Some sets of orthogonal polynomial kernel functions

Kernel methods provide high performance in a variety of machine learning tasks. However, the success of kernel methods is heavily dependent on the selection of the right kernel function and proper setting of its parameters. Several sets of kernel functions based on orthogonal polynomials have been proposed recently. Besides their good performance in the error rate, these kernel functions have only one parameter chosen from a small set of integers, and it facilitates kernel selection greatly. Two sets of orthogonal polynomial kernel functions, namely the triangularly modified Chebyshev kernels and the triangularly modified Legendre kernels, are proposed in this study. Furthermore, we compare the construction methods of some orthogonal polynomial kernels and highlight the similarities and differences among them. Experiments on 32 data sets are performed for better illustration and comparison of these kernel functions in classification and regression scenarios. In general, there is difference among these orthogonal polynomial kernels in terms of accuracy, and most orthogonal polynomial kernels can match the commonly used kernels, such as the polynomial kernel, the Gaussian kernel and the wavelet kernel. Compared with these universal kernels, the orthogonal polynomial kernels each have a unique easily optimized parameter, and they store statistically significantly less support vectors in support vector classification. New presented kernels can obtain better generalization performance both for classification tasks and regression tasks.     

一种支持向量机的组合核函数

核函数是支持向量机的核心，不同的核函数将产生不同的分类效果，核函数也是支持向量机理论中比较难理解的一部分。通过引入核函数，支持向量机可以很容易地实现非线性算法。首先探讨了核函数的本质，说明了核函数与所映射空间之间的关系，进一步给出了核函数的构成定理和构成方法，说明了核函数分为局部核函数与全局核函数两大类，并指出了两者的区别和各自的优势。最后，提出了一个新的核函数——组合核函数，并将该核函数应用于支持向量机中，并进行了人脸识别实验，实验结果也验证了该核函数的有效性。     

一种改进的显性多核支持向量机

在支持向量机(Support vector machine, SVM)中, 对核函数的定义非常重要, 不同的核会产生不同的分类结果. 如何充分利用多个不同核函数的特点, 来共同提高SVM学习的效果, 已成为一个研究热点. 于是, 多核学习(Multiple kernel learning, MKL)方法应运而生. 最近, 有的学者提出了一种简单有效的稀疏MKL算法,即GMKL (Generalized MKL)算法, 它结合了L1 范式和L2范式的优点, 形成了一个对核权重的弹性限定. 然而, GMKL算法也并没有考虑到如何在充分利用已经选用的核函数中的共有信息. 另一方面, MultiK-MHKS算法则考虑了利用典型关联分析(Canonical correlation analysis, CCA)来获取核函数之间的共有信息, 但是却没有考虑到核函数的筛选问题. 本文模型则基于这两种算法进行了一定程度的改进, 我们称我们的算法为改进的显性多核支持向量机 (Improved domain multiple kernel support vector machine, IDMK-SVM). 我们证明了本文的模型保持了GMKL 的特性, 并且证明了算法的收敛性. 最后通过模拟实验, 本文证明了本文的多核学习方法相比于传统的多核学习方法有一定的精确性优势.     

支撑向量机在航空发动机故障诊断中的应用研究

航空发动机故障样本有限,利用传统的统计识别方法故障诊断,正确率不高.支撑向量机能解决小样本的故障分类识别问题.研究Support Vector Machine(简称SVM)核函数对识别精度的影响,并把SVM与最大似然法、马氏距离法,最小距离法进行比较,结果表明SVM核函数对故障识别正确率影响不大,基于SVM的航空发动机...     

基于图的组合半监督SVM聚类核算法研究

为了在聚类假设的基础上,进一步提高支持向量机的分类精度,文中通过引入线性分段转换函数,将加权无向图上的相似矩阵重新表示,改变该图上的距离度量,使得在同一群集中两点间的距离更小,从而建立基于图的聚类核,与多项式核函数线性组合后,构造出基于图的组合半监督聚类核,并将其用于支持向量机的训练和分类。实验表明,与标准SVM算法相比,该算法分类精度较高,且高于组合前的单个核函数。随着标记样本比例的增加,该算法的分类精度也在增加,有效利用了未标记样本蕴含的信息。     

一种基于聚类核的半监督支持向量机分类方法

为了在标记样本数目有限时尽可能地提高支持向量机的分类精度,提出了一种基于聚类核的半监督支持向量机分类方法。该算法依据聚类假设,即属于同一类的样本点在聚类中被分为同一类的可能性较大的原则去对核函数进行构造。采用K-均值聚类算法对已有的标记样本和所有的无标记样本进行多次聚类,根据最终的聚类结果去构造聚类核函数,从而更好地反映样本间的相似程度,然后将其用于支持向量机的训练和分类。理论分析和计算机仿真结果表明,该方法充分利用了无标记样本信息,提高了支持向量机的分类精度。     

An enhanced Support Vector Machine classification framework by using Euclidean distance function for text document categorization

