3_Testing_Spatial_Randomness

# 3_Testing_Spatial_Randomness - NOTEBOOK FOR SPATIAL DATA...

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NOTEBOOK FOR SPATIAL DATA ANALYSIS Part I. Spatial Point Pattern Analysis ______________________________________________________________________________________ ________________________________________________________________________ ESE 502 I.3-1 Tony E. Smith 3. Testing Spatial Randomness There are at least three approaches to testing the CSR Hypothesis : the quadrat method , the nearest-neighbor method , and the method of K-functions . We shall consider each of these in turn. 3.1 Quadrat Method This simple method is essentially a direct test of the CSR Hypothesis as stated in expression 2.1(6). 1 Given a realized point pattern from a point process in a rectangular region, R , one begins by partitioning R it into congruent rectangular subcells (quadrats) 1 ,.., m CC as in Figure 1 below (where 16 m = ). Then, regardless of whether the given pattern represents trees in a forest or beetles in a field, the CSR Hypothesis asserts that the cell-count distribution for each i C must be the same, as given by 2.1(6). But rather than use this Binomial distribution, it is typically assumed that R is large enough to use the Poisson approximation in 2.1(9). In the present case, if there are n points in R , and if we let 1 () aa C = , and estimate expected point density λ by (1) ˆ n aR λ= then this common Poisson cell-count distribution has the form (2) ˆ ˆ ˆ Pr[ | ] , 0,1,2,. .. ! k a i a Nk e k k −λ λ = = Moreover, since the CSR Hypothesis also implies that each of the cell counts, ( ), 1,. ., ii NN C i k == , is independent , it follows that ( ) : 1,. ., i Ni k = must be a independent random samples from this Poisson distribution. Hence the simplest test of 1 This refers to expression (6) in section 2.1 of Part I. All other references will follow this convention. Fig. 1. Quadrat Partition of R

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NOTEBOOK FOR SPATIAL DATA ANALYSIS Part I. Spatial Point Pattern Analysis ______________________________________________________________________________________ ________________________________________________________________________ ESE 502 I.3-2 Tony E. Smith this hypothesis is to use the Pearson 2 χ goodness-of-fit test. Here the expected number of points in each cell is given by the mean of the Poisson above, which (recalling that () / aaRm = by construction) is (3) ˆˆ (|) nn EN a a aR m λ= ⋅ = Hence if the observed value of i N is denoted by i n , then the chi-square statistic (4) 2 2 1 (/ ) / n i i nnm nm = χ= is known to be asymptotically chi-square distributed with 1 m degrees of freedom, under the CSR Hypothesis. Thus one can test this hypothesis directly in these terms. Note that / is simply the sample mean , i.e., 1 /( 1 / ) m i i m n n = = = , and hence that this statistic can also be written as (5) 22 2 1 (1 ) n i i s m = = − where 1 1 1 m i i m sn n = =− is the sample variance . In this form, observe that since the variance if the Poisson distribution is exactly equal to its mean, i.e., var( ) / ( ) 1 NEN = , and since 2 / is the sample estimate of this ratio, if is often convenient to use this index of dispersion , 2 / , as the test statistic. In particular, if 2 / is significantly less than one then it can be inferred that there is too little variation among quadrat counts, suggesting possible “uniformity” rather than randomness. Similarly, if 2 / is significantly greater than one then there is too much variation among counts, suggesting possible “clustering” rather than randomness.
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3_Testing_Spatial_Randomness - NOTEBOOK FOR SPATIAL DATA...

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