Coordination of spatially balanced samples
Section 4. Coordination methods

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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 $v_{ij}\sim U\left(\mathrm{0,1}\right)$ are associated to each pair $\left(i\mathrm{,}j\right)\subseteq U\times U\mathrm{<mo>;</mo>}$
2. $s_1$ is drawn using LPM as follows: if a pair of units $\left(i\mathrm{,}j\right)$ is chosen to compete, the number $v_{ij}$ is used in the corresponding competition rule (3.1) and the pair $\left(i\mathrm{,}j\right)$ is saved into a list of pairs;
3. $s_2$ is drawn using LPM as follows: the pairs $\left(i\mathrm{,}j\right)$ are considered sequentially from the list of pairs constructed above for $s_1\mathrm{,}$ and the same numbers $v_{ij}$ are used in the corresponding competition rule (3.1). If the sample size $n_2$ is achieved using pairs from this list, the algorithm stops; if not, the selection process continues with new pairs $\left(i\mathrm{,}j\right)$ (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: 

4.2  Coordination of SCP samples

The coordination of SCP samples with PRNs is implemented as follows. Let $u_i$ be the PRN associated to unit $i\in U\mathrm{,}$ with $u_1\text{}\mathrm{,}{u}_2\text{}\mathrm{,}\dots ,u_N$ iid $U\left(\mathrm{0,}1\right)\mathrm{.}$ Let ${\pi }_{it}^{\left(i-1\right)}$ be the (updated) selection probability for unit $i$ in the selection of sample $s_t\text{}\mathrm{,}t\mathrm{<mo>=</mo>\; 1,}2.$ For positive coordination, the PRNs are introduced in the selection step similarly to Poisson sampling with PRNs: if $u_i\mathrm{<mo><</mo>}{\pi }_{it}^{\left(i-1\right)}\text{}\mathrm{,}$ unit $i$ is selected in the sample $s_t\text{}\mathrm{,}t\mathrm{<mo>=</mo>\; 1,}2.$ For negative coordination, if $u_i\mathrm{<mo><</mo>}{\pi }_{i1}^{\left(i-1\right)}\text{}\mathrm{,}$ unit $i$ is selected in $s_1\mathrm{<mo>;</mo>}$ if $1-u_i\mathrm{<mo><</mo>}{\pi }_{i2}^{\left(i-1\right)}\text{}\mathrm{,}$ unit $i$ is selected in $s_2\text{}\mathrm{.}$ 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 $w_j^{\left(i\right)}\text{}.$ 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 $w_j^{\left(i\right)}$ given by the unit $j$ to units $i\mathrm{<mo>=</mo>}j+\mathrm{1,}\dots ,N$ 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 $\alpha ,$ $0\le \alpha \le 1.$ Thus we no longer use maximal weight, but the proportion $\alpha$ 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 $\alpha \mathrm{<mo>=</mo>}$0.5, distribute the weights 0.35, 0.25, 0.1. The reach is longer but it is not guaranteed we can use all $\alpha \mathrm{.}$ 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 $\alpha ,$ $0\le \alpha \le 1.$ 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 $\alpha$ to distribute. Otherwise, this works as the maximal weight strategy, so that unit $i$ 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 $\alpha \mathrm{<mo>=</mo>}$0.5, distribute 0.5, 0, 0. The reach is shorter and it is guaranteed we can use all $\alpha .$ However, if the total weight $\alpha$ 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 $\alpha \mathrm{<mo>=</mo>\; 0},$ we get Poisson sampling and with $\alpha \mathrm{<mo>=</mo>\; 1}$ we get SCPS with maximal weights. We can scale with $\alpha$ between 0 and 1 to mix the two to any degree. Maximum coordination, worst spatial balance and highest variance of sample size for $\alpha \mathrm{<mo>=</mo>\; 0},$ and best spatial balance and guaranteed fixed sample size for $\alpha \mathrm{<mo>=</mo>\; 1}$ 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 $\alpha .$ Sample size is likely to be more stable (given the same $\alpha )$ 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 $\alpha ,$ $0\le \alpha \le 1,$ they provide a family of sampling designs. Each element in this family corresponds to a given $\alpha \mathrm{.}$ Contrary to SCPS, for any value of $\alpha \mathrm{<mo><</mo>\; 1}$ 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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