Coordination of spatially balanced samples
Section 3. Spatial balanced sampling

The two spatial sampling designs we intend to introduce coordination for are briefly recalled below for a generic sample s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGZbaaaa@32E9@ of fixed size n . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGUbGaaGOlaaaa@339C@

3.1  Local pivotal method

The local pivotal method (Grafström et al., 2012) is a spatial application of the pivotal method (Deville and Tillé, 1998). Let π = ( π 1 , π 2 , ..., π N ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWHapGaaGypamaabmaabaGaeqiWda 3aaSbaaSqaaiaaigdaaeqaaOGaaGzaVlaaiYcacaaMe8UaeqiWda3a aSbaaSqaaiaaikdaaeqaaOGaaGzaVlaaiYcacaaMe8UaaGOlaiaai6 cacaaIUaGaaGilaiaaysW7cqaHapaCdaWgaaWcbaGaamOtaaqabaaa kiaawIcacaGLPaaaaaa@49B5@ be a given vector of inclusion probabilities, with sum n , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGUbGaaiilaaaa@3394@ π i = P ( i s ) , i U . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHapaCdaWgaaWcbaGaamyAaaqaba GccaaI9aGaamiuamaabmaabaGaamyAaiabgIGiolaadohaaiaawIca caGLPaaacaaMi8UaaGilaiaaysW7caWGPbGaeyicI4Saamyvaiaai6 caaaa@4339@ The vector π MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWHapaaaa@333D@ is successively updated to become a vector with N n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGobGaeyOeI0IaamOBaaaa@34A4@ zeros and n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGUbaaaa@32E4@ ones, where the ones indicate the selected units. A unit that still has a (possibly updated) probability strictly between 0 and 1 is called undecided. In one step of the LPM, a pair of units i , j U MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbGaaGilaiaaysW7caWGQbGaey icI4Saamyvaaaa@386F@ is chosen to compete. More precisely, we choose unit i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbaaaa@32DE@ randomly among the undecided units, and unit i s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbGaaGPaVJqaaiaa=LbicaqGZb aaaa@3623@ competitor j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbaaaa@32E0@ is the nearest neighbor of i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbaaaa@32DF@ among the undecided units. Thus we apply the pivotal method locally in space. The winner receives as much probability mass as possible from the loser, so the winner ends up with π w = min ( 1, π i + π j ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHapaCdaWgaaWcbaGaam4Daaqaba GccaaI9aGaciyBaiaacMgacaGGUbWaaeWaaeaacaaIXaGaaGilaiaa ysW7cqaHapaCdaWgaaWcbaGaamyAaaqabaGccqGHRaWkcqaHapaCda WgaaWcbaGaamOAaaqabaaakiaawIcacaGLPaaaaaa@43A5@ and the loser keeps what is possibly remaining π l = π i + π j π w . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHapaCdaWgaaWcbaGaeS4eHWgabe aakiaai2dacqaHapaCdaWgaaWcbaGaamyAaaqabaGccqGHRaWkcqaH apaCdaWgaaWcbaGaamOAaaqabaGccqGHsislcqaHapaCdaWgaaWcba Gaam4DaaqabaGccaGGUaaaaa@410F@ The rules of the competition are

