A FRACTAL APPROACH TO TIME SERIES ANALYSIS WITH EMPHASIS ON INDIAN PLATE SEISMICITY

Abstract

Fractals are complex geometric shapes that exhibit self-similarity at different scales. They are ubiquitous in nature and have been used to model a wide range of phenomena, biological growth, and the behaviour of financial markets. Fractals have also been applied to the analysis of time series data. The application of fractals to time series analysis has been motivated by the observation that many natural and man-made processes exhibit fractal behaviour. One approach to analyzing fractal behaviour in time series data is using a Chaos Game representation (CGR), a geometric method that generates fractals by iteratively placing points in a defined space. This defined space can be any polygon like a triangle, square, pentagon or hexagon. A CGR on a square had unique properties, and this method was usually used to visually represent data, which made it difficult to compare CGRs. Currently, available studies lack algorithms or methods to derive statistical data from a CGR plot and to compare them statistically. Comparing CGRs statistically could give the rate of similarity /dissimilarity between respective data sequences used to plot them. The proposed research discusses a modified representation of CGR, algorithm/methods developed to study the sequence used to produce CGR, the application of the CGR in studying earthquake time series and the application of the developed methodologies in other fields of science. The current work presented can be broadly divided into three parts. The first part discusses Percentage Chaos Plots (PC Plots), the algorithms developed for analyzing PC plots, and the introduction of a new statistical tool called proximity index. Percentage Chaos Plots (PC Plots) were a modified method of representing CGR plots. From these plots, 'box addresses' were derived. Box addresses are unique addresses of each sub-square in a PC plot. The algorithms developed for deriving box addresses from PC Plots allow for the precise identification of occurrence rates of different combinations and patterns in the data sequence used to generate the CGR. Additionally, programs were developed to perform this process efficiently and quickly. The concept proximity index was also introduced as a statistical tool for comparing CGRs from different sequences, regardless of the length of the sequence used to produce them.