基于复杂网络的证券市场小波分析

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摘要

美国次贷危机引起的全球性金融危机不仅使美国的股市暴跌，其他国家的股

市也受到了巨大的冲击。作为新兴市场的我国股市受到了持续性影响，所以研究

金融危机下我国股市的动态特征具有现实意义。

本文首先采用小波变换方法分析沪深指数的波动趋势、小波经验功率谱、能

量谱密度及小波方差；发现指数序列小波经验功率谱和能量谱密度集中在大尺度

上，上证和深成指数的小波相关系数大于 0.9，证实了股票收益率序列具有长记忆

特性，沪深两市长期保持强相关联系。为了进一步分析证券的分形特征，采用多

重分形消除趋势分析和小波变换模极大值分析指数序列及收益率序列的分形，并

比较得出上证指数的分形谱宽度小，多重分形程度弱。

沪深指数的相关性研究一直是学者们研究点，本文研究沪深指数收盘价月数

据，周数据，日数据的相关关系。采用可见图算法，将数据序列映射成网络形式，

通过网络的拓扑特征分析发现所构造出的网络具有无标度特征，再次证实证券市

场的分形；网络度和聚集系数呈现幂律分布，证实了网络具有等级结构。

小波是信号局部特征分析的良好工具，是数学的显微镜。本文采用最大交叠

离散小波变换，将月数据进行 2层分解，周数据进行 4层分解，日数据进行 6层

分解，分析分解后的小波方差，小波协方差，小波相关系数。

最后从整个证券市场的宏观角度，针对部分股票波动收益率序列，采取“赢

者通吃”的原则构建无向无权复杂网络，对股市进行基本面分析。根据复杂网络

理论，证实证券市场具有无标度特征、小世界效应、社区结构和分形结构；并且

发现金融、地产行业是证券市场的核心板块；农业，林业等传统产业与市场的关

联性日趋减弱；对市场的影响力逐步降低。

关键词：离散小波变换多重分形消除趋势波动分析小波变换模极大

值法可见图算法最大交叠离散小波变换复杂网络

ABSTRACT

Global financial crisis caused by U.S. subprime mortgage crisis crash not only the

U.S. stock market, but also other country's market. Chinese stock market as an

emerging market has been affected continuously, so it is practical significance to

research the dynamic characteristics of Chinese stock market.

Firstly, we use wavelet transform to analysis fluctuation, wavelet empirical power

spectrum, power spectral density and wavelet variance of Shanghai composite index

and Shenzhen composite index. We found the wavelet empirical power spectrum and

power energy density of index series were concentrated in large scales; the wavelet

cross-correlation coefficients were greater than 0.9, confirming that the stock return

series with long memory characteristics; Shanghai market and Shenzhen market is

strongly correlative connection for a long time. To know better the fractal property, we

employed multifraetal detrended fluctuation analysis and wavelet transform modulus

maxima analysis to reveal multifractal of market, then we draw a conclusion that the

fractal spectral width of Shanghai stock index was smaller, the multifractal degree was

weaker.

Correlation between the Shanghai and Shenzhen index has been a hot research

altogether; We studied the correlation among month data, weekly data and daily data

from the closing price on the Shanghai and Shenzhen index . By using visibility graph

algorithms, we mapped data sequence of the form of networks, topology characteristics

of the network showed that constructed network were scale-free property, the stock

market was confirmed fractal once again; network degree probability distribution and

cluster coefficients showed power-law distribution, confirmed the network was

hierarchy.

Wavelet is a good tool for analysising signal local characteristics, it is aslo

mathematical microscope. Via the maximum overlap discrete wavelet transform, we

decomposed the monthly data into two levels, weekly data into four levels, and daily

data into six levels, then compared the wavelet variance, wavelet covariance and

wavelet correlation coefficient.

Above is the technical analysis of the stock market, lastly we introduced some

fundamental analysis.Based on stock return series, undirected and unweighted complex

networks were constructed with the method of “winner-take-all”. According to the

theory of complex network, we confirmed that stock market show scale-free property,

small world effect, community structure and fractal structure. We also concluded that

finance and real estate industry are the most important industry of stock market; the

connection between traditional industry such as agriculture and forestry and market, as

well as its influence on the market is getting weaker and weaker.

Key word: Discrete wavelet transform, Multifraetal detrended

fluctuation analysis, Wavelet transform modulus

maximum, Visibility graph approach, Maximal overlap

discrete wavelet transform, Complex Network

摘要

ABSTRACT

第一章绪论 ....................................................... 1

§1.1 研究背景 ................................................... 1

§1.2 研究现状 ................................................... 2

§1.2.1 证券市场小波分析研究现状 ............................... 2

§1.2.2 证券市场的分形研究现状 ................................. 2

§1.2.3 基于复杂网络理论的证券市场研究现状 ..................... 2

§1.3 研究内容 ................................................... 3

第二章证券市场分形分析 ............................................ 4

§2.1 证券市场假说 ............................................... 4

§2.1.1 有效市场假说 ........................................... 5

§2.1.2 分形市场假说 ........................................... 5

§2.2 小波变换 ................................................... 6

§2.2.1 小波变换理论知识 ....................................... 6

§2.2.2 小波变换在证券市场分析中的应用 ........................ 10

§2.2.3 实证分析 .............................................. 11

§2.3 分形分析 .................................................. 14

§2.3.1 分形维数 .............................................. 15

§2.3.2 单分形算法 ............................................ 16

§2.3.3 多重分形算法 .......................................... 19

§2.4 本章小结 .................................................. 23

第三章证券市场相关分析 ........................................... 24

§3.1 基于可见图指数序列的相关分析 .............................. 24

§3.1.1 可见图原理介绍 ........................................ 24

§3.1.2 复杂网络理论 .......................................... 25

§3.1.3 实证分析 .............................................. 25

§3.2 基于 MODWT 指数序列的相关分析 ........................... 28

§3.2.1 最大交叠离散小波变换理论 .............................. 28

§3.2.2 最大交叠离散小波方差 .................................. 29

§3.3 本章小结 .................................................. 31

第四章证券市场网络分析 ........................................... 32

§4.1 社区结构 .................................................. 32

§4.2 分形结构 .................................................. 32

§4.3 实证分析 .................................................. 32

§4.4 本章小结 .................................................. 38

第五章本文结论与展望 ............................................. 39

参考文献 .......................................................... 40

在读期间公开发表的论文和承担科研项目及取得成果 .................... 43

致谢 ............................................................. 44

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摘要：

摘要美国次贷危机引起的全球性金融危机不仅使美国的股市暴跌，其他国家的股市也受到了巨大的冲击。作为新兴市场的我国股市受到了持续性影响，所以研究金融危机下我国股市的动态特征具有现实意义。本文首先采用小波变换方法分析沪深指数的波动趋势、小波经验功率谱、能量谱密度及小波方差；发现指数序列小波经验功率谱和能量谱密度集中在大尺度上，上证和深成指数的小波相关系数大于0.9，证实了股票收益率序列具有长记忆特性，沪深两市长期保持强相关联系。为了进一步分析证券的分形特征，采用多重分形消除趋势分析和小波变换模极大值分析指数序列及收益率序列的分形，并比较得出上证指数的分形谱宽度小，多重分形程度弱。沪深指数的相关性研究...

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