What is a point pattern? For a specified, bounded region D , a set of locations s i , i = 1 , 2 ..., n Spatial Point Patterns – p. 1/24
What is a point pattern? For a specified, bounded region D , a set of locations s i , i = 1 , 2 ..., n The locations are viewed as “random” Spatial Point Patterns – p. 1/24
What is a point pattern? For a specified, bounded region D , a set of locations s i , i = 1 , 2 ..., n The locations are viewed as “random” Need not have variables at locations, just the pattern of points Spatial Point Patterns – p. 1/24
What is a point pattern? For a specified, bounded region D , a set of locations s i , i = 1 , 2 ..., n The locations are viewed as “random” Need not have variables at locations, just the pattern of points Crude features of patterns, e.g., complete randomness, clustering/attraction, inhibition/repulsion, regular/systematic Spatial Point Patterns – p. 1/24
What is a point pattern? For a specified, bounded region D , a set of locations s i , i = 1 , 2 ..., n The locations are viewed as “random” Need not have variables at locations, just the pattern of points Crude features of patterns, e.g., complete randomness, clustering/attraction, inhibition/repulsion, regular/systematic Can add “marks”, i.e., labels. Then, a point pattern for each mark; comparison of patterns Spatial Point Patterns – p. 1/24
spatial homogeneity 0.8 0.8 0.8 v v v 0.4 0.4 0.4 0.0 0.0 0.0 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 u u u 0.8 0.8 0.8 v v v 0.4 0.4 0.4 0.0 0.0 0.0 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 u u u Spatial Point Patterns – p. 2/24
cluster pattern; systematic pattern Clustered Clustered Clustered 0.8 0.8 0.8 v v v 0.4 0.4 0.4 0.0 0.0 0.0 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 u u u Regular Regular Regular 0.8 0.8 0.8 v v v 0.4 0.4 0.4 0.0 0.0 0.0 0.0 0.4 0.8 0.0 0.4 0.8 0.0 0.4 0.8 u u u Spatial Point Patterns – p. 3/24
spatial heterogeneity 20 15 lambda 10 v 5 v u 0 0 5 10 15 20 u Spatial Point Patterns – p. 4/24
spatial heterogeneity 20 20 20 15 15 15 10 10 10 v v v 5 5 5 0 0 0 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 u u u 20 20 20 15 15 15 10 10 10 v v v 5 5 5 0 0 0 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 u u u Spatial Point Patterns – p. 5/24
Examples pattern of trees in a forest, say junipers and pinions Spatial Point Patterns – p. 6/24
Examples pattern of trees in a forest, say junipers and pinions pattern of disease cases, perhaps cases and controls Spatial Point Patterns – p. 6/24
Examples pattern of trees in a forest, say junipers and pinions pattern of disease cases, perhaps cases and controls breast cancer cases; treatment option - mastectomy or radiation Spatial Point Patterns – p. 6/24
Examples pattern of trees in a forest, say junipers and pinions pattern of disease cases, perhaps cases and controls breast cancer cases; treatment option - mastectomy or radiation perhaps over time, single family homes; urban development Spatial Point Patterns – p. 6/24
Examples pattern of trees in a forest, say junipers and pinions pattern of disease cases, perhaps cases and controls breast cancer cases; treatment option - mastectomy or radiation perhaps over time, single family homes; urban development again over time, bovine tuberculosis Spatial Point Patterns – p. 6/24
Distributions N ( B ) is number of points in set B N ( B ) ∼ ?, driven by intensity surface λ ( s ) yields a Poisson Process N ( B ) ∼ Po ( λ ( B ) where λ ( B ) = � B λ ( s ) ds λ ( s ) = λ - homogeneous Poisson process, spatial homogeneity, complete spatial randomness (csr), λ ( B ) = λ | B | λ ( s ) nonconstant, fixed - nonhomogeneous Poisson process λ ( s ) random - Cox process more to follow Spatial Point Patterns – p. 7/24
Counting Measure Again, for any set B ⊂ D , let N ( B ) count the number of points in B Given for all B ’s, we call a counting measure Counting measure is equivalent to a point pattern If point pattern is random, then N ( B ) ’s are random Model for the uncountable collection of sets? Spatial Point Patterns – p. 8/24
Exploratory data analysis All directed at checking/criticizing the assumption of complete spatial randomness Spatial Point Patterns – p. 9/24
