2_Models_of_Spatial_ Randomness

2_Models_of_Spatial_ Randomness - NOTEBOOK FOR SPATIAL DATA...

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NOTEBOOK FOR SPATIAL DATA ANALYSIS Part I. Spatial Point Pattern Analysis ______________________________________________________________________________________ ________________________________________________________________________ ESE 502 I.2-1 Tony E. Smith 2. Models of Spatial Randomness As with most all statistical analyses, cluster analysis of point patterns begins by asking: What would point patterns look like if points were randomly distributed ? This requires a statistical model of randomly located points. 2.1 Spatial Laplace Principle To develop such a model, we begin by considering a square region, S , on the plane and divide it in half, as shown on the left in Figure 1 below: The Laplace Principle of probability theory asserts that if there is no information to indicate that either of two events is more likely, then they should be treated as equally likely, i.e., as having the same probability of occuring. 1 Hence by applying this principle to the case of a randomly located point in square, S , there is no reason to believe that this point is more likely to appear in either left half or the (identical) right half. So these two (mutually exclusive and collectively exhaustive) events should have the same probability, 1/2, as shown in the figure. But if these halves are in turn divided into equal quarters, then the same argument shows that each of these four “occupancy” events should have probability 1/4. If we continue in this way, then the square can be divided into a large number of n grid cells, each with the same probability, 1 n , of containing the point. Now for any subregion (or cell ), CS , the probability that C will contain this point is at least as large as the sum of probabilities of all grid cells inside C , and similarly is no greater that the sum of probabilities of all cells that intersect C . Hence by allowing n to become arbitrarily large, it is evident that these two sums will converge to the same limit – namely the fractional area of S inside C . Hence the probability, Pr( | ) that a random point in S lies in any cell is proportional to the area of C . 2 (1) () Pr( | ) aC aS = Finally, since this must hold for any pair of nested regions CRS ⊂⊂ it follows that 1 This is also known as Laplace’s “Principle of Insufficient Reason”. 2 This argument in fact simply repeats the construction of area itself in terms of Riemann sums (as for example in Bartle (1975, section 24). 1/4 1/4 1/4 1/4 1/2 1/2 C S Fig. 1. Spatial Laplace Principle
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NOTEBOOK FOR SPATIAL DATA ANALYSIS Part I. Spatial Point Pattern Analysis ______________________________________________________________________________________ ________________________________________________________________________ ESE 502 I.2-2 Tony E. Smith (2) Pr( | ) ( )/ ( ) Pr( | ) Pr( | ) Pr( | ) Pr( | ) Pr( | ) ( )/ ( ) CS aC aS CR RS R Sa R a S =⋅⇒ = = () Pr( | ) aC aR ⇒= and hence that the square in (2) can be replace by any bounded region , R , in the plane. This fundamental proportionality result, which we designate as the Spatial Laplace Principle , forms the basis for all models of spatial randomness. In probability terms, this principle induces a uniform probability distribution on R , describing the location of a single random point. With respect to any given cell, , it convenient to characterize this event as a Bernoulli ( binary ) random variable , X C , where () 1 XC = if the point is located in C and ( ) 0 = otherwise. In these terms, it
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This note was uploaded on 09/14/2011 for the course ESE 502 taught by Professor Ese502 during the Fall '08 term at UPenn.

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2_Models_of_Spatial_ Randomness - NOTEBOOK FOR SPATIAL DATA...

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