臺灣博碩士論文加值系統

近年來，分數維度布朗運動(Fractional Brownian Motion，簡稱FBM)模式已經被使用在許多不同的領域。在此模式的應用上，必須估測赫斯特參數(Hurst parameter, H)，此參數與碎型維度(fractal dimension, D)的關係為 。然而，由於離散時間分數維度布朗運動(discrete-time FBM，簡稱DFBM)的非穩固性(nonstationarity)，因此，它的被稱為離散時間分數維度高斯雜訊(discrete-time fractional Gaussian noise，簡稱DFGN)的增量(increment)程序被引用為一個估測H的輔助工具。在此論文中，離散時間分數維度高斯雜訊與離散分數維度差分高斯雜訊(discrete fractionally differenced Gaussian noise，簡稱fdGn)的關係被分析。赫斯特參數與熵(entropy)的關係被獲得。並且證明離散時間分數維度高斯雜訊是正規的(regular)。基於此正規性，一個快速且精確的估測赫斯特參數的方法被提出。此方法擁有比最大相似估測器(maximum likelihood estimator，簡稱MLE)和移動平均(moving average，簡稱MA)法更小的計算量。此外，此方法在振幅平移是強健的(robust)，且不隨時間平移而變化，並且不受功率密度函數(power spectral density，簡稱PSD)的尺度因數的影響。最後，此方法和嵌有赫斯特參數的熵數估測將被應用於外尿道括約肌(external urethral sphincter，簡稱EUS)的肌電圖(electromyogram，簡稱EMG)。這些信號來自於未受傷的成鼠和遭受脊髓損傷(spinal cord injury，簡稱SCI)的成鼠。分析顯示我們可以由此新訊息分辨未受傷與脊髓損傷的成鼠。

In recent years, fractional Brownian motion (FBM) model has been used in a large number of different disciplines. In the application of this model, it is imperative to estimate the Hurst parameter , which is directly related to fractal dimension . However, due to the nonstationarity of the discrete-time fractional Brownian motion (DFBM), its increment process, referred to as discrete-time fractional Gaussian noise (DFGN), is invoked as an auxiliary tool to estimate . In this dissertation, the relation between DFGN and discrete fractionally differenced Gaussian noise (fdGn) is analyzed. The relation between Hurst parameter and entropy is also derived. It is shown that the DFGN is regular. Based on the regularity, a fast and accurate method to estimate the parameter is proposed. This method possesses lower computational cost than maximum likelihood estimator (MLE) and moving average (MA) method. Furthermore, this method is robust under amplitude shift, invariant to time shift, and unaffected by a scaling factor in power spectral density (PSD). Finally, this method and the estimation of entropy embedded with Hurst parameter will be applied to the electromyogram (EMG) of external urethral sphincter (EUS). These signals come from intact rats and the injured ones from spinal cord injury (SCI). Analysis indicates that we can discriminate between intact and SCI rats from this new information.

Abstract i
Acknowledgements ii
List of Figures v
List of Tables ix
List of Algorithms xi
Chapter 1 Introduction 1
1.1 Fractal dimension and history.………………………......….1
1.2 Motivations………………...…………….…………………......5
1.3 Contributions of the dissertation………………………......5
1.4 Organization of the dissertation.…………………………....6
Chapter 2 Preliminaries 8
2.1 FBM and its variants.……………………………..……………..8
2.2 Levinson’s algorithm.……………………………………………10
2.3 Definitions and properties of fdGn.………………………...11
2.4 Singularity, regularity, and Wold’s decomposition………12
Chapter 3 Properties of DFGN 19
3.1 The relation between DFGN and fdGn…………………………..19
3.2 The relation between Hurst parameter and entropy………..21
3.3 The regularity of DFGN………………………………………....23
Chapter 4 Estimation of Hurst Parameter 30
4.1 Establishing a standard monotonic curve…………………...30
4.2 The performance of estimation of Hurst parameter………..33
4.3 Order selection of curve and AR model……………………….52
4.4 Applications to biomedical signal…………………………….77
Chapter 5 Conclusions and Directions for Future Research 85
Bibliography 87

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