Week 9: Evaluating Spatial Hypotheses I: Point Data

PPOL 6805 / DSAN 6750: GIS for Spatial Data Science
Fall 2025

Class Sessions

Author

Affiliation

Jeff Jacobs

jj1088@georgetown.edu

Published

Wednesday, October 23, 2024

Open slides in new tab →

Roadmap to the Midterm!

This Week (Oct 23): Evaluating Hypotheses for Point Data
Next Week (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)!

Our GIS Mystery-Solving Toolkit

Code

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()

Why Do Events Appear Where They Do?

Code

set.seed(6809)
N <- 60
r_core <- 0.05
obs_window <- square(1)
### Clustered data
clust_ppp <- rMatClust(
  kappa=6,
  scale=r_core,
  mu=10
)
clust_sf <- clust_ppp |> sf::st_as_sf()
clust_plot <- clust_sf |>
  ggplot() +
  geom_sf(size=2) +
  theme_classic(base_size=18)
ggsave("images/clust_ppp.png", clust_plot, width=3, height=3)
# Intensity fn
clust_intensity <- density(clust_ppp, sigma = 0.1)
png("images/clust_intensity.png")
par(mar=c(0,0,0,2), las=2, oma=c(0,0,0,0), cex=2)
plot(clust_intensity, main=NULL)
contour(clust_intensity, add = TRUE)
dev.off()
### PCF
clust_pcf <- spatstat.explore::pcf(
  clust_ppp, divisor="d",
  r=seq(from=0.00, to=0.50, by=0.01)
)
clust_pcf_plot <- clust_pcf |> ggplot(aes(x=r, y=iso)) +
  geom_hline(yintercept=1, linetype='dashed', linewidth=1) +
  geom_area(color='black', fill=cb_palette[1], alpha=0.75) +
  scale_x_continuous(breaks=seq(from=0.0, to=1.0, by=0.1)) +
  labs(x="Distance", y="Density") +
  theme_classic(base_size=14)
ggsave("images/clust_pcf.png", clust_pcf_plot, width=3, height=3)

Original Data	First-Order	Second-Order
\(N = 60\) Events	Events modeled individually \(\implies\) Intensity function \(\lambda(\mathbf{s})\)	Events modeled pairwise \(\implies\) \(K\)-function \(K(\vec{h})\)

What Do These Functions “Detect”?

Code

sq_base <- 16
sq_psize <- 2.5
obs_window <- square(1)
r0 <- 0.2
sq_df <- tibble::tribble(
  ~x, ~y,
  0.5-r0,0.5-r0,
  0.5+r0,0.5+r0,
  0.5-r0,0.5+r0,
  0.5+r0,0.5-r0
)
sq_sf <- sf::st_as_sf(
  sq_df,
  coords = c("x","y")
)
sq_ppp <- as.ppp(sq_sf, W=obs_window)
sq_ppp |> sf::st_as_sf() |> ggplot() +
  geom_sf(size=sq_psize) +
  theme_classic(base_size=sq_base)

Code

par(mar=c(0,0,0,2), las=2, oma=c(0,0,1,0))
sq_density <- density(sq_ppp)
plot(sq_density, main=NULL, xaxs='i', yaxs='i')
contour(sq_density, xaxs='i', yaxs='i', add = TRUE)

Code

### PCF
pcf_result <- spatstat.explore::pcf(
  sq_ppp,
  divisor="d",
  r=seq(from=0.00, to=0.8, by=0.01)
)
pcf_result |> ggplot(aes(x=r, y=iso)) +
  geom_hline(yintercept=1, linetype='dashed', linewidth=1.5) +
  geom_area(color='black', fill=cb_palette[1], alpha=0.75) +
  scale_x_continuous(breaks=seq(from=0.0, to=1.0, by=0.1)) +
  theme_classic(base_size=sq_base)

Code

csr_ppp <- spatstat.random::rpoispp(
  lambda = 60,
  win=obs_window
)
csr_ppp |> sf::st_as_sf() |> ggplot() +
  geom_sf(size=sq_psize) +
  theme_classic(base_size=sq_base)

Code

par(
  mar=c(0,0,0,2),
  las=2,
  oma=c(0,0,1,0)
)
csr_density <- density(csr_ppp)
plot(csr_density, main=NULL, xaxs="i", yaxs="i")
contour(csr_density, xaxs="i", yaxs="i", add = TRUE, lwd=1.5)

