1. Spatial Point Pattern Analysis
GRAD6104/8104 INES 8090
Spatial Statistic- Spring 2017
Spatial Point Pattern Analysis

2. Image source: http://kids. nationalgeographic

3. Image source: http://cdn4. lacdn

4. Image source: http://www.airpano.ru/files/flamingo_01_big.jpg

5. What about… Trees? Pedestrian? Cars? … Chinese New Year in China

6. What about… Trees? Pedestrian? Cars? … Traffic in Los Angeles

7. Spatial Point Patterns (SPP)
Spatial characteristics of point patterns
Random? Clustered? Regular?
Spatial scales at which these characteristics are observed

8. Spatial Point Patterns
Points are associated with events of a point process
Sampled point pattern
Events are partially observed
Mapped point pattern
All events of a realization are observed

9. Spatial Point Patterns
Complete Spatial Random (CSR) Pattern
The average number of events per unit area (intensity) is homogeneous throughout the spatial domain
Events are independent with each other

10. Spatial Point Patterns
First-order properties: intensity function λ(s)
Average number of events per unit area
where ds: an infinitesimal area at event s
N(ds): # of events in an arbitrary region
Second-order properties

11. Spatial Point Patterns
Indicator function h h

12. Spatial Point Patterns
Homogeneous Poisson process (HPP) is the stochastic representation of Complete Spatial Randomness.
Statistical test is conducted to compare an observed point pattern against that expected from a HPP

13. Homogeneous Poisson process
Attributes
Stationary and isotropic
Intensity = λ
λ2(s,t) = λ2

14. Homogeneous Poisson process
Computer Simulation (conditioned on n events in D)
For a unit square, generate 2 independent uniform random variates
Pair them up to get the coordinates of a single event
Repeat this independently n times
For the scale of a rectangle, rescale coordinates
For an irregularly shaped study area D
Simulate on a rectangle area containing D
Retain only those events that lie within D

15. Exploratory Analysis First-order properties Second-order properties
Kernel Density Estimation
Quadrat analysis
Second-order properties
Nearest neighbor distance
Ripley’s K function

16. Exploratory Analysis Area-based Approaches Distance-based Approaches
Quadrat analysis
Distance-based Approaches
Kernel Density Estimation
Nearest neighbor distance
Ripley’s K function

17. Exploratory Analysis Kernel Density Estimation h
Figure 4 a,b from Delmelle et al. 2014
Delmelle, E. M., Zhu, H., Tang, W., & Casas, I. (2014). A web-based geospatial toolkit for the monitoring of dengue fever. Applied Geography, 52,

18. Exploratory Analysis Kernel Density Estimation
Estimation of first-order property of point patterns
Nonparametric approach
where
h: bandwidth
I(.): indicator function
k(.): kernel weight function h h
Figure 1 from Gatrell et al. 1996)
Gatrell, A. C., Bailey, T. C., Diggle, P. J., & Rowlingson, B. S. (1996). Spatial point pattern analysis and its application in geographical epidemiology. Transactions of the Institute of British geographers,

19. Exploratory Analysis Kernel Density Estimation Kernel weight function
Uniform
Gaussian
Epanecknikov (1969) …
Effect of bandwidth Selection h
Source:

20. Exploratory Analysis Kernel Density Estimation Kernel weight function
Uniform
Gaussian
Epanecknikov (1969) …
Effect of bandwidth Selection h
Source:

21. Exploratory Analysis Kernel Density Estimation Kernel weight function
Uniform
Gaussian
Epanecknikov (1969) …
Effect of bandwidth Selection h
Source:

22. Quadrat Analysis
Divide the spatial domain D into non-overlapping regions (quadrats) of equal size: r by c quadrats
Count events within each quadrat for the SPP with n events
Expected #events per quadrat
n/(r*c)

