三类不确定支持向量机及其应用
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摘要
支持向量机是基于统计学习理论的VC维和结构风险最小化原则建立的一种通用机器学习算法。它较好地解决了传统学习方法难以处理的小样本、高维、非线性等问题，且具有较好的泛化能力。自提出以来，支持向量机得到了越来越多的专家和工程技术人员的青睐，并已成功应用到人脸识别、遥感图像分析、文本分类等众多模式识别领域。
尽管支持向量机在众多实际问题中得到了广泛应用，但它仍存在着一定局限性。如，支持向量机对噪声较为敏感，且对不均衡样本分类准确率不高。再如，支持向量机所处理的样本为实随机变量。在一些实际问题中，样本往往为非实随机变量（随机集和模糊等）。针对支持向量机存在的上述问题，本文分别研究了基于噪声不确定性样本、随机集不确定性输入样本和模糊等不确定性输出样本等三类不确定支持向量机。
本文的主要创新点及工作如下：
（1）提出了基于直觉模糊数的支持向量机和多类支持向量机。直觉模糊数作为模糊隶属度的一种推广，它比传统的模糊隶属度更加细腻地描述客观世界中的模糊性。基于直觉模糊数的支持向量机通过核函数在特征空间中给每个训练样本赋予一个直觉模糊数，利用直觉模糊数的得分函数描述每个训练样本的分类贡献，消除噪声对支持向量机的影响。基于上述支持向量机分别构建了一对一和一对多模式下的多类支持向量机。针对一对多模式中不均衡样本的特点，对不同类别样本赋予不同的权重。数值实验验证了该类不确定支持向量机的有效性。
（2）构建了基于随机集输入样本的支持向量机和多类支持向量机。随机集是随机变量的一种重要拓广，它能有效地处理复杂不确定环境下的模糊性和经验性数据。基于随机集输入样本的支持向量机以随机集的可测选择作为主要特征，将随机集输入样本的分类问题转化为了可测选择的分类问题。基于随机集输入样本的多类支持向量机利用模糊C-均值聚类算法将随机集输入样本转化成为实样本，进而将随机集输入样本的多类分类问题转化成实样本的多类分类问题。数值实验验证了该类不确定支持向量机的有效性。
（3）构建了可信性空间上基于模糊输出样本的支持向量机和不确定空间上基于不确定性输出样本的支持向量机。基于可信性测度和置信水平，可信性空间上基于模糊输出样本的支持向量机给出了样本类别的动态划分方法，构建了一个动态的分类超平面，有效处理了模糊输出样本类别的模糊性。同样，基于不确定测度和置信水平，不确定空间上基于不确定性输出样本的支持向量机也给出了样本类别的动态划分方法，有效处理不确定性输出样本类别的不确定性。仿真实验验证了该类不确定支持向量机的有效性。
（4）两类不确定支持向量机应用于人脸识别。由于光照、姿态和表情等因素的影响，人脸图像中存在着噪声、模糊、随机集等不确定信息。为了有效处理人脸图像中的这些不确定信息，本文分别将基于直觉模糊数的支持向量机和基于随机集输入样本的支持向量机应用于人脸识别中，利用人脸数据库验证了这两类算法的有效性。
Support vector machine is a universal machine learning algorithm, which is based on VCdimension and structural risk minimization principle of statistical learning theory. It solvesthe practical problems of traditional learning methods, such as small sample, high dimensionand nonlinearity, and has good generalization capability. Since it was proposed, supportvector machine has won the favor with many experts and engineering and technical personnel,and has been successfully applied into face recognition, remote sensing image analysis, textclassification, etc.
However, with the wide range of applications, support vector machine has somelimitations in some practical problems. For example, support vector machine is sensitive tonoises, and has low performance for imbalance data set. Another example, the trainingsamples of support vector machine are real random variables. While, in some practicalproblems, the training samples are non-real random variables (random set, fuzzy, etc.). Inorder to solve aforementioned problems, three kinds of uncertain support vector machines aregiven to handle the noise uncertainty samples, random set uncertainty input samples, andfuzzy etc. uncertainty output samples accordingly.
