﻿ 基于D-vine Copula理论的贝叶斯分类器设计
 控制与决策  2019, Vol. 34 Issue (6): 1319-1324 0

### 引用本文 [复制中英文]

[复制中文]
WANG Bei, SUN Yu-dong, JIN Jing, ZHANG Tao, WANG Xing-yu. Bayesian classifier based on D-vine Copula theory[J]. Control and Decision, 2019, 34(6): 1319-1324. DOI: 10.13195/j.kzyjc.2017.1589.
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### 文章历史

1. 华东理工大学 化工过程先进控制和优化技术教育部重点实验室，上海 200237;
2. 清华大学 自动化系，北京 100084

Bayesian classifier based on D-vine Copula theory
WANG Bei 1, SUN Yu-dong 1, JIN Jing 1, ZHANG Tao 2, WANG Xing-yu 1
1. Key Laboratory of Advanced Control and Optimization for Chemical Processes, Ministry of Education, East China University of Science and Technology, Shanghai 200237, China;
2. Department of Automation, Tsinghua University, Beijing 100084, China
Abstract: In the traditional Bayesian classifiers such as the Gaussian discriminant analysis method and the Naive Bayesian method, the correlation between variables are commonly simplified when constructing the joint probability distribution of variables. Accordingly, the estimation of the class conditional probability density would have differences with the actual data. In this study, a Bayesian classifier based on the D-vine Copula theory is developed by investigating on the correlation between variables. The main objective is to improve the accuracy of the class conditional probability density estimation. The joint probability distribution of variables is decomposed into a series of pair Copula functions and marginal probability density functions. The kernel function method is adopted to estimate the marginal probability density. The parameters of pair Copula functions are optimized by the maximum likelihood estimation. The developed method is analyzed and validated on the classification of neurophysiological signals. The obtained results show that it has better performance on several classification indexes.
Keywords: Bayesian decision    correlation analysis    class conditional probability density estimation    D-vine Copula    pattern recognition    neurophysiological signal
0 引言

Copula理论是Sklar在1959年提出的, 主要应用于金融风险管理领域[8].在控制科学相关的研究中, Copula理论对算法性能的提升也得到了关注.王丽芳等[9]将多元Copula函数应用于分布估计算法的研究中, 通过仿真实验表明, 引入Copula理论的cEDA算法能够更快地收敛于最优解.韩敏等[10]将Copula理论应用于互信息估计, 通过二维高斯数据的仿真实验表明, 基于Copula熵的互信息估计算法在计算复杂度和精度方面, 相比于核方法、k近邻方法和直方图法, 具备更高的性能.许民利等[11]将Copula函数与CVaR相结合, 构建了随机需求与随机价格之间的决策模型, 给出该模型的具体求解方法, 并证明了该模型的解的唯一性.

Vine Copula是在Copula理论的基础上发展起来的.传统的多元Copula函数[12]如椭圆Copula簇、阿基米德Copula簇等在处理高维变量时, 变量间的相关性优化问题比较复杂, 且计算量较大.早期, Joe[13]借助Vine结构将多元Copula函数分解成一系列二元Copula函数的乘积, 该模型通过一系列二元Copula函数可以构建变量间复杂的相关性, 并且减少了计算的复杂程度, 使得Vine Copula受到关注.在后来的研究中, Aas等[14]检验了Vine Copula在计算复杂度和拟合能力上比传统的多元阿基米德Copula模型具有更好的性能. Czado[15]通过三维D-vine Copula的模型, 使用不同种类的二元Copula函数描述变量间的不对称性, 并展示了Vine Copula模型的灵活性.近年来, Vine Copula函数作为随机变量相关性建模工具被广泛应用在资产收益波动[16]、化工故障诊断[17]、能源管理[18-19]等领域, 并取得了较为显著的效果.

1 D-vine Copula贝叶斯分类器 1.1 贝叶斯决策理论

 (1)

1.2 Copula函数

 (2)

 (3)

f(xi)是随机变量x的边缘概率密度函数. Copula函数的密度函数定义为

 (4)

1.3 二元Copula函数

 (5)

 (6)

1.4 D-vine Copula模型

Copula建模的本质是用样本来拟合式(4)的过程, 并通过优化准则来估计相应Copula函数的参数.对于二元样本而言, 其优化过程较易实现.然而, 随着维数的增大, 会出现“维数灾难”, 即一个m元Copula函数的待优化参数个数将远远大于m. Vine Copula是解决上述问题的有效途径[22].

 (7)

 (8)

v是标量时

 (9)

 (10)

 (11)

2 基于生物电信号的觉醒度状态识别

2.1 数据采集

2.2 特征提取

2.3 模式识别 2.3.1 D-vine Copula模型

 (12)

2.3.2 相关性分析

 图 1 样本数据散点图

2.4 结果比较

 图 2 不同分类方法的ROC曲线比较

3 结论

D-vine Copula模型的优势在于Copula函数作为连接特征的边缘分布函数, 可以从众多不同的二元Copula函数中选取最合适的函数来拟合特征之间存在的线性或非线性相关性, 且连接形式不受特征边缘分布的限制.相对于传统贝叶斯分类算法, 该方法在变量的相关性构建上更具灵活性, 能够为贝叶斯分类器中类条件概率密度的估计提供一种新的实现途径, 从而提高贝叶斯分类器在处理特征之间具有复杂相关性时的分类性能.

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