| Summary: | 博士 === 國立臺灣大學 === 物理研究所 === 102 === The main theme of this thesis is about the study of stochasticity in complex systems and its applications. First, we introduce the probability theory, Brownian motions, Ito process and multifractal random walk model, which are the basic models to describe the stochastic behavior of the time series. Then we develop methods to study the properties of the non-stationary time series, of which the mean and variance are changing with time, and quantitatively measure the deviation of the time series from a Wiener process so that the stochasticity of different time series can be compared. In addition, we make use of the properties we found in the time series of stock market to propose a trading strategy to avoid losing fortune in the bear market and the overall trading results beat the average performance of the stock markets. We apply our algorithms to study the heart rate variability(HRV) in cardiology system. Our results can be used to distinguish the difference between the HRV of the healthy people and the patients with sleep apnea symptom. We introduce some numerical methods in the appendices we used in studying the time series such as optimization methods and clustering phenomena. This thesis is organized as follows. In Chapter 1, we introduce the stochastic process and the multifractal random walk. In Chapter 2, we introduce our algorithms to measure the stochasticity in the non-stationary time series and its properties. In Chapter 3 and Chapter 4, we present the applications of our algorithms. A trading strategy which can be applied to any stock markets is introduced in Chapter 3. We define a parameter to distinguish the difference between the HRV of the healthy people and the patients with sleep apnea symptom in Chapter 4. Chapter 5 is the conclusion of our works in this thesis. In Chapter 6 (Appendix 1), we introduce a numerical method to solve optimization problems: the Genetic algorithm. In Chapter 7 (Appendix 2), we introduce the clustering effect which is the cause of the long term correlations and its applications. Chapter 8 (Appendix 3) is some miscellaneous contents which contain several useful functions and the Matlab source codes employed in the analysis and simulation in this thesis.
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