Exploratory data analysis All directed at checking/criticizing the assumption of complete spatial randomness Crude - Partition D in cells of equal area (need not exhaust D ). Compute ¯ N , the mean of the cell counts. Compute S 2 N , the sample variance of the counts. Look N / ¯ at S 2 N . Spatial Point Patterns – p. 9/24
Exploratory data analysis All directed at checking/criticizing the assumption of complete spatial randomness Crude - Partition D in cells of equal area (need not exhaust D ). Compute ¯ N , the mean of the cell counts. Compute S 2 N , the sample variance of the counts. Look N / ¯ at S 2 N . Can extend to a standard χ 2 test treating the cell counts as i.i.d. Poisson random variables under csr Spatial Point Patterns – p. 9/24
F and G functions Distance-based methods: G ( d ) is nearest neighbor distance, event to event, i.e., G ( d ) = Pr ( nearest event ≤ d ) F ( d ) is nearest neighbor distance point to event, i.e., F ( d ) = Pr ( nearest event ≤ d ) Under csr G ( d ) = F ( d ) = 1 − exp ( − λπd 2 ) ˆ G is the empirical c.d.f. of the n nearest neighbor distances (nearest neighbor distance for s 1 , for s 2 , etc. ˆ F is the empirical c.d.f. arising from the m nearest neighbor distances associated with a randomly selected set of m points in D . m is arbitrary and ˆ G � = ˆ F Edge correction if d > b i , distance from s i to edge of D compare ˆ G with G , ˆ F with F - theoretical Q-Q plot. Shorter tails - clustering/attraction, longer tails - inhibition/repulsion Spatial Point Patterns – p. 10/24
The K function The K function considers the number of points within distance d of an arbitrary point Under complete spatial randomness, expected number is the same for any point K is easy to interpret, easy to estimate Formally, K ( d ) = ( λ ) − 1 E (# of points within d of an arbitrary point) Spatial Point Patterns – p. 11/24
cont. Under complete spatial randomness, K ( d ) = λπd 2 /λ = πd 2 (area of circle of radius d ) K ( d ) = ( n ˆ ˆ λ ) − 1 � i r i j ( w ij ) − 1 I ( || s i − s j || ≤ d ) . where r i = � r i is the number of s j within d of s i ˆ λ = n/ | D | w ij is an edge correction, the proportion of the circumference of the circle centered at s i with radius || s i − s j || within D Compare ˆ K ( d ) with K ( d ) = πd 2 Regularity/inhibition implies K ( d ) < πd 2 , clustering implies K ( d ) > πd 2 Spatial Point Patterns – p. 12/24
Estimating the intensity Crude approach: Imagine a refined grid over D . Then λ ( ∂s ) = � ∂s λ ( s ) ds ≈ λ ( s ) | ∂s | So, for grid cell A l , assume λ is constant over A l and estimate with N ( A l ) / | A l | . Two dimensional step function, tile function; like a two-dimensional histogram Spatial Point Patterns – p. 13/24
cont. More sophisticated: A kernel intensity estimate (like a kernel density estimate) ˆ h ( || s − s i || /τ ) /τ 2 , s ∈ D � λ τ ( s ) = i h is a radially symmetric bivariate pdf (usually a bivariate normal), λ ). And, τ 2 not τ is “bandwidth” (controls smoothness of ˆ τ since we are in 2-dim space not 1-dim space Don’t divide by n ; we cumulate intensity Spatial Point Patterns – p. 14/24
Generating point patterns Generating a realization from a HPP with intensity λ Sample n ∼ Po ( λ | D | ) . Given n , sample n locations uniformly over D Generating a realization from a NHPP given λ ( s ) Compute λ max = max s ∈ D λ ( s ) . Sample n ∼ Po ( λ max | D | ) . Given n , sample n locations uniformly over D . “Thin” by retaining s i with probability λ ( s i ) /λ max . (Rejection method) � Easier than computing λ ( D ) = D λ ( s ) ds to draw n and then trying to sample λ ( s ) /λ ( D ) Spatial Point Patterns – p. 15/24
NHPP likelihood Two views Given N ( D ) = n , i λ ( s i ) / ( λ ( D )) n f ( s 1 , s 2 , ...s n | N ( D ) = n ) = � So, “joint density”, f ( s 1 , s 2 , ...s n , N ( D ) = n ) = i λ ( s i ) / ( λ ( D )) n ( λ ( D )) n exp ( − λ ( D )) /n ! � “likelihood” becomes � λ ( s i ) exp ( − λ ( D )) L ( λ ( s ) , s ∈ D ; s 1 , ...s n ) = i Alternatively, partition D into a fine grid. From the Poisson assumption, the likelihood will be a product l exp ( − λ ( A l ))( λ ( A l )) N ( A l ) . over the grid cells, i.e. � Product of the exponential terms is exp ( − λ ( D )) , regardless of the grid. As the grid becomes finer, N ( A l ) = 1 or 0 if s i in A l or not Spatial Point Patterns – p. 16/24
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