混沌系统最大Lyapunov指数的计算及其在脑电数据分析中的应用

【作者】 蔡聪波；

【导师】 徐慎初；

【作者基本信息】 厦门大学 ， 凝聚态物理， 2001， 硕士

【摘要】 本文在第一章首先介绍了混沌科学发展的历史和现状，并介绍了混沌在神经网络和EEG方面的应用。第二章介绍了Takens的几个重构定理，其中的第一个定理是我们工作的理论基础，本文给出了它的详细证明。 本文在第三章提出了一种新的从一维实验数据序列计算最大Lyapunov指数的方法。我们首次提出了平均周期的概念，并利用它得到两个经验公式，这两个经验公式可用来帮助确定几个重要的重构参数：重构维、重构延迟和迭代次数，并使这些参数的选择具有确定性，减少了先验性。平均周期的提出还有助于确定合理的采样频率和采样精度。本文还对Wolf算法进行改进，克服了其在计算较复杂的混沌吸引子时所遇到的困难。这主要是通过扩展角度误差的允许范围来提高有限的实验数据的利用率。该方法经过许多复杂混沌吸引子的验证，并被证明是正确和有效的。 本文在第四章讨论了对BP网络设置不同阈值范围混沌信号的初始权值对其学习效率的影响。首先论证了计算机所产生的伪随机序列实质上是混沌序列，对这个伪随机序列进行任意调整位置而不改变其混沌特性。我们用不同阈值范围的伪随机序列对BP神经网络设置初始权值，结果表明：对于XOR问题，同极性的小值混沌信号可明显提高该网络的学习效能，对于对称问题，同正极性的小值混沌信号也可提高该网络的学习效能。从另一角度揭示了混沌与神经网络间的关联。 在第五章，我们把第三章所提的方法具体运用到脑电分析中去，我们认为应依据EEG数据的特点作一些预处理。这些预处理包括：利用FFT滤掉EEG数据的高频和低频部分，用关联维来代替EEG数据的原始拓扑维，最后根据平均周期来确定三个重构参数。对EEG的分析结果表明：对应人脑海马区的EEG数据的最大Lyapunov指数较其它区域稍大，表明这早的脑活动较活跃。另外，我们还首次提出利用主成份分析来抽取各个不同通道脑电波的共同特征，并指出其潜在的临床意义。更多还原

【Abstract】 The history and the current situation of the chaos science’s development are introduced in the first chapter. Applies of the chaos in neural network and EEG analysis are also introduced in this chapter. In the second chapter, We introduce some of the Taken’s reconstruction theories, the first of which is the foundation of our works. We proved it in detail.A new method of calculating the largest Lyapunov exponent from a urn-dimensional time series was proposed in the third chapter. The conception of average period was proposed for determining some important reconstruction parameters, which includes reconstruction dimensions. reconstruction delay and iteration numbers. The average period can also be useful to determine the reasonable sampling frequency and the minimum precise. We also improve the Wolf s algorithm, and overcome its handicap in calculating the largest Lyapunov exponent from the complex chaotic attractor. We can make full use of the time series by extending the permitted range of angle errors. The new method proved to be correct and efficient after testing many complex chaotic attractors.In the fourth chapter, we discuss the affection to the learning of BP neural network by setting initial values of the BP NN with the chaotic signals which have the variable threshold value range. At first, we prove by simulations that the random numbers generated by digital computer consist of chaotic series, and resetting the numbers randomly doesn’t change its chaotic character. We use the different ranges of chaotic number as the initial connective weights. The results show: the same polarity of small chaotic numbers can improve learning efficiency of BP N7N about XOR problem, positive polarity of small chaotic numbers can improve learning efficiency of BP NN about symmetry problem.In the fifth chapter, we apply the method proposed in the third chapter to the EEG analysis. We must do some pretreatment due to the complex characteristic of EEG data. It includes: The high frequency components and the low frequency components of EEG data are filtered by FFT, The correlation dimension is calculated primarily to replace the original topological dimension. Then we can determine three reconstructed parameters in terms of average period. The results of EEG analysis show: The largest Lyapunov exponent of EEG data which corresponds to the hippocamp area is bigger than others. This manifests that the action of the hippocamp area is more active. The common feature is extracted from multi-channel EEG data by using the principal component analysis, and its potential significance is also predicted.更多还原