区间型符号数据分析理论方法及其在金融中的应用研究
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摘要
符号数据分析(Symbolic Data Analysis,简称SDA)是一种研究如何从海量数据中发掘系统知识的理论和方法。符号数据分析技术运用“数据打包”的思想,实现对庞大的样本空间的降维处理。相应的,样本数据的性质就发生了变化:由原来的“点数据”变为“符号数据”。区间数是最常用的一类符号数据。论文研究区间型符号数据分析的理论方法,并将其应用于金融中若干问题的求解,主要内容如下:
1.区间型符号数据分析基础。主要研究区间型符号变量的统计描述。首先讨论了区间型符号变量的经验密度函数的计算,在此基础上,给出了直方图的绘制方法,并研究了均值和方差、协方差和相关函数的计算。这为后面几章对区间型符号数据进行多元分析的研究奠定了基础。
2.区间型符号数据的主成分分析方法。首先对现有的区间数据主成分分析的两种主要方法——顶点法(V-PCA)和中点法(C-PCA)进行了比较研究。进一步,从公共主成分的角度出发,提出了基于公共主成分的区间PCA方法(Common PCA)。该方法还可适用于不同时间段的区间主成分分析,即进行动态的区间主成分分析。基于Hausdorff距离定义了一种效度指标,并通过模拟的方法,对V-PCA、C-PCA和Common PCA的方法有效性进行了比较研究。最后将上述方法应用于我国上海证券交易市场,研究上市公司的风险与公司规模之间的关系,以及股票的市场表现的动态评价。
3.区间型符号数据的回归分析方法。首先给出了基于区间符号数据描述统计量的回归参数估计方法。其次对于可以线性化的非线性相关的区间变量,提出了基于误差传递理论对区间型符号变量进行非线性回归的方法。接下来鉴于因变量的估计y|^~i以及残差e_i均为区间数,论文提出了矩形残差图来进行回归诊断。进一步,基于Hausdorff距离定义了评价模型优劣的指标——反映绝对误差的RMSE_H和反映相对误差的UH。最后应用区间线性回归方法对沪深300指数与中信规模风格指数的相关性进行了实证分析。
4.具区间型符号数据系数的多目标线性规划(简称区间多目标线性规划)问题的求解。首先讨论具区间系数的单目标线性规划问题的求解。在此基础上,研究区间多目标线性规划的Zimmermann模糊求解算法。最后将该算法应用于证券投资组合问题的实证分析。
Symbolic data analysis (SDA) is a new method analyzing and gleaning useful information from massive data. It is in such a way that summarizes large data to a dataset of a small size. One consequence of this is that the data may no longer be formatted as single values, but may be symbolic data. Interval number is a main type of symbolic data. This dissertation makes a study on the theory&methodology of interval-valued symbolic data analysis and its application in finance. The main points of this dissertation are as follows.
1. Foundation of interval-valued symbolic data analysis Descriptive statistics for interval-valued symbolic data is mainly studied. Firstly, the empirical density function of interval-valued symbolic variable is defined. Based on this, methods of drawing the histogram and calculating of mean, variance, covariance and correlation functions for interval-valued symbolic variable are given. All of these become fundamentals for the next studies.
2.Principal component analysis (PCA) for interval-valued symbolic data The two main methods of PCA for interval data are Vertices-PCA (V-PCA) and Centers-PCA (C-PCA). Comparative study is firstly made on them. Then a new method of PCA for interval -valued symbolic data called Common PCA from point view of common principal component. One advantage of this method is making dynamic PCA on time serial data. In order to make a further comparison on the three methods, an index which can indicate the goodness of fit of some method was defined by the Hausdorff distance. Then, comparative study of the three methods was made by means of simulation. Finally, an empirical research on Shanghai financial market is done by the given method. The relation between the risk and the company’s scale and the dynamic evaluation on several stocks’behavior on the market are studied, respectively.
3. Regression analysis for interval-valued symbolic data Firstly, estimation of regression parameters based on descriptive statistics for interval-valued symbolic data is given. On the other hand, for non-linear correlation data which can be linearized, a method of non-linear regression analysis for interval-valued symbolic data is put forward based on error transferring theory. Because of y|^_i and e_i are interval numbers, a type of rectangular residual plot is proposed to make regression diagnotics. Then, indices which can indicate the goodness of the model is defined based on the Hausdorff distance, that are called RMSE_H reflecting absolute error and U_H reflecting the relative error. Finally, it can empirical research on the correlation between CSI 300 and style indices of Chinese international trust & investment company (CITIC) is done by the given method.
