量測複雜系統隨機度與多重碎形維度之演算法開發與應用

Abstract

本篇論文主要是在研究複雜系統的隨機度分析及應用。首先我們會從機率理論、布朗運動和隨機過程出發，去探討一些基本的時間序列特性。接著我們會用提出方法去探討非穩態時間序列的特性。所謂的非穩態的時間序列是指其平均值和標準差是時間的函數，而不是常數。除此之外，我們也利用開發出的方法去分析不同種的時間序列，例如股票市場和心律變動性分析。利用一些股票市場的特性，我們提出了一種適合保守型投資者的投資策略，此策略可以避免投資者承受金融海嘯的損失並且其交易結果優於大多數的股票平均。在心律變動分析方面，我們所提出的方法可以精準地分辨出睡眠呼吸中止症的患者和一般民眾的心跳節奏的不同，因此可以在醫學上有很大的應用潛力。在本篇論文的最後，我們附上一些我們使用過的數值方法和程式碼，以方便讀者可以輕鬆地重複我們的一些研究成果。

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.