( π i , π j ) : = { ( π w , π l ) with probability ( π w π j ) / ( π w π l ) ( π l , π w ) with probability ( π w π i ) / ( π w π l ) . ( 3.1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaadaqadaqaaiabec8aWnaaBaaaleaaca WGPbaabeaakiaaygW7caaISaGaaGjbVlabec8aWnaaBaaaleaacaWG QbaabeaaaOGaayjkaiaawMcaaiaaysW7caaMe8UaaGOoaiaai2daca aMe8UaaGjbVpaaceaabaqbaeaabiGaaaqaamaabmaabaGaeqiWda3a aSbaaSqaaiaadEhaaeqaaOGaaGzaVlaaiYcacaaMe8UaeqiWda3aaS baaSqaaiabloriSbqabaaakiaawIcacaGLPaaaaeaacaqG3bGaaeyA aiaabshacaqGObGaaGjbVlaaykW7caqGWbGaaeOCaiaab+gacaqGIb GaaeyyaiaabkgacaqGPbGaaeiBaiaabMgacaqG0bGaaeyEaiaaysW7 caaMc8+aaSGbaeaadaqadaqaaiabec8aWnaaBaaaleaacaWG3baabe aakiabgkHiTiabec8aWnaaBaaaleaacaWGQbaabeaaaOGaayjkaiaa wMcaaaqaamaabmaabaGaeqiWda3aaSbaaSqaaiaadEhaaeqaaOGaey OeI0IaeqiWda3aaSbaaSqaaiabloriSbqabaaakiaawIcacaGLPaaa aaaabaWaaeWaaeaacqaHapaCdaWgaaWcbaGaeS4eHWgabeaakiaayg W7caaISaGaaGjbVlabec8aWnaaBaaaleaacaWG3baabeaaaOGaayjk aiaawMcaaaqaaiaabEhacaqGPbGaaeiDaiaabIgacaaMe8UaaGPaVl aabchacaqGYbGaae4BaiaabkgacaqGHbGaaeOyaiaabMgacaqGSbGa aeyAaiaabshacaqG5bGaaGjbVlaaykW7daWcgaqaamaabmaabaGaeq iWda3aaSbaaSqaaiaadEhaaeqaaOGaeyOeI0IaeqiWda3aaSbaaSqa aiaadMgaaeqaaaGccaGLOaGaayzkaaaabaWaaeWaaeaacqaHapaCda WgaaWcbaGaam4DaaqabaGccqGHsislcqaHapaCdaWgaaWcbaGaeS4e HWgabeaaaOGaayjkaiaawMcaaaaaaaaacaGL7baacaaIUaGaaGzbVl aaywW7caaMf8UaaGzbVlaacIcacaaIZaGaaiOlaiaaigdacaGGPaaa aa@B175@

The final outcome is decided for at least one unit each update, so the procedure has at most N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGobaaaa@32C4@ steps. Because neighboring units compete against each other for inclusion, they are unlikely to be simultaneously included in a sample.

3.2  Spatially correlated Poisson sampling

The spatially correlated Poisson sampling method (Grafström, 2012) is a spatial application of the correlated Poisson sampling method (Bondesson and Thorburn, 2008). Let π = ( π 1 , π 2 , , π N ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWHapGaaGypamaabmaabaGaeqiWda 3aaSbaaSqaaiaaigdaaeqaaOGaaGzaVlaaiYcacaaMe8UaeqiWda3a aSbaaSqaaiaaikdaaeqaaOGaaGzaVlaaiYcacaaMe8UaeSOjGSKaaG ilaiaaysW7cqaHapaCdaWgaaWcbaGaamOtaaqabaaakiaawIcacaGL Paaaaaa@48AF@ be a given vector of inclusion probabilities, with sum n , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGUbGaaiilaaaa@3394@ π i = P ( i s ) , i U . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHapaCdaWgaaWcbaGaamyAaaqaba GccaaI9aGaamiuamaabmaabaGaamyAaiabgIGiolaadohaaiaawIca caGLPaaacaaMi8UaaGilaiaaysW7caWGPbGaeyicI4Saamyvaiaai6 caaaa@4339@ The vector π MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWHapaaaa@333D@ is sequentially updated to become a vector with N n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGobGaeyOeI0IaamOBaaaa@34A4@ zeros and n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGUbaaaa@32E4@ ones, where the ones indicate the selected units. First unit 1 is included with probability π 1 ( 0 ) = π 1 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHapaCdaqhaaWcbaGaaGymaaqaam aabmaabaGaaGimaaGaayjkaiaawMcaaaaakiaai2dacqaHapaCdaWg aaWcbaGaaGymaaqabaGccaGGUaaaaa@3B0A@ If unit 1 was included, we set I 1 = 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGjbWaaSbaaSqaaiaaigdaaeqaaO GaaGypaiaaigdaaaa@3532@ and otherwise I 1 = 0. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGjbWaaSbaaSqaaiaaigdaaeqaaO GaaGypaiaaicdacaGGUaaaaa@35E3@ Generally at step j , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbGaaiilaaaa@3390@ when the values for I 1 , , I j 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGjbWaaSbaaSqaaiaaigdaaeqaaO GaaGzaVlaaiYcacaaMe8UaeSOjGSKaaGilaiaaysW7caWGjbWaaSba aSqaaiaadQgacqGHsislcaaIXaaabeaaaaa@3E73@ have been recorded, unit j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbaaaa@32E0@ is included with probability π j ( j 1 ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHapaCdaqhaaWcbaGaamOAaaqaam aabmaabaGaamOAaiabgkHiTiaaigdaaiaawIcacaGLPaaaaaGccaGG Uaaaaa@39A6@ Then the inclusion probabilities are updated for the units i = j + 1, , N , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbGaaGypaiaadQgacqGHRaWkca aIXaGaaGilaiaaysW7cqWIMaYscaaISaGaaGjbVlaad6eacaaMb8Ua aiilaaaa@3EE7@ according to

