Week 10: Evaluating Spatial Hypotheses II: Areal Data

PPOL 6805 / DSAN 6750: GIS for Spatial Data Science
Fall 2024

Author

Affiliation

Jeff Jacobs

jj1088@georgetown.edu

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Roadmap to the Midterm!

• Last Week (Oct 23): Evaluating Hypotheses for Point Data
• Today (Oct 30): Evaluating Hypotheses for Areal Data
• In-Class Midterm (Nov 6): Basically a “mini-homework” on Spatial Data Science unit (topics from HW4 onwards)!

Point Processes $\to$ Areal Processes

source("../dsan-globals/_globals.r")
set.seed(6805)
library(tidyverse) |> suppressPackageStartupMessages()
library(sf) |> suppressPackageStartupMessages()
library(spatstat) |> suppressPackageStartupMessages()
library(mdthemes) |> suppressPackageStartupMessages()
library(mapview) |> suppressPackageStartupMessages()
library(leaflet) |> suppressPackageStartupMessages()

• The “Complete” Part of Complete Spatial Randomness (CSR)
• Bringing Neighbors Back In
• Spatial Regression

Points $\to$ Areas

Aggregating Point Processes

• Recall the two “stages” of our Poisson-based point processes
  • A Poisson-distributed number of points, then
  • Uniformly-distributed coordinates for each point
• One way to see areal data modeling: we keep the Poisson part, but we no longer observe the coordinates for the points!
• So… why is it still helpful for you to have sat through those lectures on Point Processes? Two reasons…

(1) Areal Patterns = Aggregated Point Patterns

• One application, mentioned last week: areal-weighted interpolation using actual models of how the points are distributed within the area!
  • Regularly-spaced points is rarely a good “default” model!
  • Humans, for example, rarely live at perfect evenly-spaced intervals… they form households, villages, cities
  • Regularly-spaced points (we now know) $⟹$ negative autocorrelation $⟹$ typically due to inhibition process (competition)

(2) Spatial Scan Statistics

• Areal regions often the result of artificial / “arbitrary” human divisions
  • (Particulate matter doesn’t pass through customs)
• $⟹$ If we care about processes which don’t “adhere to” borders (like disease spread), we want to “scan” buffers around points regardless of areal borders

(3) Hierarchical Bayesian Smoothing

• The issue might just be not enough data for some areas, while we have an abundance of data in nearby areas…

From Kramer (2023), Spatial Epidemiology

Autocorrelation on Networks

Reminder: Weight Matrix Defines “Neighbors”

From Canadian Rockies Exploration Guide

Choices Available in spdep

• $w_{ij}=1$ if $i$ and $j$ “overlap”
  • Rook: Overlap is 1 or 2-dimensional
  • Queen: Overlap is 0, 1, or 2-dimensional
• $w_{ij}=\frac{1}{\text{dist}(i,j)}$
• $w_{ij}=1$ if $dist(i,j)<\bar{D}$
• $w_{ij}=1$ for $K$ nearest neighbors

Moran’s $I$ One More Time

• Round 1 (Week 6): Moran’s $I$ as “thermometer”
• Round 2 (Today): Could an $I$ value this extreme occur due to random noise?

Example: Bolshevik Revolution $\to$ “Bipolar” World

• Null hypothesis: no spatial effect of democratization/de-democratization on neighboring countries
• If null hypothesis is true, and countries democratize/de-democratize independently… could this pattern still arise?

Spatial Regression

• (We made it! This is the last Spatial Data Science topic!)
• (Nearly all extremely-fancy applied GIS models can be implemented via Spatial Regression!)

Motivation: When Does Non-Spatial Regression “Work”?

$$Y_i={\beta }_0+{\beta }_1{X}_{i,1}+{\beta }_2{X}_{i,2}+\cdots +{\beta }_M{X}_{i,M}+{\epsilon }_i$$

• Importance of OLS regression: can give us the Best Linear Unbiased Estimator (BLUE)

• This is only true if the Gauss-Markov assumptions hold—one of these is that the error terms are uncorrelated:

  $$\text{Cov}[{\epsilon }_i,{\epsilon }_j]=0\mathrm{\forall }i\ne j$$

Spatial Autocorrelation in Residuals

• We have now seen several models / datasets where effect of some variable $X$ (say, population) on another variable $Y$ (say, disease count) is spatial! (Kind of the whole point of the class 😜)
• So, to see when OLS will “work”, vs. when you need to incorporate GIS, key step is plotting the spatial distribution of regression residuals!

Example: Italian Elections

Ward and Gleditsch (2018), Figure 2.3

Ward and Gleditsch (2018), Figure 2.4

Will OLS “Work” Here?

• Can we use OLS to derive the BLUE of the effect of GDP on voting?

Plain OLS

$N=477$	$\hat{\beta }$	SE	$t$
Intercept	35.30	2.21	15.96
Log GDP per cap	13.46	0.65	20.84

Moran’s $I$ for residuals = 0.47(!)

 

Spatial Regression

$N=477$	$\hat{\beta }$	SE	$t$
Intercept	4.70	1.66	2.80
Log GDP per cap	1.77	0.48	3.66
$\rho$	0.87	0.02	36.7

What Does it Mean that “Spatial Effect” is Significant?

Ward and Gleditsch (2018), Figure 2.5

Case 1: No Residual Spatial Autocorrelation

• Example: Residuals have Moran’s $I$ near 0…
• …Don’t need GIS at all!

Case 2: Conditional Autoregressive Model (CAR)

• GIS, but… only for “fixing” your regression

$$Y_i=\underset{\text{Non-spatial model}}{\underbrace{{\mu }_i}}+\underset{\text{Spatial Autocorrelation}}{\underbrace{\frac{1}{w_{i,cdot}}\sum _{j\ne i}(Y_j-{\mu }_j)}}+{\epsilon }_i$$

Case 3: Simultaneous Autoregressive Model (SAR)

• The main event! Here we are explicitly modeling “spatially lagged” versions of our dependent variable $Y$!

$$Y=\mathbf{X}\beta +\rho \underset{\text{Spatially-lagged }Y}{\underbrace{\mathbf{W}Y}}+\mathit{\epsilon }$$

References

Ward, Michael D., and Kristian Skrede Gleditsch. 2018. Spatial Regression Models. SAGE Publications.