On certain applications of Fractal and Wavelet Methods for data analysis

Abstract

In the last two decades, the study of wavelets and the exploration of the principles governing their behaviour have brought about sweeping changes in the disciplines of pure and applied mathematics and sciences. One of the most significant developments is the realization that, in addition to the canonical tool of representing a function by its Fourier series, there is a different representation more adapted to certain problems in data compression, noise removal and fast scientific computation. Fourier analysis does not provide appropriate method to study trends, abrupt changes, drift, breakdown points and self-similarity. Further, in transforming to frequency domain, time information is lost. Keeping these facts in mind, researchers made serious efforts to develop some methods and techniques to determine correctly these changes, in a time series, representing physical phenomenon. The thesis is related to time series analysis of Indian rainfall data and data of Indian stock markets BSE 100 and Nifty 50. We have used wavelets and wavelet based multifractal analysis for this study. We have divided this thesis into six chapters. In the first chapter, we give a brief introduction to the topic of this thesis. We present historical development, review of literature, a brief introduction to the wavelet theory and motivation for the work. Basic definitions and concepts of wavelet like wavelet coefficients, scaling functions, decomposition and reconstruction algorithms and multiresolution analysis, the heart of wavelet theory and some examples are described. We also give some aspects of fractal geometry like fractals, multifractals, and various measures of fractal dimension such as box dimension, fractal dimension and Hausdorff dimension here. We continue with the discussion on time series, time series analysis, Hurst exponent and fractal dimension, meteorology, stocks, stock market and ANFIS (adaptive network based fuzzy inference system). India is one of the largest countries whose economy and politic.