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Fractal Dimension for Fractal Structures

With Applications to Finance

  • Manuel Fernández-Martínez
  • Juan Luis García Guirao
  • Miguel Ángel Sánchez-Granero
  • Juan Evangelista Trinidad Segovia
Book

Part of the SEMA SIMAI Springer Series book series (SEMA SIMAI, volume 19)

Table of contents

  1. Front Matter
    Pages i-xvii
  2. Manuel Fernández-Martínez, Juan Luis García Guirao, Miguel Ángel Sánchez-Granero, Juan Evangelista Trinidad Segovia
    Pages 1-48
  3. Manuel Fernández-Martínez, Juan Luis García Guirao, Miguel Ángel Sánchez-Granero, Juan Evangelista Trinidad Segovia
    Pages 49-83
  4. Manuel Fernández-Martínez, Juan Luis García Guirao, Miguel Ángel Sánchez-Granero, Juan Evangelista Trinidad Segovia
    Pages 85-147
  5. Manuel Fernández-Martínez, Juan Luis García Guirao, Miguel Ángel Sánchez-Granero, Juan Evangelista Trinidad Segovia
    Pages 149-195
  6. Back Matter
    Pages 197-204

About this book

Introduction

This book provides a generalised approach to fractal dimension theory from the standpoint of asymmetric topology by employing the concept of a fractal structure. The fractal dimension is the main invariant of a fractal set, and provides useful information regarding the irregularities it presents when examined at a suitable level of detail. New theoretical models for calculating the fractal dimension of any subset with respect to a fractal structure are posed to generalise both the Hausdorff and box-counting dimensions. Some specific results for self-similar sets are also proved. Unlike classical fractal dimensions, these new models can be used with empirical applications of fractal dimension including non-Euclidean contexts.

In addition, the book applies these fractal dimensions to explore long-memory in financial markets. In particular, novel results linking both fractal dimension and the Hurst exponent are provided. As such, the book provides a number of algorithms for properly calculating the self-similarity exponent of a wide range of processes, including (fractional) Brownian motion and Lévy stable processes. The algorithms also make it possible to analyse long-memory in real stocks and international indexes.

This book is addressed to those researchers interested in fractal geometry, self-similarity patterns, and computational applications involving fractal dimension and Hurst exponent.


Keywords

fractal fractal structure fractal dimension Hausdorff dimension Hurst exponent

Authors and affiliations

  • Manuel Fernández-Martínez
    • 1
  • Juan Luis García Guirao
    • 2
  • Miguel Ángel Sánchez-Granero
    • 3
  • Juan Evangelista Trinidad Segovia
    • 4
  1. 1.Department of Sciences and ComputationUniversity Centre of Defence at Spanish Air Force AcademySantiago de la RiberaSpain
  2. 2.Departamento de Matemática Aplicada y EstadísticaUniversidad Politécnica de CartagenaCartagenaSpain
  3. 3.Departamento de MatemáticasUniversidad de AlmeríaLa Cañada de San UrbanoSpain
  4. 4.Departamento de Ciencias Económicas y EmpresarialesUniversidad de AlmeríaLa Cañada de San UrbanoSpain

Bibliographic information

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