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

2. Multivariate Point Pattern Analysis

3. Analysis of Multivariate Point Patterns
Bramble Canes
(dataset from spatstat)
Nests of two ant species (Greece)

4. Analysis of Multivariate Point Patterns
Nests of two ant species (Greece)
Nests of two ant species (Greece)
(dataset from spatstat)

5. Analysis of Multivariate Point Patterns
Each event can be classified into one of a finite number of categories
Marked
Nests of two ant species (Greece)
Pine
Maple
Oak
Nests of two ant species (Greece)

6. Analysis of Multivariate Point Patterns
Each event can be classified into one of a finite number of categories
Marked
Pine
Nests of two ant species (Greece)
Maple
Oak

7. Analysis of Multivariate Point Patterns
Bivariate point pattern
Pattern of Type 1 events only
Pattern of Type 2 events only
Combined pattern of intermingled Type 1 and Type 2 events
Interrelation between Type 1 and Type 2 events
Pine
Maple

8. Analysis of Multivariate Point Patterns
Bivariate point pattern
Interrelation (association) between Type 1 and Type 2 events
Attraction
Independence
Repulsion
Pine
Maple

9. Bivariate Point Pattern
Models
Linking univariate models independently or with either positive or negative dependence
Bivariate Poisson process
Mutual inhibition process

10. Bivariate Point Pattern
Models
Bivariate Poisson process
Each process is an HPP (Homogeneous Poisson Process)
Independence model: superposition of the two HPPs
Positive dependence
Displacing each Type 2 event randomly about a Type 1 event, according to a radially symmetric bivariate distribution with model (0,0), i.e.,
Type 1 event =parent
Type 2 event = offspring

11. Bivariate Point Pattern
Models
Mutual inhibition process
Events of each type are generated in an alternating sequence over A.
At each stage, the next event of a given type is realized from a uniform distribution over that portion of D that is at least distance δ away from any previously realized events of the opposite type δ

12. Theoretical Concepts and Terms
Intensities: λ1, λ2
Second-order intensity function λij= λ2(si,sj)
Note that λ12 = λ21

13. Theoretical Concepts and Terms
Second-moment cumulative functions
Large K12(t): Positive association
Small K12(t): Negative association
K12(t)=K21(t)

14. Theoretical Concepts and Terms
Cdf’s
Gij(y)= cdf of distance to nearest Type j event from an arbitrary Type i event
Fj(x) = cdf of distance to nearest Type j event from an arbitrary point
G12(y) <> G21(y) and F1(x) <> F2(x)

15. Theoretical Concepts and Terms
Under independence
K12(t) = K21(t) = πt2
F1(x) = G21(x)
F2(x) = G12(x)

16. Testing for Independence
Random Labelling Hypothesis
Locations arise from a univariate spatial point process
A second random mechanism determines the marks

17. Testing for Independence
Bivariate Spatial Point Patterns
Quadrat Methods
Distance Methods
Nearest Neighborhood Distance
Ripley’s K

18. Testing for Independence
Bivariate Spatial Point Patterns
Quadrat approach
Presence-absence table (Greig-Smith, 1983, 3rd edition)
{(ni1,ni2)} where nij: #of type j event in quadrat i
Present
Absent a b
a+b c d
C+d
a+c
b+d m
m=a+b+c+d
m=a+b+c+d
Greig-Smith, P. (1983). Quantitative plant ecology (Vol. 9). Univ of California Press.

19. Testing for Independence
Bivariate Spatial Point Patterns
Quadrat approach
Presence-absence table (Greig-Smith, 1983, 3rd edition)
Chi-square test
For large m, under independence
X2 ~ χ12
Present
Absent a b
a+b c d
c+d
a+c
b+d m
m=a+b+c+d
m=a+b+c+d
Greig-Smith, P. (1983). Quantitative plant ecology (Vol. 9). Univ of California Press.

20. Testing for Independence
Bivariate Spatial Point Patterns
Quadrat approach
Correlation coefficient
r(ni1,ni2) where nij: #of type j event in quadrat i
R > 0: Attraction
R < 0 : Repulsion
Test for significance
m=a+b+c+d
Greig-Smith, P. (1983). Quantitative plant ecology (Vol. 9). Univ of California Press.

21. Testing for Independence
Bivariate Spatial Point Patterns
Nearest Neighborhood Distance
Nearest-neighbor table (Pielou, 1961)
Comparison of point-to-nearest-event and NN distance distributions (Goodall, 1965)
Correlation of paired point-to-nearest event distances (Diggle and Cox, 1983)

22. Testing for Independence
Bivariate Spatial Point Patterns
Nearest Neighborhood Distance
Nearest-neighbor table (Pielou, 1961)
NN table records the information of frequency for the type of the NN from each of the events (type 1 and type 2)
Significance testing is similar to presence-absence table for quadrat approach NN
Type 1
Type 2
Original Type 1 a b
a+b
Original Type 2 c d
c+d
a+c
b+d m

23. Testing for Independence
Bivariate Spatial Point Patterns
Nearest Neighborhood Distance
Comparison of point-to-nearest-event and NN distance distributions (Goodall, 1965)
Under independence
F1(x) = G21(x) and F2(x) = G12(x)
Compare sample means of the distances
t-test (t.test() in R)
A two-sample Kolmogorov-Smirnov type of statistic
ks.test() in R

24. Testing for Independence
Bivariate Spatial Point Patterns
Nearest Neighborhood Distance
Correlation of paired point-to-nearest event distances (Diggle and Cox, 1983)
Distances from arbitrary points to the nearest events of each type, Xi1, Xi2
Under independence, the correlation between Xi1 and Xi2 is o.
Positive correlation: attraction
Negative correlation: repulsion

25. Testing for Independence
Bivariate Spatial Point Patterns
Nearest Neighborhood Distance
Correlation of paired point-to-nearest event distances (Diggle and Cox, 1983)
Distances from arbitrary points to the nearest events of each type, Xi1, Xi2
Rank-based correlation could be used
Spearman’s rank correlation
Kendall’s tau
Rank
Type 1
Type 2 2 5 6 3 4 1
Rank

26. Testing for Independence
Bivariate Spatial Point Patterns
Ripley’s K
where w(sim-sjl): the proportion of the circumference of a circle within A that passes through sjl and is centered at sim.

27. Testing for Independence
Bivariate Spatial Point Patterns
Ripley’s K
Oftentime, we could use L function.
L12(t) < 0 : repulsion at that t
L12(t) > 0 : attraction at that t
Testing: similar to univariate K function
Simulation envelope

28. Toroidal Shifts
Monte Carlo testing approach that preserves the observed patterns of each type
Perturb each event of one type a random amount (δx, δy )

29. Rotate
Monte Carlo testing approach that preserves the observed patterns of each type
Perturb a random direction (choice of origins)

30. Reading Assignment Chapter 4 by Schabenberger and Gotway (2005)
Semivariogram and Covariance Function Analysis and Estimation