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| Autori principali: | , , |
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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2512.02397 |
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| _version_ | 1866909965989445632 |
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| author | Bossi, Emanuele Diggans, C. Tyler AlMomani, Abd AlRahman R. |
| author_facet | Bossi, Emanuele Diggans, C. Tyler AlMomani, Abd AlRahman R. |
| contents | The Boltzmann-Shannon Index (BSI) for clustered continuous data is introduced as a normalized measure that captures the relationship between geometry-based and frequency-based probability distributions defined over the clusters. In essence, it quantifies the similarity across densities of the clusters, which are defined by a given labeling. This labeling may originate from a geometric partitioning of the state space itself, but need not in general. We illustrate its performance on synthetic Gaussian mixtures, the Iris benchmark data set, and a high-imbalance resource-allocation scenario, showing that the BSI provides a coherent assessment in cases where traditional metrics give incomplete or misleading signals. Moreover, in the resource-allocation setting where equal density may be associated with a "fair" distribution, we demonstrate that BSI not only detects inequality with high sensitivity, but also offers a numerically smooth measure that can be easily embedded in optimization frameworks as a regularization term for modern policy-making. Finally, the BSI also offers a new measure of the effectiveness for a given symbolic representation, i.e. coarse-grain states, for continuous-valued data recorded from complex dynamical systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_02397 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Boltzmann-Shannon Index: A Geometric-Aware Measure of Clustering Balance Bossi, Emanuele Diggans, C. Tyler AlMomani, Abd AlRahman R. Information Theory The Boltzmann-Shannon Index (BSI) for clustered continuous data is introduced as a normalized measure that captures the relationship between geometry-based and frequency-based probability distributions defined over the clusters. In essence, it quantifies the similarity across densities of the clusters, which are defined by a given labeling. This labeling may originate from a geometric partitioning of the state space itself, but need not in general. We illustrate its performance on synthetic Gaussian mixtures, the Iris benchmark data set, and a high-imbalance resource-allocation scenario, showing that the BSI provides a coherent assessment in cases where traditional metrics give incomplete or misleading signals. Moreover, in the resource-allocation setting where equal density may be associated with a "fair" distribution, we demonstrate that BSI not only detects inequality with high sensitivity, but also offers a numerically smooth measure that can be easily embedded in optimization frameworks as a regularization term for modern policy-making. Finally, the BSI also offers a new measure of the effectiveness for a given symbolic representation, i.e. coarse-grain states, for continuous-valued data recorded from complex dynamical systems. |
| title | Boltzmann-Shannon Index: A Geometric-Aware Measure of Clustering Balance |
| topic | Information Theory |
| url | https://arxiv.org/abs/2512.02397 |