分数阶高斯随机场中的长记忆性研究

发布时间：2019-03-29 11:53

【摘要】：长记忆性在现今各领域研究中有着十分重要的地位。分数布朗单是具有长记忆性与自相似性的各向异性高斯随机场的重要模型。本文综合运用多维小波分析、回归分析及概率极限理论,基于分数布朗单模型,进行长记忆性研究,包括分数布朗单Hurst参数的估计(基于分数布朗单的长记忆检测)与分数布朗单的表示与模拟(多维长记忆信号的刻画)。这些工作对于大脑核磁共振扩散张量成像的改进和研究由分数布朗单驱动的随机微分方程具有重要意义。在前两章中,叙述了问题的研究背景与现状以及论文的主要内容,并介绍了分数阶高斯随机场长记忆研究的基础知识。在第三章中,针对一维情形(具有平稳增量的自相似过程),介绍了两种基于小波的Hurst参数估计量,并在分数布朗运动的情形,进行了相关的模拟计算研究。其中包括这两种估计量的比较,参数的选择,经验偏差的分析。这些研究结果补充了现有的一维情形Hurst参数小波估计研究未涉及的地方,为拓展小波估计量到多维情形提供了相关指导。在第四章中,利用多维小波分析对分数布朗单的Hurst参数进行估计。首先在加入小波系数之间独立性假设后,构建了分数布朗单Hurst参数小波估计量,并利用模拟产生的分数布朗单进行验证。数值模拟结果显示该估计量在标准差与均方根误差上都表现很好。然后我们去掉小波系数之间的独立性假设,在严格数学意义下,重新构建小波估计量,并且证明了该估计量的渐近正态性。最后通过两步法实现了重新构建的估计量,并通过数值模拟验证了其精确程度。在第五章中,利用小波分析方法模拟分数布朗单。首先得到广义函数意义下多参数高斯白噪声的小波表示。然后根据分数布朗单的积分定义,对多参数高斯白噪声小波表示进行分数阶积分得到分数布朗单的小波表示,同时证明了该小波表示的几乎处处收敛性。最后根据得到的小波表示,利用快速塔式滤波组算法实现了分数布朗单的模拟,并通过评估模拟的相关系数与理论相关系数之间的差异验证了该模拟方法的精确程度。第六章总结了全文内容,并列出本文研究的后续工作。
[Abstract]:Long memory plays a very important role in many fields nowadays. Fractional Brownian list is an important model of anisotropic Gao Si random field with long memory and self-similarity. In this paper, multi-dimensional wavelet analysis, regression analysis and probability limit theory are used to study long memory based on fractional Brownian list model. It includes the estimation of Hurst parameters of fractional Brownian list (long memory detection based on fractional Brownian list) and the representation and simulation of fractional Brownian list (characterization of multi-dimensional long memory signal). These work are of great significance to the improvement of brain diffusion-diffusion Zhang Liang imaging and the study of stochastic differential equations driven by fractional Brownian sheets. In the first two chapters, the research background and present situation of the problem and the main contents of the thesis are described, and the basic knowledge of fractional Gaussian random field long memory is introduced. In the third chapter, two wavelet-based Hurst parameter estimators are introduced for one-dimensional cases (self-similar processes with stationary increments), and related simulation studies are carried out in the case of fractional Brownian motion. These include the comparison of these two estimators, the selection of parameters, and the analysis of empirical deviations. These results supplement the existing research on Hurst parameter wavelet estimation in one-dimensional case, and provide some guidance for extending wavelet estimation to multidimensional case. In chapter 4, the Hurst parameters of the fractional Brownian list are estimated by using multi-dimensional wavelet analysis. Firstly, after adding the independence hypothesis between wavelet coefficients, the wavelet estimator of fractional Brownian list Hurst parameter is constructed and verified by the simulated fractional Brownian list. The numerical simulation results show that the estimator performs well in both the standard deviation and the root mean square error. Then we remove the independence hypothesis between wavelet coefficients and reconstruct the wavelet estimator in the strict mathematical sense and prove the asymptotic normality of the estimator. Finally, the reconstructed estimator is realized by two-step method, and its accuracy is verified by numerical simulation. In the fifth chapter, the wavelet analysis method is used to simulate the fractional Brownian list. Firstly, the wavelet representation of multi-parameter Gaussian white noise in the sense of generalized function is obtained. Then according to the integral definition of fractional Brownian list, the wavelet representation of fractional Brownian list is obtained by fractional integration of multi-parameter Gaussian white noise wavelet representation, and the almost everywhere convergence of the wavelet representation is proved. Finally, according to the obtained wavelet representation, the fractional Brownian list is simulated by fast tower filter bank algorithm, and the accuracy of the simulation method is verified by evaluating the difference between the simulated correlation coefficient and the theoretical correlation coefficient. The sixth chapter summarizes the full text, and lists the follow-up work of this paper.
【学位授予单位】：中国科学院研究生院(武汉物理与数学研究所)
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O211

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