近來有些研究報告顯示生理資料具有長相關和自我相似的特性。此二特性可分別用長相關參數 d 和自我相關係數 H 量化來表示。Peng（1995）藉由分類所得到的心律資料進行分析，研究具有致命病變者其長相關係數的特性。分數布朗運動（Fractional Brownian Motion，簡稱 FBM）和分數差分ARMA（Fractional ARIMA，簡稱 FARIMA）是兩個著名具有自我相似特性的隨機過程，我們有興趣了解自我相似過程，是否適用於對心率的資料建模，
以用來了解病人的健康狀況。本文利用 Jones 和 Shen（2004）所提出的嵌入分歧過程（Embedded Branching Process，簡稱 EBP）方法估計參數 H，以及利用自我相似過程最適度檢定，對模擬的 FBM 和 FARIMA 過程做檢定，來討論其適用性，並進一步修訂此檢定量之分佈。最後，針對模擬的 FARIMA 過程和從高雄榮總醫院得到的心率資料，比較不同估計方法求得的參數 H。

Recently there have been reports that certain physiological data seem to have the properties of long-range correlation and self-similarity. These two properties can be characterized by a long-range dependent parameter d, as well as a self-similar parameter H. In Peng et al (1995), the alteration of long-range correlations with life-threatening pathologies are studied by analyzing the heart rate data of different groups of subjects. The self-similarity properties of two well-known processes, namely the Fractional Brownian Motion (FBM) and the Fractional ARIMA (FARIMA), are of interest to see if it is suitable to be used to model the heart rate data in order to examine the health conditions of some patients. The Embedded Branching Process (EBP) method for estimating parameter $H$ and a goodness of fit test for examining the self-similarity of a process based on the EBP method are proposed in Jones and Shen (2004). In this work, the performance of the goodness of fit test are examined using simulated data from the FBM and FARIMA processes. A modification of the distribution of the test statistics under null hypothesis is proposed and has been modified to be more appropriate. Some simulation comparisons of different estimation methods of the parameter $H$ for some FARIMA processes are also presented and applied to heart rate data obtained from Kaohsiung Veterans General Hospital.

1. Introduction ........................................... 1
2. Self-similar processes .............................. 2
2.1. Properties of stationary increment of self-similar processes .... ..........................................3
2.2. Processes with self-similar properties ............ 4
2.2.1. Brownian Motion and Fractional Brownian Motion .. 4
2.2.2. Gaussian Fractional ARIMA (FARIMA) .............. 6
3. Method for estimation of the Hurst parameter ........ 7
3.1 Embedded Branching Process (EBP) or Crossing Tree .. 8
4. Goodness of fit test for examining a self-similar process ................................................ 11
4.1 The method of goodness of fit test and an example .. 11
4.2. A modi‾cation of the goodness of fit test ......... 13
5. Numerical Comparison and Application ................ 20
5.1. Numerical Comparison .............................. 20
5.2. Application ....................................... 21
6. Conclusion .......................................... 22
References ............................................. 23
Appendix ............................................... 25

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