The main contributions are as follows:
(1) The support vector machine and multi-class support vector machine based onintuitionistic fuzzy numbers are proposed. Intuitionistic fuzzy number as an extension offuzzy membership can describe the fuzziness of the objective world more accurately. Thesupport vector machine based on intuitionistic fuzzy numbers assign each training samplewith a corresponding intuitionistic fuzzy number by the use of kernel function, and a newscore function of the intuitionistic fuzzy number is proposed to measure the contribution ofeach training sample and distinguish from the noises and support vectors. Then, themulti-class support vector machine of one-against-one and one-against-all are given based onthe above support vector machine. In order to solve the misclassification problem resultedfrom the imbalance samples of different classes in the construction of one-against-all supportvector machine, different weights are assigned to differed class sample. The effectiveness ofthe above uncertain support vector machines are verified by numerical experiments.
(2) The support vector machine and multi-class support vector machine based on randomset input samples are constructed. Random set as an important extension of real valued random variable, can well deal with the fuzzy and experimental data of complicated uncertainenvironments. The support vector machine based on random set input samples takesmeasurable selections are taken as the main feature of random set. Then the classification ofrandom set input samples is transformed into the classification of measurable selections. Themulti-class support vector machine transform the multi-classification of random input samplesinto the multi-classification of real-valued samples by the use of fuzzy C-means clusteringalgorithm. Numerical experiments show the effective of this kind of uncertain support vectormachine
(3) The support vector machine based on fuzzy output samples in credibility space andthe support vector machine based on uncertainty output samples in uncertainty space aregiven. In credibility space, the support vector machine based on fuzzy output samples proposea dynamic division method of class label by the use of credibility measure and the confidencelevel, and construct a dynamic classification hyperplane to effectively deal with thefuzziness of fuzzy output samples. Similarly, in uncertain space, the support vector machinebased on uncertain output samples also give a dynamic division method of class label by theuse of uncertain measure and the confidence level, and then construct a correspondingdynamic classification hyperplane to handle the uncertainty of uncertain output samples. Thesimulation experiments show the effective of this kind of uncertain support vector machine.
(4) Two kinds of uncertain support vector machine are applied into the face recognition.Due to the impact of lighting condition, gesture and facial expression, there are muchuncertain information such as noise, fuzzy, intuitionistic fuzzy and random set. In order todeal with the above uncertain information, two kinds of uncertain support vector machineproposed in this paper are applied into the face recognition, and experiments show theeffective of the two algorithms.
尽管支持向量机在众多实际问题中得到了广泛应用，但它仍存在着一定局限性。如，支持向量机对噪声较为敏感，且对不均衡样本分类准确率不高。再如，支持向量机所处理的样本为实随机变量。在一些实际问题中，样本往往为非实随机变量（随机集和模糊等）。针对支持向量机存在的上述问题，本文分别研究了基于噪声不确定性样本、随机集不确定性输入样本和模糊等不确定性输出样本等三类不确定支持向量机。
本文的主要创新点及工作如下：
（1）提出了基于直觉模糊数的支持向量机和多类支持向量机。直觉模糊数作为模糊隶属度的一种推广，它比传统的模糊隶属度更加细腻地描述客观世界中的模糊性。基于直觉模糊数的支持向量机通过核函数在特征空间中给每个训练样本赋予一个直觉模糊数，利用直觉模糊数的得分函数描述每个训练样本的分类贡献，消除噪声对支持向量机的影响。基于上述支持向量机分别构建了一对一和一对多模式下的多类支持向量机。针对一对多模式中不均衡样本的特点，对不同类别样本赋予不同的权重。数值实验验证了该类不确定支持向量机的有效性。
（2）构建了基于随机集输入样本的支持向量机和多类支持向量机。随机集是随机变量的一种重要拓广，它能有效地处理复杂不确定环境下的模糊性和经验性数据。基于随机集输入样本的支持向量机以随机集的可测选择作为主要特征，将随机集输入样本的分类问题转化为了可测选择的分类问题。基于随机集输入样本的多类支持向量机利用模糊C-均值聚类算法将随机集输入样本转化成为实样本，进而将随机集输入样本的多类分类问题转化成实样本的多类分类问题。数值实验验证了该类不确定支持向量机的有效性。
（3）构建了可信性空间上基于模糊输出样本的支持向量机和不确定空间上基于不确定性输出样本的支持向量机。基于可信性测度和置信水平，可信性空间上基于模糊输出样本的支持向量机给出了样本类别的动态划分方法，构建了一个动态的分类超平面，有效处理了模糊输出样本类别的模糊性。同样，基于不确定测度和置信水平，不确定空间上基于不确定性输出样本的支持向量机也给出了样本类别的动态划分方法，有效处理不确定性输出样本类别的不确定性。仿真实验验证了该类不确定支持向量机的有效性。
（4）两类不确定支持向量机应用于人脸识别。由于光照、姿态和表情等因素的影响，人脸图像中存在着噪声、模糊、随机集等不确定信息。为了有效处理人脸图像中的这些不确定信息，本文分别将基于直觉模糊数的支持向量机和基于随机集输入样本的支持向量机应用于人脸识别中，利用人脸数据库验证了这两类算法的有效性。
Support vector machine is a universal machine learning algorithm, which is based on VCdimension and structural risk minimization principle of statistical learning theory. It solvesthe practical problems of traditional learning methods, such as small sample, high dimensionand nonlinearity, and has good generalization capability. Since it was proposed, supportvector machine has won the favor with many experts and engineering and technical personnel,and has been successfully applied into face recognition, remote sensing image analysis, textclassification, etc.