Code

csr_pcf_result <- spatstat.explore::pcf(
  csr_ppp,
  divisor="d",
  r=seq(from=0.00, to=0.8, by=0.01)
)
csr_pcf_result |> ggplot(aes(x=r, y=iso)) +
  geom_hline(yintercept=1, linetype='dashed', linewidth=1.5) +
  geom_area(color='black', fill=cb_palette[1], alpha=0.75) +
  scale_x_continuous(breaks=seq(from=0.0, to=1.0, by=0.1)) +
  theme_classic(base_size=sq_base)

The Point of All This: Null Models via Base Rates!

Null Models

Given an intensity function, we can compute a bunch of simulated point patterns (inhomogeneous Poisson Point Process)…
How does an observed point pattern differ from the simulated point patterns?

Constant Risk Hypothesis: Motivation

Here is a (fictional) map of flu cases in the US “Midwest”
Are people in Chicago and Detroit equally “at risk”?

Code

city_df <- tibble::tribble(
  ~city, ~lon, ~lat, ~pop,
  "Chicago", 41.950567516553896, -87.93011127491978, 2746388,
  "Detroit", 42.45123004999075, -83.18402319217698, 631524
)
city_sf <- sf::st_as_sf(
  city_df,
  coords = c("lat", "lon"),
  crs=4326
)
city_buf_sf <- city_sf |> sf::st_buffer(20000)
city_cases_sf <- city_buf_sf |> sf::st_sample(size=rep(10,2)) |> sf::st_as_sf()
city_cases_sf$city <- "Detroit (10 Cases)"
city_cases_sf[1:10, 'city'] <- "Chicago (10 Cases)"
city_cases_sf$sample <- "Flu"
mapview(city_cases_sf, zcol="city")

Constant Risk Hypothesis: Base Rates

	Chicago	Detroit
Population	2,746,388	631,524

Code

city_pop_sf <- city_buf_sf |> sf::st_sample(size=c(16, 4)) |> sf::st_as_sf()
city_pop_sf$city <- "Detroit"
city_pop_sf[1:16, 'city'] <- "Chicago"
city_pop_sf$sample <- "People"
city_combined_sf <- bind_rows(city_cases_sf, city_pop_sf)
# mapview(city_combined_sf, zcol="city", marker="sample")
Flu = makeIcon(
    "https://upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Maki2-danger-24.svg/240px-Maki2-danger-24.svg.png",
    "https://upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Maki2-danger-24.svg/24px-Maki2-danger-24.svg.png",
    20,
    20
)
People = makeIcon(
  "https://upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Maki2-pitch-24.svg/24px-Maki2-pitch-24.svg.png",
  "https://upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Maki2-pitch-24.svg/24px-Maki2-pitch-24.svg.png",
  20,
  20
)

city_combined_sf$r <- ifelse(city_combined_sf$sample == "Flu", 0, 4)
city_flu_sf <- city_combined_sf |> filter(sample == "Flu")
city_ppl_sf <- city_combined_sf |> filter(sample == "People")
leaflet(city_flu_sf) |>
  addProviderTiles("CartoDB.Positron") |>
  addMarkers(data=city_flu_sf, icon = Flu) |>
  addMarkers(data=city_ppl_sf, icon = People)

Meaningful Results \(\leftrightarrow\) Comparisons with Null Model(s)

Cases of disease form an intensity function \(\lambda(\mathbf{s})\)
Controls form an “ambient” intensity function \(\lambda_0(\mathbf{s})\)
\(\implies\) The following model allows us to separate the quantities we really care about from the base rate information:

\[ \lambda(\mathbf{s}) = \alpha \overbrace{\lambda_0(\mathbf{s})}^{\mathclap{\text{Base Rate}}} \underbrace{\rho(\mathbf{s})}_{\mathclap{\text{Relative Risk}}}, \]

where

\[ \alpha = \frac{\# \text{Cases}}{\# \text{Controls}} \]
(Remember: unit of \(\lambda(\mathbf{s})\) is number of cases; not a probability density!)

