Point Pattern Analysis

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 (?)
• -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