1. Summary of Prev. Lecture
Points and point pattern
Locations and distribution of locations
Mean center: spatial mean
Weighted mean center
Spatial mean considering attributes
Median center
Min. distance center from all points
Standard distance
Average distribution of points from the mean center
Standard deviational ellipses
Spatial distribution of points considering directional bias
GIS and Spatial Analysis

2. Point Pattern Analyzers
Geography KHU Jinmu Choi
Limitations (3)
Scale, extent, projection
Density-based Measurement (6)
Density, Quadrat Count, Kernel-Density Estimation
Distance-based Measurement (10)
Nearest Neighbor Distance, Function G, Function F, Function K
Summary and Next…
GIS and Spatial Analysis

3. GIS and Spatial Analysis
Point Pattern
GIS and Spatial Analysis

4. GIS and Spatial Analysis
Scale & Extent
Different spatial resolution or scales
Changes in spatial resolution
-> Changes in spatial pattern
-> Changes in statistical results
Edge effect
Spatial continuity
GIS and Spatial Analysis

5. Distortion with Projection
All maps contain errors because of the transformation process from the spherical surface to a plane
Distortion types
Tearing
Shearing
Compression
Alterations
Shape
Area
Distance
Direction
GIS and Spatial Analysis

6. Density-based Measure
Overall Intensity of a point pattern
High density -> high trend
Problem: sensitivity
to the definition of area
Quadrat count method
Kernel Density Estimation
GIS and Spatial Analysis

7. GIS and Spatial Analysis
Quadrat Count
Number of events in the pattern in a set of cells (or quadrat)
Size: 2*Area/#of points
Relative frequency table
Many quadrat with many events -> high density
Compared to the expected(x2) distribution
Problem: size of quadrat has an effect on the frequency
GIS and Spatial Analysis

8. GIS and Spatial Analysis
Test Quadrat Count
Chi-square (x2) distribution
Sum-of-square differences by the mean
Null H: the observed distribution is produced by IRP/CSR
Comparison to x2 distribution
DOF = n-1 = 40-1=39
5%: x2 = , 1%: x2 =
0.1%: x2 =
(X2=73) > 0.1% level
Reject Null hypothesis at 0.1%
Less than 0.1% chance
GIS and Spatial Analysis

9. GIS and Spatial Analysis
Assess Quadrat Count
Expected Probability distribution for a quadrat count is binomial (Poisson) distribution
How well an observed distribution fits a Poisson prediction
VMR = variance/mean ratio
n= 47, x = 40, u = 47/40 = 1.175
S2 = /40=2.1444
U=1.175
VMR = /1.175 = 1.825
VMR>1, clustered:
High variability
among quadrat count
If VMR<1, evenly distributed
GIS and Spatial Analysis

10. GIS and Spatial Analysis
Density Estimation
Kernel-density estimation (KDE): local density
Basic idea: pattern has a density at any location, not just at event
Simple or Naive method: intensity at point p
Result: from discrete points
to continuous density
estimate
High density -> high trend
GIS and Spatial Analysis

11. GIS and Spatial Analysis
Advantage of KDE
Good visualization to detect hot spots
Good to show first order stationary (local variations)
No trend in density through study area
Linking various geographic data
Compare death point density surface with air pollutant concentration
Problem: radius (r, bandwidth) affects density surface
Too large r, similar everywhere
Too small r, too many no density locations
Kernel bandwidth have meaning in the context of the study
Crime point density: bandwidth: patrol vehicle response time
GIS and Spatial Analysis

12. Nearest-Neighbor Distance
Distance between events in a point pattern
The second-order property of the pattern
NND: Distance from nearest neighbor event
Mean NND
Clustered: Small Mean D
Drawback: too concise
GIS and Spatial Analysis

13. GIS and Spatial Analysis
Assess NND
Clark and Evans’ R statistic
Expected value for mean NND under IRP/CSR
R: Ratio of observed mean NND to expected value
R<1, clustered
Observed NND shorter than expected
R>1, evenly spaced events
GIS and Spatial Analysis

14. GIS and Spatial Analysis
Distance Function G
G is the simplest distance function
Cumulative frequency distribution of NND
G(d) is a fraction of all NND in the pattern in d
Clustered events
Rapid increase at short d
Evenly spaced events
Rapid at longest d
GIS and Spatial Analysis

15. GIS and Spatial Analysis
Distance Function F
Distance between event and random point
Cumulative frequency distribution of NND
Advantage over G is to increase sample size to m: smoother frequency curve
GIS and Spatial Analysis

16. GIS and Spatial Analysis
Result of F functions
Clustered: rapid G in short distance but rapid F at longer distance
Evenly spaced: rapid G in longer distance but rapid F at short distance
GIS and Spatial Analysis

17. Assess G and F functions
Expected value for G and F under IRP/CSR
How unusual the observed pattern is ?
Expected function is the smooth curve
If clustered,
G: Events are closer together than expected
F: Locations are farther from event than expected
GIS and Spatial Analysis

18. GIS and Spatial Analysis
Distance Function K
Not only NND, use all distances between events
GIS and Spatial Analysis

19. GIS and Spatial Analysis
Interpretation of K
Cluster size ≈0.2, cluster separation ≈0.6
Smooth increase entirely => evenly spaced
GIS and Spatial Analysis

20. GIS and Spatial Analysis
Assess K functions
Expected value for K under IRP/CSR
d2 makes difficult to compare observed to expected
L function makes K(d) to 0
GIS and Spatial Analysis

21. GIS and Spatial Analysis
Assess L function
L(d) > 0, clustered
L(d) > 0 at certain d range
More events at this short d range than expected
L(d) < 0, evenly distributed
L(d) < 0, drop at a certain d range and then sharply rise
Events are evenly distributed at d range
Figure 4.19
GIS and Spatial Analysis

22. GIS and Spatial Analysis
Summary
Density-based measurement
Density, Quadrat Count, KDE
Distance-based measurement
NND, G, F, K (L) functions
GIS and Spatial Analysis

23. GIS and Spatial Analysis
Next …
Lab: Point Pattern Analysis
Final Presentation
Final Review
Final Exam
GIS and Spatial Analysis