However, with the wide range of applications, support vector machine has somelimitations in some practical problems. For example, support vector machine is sensitive tonoises, and has low performance for imbalance data set. Another example, the trainingsamples of support vector machine are real random variables. While, in some practicalproblems, the training samples are non-real random variables (random set, fuzzy, etc.). Inorder to solve aforementioned problems, three kinds of uncertain support vector machines aregiven to handle the noise uncertainty samples, random set uncertainty input samples, andfuzzy etc. uncertainty output samples accordingly.
The main contributions are as follows:
(1) The support vector machine and multi-class support vector machine based onintuitionistic fuzzy numbers are proposed. Intuitionistic fuzzy number as an extension offuzzy membership can describe the fuzziness of the objective world more accurately. Thesupport vector machine based on intuitionistic fuzzy numbers assign each training samplewith a corresponding intuitionistic fuzzy number by the use of kernel function, and a newscore function of the intuitionistic fuzzy number is proposed to measure the contribution ofeach training sample and distinguish from the noises and support vectors. Then, themulti-class support vector machine of one-against-one and one-against-all are given based onthe above support vector machine. In order to solve the misclassification problem resultedfrom the imbalance samples of different classes in the construction of one-against-all supportvector machine, different weights are assigned to differed class sample. The effectiveness ofthe above uncertain support vector machines are verified by numerical experiments.
(2) The support vector machine and multi-class support vector machine based on randomset input samples are constructed. Random set as an important extension of real valued random variable, can well deal with the fuzzy and experimental data of complicated uncertainenvironments. The support vector machine based on random set input samples takesmeasurable selections are taken as the main feature of random set. Then the classification ofrandom set input samples is transformed into the classification of measurable selections. Themulti-class support vector machine transform the multi-classification of random input samplesinto the multi-classification of real-valued samples by the use of fuzzy C-means clusteringalgorithm. Numerical experiments show the effective of this kind of uncertain support vectormachine
(3) The support vector machine based on fuzzy output samples in credibility space andthe support vector machine based on uncertainty output samples in uncertainty space aregiven. In credibility space, the support vector machine based on fuzzy output samples proposea dynamic division method of class label by the use of credibility measure and the confidencelevel, and construct a dynamic classification hyperplane to effectively deal with thefuzziness of fuzzy output samples. Similarly, in uncertain space, the support vector machinebased on uncertain output samples also give a dynamic division method of class label by theuse of uncertain measure and the confidence level, and then construct a correspondingdynamic classification hyperplane to handle the uncertainty of uncertain output samples. Thesimulation experiments show the effective of this kind of uncertain support vector machine.
(4) Two kinds of uncertain support vector machine are applied into the face recognition.Due to the impact of lighting condition, gesture and facial expression, there are muchuncertain information such as noise, fuzzy, intuitionistic fuzzy and random set. In order todeal with the above uncertain information, two kinds of uncertain support vector machineproposed in this paper are applied into the face recognition, and experiments show theeffective of the two algorithms.
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