Index spaces serve as valuable metric models for studying properties relevant to various applications, such as social science or economics. These properties are represented by real Lipschitz functions that describe the degree of association with each element within the underlying metric space. After determining the index value within a given sample subset, the classic McShane and Whitney formulas allow a Lipschitz regression procedure to be performed to extend the index values over the entire metric space. To improve the adaptability of the metric model to specific scenarios, this paper introduces the concept of a composition metric, which involves composing a metric with an increasing, positive and subadditive function $\phi$. The results presented here extend well-established results for Lipschitz indices on metric spaces to composition metrics. In addition, we establish the corresponding approximation properties that facilitate the use of this functional structure. To illustrate the power and simplicity of this mathematical framework, we provide a concrete application involving the modelling of livability indices in North American cities.

翻译：指数空间作为研究社会科学或经济学等应用中相关属性的有效度量模型，这些属性由描述底层度量空间内各元素关联程度的实值Lipschitz函数表示。在给定样本子集内确定指数值后，可通过经典的McShane与Whitney公式执行Lipschitz回归过程，将指数值扩展到整个度量空间。为提升度量模型对特定场景的适应性，本文引入合成度量的概念——即通过递增、正且次可加的函数$\phi$对度量进行复合运算。本文结果将度量空间上Lipschitz指数的既有结论推广至合成度量情形，同时建立相应的逼近性质，以促进该函数结构的应用。为展示这一数学框架的简洁性与有效性，我们提供了涉及北美城市宜居性指数建模的具体应用案例。