3_Testing_Spatial_Randomness

3_Testing_Spatial_Randomness - NOTEBOOK FOR SPATIAL DATA...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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.
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 27

3_Testing_Spatial_Randomness - NOTEBOOK FOR SPATIAL DATA...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online