Coordination of spatially balanced samples
Section 1. Introduction

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In the classical survey sampling framework, a random sample is selected from a finite population with a probability provided by the sampling design. The sampling design can be extended to the case of several samples, defining a joint probability to select them. On the other hand, two or more samples can be drawn from the same population or from overlapping populations, independently or not. Sample coordination applies to the latter case and seeks to create a probabilistic dependence between samples’ selections based on a joint sampling design. It is used in the case of repeated surveys or of several surveys. Two types of coordination are defined in the literature: positive and negative. In the former case, the goal is to maximize the overlap between different samples. In the latter, one wants to minimize it. Positive coordination can be used to reduce the survey costs or to induce a positive covariance between successive estimators of state in repeated surveys, and thus reduce the variance of an estimator of change. Negative coordination may be applied to reduce the response burden of units that have a risk of being selected for several surveys.

When updating a sample in repeated surveys over time (a panel), deaths, births or merge of the units can appear in the population. Thus, the population changes over time and the same sample can not be used at each time occasion. New samples are drawn at different time occasions, but a certain degree of overlap between samples can be required. This can be achieved using positive coordination. On the other hand, negative coordination is usually used to draw samples in several surveys, involving thus different but overlapping populations. Due to births, deaths, changes in activity or size, splits, mergers, etc. of units in the same population or due to the use of different overlapping populations, an important problem in sample coordination is the difficulty to manage the population changes over time or different overlapping populations. Usually, to overcome this problem, an overall population is constructed as a union of all units that ever existed, or as a union of different overlapping populations.

Various methods to provide sample coordination have been introduced in the literature. A summary of such methods is given for instance in Grafström and Matei (2015). An easy method to provide sample coordination is based on the use of so-called permanent random numbers introduced by Brewer, Early and Joyce (1972) for Poisson samples: one associates to each unit in the overall population an $U\left(\mathrm{0,1}\right)$ random number. Such a number is called a permanent random number (PRN); these numbers are independent and are used in all sample selections. The probabilistic dependence of the samples’ selection is thus created based on the use of permanent random numbers. Versions of the PRN method of Brewer et al. (1972) have been introduced in the literature (see Kröger, Särndal and Teikari, 1999; Kröger, Särndal and Teikari, 2003, for instance) and are widely used in different contexts. A recent example of a PRN method is the new system to coordinate business surveys by Statistics Canada. A two-phase stratified sampling design is used. The first-phase is a stratified sampling by Geography $\times$ Industry type $\times$ Business size and a Bernoulli sample is selected in each stratum by the use of PRNs. The main goal of the first-phase is to select a large sample covering all industries. For two consecutive first-phase waves a positive coordination is employed. In the second-phase, a sample is selected from the first-phase sample. For two consecutive second-phase waves, a negative coordination is applied to control the response burden of the business units (Haziza, 2013).

Our interest is to provide solutions to coordinate spatially balanced samples (for an overview on spatially balanced samples see Benedetti, Piersimoni and Postiglione, 2017). Usually, spatial sampling uses a space discretization, leading to the use of the classical sampling definition for finite populations. Thus, a population is defined as a finite set of units or locations having associated geographical coordinates. In most of the cases data are spatially autocorrelated and nearby locations tend to provide similar information. Consequently, it is desirable to sample units spread across the whole area of interest and to obtain a spatially balanced sample. The intuitive idea behind this is to cover through sampling the entire area of interest in order to obtain some representativeness. The selected sample should thus provide a full spatial coverage. Spatially balanced samples are efficient if a spatial trend is present in the variable of interest, denoted by $y\mathrm{.}$ Benedetti et al. (2017, page 447) note that “The motivation for the choice of selecting spatial well-spread samples is surely realistic if it is considered to be acceptable that increasing the distance between two units $k$ and $l$ increases the difference, observed at units $k$ and $l,$ namely, $\left|y_k-y_l\right|\mathrm{.}$ In this situation, it is evident that the variance of the Horvitz-Thompson estimator will necessarily decrease if we set high joint inclusion probabilities to pairs that have very different $y$ values.” Two spatial schemes useful for these goals are the local pivotal method (Grafström, Lundström and Schelin, 2012) and the spatially correlated Poisson sampling (Grafström, 2012). It was empirically found that both sampling schemes provide a good degree of spatial spreading, measured using Voronoi polytopes (see for instance Grafström et al., 2012, for some results).

We focus on coordination of spatially balanced samples using PRN methods, where sample selection is ensured using the local pivotal method (LPM) and the spatially correlated Poisson sampling (SCPS). Spatial sampling is used in many applications in environmental studies, forestry, agricultural surveys, but also in official statistics. We motivate the introduction of the coordinated spatially balanced samples by giving the following examples: 

• In ecological monitoring, it is important to preserve over time the same sampled spatial locations in order to measure the changes in species abundance. However, over time, these locations may disappear. Positive coordination can be applied in this case to ensure a significant overlap of the selected locations.
• In national forest inventories, both current state and change of several parameters, such as growing stock volume for different tree species, are of interest. The methodology we present can be used to make sure the sample is continuously updated, e.g., yearly, to be well-spread geographically or in auxiliary variables available from remote sensing (to improve estimates of current state); a high positive coordination would guarantee good estimates of change as well.
• Different official national business registers contain spatial coordinates of business units (e.g., US Census Bureau’s Longitudinal Business Database, the Swiss GeoStat, the Italian Statistical Archive of Active Enterprisers). The business units can be selected based on their geographical coordinates (Dickson, Benedetti, Giuliani and Espa, 2014). A negative coordination would be useful in this case to control the response burden of the units sampled by different surveys.

Note that methods to coordinate spatial samples have not yet been introduced in the literature. The novelty of the paper consists in introducing methods to coordinate spatially balanced samples. All the benefits of the sample coordination described above are provided for spatially balanced samples. In both types of coordination, the proposed methods preserve the spatial balancing property of the selected samples. Note that our goal is to control the overlap size between balanced samples, and not to improve sample coordination in general.

The paper is organized as follows. Section 2 introduces the notation. Sections 3.1 and 3.2 remind the local pivotal (LP) method and spatially correlated Poisson (SCP) sampling, respectively, while Section 3.3 a measure of spatial balance based on the Voronoi polytopes. We introduce methods to coordinate LP samples and SCP samples in Section 4. The same section introduces a new family of balanced sampling designs derived from SCP sampling, that provides good results for sample coordination. The coordination performances of the methods are presented in Section 5.1. Section 5.2 compares the new family of balanced sampling designs with Poisson sampling, while Section 5.3 provides simulation results for two typical estimators in repeated surveys. Section 6 shows an application of the proposed methods on real data. Discussion of the proposed methods and conclusions are provided in Section 7.

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