π i ( j ) = π i ( j 1 ) ( I j π j ( j 1 ) ) w j ( i ) , ( 3.2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHapaCdaqhaaWcbaGaamyAaaqaam aabmaabaGaamOAaaGaayjkaiaawMcaaaaakiaaysW7caaMe8UaaGyp aiaaysW7caaMe8UaeqiWda3aa0baaSqaaiaadMgaaeaadaqadaqaai aadQgacqGHsislcaaIXaaacaGLOaGaayzkaaaaaOGaaGjbVlabgkHi TiaaysW7caaMe8+aaeWaaeaacaWGjbWaaSbaaSqaaiaadQgaaeqaaO GaeyOeI0IaeqiWda3aa0baaSqaaiaadQgaaeaadaqadaqaaiaadQga cqGHsislcaaIXaaacaGLOaGaayzkaaaaaaGccaGLOaGaayzkaaGaam 4DamaaDaaaleaacaWGQbaabaWaaeWaaeaacaWGPbaacaGLOaGaayzk aaaaaOGaaGzaVlaaiYcacaaMf8UaaGzbVlaaywW7caaMf8Uaaiikai aaiodacaGGUaGaaGOmaiaacMcaaaa@66DC@

where w j ( i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG3bWaa0baaSqaaiaadQgaaeaada qadaqaaiaadMgaaiaawIcacaGLPaaaaaaaaa@3680@ are weights given by unit j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbaaaa@32E0@ to the units i = j + 1, j + 2, , N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbGaaGypaiaadQgacqGHRaWkca aIXaGaaGilaiaaysW7caWGQbGaey4kaSIaaGOmaiaaiYcacaaMe8Ua eSOjGSKaaiilaiaaysW7caWGobaaaa@4177@ and π i ( 0 ) = π i . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHapaCdaqhaaWcbaGaamyAaaqaam aabmaabaGaaGimaaGaayjkaiaawMcaaaaakiaai2dacqaHapaCdaWg aaWcbaGaamyAaaqabaGccaGGUaaaaa@3B70@ The weight w j ( i ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG3bWaa0baaSqaaiaadQgaaeaada qadaqaaiaadMgaaiaawIcacaGLPaaaaaGccaaMb8Uaaiilaaaa@38C4@ j < i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbGaaGipaiaadMgacaGGSaaaaa@3544@ determine how the inclusion probability for unit i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbaaaa@32DF@ should be affected by the sampling outcome of unit j . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbGaaiOlaaaa@3392@ More precisely, the weight w j ( i ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG3bWaa0baaSqaaiaadQgaaeaada qadaqaaiaadMgaaiaawIcacaGLPaaaaaGccaaMb8Uaaiilaaaa@38C4@ j < i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbGaaGipaiaadMgacaGGSaaaaa@3544@ may depend on the previous sampling outcome I 1 , I 2 , , I j 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGjbWaaSbaaSqaaiaaigdaaeqaaO GaaGzaVlaaiYcacaaMe8UaamysamaaBaaaleaacaaIYaaabeaakiaa ygW7caaISaGaaGjbVlablAciljaacYcacaaMe8UaamysamaaBaaale aacaWGQbGaeyOeI0IaaGymaaqabaaaaa@43FA@ but not on the future outcomes I j , I j + 1 , , I N . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGjbWaaSbaaSqaaiaadQgaaeqaaO GaaGzaVlaaiYcacaaMe8UaamysamaaBaaaleaacaWGQbGaey4kaSIa aGymaaqabaGccaaMb8UaaGilaiaaysW7cqWIMaYscaGGSaGaaGjbVl aadMeadaWgaaWcbaGaamOtaaqabaGccaGGUaaaaa@44F6@ The weights should also satisfy the following restrictions

