Beyond Two Barycenters: Adaptive Complexity in Distributional Regime Geometry

Abstract

Distributional-regime methods often summarize market geometry with a small set of reference distributions. In the Distributional Regime Persistence (DRP) framework, the maintained representation has been a calm/stress pair of Wasserstein barycenters. This paper asks whether that two-anchor geometry is empirically sufficient. Using frozen pre-2007 calibration samples for the S&P 500, FTSE 100, Nikkei 225, and Hang Seng Index, I fit one-dimensional Wasserstein-1 k-medians representations with one to four prototypes and evaluate them strictly out of sample from January 2007 through June 2026. The analysis is repeated in raw return-distribution space and in independently median-centered, IQR-scaled shape space. Additional prototypes materially reduce out-of-sample reconstruction loss in every market and both spaces: moving from two to three prototypes lowers loss by 6.4%–9.6% in raw space and 7.9%–9.1% in shape space, while moving from three to four still lowers loss by 4.1%–6.4% and 2.8%–4.9%, respectively. Yet lower loss does not imply stable regime identity. Calibration-block bootstrap recurrence rejects several apparently useful extra prototypes, and the number of stable additional anchors differs by market and representation. Stable extra prototypes do, however, recur across markets under a frozen analogue criterion. The main result is therefore a dimensionality distinction: two anchors underfit distributional geometry, but approximation dimension exceeds stable identity dimension. Shape normalization does not eliminate the extra geometry, so the result cannot be reduced to finer volatility-scale partitioning.

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