Study On The Fractal-like Self-similar Structures In Financial Data Recurrence Plot

Since the late 20 th century, the emerging of nonlinear dynamics, especially the emerging of the chaos and fractal theory, has brought about not only changes of ideology in the research of economics and finance but also exploratory and constructive analysis tools. The application of the chaos and fractal theory has observably changed our research tradition of the well-known time series analysis and provides new vision for the research method of using chaotic time series to study financial data. In recent years, study on the dynamic mechanism behind fractal and specific connotation of physics and economics has been an arresting cutting-edge topic.The basis of chaotic time series analysis is the phase-space reconstruction theory, and the recurrence plot method is an effective way for analyzing the nonlinear structure of time series. In this paper, the phase-space reconstruction theory and the recurrence plot method are adopted to analyze and study the inner structure of a variety of financial data, and the peripheral point analysis method and the sandpile evolution process model for the fractal-like self-similar structures are innovatively proposed on this basis. The paper is aimed at exploring the complexity of financial data from a new angle.In China, many scholars place extra emphasis on the study of phase-space reconstruction theory and technological improvement in such aspects as selection of best embedding dimensions, delay time, CAO method etc; many other scholars focus on the analytical application of financial market by mainly using such methods as correlation dimension, maximum Lyapunov index, R/S analysis, Hurst index etc; some scholars combine wavelet transformation, neural network forecast technology and other methods to conduct chaotic feature research on financial prices(indices), verify that they are complex nonlinear systems and explore possible forecasting methods. However, there are few studies involving the use of recurrence plot for analysis.Different from general research thoughts of domestic and foreign scholars, this paper ignores the research thoughts on any assumed conditions but completely conducts analysis starting with the intuitive structure of financial data recurrence plot. In terms of data analysis, a method similar to data mining is adopted. On the premise of there is no specific assumed condition, the structural features reflected in the plot are analyzed, and the statistical regularity reflected in data is analyzed.Regarding the research results, first, this paper conducts in-depth study on the fractal-like self-similar structures in the recurrence plot, analyzes in detail the fractal-like self-similar structures in the financial data recurrence plot from multiple angles of multi-group data, various cycles and multi-group parameter combination, and has obtained lots of useful results. Specifically, it relatively analyzes in detail the fractal-like self-similar structures in the financial data recurrence plot; proposes the image identification method to identify the fractal-like self-similar structures in the financial data recurrence plot; has found that fractal-like self-similar structures is ubiquitous in the financial data recurrence plot; studies the stability of the fractal-like self-similar structures in the same kind of financial data in different cycle data; studies the stability of the fractal-like self-similar structures under the situation that the phase-space dimension and the distance parameter are changed and determines parameter variation range for the stable existence of the structure; studies the similarities between the fractal-like self-similar structures in different financial data recurrence plots. In addition, in the process of calculating the correlation dimensions of time series, this paper uses intuitive analysis method through judging the sharpness of the structures in the recurrence plot to ensure the choosing of phase-space dimension and the distance parameter be set on the premise that fractal-like self-similar structures in the financial data recurrence plot must be clear and complete. The relevant dimensions obtained by using this method tend to be more reasonable.Second, this paper innovatively proposes a new method and a new model for studying the time series of financial data, namely the peripheral point analysis method and the sandpile evolution model for the fractal-like self-similar structures. The former is a method used to measure the correlation between historical events about finance. Through quantitative analysis on the correlation between numerous data of the points in the financial data recurrence plot and with the help of a series of new conceptual analysis tools including the starting point, farthest distance point, peripheral point, farthest peripheral point etc, the paper studies the major historical events(groups) influencing the financial market and the logic of market development and evolution. The paper analyzes the major historical events(groups) influencing the historical course of China stock markets with Shanghai composite index as an example and has obtained the results in conformity with the reality. Moreover, through contrastive analysis on the similarities between the drawing process of the fractal-like self-similar structures in the recurrence plot and the sandpile stacking and collapse process, the paper puts forward the sandpile evolution model of the fractal-like self-similar structures to analyze the sand avalanche and zooming phenomena contained in financial data, explore the evolution process of the complexity in financial data and verify the effectiveness of the method with Shanghai composite index as the example.The research methods of this paper can be used for the research on the chaotic time series in other fields, such as earthquake research, meteorological research etc.