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Generating fractal functions associated with Suzuki iterated function systems

Mridul Patel, G. Verma, A. Eberhard, A. Rao

TL;DR

The paper addresses constructing fractal interpolation functions by leveraging Suzuki-type generalized $\varphi$-contraction mappings (STGPC) to define $\alpha$-fractal functions within an iterated function system (IFS). It develops the $\alpha$-FIF framework with a base function $b=Lg$ and a fractal operator $\mathcal{G}^\alpha$, proving existence and uniqueness of a fixed point and providing an error bound $\|g^\alpha-g\|_\infty \le \frac{\|\alpha\|_\infty}{1-\|\alpha\|_\infty}\|g-b\|_\infty$. A case study on spinach price volatility in the Azadpur market demonstrates how varying the scaling vector $\alpha$ yields graphs of different fractal complexity and box-dimension values via Theorem 5. The work offers a general, non-Banach approach to modeling irregular, self-similar data with potential applications in finance forecasting and time-series analysis.

Abstract

This article constructs a fractal interpolation function, also referred to as $α$-fractal function, using Suzuki-type generalized $\varphi$-contraction mappings (STGPC). The STGPC is a generalization of $\varphi$-contraction mappings. The process of constructing $α$-fractal functions using the STGPC is detailed, and examples of STGPC are given. The FIF has broad applications in data analysis, finance and price prediction. We have included a case study analyzing the price volatility of spinach in the Azadpur vegetable market in New Delhi. The fractal analysis gives a unique perspective on understanding price fluctuations over a period. Finally, the box-dimensional analysis is presented to comprehend the complexity of price fluctuations.

Generating fractal functions associated with Suzuki iterated function systems

TL;DR

The paper addresses constructing fractal interpolation functions by leveraging Suzuki-type generalized -contraction mappings (STGPC) to define -fractal functions within an iterated function system (IFS). It develops the -FIF framework with a base function and a fractal operator , proving existence and uniqueness of a fixed point and providing an error bound . A case study on spinach price volatility in the Azadpur market demonstrates how varying the scaling vector yields graphs of different fractal complexity and box-dimension values via Theorem 5. The work offers a general, non-Banach approach to modeling irregular, self-similar data with potential applications in finance forecasting and time-series analysis.

Abstract

This article constructs a fractal interpolation function, also referred to as -fractal function, using Suzuki-type generalized -contraction mappings (STGPC). The STGPC is a generalization of -contraction mappings. The process of constructing -fractal functions using the STGPC is detailed, and examples of STGPC are given. The FIF has broad applications in data analysis, finance and price prediction. We have included a case study analyzing the price volatility of spinach in the Azadpur vegetable market in New Delhi. The fractal analysis gives a unique perspective on understanding price fluctuations over a period. Finally, the box-dimensional analysis is presented to comprehend the complexity of price fluctuations.

Paper Structure

This paper contains 13 sections, 10 theorems, 55 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Background and Preliminaries
  3. Iterated function systems (IFS)
  4. Fractal interpolation function using Suzuki type generalized $\varphi$- contractions
  5. Iterated function system using Suzuki type generalized $\varphi$-contractive mapping
  6. Construction of $\alpha$-fractal function
  7. Dimensional Analysis
  8. Case study
  9. Methodology
  10. Case study data
  11. Fractal Analysis
  12. Box Dimension Analysis
  13. Conclusion and future work

Key Result

Theorem 1

Let $\mathcal{C}$ be the space of continuous functions and $\lbrace K;w_p: p=1,2,\ldots, P \rbrace$ be an IFS as defined in IFS. Define $g:A\rightarrow \mathbb{R}$ with $h(y_0)=z_0, h(y_P)=z_P$ associated with the metric $d(h,f)=\max\lbrace|h(y)-f(y)|:y\in A \rbrace$. Then $(\mathcal{C},d)$ is a com The function $g$ in FIF is called the FIF associated with $\lbrace (y_p,z_p) \in A\times \mathbb{R

Figures (3)

  • Figure 1: Graph of $\alpha$-FIF through STGPC for $\alpha=0.5$ for the given data set
  • Figure 2: Methodlogy flowchart
  • Figure 3: $\alpha$-FIF for the data set $(y_i,z_i)$ for the different values of $\alpha$

Theorems & Definitions (18)

  • Theorem 1: Theorem 1, barnsley1986fractal
  • Theorem 2: Theorem 2, suzuki2008generalized
  • Definition 1: $\varphi$-contraction mapping, jachymski1997equivalence
  • Theorem 3: Theorem 1, boyd1969nonlinear
  • Definition 2: Suzuki-type generalized $\varphi$-contraction mapping, pant2018fixed
  • Theorem 4: Theorem 2.2, pant2018fixed
  • Example 1
  • Example 2
  • Lemma 1: Lemma 3.1, pant2018fixed
  • Lemma 2: Lemma 3.2pant2018fixed
  • ...and 8 more