1. Point Pattern Analysis
Chapter 4
Geographic Information Analysis
By David O’ Sullivan and David J. Unwin

2. Introduction to Point Pattern Analysis
Simplest Possible Spatial Data
-A point pattern is a set of events in a study region
-Each event is symbolized by a point object
-Data are the locations of a set of point objects
Applications
-Hot-spot analysis (crime, disease)
-Vegetation, archaeological studies

3. Introduction to Point Pattern Analysis
Requirements for a set of events to constitute a point pattern
-Pattern should be mapped on a plane
-Study area determined objectively
-Pattern is a census of the entities of interest
-One-to-one correspondence between objects and
events
-Event locations are proper

4. Introduction to Point Pattern Analysis
Describing a point pattern
Point Density
-First-order effect: Variation
of intensity of a process
across space
-Number of events per unit
area
-Absolute location
Point Separation
-Second-order effect:
Interaction between
locations based on distance
between them
-Relative location

5. Introduction to Point Pattern Analysis
Descriptive statistics to provide summary descriptions of point patterns
-Mean center
-Standard Distance

6. Density-Based Point Pattern Measures
First-order effect
Sensitive to the definition of the study area

7. Density-Based Point Pattern Measures
Quadrant count methods
-Record number of events of a pattern in a set of
cells of a fixed size
-Census vs. Random

8. Density-Based Point Pattern Measures
Kernel-density estimation
-Pattern has a density at any location in the study
region
-Good for hot-spot analysis, checking first-order
stationary process, and linking point objects to
other geographic data
Naive method

9. Distance-Based Point Pattern Measures
Second-order effect
Nearest-neighbor distance
-The distance from an event to the nearest event in
the point pattern
Mean nearest-neighbor distance
-Summarizes all the nearest-neighbor distances by a
single mean value
-Throws away much of the information about the
pattern

10. Distance-Based Point Pattern Measures
G function
-Simplest
-Examines the cumulative frequency distribution of
the nearest-neighbor distances
-The value of G for any distance tells you what
fraction of all the nearest-neighbor distances in the
pattern are less than that distance

11. Distance-Based Point Pattern Measures
F function
-Point locations are selected at random in the study
region and minimum distance from point location to
event is determined
-The F function is the cumulative frequency
distribution
-Advantage over G function: Increased sample size
for smoother curve

12. Distance-Based Point Pattern Measures
K function
-Based on all distances between events
-Provides the most information about the pattern

13. Distance-Based Point Pattern Measures
Problem with all distance functions are edge effects
Solution is to implement a guard zone

14. Assessing Point Patterns Statistically
Null hypothesis
-A particular spatial process produced the observed
pattern (IRP/CSR)
Sample
-A set of spatial data from the set of all possible
realizations of the hypothesized process
Testing
-Using a test to illustrate how probable an observed
value of a pattern is relative to the distribution of values
in a sampling distribution

15. Assessing Point Patterns Statistically

16. Assessing Point Patterns Statistically
Quadrant counts
-Probability distribution for a quadrant count
description of a point pattern is given by a Poisson
distribution
-Null hypothesis: (IRP/CSR)
-Test statistic: Intensity (λ)
-Tests: Variance/mean ratio, Chi-square
Nearest-neighbor distances
-R statistic

17. Assessing Point Patterns Statistically
G and F functions
-Plot observed pattern and IRP/CSR pattern

18. Assessing Point Patterns Statistically
K function
-Difficult to see small differences between expected
and observed patterns when plotted
-Develop another function L(d) that should equal
zero if K(d) is IRP/CSR
-Use computer simulations to generate IRP/CSR
(Monte Carlo procedure)

19. Critiques of Spatial Statistical Analysis
Peter Gould
-Geographical data sets are not samples
-Geographical data are not random
-Geographical data are not independent random
-n is always large so results are almost always
statistically significant
-A null hypothesis of IRP/CSR being rejected means
any other process is the alternative hypothesis
David Harvey
-Altering parameter estimates by changing study
region size often can alter conclusions