23. Quadrat Analysis Chi-square statistic for goodness-of-fit test
A.k.a.: Index of dispersion
The distribution of X2 is χ2(rc-1)
Provided that is not too small, say ≥ 5
Obtain Pr(X2(rc-1) ≤ X2)
In R, use pchisq(q,df) {q = X2; df=rc-1}
The test is two-sided
Large X2: aggregation
Small X2: regularity

24. Quadrat Analysis
Example from the textbook

25. Quadrat Analysis Criticisms Insensitive to regular departures from CSR
Conclusion may depend on quadrat size and shape
MAUP (Modifiable Areal Unit Problem)
Too much information is lost by reducing the pattern to areal counts

26. Quadrat Analysis Scale analysis: Vary quadrat size
Plot X2 against block size
Peaks or toughs: evidence of scales of pattern

27. Distance-based Methods
Clark-Evans test
Diggle’s refined NN analysis
Ripley’s K function

28. Distance-based Methods
Clark-Evans test (Clark and Evans, 1954)
Based on the mean nearest-neighbor (NN) distance:
Small: Aggregation
Large: Regularity
Clark, P. J., & Evans, F. C. (1954). Distance to nearest neighbor as a measure of spatial relationships in populations. Ecology,

29. Distance-based Methods
Clark-Evans test (Clark and Evans, 1954)
Test statistic:
where λ = n/|D| D: area
Under CSR, the distribution of CE: N(0,1)
Assume edge and overlap effects are ignored
Powerful for detecting aggregation and regularity
Weak at detecting heterogeneity
Clark, P. J., & Evans, F. C. (1954). Distance to nearest neighbor as a measure of spatial relationships in populations. Ecology,

30. Distance-based Methods
Diggle’s refined NN analysis
A test based on the entire empirical distribution function (EDF) of the NN distances
where I(.) is indicator function
G(h)= 1 - exp(-λπy2), y ≥ 0
Plot of Ghat(.) against G(.) is a straight line under independence
Ghat(h) > G(h) for small h: aggregation
Ghat(h) < G(h) for small h: regularity

31. Distance-based Methods
Diggle’s refined NN analysis
A test based on the entire empirical distribution function (EDF) of the NN distances
Significance testing: Monte Carlo test
#runs (recommended): 99 for 5%; for 1%
Simulation envelopes
Indication of the distance of departure from CSR

32. Distance-based Methods
Diggle’s refined NN analysis
Figure 3.2 Location of lightning strikes

33. Distance-based Methods
Diggle’s refined NN analysis
Similarly, tests using point-to-nearest event distances can be applied
Sample points (m) randomly or systematically generated. F

34. Distance-based Methods
Diggle’s refined NN analysis
Both and are Diggle’s refined NN analysis F

35. Distance-based Methods
Ripley’s K Function
Second-moment cumulative function
For a HPP, K(h)=πh2 and
L(h) < 0 (K(h) > πh2 ) for small h: aggregation
L(h) > 0 (K(h) < πh2 ) for small h: regularity
Note that L(h) function may be defined differently

36. Distance-based Methods
Ripley’s K Function
Estimator
where A: the area of study area
n: #of events
Iij(d): indicator function (1 if dij<d; otherwise 0)
wij: edge-corrected weights

37. Distance-based Methods
Ripley’s K Function
Estimator
Correction of edge effect
Ripley’s circumference
Toroidal shift
Guard

38. Distance-based Methods
Ripley’s K Function
Estimator
Correction of edge effect
Ripley’s circumference
Toroidal shift
Guard

39. Distance-based Methods
Ripley’s K Function
Estimator
Correction of edge effect
Ripley’s circumference
Toroidal shift
Guard (Inner|Outer)

40. Distance-based Methods
Ripley’s K Function
Estimator
Significance test
Monte Carlo
Source:

41. Reading Assignment
Gatrell, A. C., Bailey, T. C., Diggle, P. J., & Rowlingson, B. S. (1996). Spatial point pattern analysis and its application in geographical epidemiology. Transactions of the Institute of British geographers,
Chapter 2: Preliminary testing for mapped patterns, by Diggle (1983): Statistical Analysis of Spatial Point Patterns, Academic Press, London