min ( 1 π i ( j 1 ) 1 π j ( j 1 ) , π i ( j 1 ) π j ( j 1 ) ) w j ( i ) min ( π i ( j 1 ) 1 π j ( j 1 ) , 1 π i ( j 1 ) π j ( j 1 ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqGHsislciGGTbGaaiyAaiaac6gada qadaqaamaalaaabaGaaGymaiabgkHiTiabec8aWnaaDaaaleaacaWG PbaabaWaaeWaaeaacaWGQbGaeyOeI0IaaGymaaGaayjkaiaawMcaaa aaaOqaaiaaigdacqGHsislcqaHapaCdaqhaaWcbaGaamOAaaqaamaa bmaabaGaamOAaiabgkHiTiaaigdaaiaawIcacaGLPaaaaaaaaOGaaG ilaiaaysW7caaMe8+aaSaaaeaacqaHapaCdaqhaaWcbaGaamyAaaqa amaabmaabaGaamOAaiabgkHiTiaaigdaaiaawIcacaGLPaaaaaaake aacqaHapaCdaqhaaWcbaGaamOAaaqaamaabmaabaGaamOAaiabgkHi TiaaigdaaiaawIcacaGLPaaaaaaaaaGccaGLOaGaayzkaaGaaGjbVl aaysW7cqGHKjYOcaaMe8UaaGjbVlaadEhadaqhaaWcbaGaamOAaaqa amaabmaabaGaamyAaaGaayjkaiaawMcaaaaakiaaysW7caaMe8Uaey izImQaaGjbVlaaysW7ciGGTbGaaiyAaiaac6gadaqadaqaamaalaaa baGaeqiWda3aa0baaSqaaiaadMgaaeaadaqadaqaaiaadQgacqGHsi slcaaIXaaacaGLOaGaayzkaaaaaaGcbaGaaGymaiabgkHiTiabec8a WnaaDaaaleaacaWGQbaabaWaaeWaaeaacaWGQbGaeyOeI0IaaGymaa GaayjkaiaawMcaaaaaaaGccaaISaGaaGjbVlaaysW7daWcaaqaaiaa igdacqGHsislcqaHapaCdaqhaaWcbaGaamyAaaqaamaabmaabaGaam OAaiabgkHiTiaaigdaaiaawIcacaGLPaaaaaaakeaacqaHapaCdaqh aaWcbaGaamOAaaqaamaabmaabaGaamOAaiabgkHiTiaaigdaaiaawI cacaGLPaaaaaaaaaGccaGLOaGaayzkaaaaaa@9692@

in order for 0 π i ( j 1 ) 1 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaaIWaGaaGjbVlaaykW7cqGHKjYOca aMe8UaaGPaVlabec8aWnaaDaaaleaacaWGPbaabaWaaeWaaeaacaWG QbGaeyOeI0IaaGymaaGaayjkaiaawMcaaaaakiaaysW7caaMc8Uaey izImQaaGjbVlaaykW7caaIXaGaaiilaaaa@4AE2@ i = j , j + 1, , N , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbGaaGypaiaadQgacaaMb8UaaG ilaiaaysW7caWGQbGaey4kaSIaaGymaiaaiYcacaaMe8UaeSOjGSKa aiilaiaaysW7caWGobGaaGzaVlaacYcaaaa@439D@ to hold. The unconditional inclusion probabilities are not affected by the weights since the updating rule (3.2) gives

E ( π i ( i 1 ) ) = E ( E ( π i ( i 1 ) | π i ( i 2 ) ) ) = E ( π i ( i 2 ) ) = = π i . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGfbWaaeWaaeaacqaHapaCdaqhaa WcbaGaamyAaaqaamaabmaabaGaamyAaiabgkHiTiaaigdaaiaawIca caGLPaaaaaaakiaawIcacaGLPaaacaaMe8UaaGjbVlaai2dacaaMe8 UaaGjbVlaadweadaqadaqaaiaadweadaqadaqaamaaeiaabaGaeqiW da3aa0baaSqaaiaadMgaaeaadaqadaqaaiaadMgacqGHsislcaaIXa aacaGLOaGaayzkaaaaaOGaaGPaVdGaayjcSdGaaGPaVlabec8aWnaa DaaaleaacaWGPbaabaWaaeWaaeaacaWGPbGaeyOeI0IaaGOmaaGaay jkaiaawMcaaaaaaOGaayjkaiaawMcaaaGaayjkaiaawMcaaiaaysW7 caaMe8UaaGypaiaaysW7caaMe8UaamyramaabmaabaGaeqiWda3aa0 baaSqaaiaadMgaaeaadaqadaqaaiaadMgacqGHsislcaaIYaaacaGL OaGaayzkaaaaaaGccaGLOaGaayzkaaGaaGjbVlaaysW7caaI9aGaaG jbVlaaysW7cqWIMaYscaaMe8UaaGjbVlaai2dacaaMe8UaaGjbVlab ec8aWnaaBaaaleaacaWGPbaabeaakiaai6caaaa@7C95@

