2026-11-092026-11-09https://scholars.csmu.edu.tw/handle/123456789/38033Recently special signals are more and more numerous and paid an attention by many fields. Particularly after a biomedical and bioinformatic development is heavily valued, makes a random process of a description to these phenomena more important. Among these, Brownian motion (BM) is always widely applied on relative problems. This process is further defined in generalized parameter space, whose parameter is referred to as fractional Brownian motion (FBM) and space is between 0 and 1. In order to analyze practical data, we must sample a real signal; these sampled signals are called as discrete-time fractional Brownian motion (DFBM). On the other hand, due to increment processes of these signals are stationary, hence they are usually used to analyze and estimate Hurst parameter. This increment process is named as discrete-time fractional Gaussian noise (DFGN). Because general signals are measured by electrical signal transformation, this makes measured signals possess some constant frequencies, for example a signal with 60Hz, which is a kind of singular process. For this purpose, this proposal is to study the estimation of Hurst parameter to a DFGN affected by some singular processes; even estimate the Hurst parameter after getting rid of singular processes. Finally, compare the performance with other estimators with more accurate or faster algorithm.近年來由於特殊訊號越來越多，也越來越受到許多領域的重視，特別是生物醫學資 訊的發展受到高度的關注後，使得一些可用來描述這些現象的隨機程序(random process) 變得非常重要，其中的布朗運動(Brownian motion, BM)一直被廣泛應用在相關的問題 上。此程序被進一步的定義在較廣的參數空間上，此參數稱為赫斯特參數(Hurst parameter)，它的範圍在0 至1 間。此廣義的布朗運動被稱為分數維度布朗運動(fractional Brownian motion, FBM)，為分析實際的數據我們經常需要取樣真實的訊號，所取樣的訊 號稱為離散時間分數維度布朗運動(discrete-time fractional Brownian motion, DFBM)。另 一方面，由於此訊號的增量程序(increment process)具有穩固的(stationary)特性，因此經 常被用來分析和估測赫斯特參數。此增量程序稱為離散時間分數維度高斯雜訊 (discrete-time fractional Gaussian nosie, DFGN)。由於一般訊號量測大都使用電訊號轉 換，使得所測得的訊號中經常出現固定頻率的訊號，例如60Hz 的訊號，此訊號正是奇 異(singular)訊號的一種。因此本計劃要來研究離散時間分數維度高斯雜訊遭受奇異訊號 影響後其赫斯特參數的估測，甚至消除奇異訊號後赫斯特參數的估測，最後比較一些較 精確或較快速的估測法的性能。布朗運動分數維度布朗運動離散時間分數維度布朗運動離散時間分數維度高斯雜訊奇異的訊號和正規的訊號Brownian motionfractional Brownian motiondiscrete-time fractional Brownian motiondiscrete-time fractional Gaussian noisesingular process and regular processProject