Example from Moraga (2024)

Data: 761 primary biliary cirrhosis (PBC) cases, 3020 controls, England 1987-94

Code

library(sparr) |> suppressPackageStartupMessages()
data(pbc)
controls <- unmark(pbc[which(pbc$marks == "control"), ])
controls |> sf::st_as_sf() |> ggplot() +
  geom_sf() +
  theme_classic() +
  labs(title="Controls (Population Sample)")

Code

cases <- unmark(pbc[which(pbc$marks == "case"), ])
cases |> sf::st_as_sf() |> ggplot() +
  geom_sf() +
  theme_classic() +
  labs(title="PBC Cases")

Estimating Comparable Intensity Surfaces

To ensure comparable intensity functions, we need a common bandwidth parameter.
density() from spatstat makes “smart” choices for this bandwidth, so for now we just average the bandwidths estimated for cases and controls:

Code

library(sparr)
data(pbc)
cases <- unmark(pbc[which(pbc$marks == "case"), ])
controls <- unmark(pbc[which(pbc$marks == "control"), ])
bwcases <- attr(density(cases), "sigma")
bwcontr <- attr(density(controls), "sigma")
(bw <- (bwcases + bwcontr)/2)

[1] 11.45837

Code

int_controls <- density(controls, sigma = bw, eps=0.25)
plot(int_controls, main = NULL)

Code

int_cases <- density(cases, sigma = bw, eps=0.25)
plot(int_cases, main = NULL)

Visualizing Relative Risk Surface

All that’s left is \(\alpha = \# \text{Controls} / \# \text{Cases}\)!

Code

library(fields)

Loading required package: spam

Spam version 2.10-0 (2023-10-23) is loaded.
Type 'help( Spam)' or 'demo( spam)' for a short introduction 
and overview of this package.
Help for individual functions is also obtained by adding the
suffix '.spam' to the function name, e.g. 'help( chol.spam)'.


Attaching package: 'spam'

The following objects are masked from 'package:base':

    backsolve, forwardsolve

Loading required package: viridisLite


Try help(fields) to get started.


Attaching package: 'fields'

The following object is masked from 'package:leaflet':

    addLegend

Code

(alpha_hat <- cases$n/controls$n)

[1] 0.2519868

Code

x <- int_cases$xcol
y <- int_cases$yrow
rr <- t(int_cases$v)/t(alpha_hat * int_controls$v)
image.plot(x, y, rr, asp = 1)
title(xlab = "Easting", ylab = "Northing")

From Exploratory to Confirmatory Data Analysis

How do we know whether this raised risk around North Newcastle could be attributed to “chance”? Maybe relative risk is actually constant, and people in Newcastle are just unlucky…
Approach: compare with null model

Null Model: CSR

The Random Labeling Hypothesis

Randomly apply 761 “sick” labels to the 3781 points!

Code

set.seed(6810)
pbc_1 <- rlabel(pbc)
plot(pbc_1)

Code

set.seed(6811)
pbc_2 <- rlabel(pbc)
plot(pbc_2)

Code

set.seed(6812)
pbc_3 <- rlabel(pbc)
plot(pbc_3)

The Constant Risk Hypothesis

Everyone equally at risk of contracting disease, regardless of location: \(\rho(\mathbf{s}) = 1\)
In this case: \(\lambda_{CR}(\mathbf{s}) = \alpha \lambda_0(\mathbf{s})\)

Code

pbc_win_sf <- pbc |> sf::st_as_sf() |> filter(label == "window")
plot_pbc_sim <- function() {
    pbc_sim <- spatstat.random::rpoispp(
        lambda = alpha_hat * int_controls
    )
    # Separate window and points
    pbc_sf <- pbc_sim |> sf::st_as_sf()
    pbc_point_sf <- pbc_sf |> filter(label == "point")
    pbc_win_sf <- pbc_sf |> filter(label == "window")
    pbc_plot <- ggplot() +
      geom_sf(data=pbc_win_sf) +
      geom_sf(data=pbc_point_sf, size=0.5, alpha=0.5) +
      theme_classic()
    return(pbc_plot)
}
plot_pbc_sim()

Code

plot_pbc_sim()

Code

plot_pbc_sim()

References

Gimond, Manuel. 2024. Introduction to GIS and Spatial Analysis. Colby College (ES214). https://mgimond.github.io/Spatial/.