Graphical Finite Population Sampling

TL;DR

This work addresses the challenge of designing finite population samples when first-order inclusion probabilities (FIP) are fixed but second-order inclusion probabilities (SIP) can be tuned for efficiency. It introduces Graphical Finite Population-Sampling (GFS), a visual framework that represents $\pi_k$ as bars on a plane, enabling the generation of a broad family of designs while preserving $FIP$ and allowing exact computation of SIPs; it also presents Fixed-size GFS and Chaotic GFS variants to accommodate fixed sample sizes and to expand the space of feasible SIP patterns. To operationalize design optimization within GFS, the paper proposes OGFS (Optimal GFS) — a greedy best-first search that iteratively refines designs to minimize a chosen criterion $C(\theta_z=Z,\mathbf p)$, using auxiliary variables to balance information about the main variable $Y$ and an auxiliary $Z$. Through simulations on synthetic data and the MU284 real dataset, OGFS demonstrates robust improvements in efficiency over SRS, Cube Method, and DSD in many settings, illustrating the practical value of a flexible, graphical approach to sampling design. Overall, GFS offers a versatile, integrative pathway to explore, compare, and optimize sampling designs within a single construction, with potential extensions to broader intelligent-search methods and standard designs, while acknowledging scalability challenges for large $N$.

Abstract

This paper introduces an innovative and intuitive finite population sampling method that has been developed using a unique graphical framework. In this approach, first-order inclusion probabilities are represented as bars on a two-dimensional graph. By manipulating the positions of these bars, researchers can create a wide range of different sampling designs. This graphical visualization of sampling designs facilitates the exploration of alternative designs and may simplify certain aspects of the implementation compared to traditional mathematical algorithms. This novel approach holds significant promise for tackling complex challenges in sampling, such as achieving an optimal design. By applying a version of the greedy best-first search algorithm to this graphical approach, the potential for integrating intelligent algorithms into finite population sampling is demonstrated.

Figures (8)

• Figure 1: The top two plots depict different arbitrary arrangements of bars with $\bm{\pi} = \{.38, .30, .42, .65, .25, .10, .90\}$ shown in different colors based on Algorithm \ref{['Al00']}, in which both arrangements respect the specified FIP. The designs ($\bm{p}$ and $\bm{p^*}$) are calculated and presented to the left of the plots, showing that any change in the position of the bars results in creating a new design. The bottom plot illustrates two examples: (1) The calculation of the SIP for units $k=2$ and $\ell=3$, highlighted in gray, resulting in $\pi_{k\ell}=.12$, (2) A random line (horizontal black line) is drawn, selecting units $1$, $5$ and $7$ as the final sample, $s=\{1,5,7\}$.
• Figure 2: An example of generating a new design $\bm{p^*}$ (the middle plot of Figure \ref{['fig:3random']}). By selecting $\Delta_2$ and $\Delta_5$ with heights $p^*_2 = 0.2$ and $p^*_5 = 0.15$, and choosing substrips $\delta_2$ and $\delta_5$ of size $v_{2,5}(7/15)=0.07$, it is possible to interchange segments between substrips $\delta_2$ and $\delta_5$ for unit $k=4$, which yields four samples: $s_{2,1}$, $s_{2,2}$, $s_{5,1}$, and $s_{5,2}$.
• Figure 3: The plot depicts a fixed-size version (Algorithm \ref{['Alfixed']}) of GFS with $\bm{\pi} = \{.38, .3, .42, .65, .25, .1, .9\}$, arranged sequentially to create a fixed-size design.
• Figure 4: The top plot depicts a fixed-size version of GFS with $\bm{\pi} = \{0.38, 0.30, 0.42, 0.65, 0.25, 0.10, 0.90\}$, arranged sequentially to create a fixed-size design in which two strips, $\Delta_2$ and $\Delta_5$, are selected. Within these strips, two substrips of size $v_{2,5} = v_{2,5}(10/25) = 0.10$ are indicated by dashed lines. Since units $1$ and $5$ are interchangeable in this situation, they are interchanged, resulting in two new samples, $s_{2,2}$ and $s_{5,2}$, alongside the original samples, $s_{2,1}$ and $s_{5,1}$, depicted in the bottom plot.
• Figure 5: Implementing of Algorithm \ref{['Al2']} on the data of population \ref{['FIPexample']} with $\alpha = .5$ and $M = 10,000$.

Theorems & Definitions (8)

• Theorem 2.1
• proof
• Conjecture 3.1
• Definition 4.1
• Definition 4.2
• Theorem 4.3
• proof
• Conjecture 4.4