ways to classify (caution with that!)
- Unique values (categorical)
- Equal interval
- Quantiles (equal count)
- Fisher-Jenks
- ...

Thematic map in which values of a variable are displayed by distorting the shape of their locations
Useful, for instance, when observations greatly differ in the geographic size but want to visualize a different variable (e.g. employment)
Many approaches and algorithms, from transforming polygons into circles to very intricate polygon deformations

Drawing

Visual device to explore spatial autocorrelation
Simple scatter cloud of:
- [X axis] Variable of interest (y)
- [Y axis] Spatial lag of y

[Demo with notebook]

$I = \dfrac{N} {\sum_{i} \sum_{j} w_{ij}} \dfrac {\sum_{i} \sum_{j} w_{ij}(Z_i) (Z_j)} {\sum_{i} (Z_i)^2}$

with $Z_i = X_i-\bar X$

Note: first part goes away with row-standardized W and, in matrix notation:

$I = \dfrac{ZWZ}{ZZ}$

Similar measure: Geary's C, Getis & Ord

Local Moran's I (1996)

$I_i = \frac{Z_i}{m_2} \sum_j W_{ij} Z_j \; \; ; \; \; m_2= \frac{\sum_i Z_i^2 }{N}$ This makes I_i nicely additive into I: $I= \sum_i \frac{I_i}{N}$

Similar measures: Local Geary's C, Local Getis & Ord

Statistic to detect pockets of spatial instability
It does not summarize but extends the amount of insight from the data
Since it produces large outputs, it is usually visualized through maps:
- Clusters
- Significance