4. Solving of multi-objective linear programming with interval-valued symbolic coefficient Firstly, method of solving single-objective linear programming with interval-valued symbolic coefficient is discussed. Then, Zimmermann fuzzy method of solving of multi-objective linear programming with interval-valued symbolic coefficient is put forward. Finally, an empirical research on the portfolio investment is made.
1.区间型符号数据分析基础。主要研究区间型符号变量的统计描述。首先讨论了区间型符号变量的经验密度函数的计算,在此基础上,给出了直方图的绘制方法,并研究了均值和方差、协方差和相关函数的计算。这为后面几章对区间型符号数据进行多元分析的研究奠定了基础。
2.区间型符号数据的主成分分析方法。首先对现有的区间数据主成分分析的两种主要方法——顶点法(V-PCA)和中点法(C-PCA)进行了比较研究。进一步,从公共主成分的角度出发,提出了基于公共主成分的区间PCA方法(Common PCA)。该方法还可适用于不同时间段的区间主成分分析,即进行动态的区间主成分分析。基于Hausdorff距离定义了一种效度指标,并通过模拟的方法,对V-PCA、C-PCA和Common PCA的方法有效性进行了比较研究。最后将上述方法应用于我国上海证券交易市场,研究上市公司的风险与公司规模之间的关系,以及股票的市场表现的动态评价。
3.区间型符号数据的回归分析方法。首先给出了基于区间符号数据描述统计量的回归参数估计方法。其次对于可以线性化的非线性相关的区间变量,提出了基于误差传递理论对区间型符号变量进行非线性回归的方法。接下来鉴于因变量的估计y|^~i以及残差e_i均为区间数,论文提出了矩形残差图来进行回归诊断。进一步,基于Hausdorff距离定义了评价模型优劣的指标——反映绝对误差的RMSE_H和反映相对误差的UH。最后应用区间线性回归方法对沪深300指数与中信规模风格指数的相关性进行了实证分析。
4.具区间型符号数据系数的多目标线性规划(简称区间多目标线性规划)问题的求解。首先讨论具区间系数的单目标线性规划问题的求解。在此基础上,研究区间多目标线性规划的Zimmermann模糊求解算法。最后将该算法应用于证券投资组合问题的实证分析。
Symbolic data analysis (SDA) is a new method analyzing and gleaning useful information from massive data. It is in such a way that summarizes large data to a dataset of a small size. One consequence of this is that the data may no longer be formatted as single values, but may be symbolic data. Interval number is a main type of symbolic data. This dissertation makes a study on the theory&methodology of interval-valued symbolic data analysis and its application in finance. The main points of this dissertation are as follows.
1. Foundation of interval-valued symbolic data analysis Descriptive statistics for interval-valued symbolic data is mainly studied. Firstly, the empirical density function of interval-valued symbolic variable is defined. Based on this, methods of drawing the histogram and calculating of mean, variance, covariance and correlation functions for interval-valued symbolic variable are given. All of these become fundamentals for the next studies.
2.Principal component analysis (PCA) for interval-valued symbolic data The two main methods of PCA for interval data are Vertices-PCA (V-PCA) and Centers-PCA (C-PCA). Comparative study is firstly made on them. Then a new method of PCA for interval -valued symbolic data called Common PCA from point view of common principal component. One advantage of this method is making dynamic PCA on time serial data. In order to make a further comparison on the three methods, an index which can indicate the goodness of fit of some method was defined by the Hausdorff distance. Then, comparative study of the three methods was made by means of simulation. Finally, an empirical research on Shanghai financial market is done by the given method. The relation between the risk and the company’s scale and the dynamic evaluation on several stocks’behavior on the market are studied, respectively.
3. Regression analysis for interval-valued symbolic data Firstly, estimation of regression parameters based on descriptive statistics for interval-valued symbolic data is given. On the other hand, for non-linear correlation data which can be linearized, a method of non-linear regression analysis for interval-valued symbolic data is put forward based on error transferring theory. Because of y|^_i and e_i are interval numbers, a type of rectangular residual plot is proposed to make regression diagnotics. Then, indices which can indicate the goodness of the model is defined based on the Hausdorff distance, that are called RMSE_H reflecting absolute error and U_H reflecting the relative error. Finally, it can empirical research on the correlation between CSI 300 and style indices of Chinese international trust & investment company (CITIC) is done by the given method.
4. Solving of multi-objective linear programming with interval-valued symbolic coefficient Firstly, method of solving single-objective linear programming with interval-valued symbolic coefficient is discussed. Then, Zimmermann fuzzy method of solving of multi-objective linear programming with interval-valued symbolic coefficient is put forward. Finally, an empirical research on the portfolio investment is made.
引文
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