本文選題：分?jǐn)?shù)布朗運動 + 水平可視圖　； 參考：《湘潭大學(xué)》2015年碩士論文

【摘要】：本文針對不同Hurst指數(shù)的分?jǐn)?shù)布朗運動時間序列,運用水平可視圖方法構(gòu)建了水平可視網(wǎng)絡(luò),并對這些網(wǎng)絡(luò)進(jìn)行了研究。已有研究表明水平可視網(wǎng)絡(luò)繼承了原始對應(yīng)分?jǐn)?shù)布朗運動時間序列的分形性質(zhì),我們的工作是探究這些水平可視網(wǎng)絡(luò)是否具有多重分形性質(zhì),并試圖從圖論的Laplace譜分析方面分析水平可視網(wǎng)絡(luò)的性質(zhì)。首先,我們利用沙箱法對水平可視網(wǎng)絡(luò)進(jìn)行重分形分析,試圖找出這些是否具有重分形性質(zhì),并探究這種性質(zhì)與Hurst指數(shù)H的關(guān)系。利用沙箱法我們首先得到了水平可視網(wǎng)絡(luò)的平均分形維數(shù)與Hurst指數(shù)H的依賴關(guān)系dB≈2-H,接近于理論解,與已有研究結(jié)果相符。接下來,我們得到的數(shù)值結(jié)果表明我們的水平可視網(wǎng)絡(luò)具有重分形性質(zhì),并且這些水平可視網(wǎng)絡(luò)的重分形性質(zhì)隨著指數(shù)H的增大而大致趨于減弱,而網(wǎng)絡(luò)的平均信息維數(shù)D(1)和平均關(guān)聯(lián)維數(shù)D(2)隨著指數(shù)H的增大而趨于減小。其次,我們進(jìn)一步對所構(gòu)建水平可視網(wǎng)絡(luò)的經(jīng)典的和改進(jìn)的Lapalce次小特征值、第三小特征值及最大特征值和它的Lapalce能量進(jìn)行了研究。我們發(fā)現(xiàn)無論是經(jīng)典的還是改進(jìn)的Lapalce次小特征值、第三小特征值都隨Hurst指數(shù)H的增大而減小,并與H呈線性關(guān)系,而最大特征值與H的關(guān)系隨Lapalce矩陣定義的不司而不同。根據(jù)實驗結(jié)果發(fā)現(xiàn),經(jīng)典Laplace最大特征值隨H的增大而減小,而改進(jìn)的Laplace最大特征值卻是隨H的增大而增大。同時,這些水平可視網(wǎng)絡(luò)的Lapalce能量也隨著Hurst指數(shù)H的增大而減少。
[Abstract]:In this paper, based on the fractional Brownian motion time series with different Hurst exponents, the horizontal visual networks are constructed by using the horizontal visual graph method, and these networks are studied. It has been shown that the horizontal visual networks inherit the fractal properties of the original fractional Brownian motion time series. Our work is to find out whether these horizontal visual networks have multifractal properties. This paper attempts to analyze the properties of horizontal visual networks from the point of view of Laplace spectrum analysis of graph theory. Firstly, we use the sandbox method to analyze the multifractal of horizontal visual networks, and try to find out whether these networks have multifractal properties, and explore the relationship between these properties and Hurst exponent H. By using the sandbox method, we first obtain the dependence of the average fractal dimension of the horizontal visual network on the Hurst exponent H, dB 鈮，

本文編號：1779538