将Fisher判别分析拓展到区间数据，提升分类可解释性。
Interval Fisher's Discriminant Analysis and Visualisation
- 用摩尔区间算术和马洛斯距离扩展Fisher判别函数，同时考虑区间中心与范围。
- 在真实数据集上实现高精度分类，判别方向能有效区分不同类别区间特征。
- 配套可视化工具帮助理解分类结果，适合处理带不确定性的区间数据场景。
在数据科学中，实体通常由单一数值表示。符号数据分析将其扩展至区间、直方图等更复杂的结构，以表达内部变异性。本文提出将多类Fisher判别分析推广至区间值数据，采用摩尔区间算术和马洛斯距离，将Fisher目标函数扩展为同时考虑区间中心与范围的贡献，并通过数值最大化求解。所得判别方向用于对区间值观测进行分类。为支持视觉评估，我们改编了原始用于常规数据的类别地图，适用于基于最小距离规则的分类器；同时扩展轮廓图（silhouette plot）并使用堆叠马赛克图补充类别分配的可视化展示。这些图形工具共同揭示分类器性能及类别归属强度。在真实数据集上的应用验证了该方法的有效性，显著提升了区间值数据分类结果的可解释性。
原文摘要 · Abstract (English)
In Data Science, entities are typically represented by single valued measurements. Symbolic Data Analysis extends this framework to more complex structures, such as intervals and histograms, that express internal variability. We propose an extension of multiclass Fisher's Discriminant Analysis to interval-valued data, using Moore's interval arithmetic and the Mallows' distance. Fisher's objective function is generalised to consider simultaneously the contributions of the centres and the ranges of intervals and is numerically maximised. The resulting discriminant directions are then used to classify interval-valued observations.To support visual assessment, we adapt the class map, originally introduced for conventional data, to classifiers that assign labels through minimum distance rules. We also extend the silhouette plot to this setting and use stacked mosaic plots to complement the visual display of class assignments. Together, these graphical tools provide insight into classifier performance and the strength of class membership. Applications to real datasets illustrate the proposed methodology and demonstrate its value in interpreting classification results for interval-valued data.
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