Coordination of spatially balanced samples
Section 3. Spatial balanced sampling

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The two spatial sampling designs we intend to introduce coordination for are briefly recalled below for a generic sample $s$ of fixed size $n\mathrm{.}$

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 $\pi =\left({\pi }_1\text{}\mathrm{,}{\pi }_2\text{}\mathrm{,}\mathrm{...,}{\pi }_N\right)$ be a given vector of inclusion probabilities, with sum $n,$ ${\pi }_i\mathrm{<mo>=</mo>}P\left(i\in s\right)\mathrm{,}i\in U\mathrm{.}$ The vector $\pi$ is successively updated to become a vector with $N-n$ zeros and $n$ 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 $i\mathrm{,}j\in U$ is chosen to compete. More precisely, we choose unit $i$ randomly among the undecided units, and unit $i’\text{s}$ competitor $j$ is the nearest neighbor of $i$ 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 ${\pi }_w\mathrm{<mo>=</mo>}\min \left(\mathrm{1,}{\pi }_i+{\pi }_j\right)$ and the loser keeps what is possibly remaining ${\pi }_l\mathrm{<mo>=</mo>}{\pi }_i+{\pi }_j-{\pi }_w.$ The rules of the competition are

$$\left({\pi }_i\text{}\mathrm{,}{\pi }_j\right)\mathrm{<mo>:</mo>\; <mo>=</mo>}\{\begin{array}{ll}\left({\pi }_w\text{}\mathrm{,}{\pi }_l\right)\hfill & \text{with}\text{probability}\left({\pi }_w-{\pi }_j\right)/\left({\pi }_w-{\pi }_l\right)\hfill \\ \left({\pi }_l\text{}\mathrm{,}{\pi }_w\right)\hfill & \text{with}\text{probability}\left({\pi }_w-{\pi }_i\right)/\left({\pi }_w-{\pi }_l\right)\hfill \end{array}\mathrm{.}(3.1)$$

The final outcome is decided for at least one unit each update, so the procedure has at most $N$ 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 $\pi \mathrm{<mo>=</mo>}\left({\pi }_1\text{}\mathrm{,}{\pi }_2\text{}\mathrm{,}\dots \mathrm{,}{\pi }_N\right)$ be a given vector of inclusion probabilities, with sum $n,$ ${\pi }_i\mathrm{<mo>=</mo>}P\left(i\in s\right)\mathrm{,}i\in U\mathrm{.}$ The vector $\pi$ is sequentially updated to become a vector with $N-n$ zeros and $n$ ones, where the ones indicate the selected units. First unit 1 is included with probability ${\pi }_1^{\left(0\right)}\mathrm{<mo>=</mo>}{\pi }_1.$ If unit 1 was included, we set $I_1\mathrm{<mo>=</mo>\; 1}$ and otherwise $I_1\mathrm{<mo>=</mo>\; 0.}$ Generally at step $j,$ when the values for $I_1\text{}\mathrm{,}\dots \mathrm{,}{I}_{j-1}$ have been recorded, unit $j$ is included with probability ${\pi }_j^{\left(j-1\right)}.$ Then the inclusion probabilities are updated for the units $i\mathrm{<mo>=</mo>}j+\mathrm{1,}\dots \mathrm{,}N\text{},$ according to

$${\pi }_i^{\left(j\right)}\mathrm{<mo>=</mo>}{\pi }_i^{\left(j-1\right)}-\left(I_j-{\pi }_j^{\left(j-1\right)}\right)w_j^{\left(i\right)}\text{}\mathrm{,}(3.2)$$

where $w_j^{\left(i\right)}$ are weights given by unit $j$ to the units $i\mathrm{<mo>=</mo>}j+\mathrm{1,}j+\mathrm{2,}\dots ,N$ and ${\pi }_i^{\left(0\right)}\mathrm{<mo>=</mo>}{\pi }_i.$ The weight $w_j^{\left(i\right)}\text{},$ $j\mathrm{<mo><</mo>}i,$ determine how the inclusion probability for unit $i$ should be affected by the sampling outcome of unit $j.$ More precisely, the weight $w_j^{\left(i\right)}\text{},$ $j\mathrm{<mo><</mo>}i,$ may depend on the previous sampling outcome $I_1\text{}\mathrm{,}{I}_2\text{}\mathrm{,}\dots ,I_{j-1}$ but not on the future outcomes $I_j\text{}\mathrm{,}{I}_{j+1}\text{}\mathrm{,}\dots ,I_N.$ The weights should also satisfy the following restrictions

$$-\min \left(\frac{1-{\pi }_i^{\left(j-1\right)}}{1-{\pi }_j^{\left(j-1\right)}}\mathrm{,}\frac{{\pi }_i^{\left(j-1\right)}}{{\pi }_j^{\left(j-1\right)}}\right)\le w_j^{\left(i\right)}\le \min \left(\frac{{\pi }_i^{\left(j-1\right)}}{1-{\pi }_j^{\left(j-1\right)}}\mathrm{,}\frac{1-{\pi }_i^{\left(j-1\right)}}{{\pi }_j^{\left(j-1\right)}}\right)$$

in order for $0\le {\pi }_i^{\left(j-1\right)}\le 1,$ $i\mathrm{<mo>=</mo>}j\text{}\mathrm{,}j+\mathrm{1,}\dots ,N\text{},$ to hold. The unconditional inclusion probabilities are not affected by the weights since the updating rule (3.2) gives

$$E\left({\pi }_i^{\left(i-1\right)}\right)\mathrm{<mo>=</mo>}E\left(E\left({\pi }_i^{\left(i-1\right)}|{\pi }_i^{\left(i-2\right)}\right)\right)\mathrm{<mo>=</mo>}E\left({\pi }_i^{\left(i-2\right)}\right)\mathrm{<mo>=</mo>}\dots \mathrm{<mo>=</mo>}{\pi }_i\mathrm{.}$$

Thus the method always gives the prescribed inclusion probabilities ${\pi }_i\text{}\mathrm{,}i\mathrm{<mo>=</mo>\; 1,}\mathrm{2,}\dots ,N\mathrm{.}$

Bondesson and Thorburn (2008) showed that a fixed size sampling is obtained only if ${\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^{}{\pi }_i}\mathrm{<mo>=</mo>}n$ and the weights are chosen such that ${\displaystyle {\sum }_{i\mathrm{<mo>=</mo>}j+1}^{}{w}_j^{\left(i\right)}}\mathrm{<mo>=</mo>\; 1,}j\in U\text{}\mathrm{.}$

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 $j$ first gives as much weight as possible to the closest unit (in distance) among the units $i\mathrm{<mo>=</mo>}j+\mathrm{1,}j+\mathrm{2,}\dots ,N\text{},$ 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 $P_i$ is constructed for each unit $i\in s,$ and $P_i$ includes all population units closer to unit $i$ than to any other sample unit $j\in s\mathrm{,}j\ne i.$ Optimally, each polytope should have a probability mass that is equal to 1. A measure of spatial balance of a realised sample $s$ is (see Stevens and Olsen, 2004)

$$B\mathrm{<mo>=</mo>}\frac{1}{n}{\displaystyle \sum _{i\in s}{\left(v_i-1\right)}^2}\text{}\mathrm{,}(3.3)$$

where $v_i$ is the sum of the inclusion probabilities of the units in $P_i.$ The expected value of $B$ 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.

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