Intelligent n-Means Spatial Sampling

Well-spread samples are desirable in many disciplines because they improve estimation when target variables exhibit spatial structure. This paper introduces an integrated methodological framework for spreading samples over the population's spatial coordinates. First, we propose a new, translation-invariant spreadness index that quantifies spatial balance with a clear interpretation. Second, we develop a clustering method that balances clusters with respect to an auxiliary variable; when the auxiliary variable is the inclusion probability, the procedure yields clusters whose totals are one, so that a single draw per cluster is, in principle, representative and produces units optimally spread along the population coordinates, an attractive feature for finite population sampling. Third, building on the graphical sampling framework, we design an efficient sampling scheme that further enhances spatial balance. At its core lies an intelligent, computationally efficient search layer that adapts to the population's spatial structure and inclusion probabilities, tailoring a design to each specific population to maximize spread. Across diverse spatial patterns and both equal- and unequal-probability regimes, this intelligent coupling consistently outperformed all rival spread-oriented designs on dispersion metrics, while the spreadness index remained informative and the clustering step improved representativeness.