NOTEBOOK FOR SPATIAL DATA ANALYSIS Part II. Continuous Spatial Data Analysis ______________________________________________________________________________________ ________________________________________________________________________ ESE 502 A2-1 Tony E. Smith APPENDIX TO PART II This Appendix, designated as A2, contains additional analytical results for Part II of the NOTEBOOK, and follows the notational conventions in Appendix A1. A2.1. Covariograms for Sums of Independent Spatial Processes First recall that the covariance of any random variables, 1Z and 2Z, with respective means, 1 and 2, is given by (A2.1.1) 1 2 1 1 2 2 12 12 12 12cov( , ) [( )( )] ( )ZZ EZ Z EZZ Z Z   12 1 2 1 2 12()() ()EZZ EZ EZ 12 12 12 12()EZZ 12 12()EZZ so that if 1Z and 2Z are independent then (A2.1.2) 12 1 2 12 1 2()()() cov(,)0EZZ EZ EZ Z Z Hence if a given covariance stationary stochastic process,{(): }Ys s R, with mean, , is the sum of two independent covariance stationary components (A2.1.3) 12() () (),Ys Ys Y s s R  , with respective means, 1 and 2, then it follows by definition that 12, and that 1()Ys and 2()Yv are independent for all ,sv R. Hence for any 0h  and ,sv R with sv h, we see that the covariogram, C , of the Y-process must satisfy, (A2.1.1) () cov[(), ()]Ch Ys Yv 2[() ()] [()] [()] [() ()]EY s Y v EY s EY v EY s Y v  21212 12() () () () ( )EYsYsYvYv  11 1 2 21 2 2[()() () () ()() ()()]EY sY v Y sY v Y sY v Y sY v  221122(2 )  NOTEBOOK FOR SPATIAL DATA ANALYSIS Part II. Continuous Spatial Data Analysis ______________________________________________________________________________________ ________________________________________________________________________ ESE 502 A2-2 Tony E. Smith 11 1 2 2 1 2 2[()()] [()][ ()] [ ()][()] [ ()()]EY sY v EY s EY v EY s EY v EY sY v 22112212()   11 1221 2 2[()()] [() ()]EY sY v EY sY v 22112212()   2211 1 2 2 2[()()] [() ()]EY sY v EY sY v  11 2 2cov[ ( ), ( )] cov[ ( ), ( )]YsYv YsYv 12() ()Ch Ch where 1C and 2C are the respective covariograms for the 1Y and 2Y components of Y . A2.2. Expectation of the Sample Covariance Estimator under Spatial Dependence Given any collection of 2n jointly distributed random variables, {12( , ) , 1,..,iiYY i n } where the pairs 12(, )iiYY have common means 112 2() ,( )iiEY EY and covariance 12 12cov( , )iiYY for all 1,..,in , consider the following estimator of 12, (A2.2.1) 12 1 1 2 2111ˆ()( )niiinYYYY where 11,1,2njjiinYYj. Here 12ˆ and 12 are taken to correspond to the estimator ˆ()Ch of the covariance ()Ch in expressions (4.10.2) and (4.10.1), respectively. To analyze this estimator, it is convenient to begin with the rescaled version (A2.2.2) 12 12 1 1 2 2111ˆ()( )niiinnnYYYY  and recall the following standard decomposition of sums of squares: (A2.2.3) 12 12 12 12 1211()nii i iinYY YY YY YY  12 1 2 1 2 1211 111 1 1nn nii i iii inn n nYY Y Y Y Y n YY   1 2 12 12 1211niiinY Y YY YY YY 12 12111niiinYY YYNOTEBOOK FOR SPATIAL DATA ANALYSIS Part II. Continuous Spatial Data Analysis ______________________________________________________________________________________ ________________________________________________________________________ ESE 502 A2-3 Tony E. Smith But since (A2.2.4) 12 1 21111nniiiinnYY Y Y212111nnijijnYY it follows from (A2.2.2) through (A2.2.4) that (A2.2.5) 12 12 1 2 1 21111ˆ() () ( ) ( )niiinnnnnEEEYYEYY  212 12111111() ( )nnnii i jiijnnnnEYY EYY 2212 12 121111111() () ( )nnnii ii i jiiijinnnnnEYY EYY EYY 2212 1211111() ( )nnii i jiijinnnnnEYY EYY 12 121111(1)() ( )nnii i jiijinnnEYY EYY Finally, if we let ( ), 1,2,jjiEY jthen since by definition, 12 12 12()iiEYY and 12 1 2 12()cov(,)ii i jEYY Y Y both hold for all 1,..,in and ji, it follows from (A2.2.5) that (A2.2.6) 12 12 1 2 1 2 1 2111(1)ˆ() ( ) cov(,)nnijijnnnnEYY   12 12 1 2 121(1)1(1) (1)cov( , )nijijinnnn nnYY    12 1 211(1)cov( , )nijijinnYY A2.3. A Bound on the Binning Bias of Empirical Variogram Estimates Here it suffices to consider the variogram, ( )h, on the interval of distance values, 1kkdhd , for a typical bin k . Recall from (4.7.1) that for a given sample of values ( ) : 1,..,iYs i n , if kN denotes the set of distance pairs, ( , )ijss , in bin k, and if the distance between each such pair is denoted by ij i jhss , then the lag distance, kh , for bin k is defined to be (A2.3.1) (, )1ij kkijss NkhhNNOTEBOOK FOR SPATIAL DATA ANALYSIS Part II. Continuous Spatial Data Analysis ______________________________________________________________________________________ ________________________________________________________________________ ESE 502 A2-4 Tony E. Smith Recall also that if the k-linear approximation to ( )h on this interval is denoted by (A2.3.2) ( )kkklh a hb then by definition, (A2.3.3) 1[,) ()()kk k k khdd hlh In this context we