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 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 2.1 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( | )CS that a random point in S lies in any cell CS is proportional to the area of C .2 (2.1.1) ()Pr( | )()aCCSaS Finally, since this must hold for any pair of nested regions CRS it follows that3 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)]. 3 Expression (2.1.2) refers to equation (2) in section 2.1. This convention will be followed throughout. 1/4 1/4 1/4 1/4 1/2 1/2 - - - C S Fig. 2.1. Spatial Laplace PrincipleNOTEBOOK FOR SPATIAL DATA ANALYSIS Part I. Spatial Point Pattern Analysis ______________________________________________________________________________________ ________________________________________________________________________ ESE 502 I.2-2 Tony E. Smith (2.1.2) Pr( | ) ( )/ ( )Pr( | ) Pr( | ) Pr( | ) Pr( | )Pr( | ) ( )/ ( )CS aC aSCS CR RS CRRSaRaS   ()Pr( | )()aCCRaR and hence that the square in Figure 2.1 can be replaced by any bounded region, R, in the plane. This fundamental proportionality result, which we designate as the Spatial Laplace Principle, forms the basis for almost 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, CR , it convenient to characterize this event as a Bernoulli (binary) random variable, ()XC , where () 1XC if the point is located in C and ( ) 0XC otherwise. In these terms, it follows from (2.1.2) that the conditional probability of this event (given that the point is located in R) must be (2.1.3) Pr ( ) 1| ( )/ ( )XCRaCaR , so that Pr ( ) 0| 1 Pr ( ) 1| 1 [ ( )/ ( )]XCR XCR aCaR . 2.2 Complete Spatial Randomness In this context, suppose now that n points are each located randomly in region R. Then the second key assumption of spatial randomness is that the locations of these points have no influence on one another. Hence if for each 1,..,in, the Bernoulli variable, ( )iXC , now denotes the event that point i is located in region C , then under spatial randomness the random variables { ( ): 1,.., }iXCi n are assumed to be statistically independent for each region C . This together with the Spatial Laplace Principle above defines the fundamental hypothesis of complete spatial randomness (CSR), which we shall usually refer to as the CSR Hypothesis. Observe next that in terms of the individual variables, ( )iXC , the total number of points appearing in C , designated as the cell count, ( )NC , for C , must be given by the random sum (2.2.1) 1() ()niiNC X C [It is this additive representation of cell counts that in fact motivates the Bernoulli (0-1) characterization of location events above.] Note in particular that since the expectedNOTEBOOK FOR SPATIAL DATA ANALYSIS Part I. Spatial Point Pattern Analysis ______________________________________________________________________________________ ________________________________________________________________________ ESE 502 I.2-3 Tony E. Smith value of a Bernoulli random variable, X, is simply (1)PX,4 it follows (from the linearity of expectations) that the expected number of points in C must be (2.2.2) 11()|, [ ()|] Pr[ () 1|]nniiiiENC nR EXC R XC R 1() ()()() () ()niaC aC nnaCaR aR aR Finally, it follows from expression (2.1.3) that the under the CSR Hypothesis, the sum of independent Bernoulli variables in (2.2.1) is by definition a Binomial random variable with distribution given by (2.2.3) !() ()Pr[ ( ) | , ] 1 , 0,1,..,!( )! ( ) ( )knknaC aCNC k nR k nknk aR aR      For most practical purposes, this conditional cell-count distribution for the number of points in cell, CR (given that n points are