On Grouping for Maximum Homogeneity

Fisher1958Vector quantization and Voronoi theory

doi:10.1080/01621459.1958.10501479

Scalar grouping and 1-D contiguity (30 Aug 2026)

Use: the classical anchors for DS8 (scalar efficient-score DP) and the $\mathrm{btw}\le\hat v_K$ leg of DS15's Proposition 4; the DS15 audit verified contiguity against full partition lattices, including duplicate values and unequal weights.
Does not transfer: one-dimensional SSE only; nothing about matrix criteria or feasibility constraints.
Sources: W. D. Fisher (1958, JASA 53:789–798, verbatim): optimal one-dimensional weighted least-squares partitions are contiguous in the sorted order (appendix proof), reducing the search to $\binom{K-1}{G-1}$ contiguous partitions; credits the continuous-distribution analogue to Dalenius (1950) and Cox (1957). Bellman (1961, CACM 4:284): dynamic programming for segmented least squares — the exact-DP citation. Wang & Song (2011, R Journal 3(2):29–33): Ckmeans.1d.dp, modern guaranteed-optimal 1-D DP.

JASA 53(284):789-798 (verbatim-verified 30 Aug 2026). Optimal 1-D weighted SSE partitions are contiguous (appendix proof); reduction to C(K-1,G-1) contiguous partitions - the classical anchor for the scalar interval optimum and exact DP step (DS8, DS15 Proposition 4). Credits the continuous analogue to Dalenius 1950 and Cox 1957.

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Publication-grade audit of the DS18 exact off-(L) basin and empirical transferAUDIT-DS-NONCENTERED-GLOBAL-BASIN-TRANSFER