A Hybrid Fourier-Fuzzy-Fibonacci Framework for Modeling Chaotic Financial Time Series

Abstract

Problem: Modeling non-periodic, chaotic signals, particularly within volatile financial markets such as the stock market, presents a formidable challenge. Classical analytical tools, including Fourier series, often prove inadequate when confronted with the inherent irregularity and non-stationarity of financial time series. Conversely, while computational intelligence methodologies like fuzzy logic systems and machine learning models offer powerful alternatives, they can suffer from a lack of theoretical robustness, interpretability, or may require extensive data for training. Consequently, traditional financial models frequently oversimplify complex market dynamics or fail to accurately capture underlying structural patterns, especially at critical junctures such as market peaks and troughs.

Proposed Solution: This paper introduces and evaluates a novel hybrid modeling framework designed to address these limitations. The proposed framework synergistically integrates three distinct yet complementary analytical techniques: the Fourier Transform, Fuzzy Logic, and the Fibonacci Sequence.

Methodology: The Fourier Transform is employed for its efficacy in frequency domain decomposition, enabling the identification of dominant cyclical components and underlying trends within financial data. Fuzzy Logic is incorporated to manage the inherent uncertainty and to model the imprecise, often ambiguous, transitional states characteristic of market behavior. Finally, the Fibonacci Sequence, a tool frequently utilized in technical analysis, is integrated for its potential in aligning and optimizing the identification of patterns within observed oscillatory behavior, with a particular focus on its application at critical price inflection points.

Expected Contributions: The proposed hybrid framework is anticipated to deliver several key contributions. Firstly, it is expected to offer improved accuracy in modeling and potentially predicting chaotic financial time series compared to standalone models or simpler hybrid approaches. Secondly, it aims to enhance the interpretability of model outputs, offering more transparent insights into market dynamics than conventional "black-box" models. Lastly, the framework is designed to provide more robust and nuanced insights into the structural characteristics of financial markets, particularly concerning the identification and understanding of behavior around critical turning points.