Thus the method always gives the prescribed inclusion probabilities π i , i = 1, 2, , N . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHapaCdaWgaaWcbaGaamyAaaqaba GccaaMb8UaaGilaiaaysW7caWGPbGaaGypaiaaigdacaaISaGaaGjb VlaaikdacaaISaGaaGjbVlablAciljaacYcacaaMe8UaamOtaiaai6 caaaa@453B@

Bondesson and Thorburn (2008) showed that a fixed size sampling is obtained only if i = 1 N π i = n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaadaaeWaqaaiabec8aWnaaBaaaleaaca WGPbaabeaaaeaacaWGPbGaaGypaiaaigdaaeaacaWGobaaniabggHi LdGccaaI9aGaamOBaaaa@3BC6@ and the weights are chosen such that i = j + 1 N w j ( i ) = 1, j U . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaadaaeWaqaaiaadEhadaqhaaWcbaGaam OAaaqaamaabmaabaGaamyAaaGaayjkaiaawMcaaaaaaeaacaWGPbGa aGypaiaadQgacqGHRaWkcaaIXaaabaGaamOtaaqdcqGHris5aOGaaG ypaiaaigdacaaISaGaaGjbVlaaykW7caWGQbGaeyicI4Saamyvaiaa ygW7caaIUaaaaa@4874@

To achieve spatial balance, the weights should be decided on the basis of the distance between units. The most common approach to choose weights in SCPS is that unit j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbaaaa@32E0@ first gives as much weight as possible to the closest unit (in distance) among the units i = j + 1, j + 2, , N , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbGaaGypaiaadQgacqGHRaWkca aIXaGaaGilaiaaysW7caWGQbGaey4kaSIaaGOmaiaaiYcacaaMe8Ua eSOjGSKaaiilaiaaysW7caWGobGaaGzaVlaacYcaaaa@43B1@ then as much weight as possible to the second closest unit etc. with the restriction that the weights are non-negative and sum up to 1. This strategy is called the maximal weight strategy. If distances are equal, then the weight is distributed equally on those units that have equal distance if possible. The first priority is that weight is not put on a unit if it is possible to put the weight on a closer unit. The maximal weight strategy always produces samples of fixed size if the inclusion probabilities sum up to an integer. In what follows, when we refer to SCPS, the “maximal weight strategy” is used.

3.3  Voronoi polytopes

Voronoi polytopes are used to measure the level of spatial balance (or spread) with respect to the inclusion probabilities (Stevens and Olsen, 2004). A polytope P i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGqbWaaSbaaSqaaiaadMgaaeqaaa aa@33E0@ is constructed for each unit i s , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbGaeyicI4Saam4CaiaacYcaaa a@360B@ and P i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGqbWaaSbaaSqaaiaadMgaaeqaaa aa@33E0@ includes all population units closer to unit i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbaaaa@32DF@ than to any other sample unit j s , j i . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbGaeyicI4Saam4CaiaaiYcaca aMe8UaamOAaiabgcMi5kaadMgacaGGUaaaaa@3BF5@ Optimally, each polytope should have a probability mass that is equal to 1. A measure of spatial balance of a realised sample s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGZbaaaa@32E9@ is (see Stevens and Olsen, 2004)

B = 1 n i s ( v i 1 ) 2 , ( 3.3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGcbGaaGjbVlaaysW7caaI9aGaaG jbVlaaysW7daWcaaqaaiaaigdaaeaacaWGUbaaamaaqafabaWaaeWa aeaacaWG2bWaaSbaaSqaaiaadMgaaeqaaOGaeyOeI0IaaGymaaGaay jkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaeaacaWGPbGaeyicI4Sa am4Caaqab0GaeyyeIuoakiaaygW7caaISaGaaGzbVlaaywW7caaMf8 UaaGzbVlaacIcacaaIZaGaaiOlaiaaiodacaGGPaaaaa@5332@

where v i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG2bWaaSbaaSqaaiaadMgaaeqaaa aa@3406@ is the sum of the inclusion probabilities of the units in P i . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGqbWaaSbaaSqaaiaadMgaaeqaaO GaaiOlaaaa@349C@ The expected value of B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGcbaaaa@32B8@ under repeated sampling is a measure of how well a design succeeds in selecting spatially balanced samples. The smaller the value the better the spread of the selected samples.


Report a problem on this page

Is something not working? Is there information outdated? Can't find what you're looking for?

Please contact us and let us know how we can help you.

Privacy notice

Date modified: