Coordination of spatially balanced samples
Section 4. Coordination methods

We present below PRN methods based on the local pivotal (LP) method and the spatially correlated Poisson (SCP) sampling.

4.1  Coordination of LP samples

The positive coordination of LP samples with PRNs is implemented as follows: 

  1. independent permanent random numbers v i j U ( 0,1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG2bWaaSbaaSqaaiaadMgacaWGQb aabeaarqqr1ngBPrgifHhDYfgaiqaakiab=XJi6iaadwfadaqadaqa aiaaicdacaaISaGaaGymaaGaayjkaiaawMcaaaaa@3F45@ are associated to each pair ( i , j ) U × U ; MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaadaqadaqaaiaadMgacaaISaGaamOAaa GaayjkaiaawMcaaiabgAOinlaadwfacqGHxdaTcaWGvbGaaG4oaaaa @3C9E@
  2. s 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGZbWaaSbaaSqaaiaaigdaaeqaaa aa@33D0@ is drawn using LPM as follows: if a pair of units ( i , j ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaadaqadaqaaiaadMgacaaISaGaamOAaa GaayjkaiaawMcaaaaa@360D@ is chosen to compete, the number v i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG2bWaaSbaaSqaaiaadMgacaWGQb aabeaaaaa@34F5@ is used in the corresponding competition rule (3.1) and the pair ( i , j ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaadaqadaqaaiaadMgacaaISaGaamOAaa GaayjkaiaawMcaaaaa@360D@ is saved into a list of pairs;
  3. s 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGZbWaaSbaaSqaaiaaikdaaeqaaa aa@33D1@ is drawn using LPM as follows: the pairs ( i , j ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaadaqadaqaaiaadMgacaaISaGaamOAaa GaayjkaiaawMcaaaaa@360D@ are considered sequentially from the list of pairs constructed above for s 1 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGZbWaaSbaaSqaaiaaigdaaeqaaO GaaGilaaaa@3490@ and the same numbers v i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG2bWaaSbaaSqaaiaadMgacaWGQb aabeaaaaa@34F5@ are used in the corresponding competition rule (3.1). If the sample size n 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGUbWaaSbaaSqaaiaaikdaaeqaaa aa@33CC@ is achieved using pairs from this list, the algorithm stops; if not, the selection process continues with new pairs ( i , j ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaadaqadaqaaiaadMgacaaISaGaamOAaa GaayjkaiaawMcaaaaa@360D@ (not included in this list) and selected as described in Section 3.1.

For negative coordination, the first two steps are the same, but the last step becomes: 

3’.
s 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGZbWaaSbaaSqaaiaaikdaaeqaaa aa@33D1@ is drawn using LPM as follows: the pairs ( i , j ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaadaqadaqaaiaadMgacaaISaGaaGjbVl aadQgaaiaawIcacaGLPaaaaaa@379A@ are considered sequentially from the list of pairs constructed above for s 1 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGZbWaaSbaaSqaaiaaigdaaeqaaO GaaGilaaaa@3490@ and the numbers 1 v i j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaaIXaGaeyOeI0IaamODamaaBaaale aacaWGPbGaamOAaaqabaaaaa@369D@ are used in the corresponding competition rule (3.1). If the sample size n 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGUbWaaSbaaSqaaiaaikdaaeqaaa aa@33CC@ is achieved using pairs from this list, the algorithm stops; if not, the selection process continues with new pairs ( i , j ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaadaqadaqaaiaadMgacaaISaGaaGjbVl aadQgaaiaawIcacaGLPaaaaaa@379A@ (not included in this list) and selected as described in Section 3.1.

4.2  Coordination of SCP samples

The coordination of SCP samples with PRNs is implemented as follows. Let u i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG1bWaaSbaaSqaaiaadMgaaeqaaa aa@3405@ be the PRN associated to unit i U , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbGaeyicI4SaamyvaiaaiYcaaa a@35F3@ with u 1 , u 2 , , u N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG1bWaaSbaaSqaaiaaigdaaeqaaO GaaGzaVlaaiYcacaaMe8UaamyDamaaBaaaleaacaaIYaaabeaakiaa ygW7caaISaGaaGjbVlablAciljaacYcacaaMe8UaamyDamaaBaaale aacaWGobaabeaaaaa@42BA@ iid U ( 0, 1 ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGvbWaaeWaaeaacaaIWaGaaGilai aaysW7caaIXaaacaGLOaGaayzkaaGaaGOlaaaa@38C4@ Let π i t ( i 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHapaCdaqhaaWcbaGaamyAaiaads haaeaadaqadaqaaiaadMgacqGHsislcaaIXaaacaGLOaGaayzkaaaa aaaa@39E1@ be the (updated) selection probability for unit i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbaaaa@32DF@ in the selection of sample s t , t = 1, 2. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGZbWaaSbaaSqaaiaadshaaeqaaO GaaGzaVlaaiYcacaaMe8UaamiDaiaai2dacaaIXaGaaGilaiaaysW7 caaIYaGaaGOlaaaa@3E17@ For positive coordination, the PRNs are introduced in the selection step similarly to Poisson sampling with PRNs: if u i < π i t ( i 1 ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG1bWaaSbaaSqaaiaadMgaaeqaaO GaaGipaiabec8aWnaaDaaaleaacaWGPbGaamiDaaqaamaabmaabaGa amyAaiabgkHiTiaaigdaaiaawIcacaGLPaaaaaGccaaMb8UaaGilaa aa@3F0F@ unit i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbaaaa@32DF@ is selected in the sample s t , t = 1, 2. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGZbWaaSbaaSqaaiaadshaaeqaaO GaaGzaVlaaiYcacaaMe8UaamiDaiaai2dacaaIXaGaaGilaiaaysW7 caaIYaGaaGOlaaaa@3E17@ For negative coordination, if u i < π i 1 ( i 1 ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG1bWaaSbaaSqaaiaadMgaaeqaaO GaaGipaiabec8aWnaaDaaaleaacaWGPbGaaGymaaqaamaabmaabaGa amyAaiabgkHiTiaaigdaaiaawIcacaGLPaaaaaGccaaMb8UaaGilaa aa@3ED1@ unit i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbaaaa@32DF@ is selected in s 1 ; MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGZbWaaSbaaSqaaiaaigdaaeqaaO GaaG4oaaaa@349F@ if 1 u i < π i 2 ( i 1 ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaaIXaGaeyOeI0IaamyDamaaBaaale aacaWGPbaabeaakiaaiYdacqaHapaCdaqhaaWcbaGaamyAaiaaikda aeaadaqadaqaaiaadMgacqGHsislcaaIXaaacaGLOaGaayzkaaaaaO GaaGzaVlaaiYcaaaa@407A@ unit i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbaaaa@32DF@ is selected in s 2 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGZbWaaSbaaSqaaiaaikdaaeqaaO GaaGzaVlaai6caaaa@361D@ This coordination method is general for spatially correlated Poisson sampling and can be used no matter what weights are applied within the method.

We utilize the maximal weight strategy advocated in Section 3.2 as the main alternative, but we also introduce two new alternative strategies to compute the weights w j ( i ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG3bWaa0baaSqaaiaadQgaaeaada qadaqaaiaadMgaaiaawIcacaGLPaaaaaGccaaMb8UaaiOlaaaa@38C6@ The new strategies are intended to provide a good compromise between the degrees of spatial balance and coordination. By reducing the amount of spatial correlation in SCPS we can achieve any level of mixing between SCPS and Poisson sampling. Both of the new strategies are similar to the SCPS with maximal weights, but the weights w j ( i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG3bWaa0baaSqaaiaadQgaaeaada qadaqaaiaadMgaaiaawIcacaGLPaaaaaaaaa@3680@ given by the unit j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGQbaaaa@32E0@ to units i = j + 1, , N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbGaaGypaiaadQgacqGHRaWkca aIXaGaaGilaiaaysW7cqWIMaYscaGGSaGaaGjbVlaad6eaaaa@3CA7@ do not sum up to 1 any more. Consequently, the result of Bondesson and Thorburn (2008) advocated in Section 3.2 does not apply and the new sampling designs do not any more provide fixed sample sizes. We denote the resulting  family of designs Transformed Spatially Correlated Poisson Sampling (TSCPS).

The first mixing strategy is to modify SCPS by multiplying the maximal weight by a given scalar α , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHXoqycaGGSaaaaa@3440@ 0 α 1. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaaIWaGaaGjbVlaaykW7cqGHKjYOca aMe8UaaGPaVlabeg7aHjaaysW7caaMc8UaeyizImQaaGjbVlaaykW7 caaIXaGaaiOlaaaa@4581@ Thus we no longer use maximal weight, but the proportion α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHXoqyaaa@3390@ of the maximal weight is the limit for the applied weight. This method is denoted TSCPS 1. With this method the positive weights will reach longer (more neighbors) than in SCPS. Each unit would distribute a total weight of maximum 1, starting with the nearest unit and then the second nearest etc. Say the maximal weights for the three nearest neighbors of a unit are 0.7, 0.5, 0.2. Then, in standard SCPS (with maximal weights) the unit would distribute the weights 0.7, 0.3, 0, and the new modified version would, with α = MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHXoqycaaI9aaaaa@3457@ 0.5, distribute the weights 0.35, 0.25, 0.1. The reach is longer but it is not guaranteed we can use all α . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHXoqycaaIUaaaaa@3448@ As a result, the total weight is not necessary 1, and the sample size becomes random.

The second mixing strategy is achieved by limiting the weights that a unit distributes to sum to a fixed scalar α , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHXoqycaGGSaaaaa@3440@ 0 α 1. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaaIWaGaaGjbVlaaykW7cqGHKjYOca aMe8UaaGPaVlabeg7aHjaaysW7caaMc8UaeyizImQaaGjbVlaaykW7 caaIXaGaaiOlaaaa@4581@ This method is denoted TSCPS 2. In SCPS with maximal weight strategy, each unit is given a total weight 1 (the sum of the weights) to distribute on remaining units in the list. Instead, each unit is given a total weight α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHXoqyaaa@3390@ to distribute. Otherwise, this works as the maximal weight strategy, so that unit i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbaaaa@32DF@ first gives as much weight as possible to the nearest, then the second nearest etc. With this strategy the weights will reach a shorter distance (fewer neighbors). Say the maximal weights for the three nearest neighbors of a unit are 0.7, 0.5, 0.2. Then standard SCPS (with maximal weights) would distribute the weights 0.7, 0.3, 0, and the new modified version would, with α = MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHXoqycaaI9aaaaa@3457@ 0.5, distribute 0.5, 0, 0. The reach is shorter and it is guaranteed we can use all α . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHXoqycaGGUaaaaa@3442@ However, if the total weight α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHXoqyaaa@3390@ is less than 1, there will be a random sample size.

Note that for both TSCPS 1 and 2 we have the following result. With α = 0 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHXoqycaaI9aGaaGimaiaacYcaaa a@35C1@ we get Poisson sampling and with α = 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHXoqycaaI9aGaaGymaaaa@3512@ we get SCPS with maximal weights. We can scale with α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHXoqyaaa@3390@ between 0 and 1 to mix the two to any degree. Maximum coordination, worst spatial balance and highest variance of sample size for α = 0 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHXoqycaaI9aGaaGimaiaacYcaaa a@35C1@ and best spatial balance and guaranteed fixed sample size for α = 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHXoqycaaI9aGaaGymaaaa@3512@ while level of coordination will be to some extent worse. Both TSCPS 1 and 2 offer the possibility to make a trade-off between the Poisson and SCPS designs. Degree of spatial balance and coordination, as well as variance of achieved sample size depend on the parameter α . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHXoqycaGGUaaaaa@3442@ Sample size is likely to be more stable (given the same α ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHXoqycaGGPaaaaa@343D@ for TSCPS 1 than for TSCPS 2, as more weight is likely to be distributed with TSCPS 1. Since both TSCPS 1 and 2 use a given scalar α , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHXoqycaGGSaaaaa@3440@ 0 α 1 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaaIWaGaaGjbVlaaykW7cqGHKjYOca aMe8UaaGPaVlabeg7aHjaaysW7caaMc8UaeyizImQaaGjbVlaaykW7 caaIXaGaaiilaaaa@457F@ they provide a family of sampling designs. Each element in this family corresponds to a given α . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHXoqycaaIUaaaaa@3448@ Contrary to SCPS, for any value of α < 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8peuj0lXxdrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHXoqycaaI8aGaaGymaaaa@3511@ both TSCPS 1 and 2 involve random sample sizes. The consequences of having random sample sizes on coordination is empirically studied in Section 5.1, on spatial balance degree in Section 5.2 and on variance estimation in Section 5.3.


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