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Moduli of Continuity in Metric Models and Extension of Liveability Indices

R. Arnau, J. M. Calabuig, Álvaro González, Enrique A. Sánchez Pérez

Abstract

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 $φ$. 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.

Moduli of Continuity in Metric Models and Extension of Liveability Indices

Abstract

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 . 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.
Paper Structure (13 sections, 7 theorems, 33 equations, 5 figures, 5 tables)

This paper contains 13 sections, 7 theorems, 33 equations, 5 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Basic definitions and concepts
  3. Composition metrics and modulus of continuity
  4. Index extension
  5. Approximation through standard indices
  6. Extension theorems: McShane and Whitney formulas
  7. Applications: the liveability index for cities
  8. Methodology
  9. Extension of a liveability index
  10. Conclusions
  11. Contribution
  12. Funding
  13. Data availability

Key Result

Theorem 2.1

If $f\colon S \subset D\to\mathbb{R}$ is an $L$-Lipschitz function, then there exists an $L$-Lipschitz function $F\colon D\to\mathbb{R}$ such that $F\restriction_{S}=f$. This function $F$ is not unique, and two possible formulae are which are known as the McShane and Whitney extensions respectively.

Figures (5)

  • Figure 1: Comparative between $d(x,y)$, in pink, and $d_{\phi}(x,y)$, in blue.
  • Figure 2: Representation of the triangular inequality of $d$ and $d_{\phi}$.
  • Figure 3: Comparison of extension errors using McShane-Whitney formulas, for different metrics.
  • Figure 4: Comparison of extension errors using standard indices, for different metrics.
  • Figure 5: Comparison of extension errors using McShane-Whitney formulas, for different metrics.

Theorems & Definitions (20)

  • Theorem 2.1
  • remark 2.2
  • Definition 2.3
  • Definition 2.4
  • proposition 2.5
  • proof
  • example 2.6
  • proposition 2.7
  • Definition 3.1
  • Definition 3.2
  • ...and 10 more