This paper presents the implementation of a new text document classification framework that uses the Support Vector Machine (SVM) approach in the training phase and the Euclidean distance function in the classification phase, coined as Euclidean-SVM. The SVM constructs a classifier by generating a decision surface, namely the optimal separating hyper-plane, to partition different categories of data points in the vector space. The concept of the optimal separating hyper-plane can be generalized for the non-linearly separable cases by introducing kernel functions to map the data points from the input space into a high dimensional feature space so that they could be separated by a linear hyper-plane. This characteristic causes the implementation of different kernel functions to have a high impact on the classification accuracy of the SVM. Other than the kernel functions, the value of soft margin parameter, C is another critical component in determining the performance of the SVM classifier. Hence, one of the critical problems of the conventional SVM classification framework is the necessity of determining the appropriate kernel function and the appropriate value of parameter C for different datasets of varying characteristics, in order to guarantee high accuracy of the classifier. In this paper, we introduce a distance measurement technique, using the Euclidean distance function to replace the optimal separating hyper-plane as the classification decision making function in the SVM. In our approach, the support vectors for each category are identified from the training data points during training phase using the SVM. In the classification phase, when a new data point is mapped into the original vector space, the average distances between the new data point and the support vectors from different categories are measured using the Euclidean distance function. The classification decision is made based on the category of support vectors which has the lowest average distance with the new data point, and this makes the classification decision irrespective of the efficacy of hyper-plane formed by applying the particular kernel function and soft margin parameter. We tested our proposed framework using several text datasets. The experimental results show that this approach makes the accuracy of the Euclidean-SVM text classifier to have a low impact on the implementation of kernel functions and soft margin parameter C.     

多项式核函数SVM快速分类算法

标准的SVM分类计算过程中有大量的支持向量参与了计算，导致了分类速度缓慢。该文为提高SVM的分类速度，提出了一种快速的多项式核函数SVM分类算法，即将使用多项式核的SVM分类决策函数展开为关于待分类向量各分量的多项式，分类时通过计算各个多项式的值而得到分类结果，使分类计算量和支持向量数量无关，又保留了全部支持向量的信息。当多项式核函数的阶数或待分类向量的维数较低而支持向量数量较多时，使用该算法可以使SVM 分类的速度得到极大的提高。针对实际数据集的实验表明了该算法的有效性。     

Relationship Between Support Vector Set and Kernel Functions in SVM

Based on a constructive learning approach,covering algorithms,we investigate the relationship between support vector sets and kernel functions in support vector machines (SVM).An interesting result is obtained.That is,in the linearly non-separable case,any sample of a given sample set K can become a support vector under a certain kernel function.The result shows that when the sample set K is linearly non-separable,although the chosen kernel function satisfies Mercer‘s condition its corresponding support vector set is not necessarily the subset of K that plays a crucial role in classifying K.For a given sample set,what is the subset that plays the crucial role in classification?In order to explore the problem,a new concept,boundary or boundary points,is defined and its properties are discussed.Given a sample set K,we show that the decision functions for classifying the boundary points of K are the same as that for classifying the K itself.And the boundary points of K only depend on K and the structure of the space at which k is located and independent of the chosen approach for finding the boundary.Therefore,the boundary point set may become the subset of K that plays a crucial role in classification.These results are of importance to understand the principle of the support vector machine(SVM) and to develop new learning algorithms.     

一种推广高斯核模型设计及其优化

支持向量机分类中,不同属性对分类的贡献往往不同,针对此问题,在核函数中引入属性权重,提出一种推广的高斯核模型,同时以最小化k-fold交叉验证误差为目标,采用粒子群算法进行推广高斯核的模型选择。最后通过UCI上标准数据集进行实验,证实该方法能够有效提高支持向量机的推广能力。     

一种支持向量机的混合核函数

核函数是支持向量机的核心,不同的核函数将产生不同的分类效果.由于普通核函数各有其利弊,为了得到学习能力和泛化能力较强的核函数,根据核函数的基本性质,两个核函数之和仍然是核函数,将局部核函数和全局核函数线性组合构成新的核函数--混合核函数.该核函数吸取了局部核函数和全局核函数的优点.利用混合核函数进行流程企业供应链预测实验,仿真结果验证了该核函数的有效性和正确性.     

Evolutionary strategies for hyperparameters of support vector machines based on multi-scale radial basis function kernels

Kernel functions are used in support vector machines (SVM) to compute inner product in a higher dimensional feature space. SVM classification performance depends on the chosen kernel. The radial basis function (RBF) kernel is a distance-based kernel that has been successfully applied in many tasks. This paper focuses on improving the accuracy of SVM by proposing a non-linear combination of multiple RBF kernels to obtain more flexible kernel functions. Multi-scale RBF kernels are weighted and combined. The proposed kernel allows better discrimination in the feature space. This new kernel is proved to be a Mercer’s kernel. Furthermore, evolutionary strategies (ESs) are used for adjusting the hyperparameters of SVM. Training accuracy, the bound of generalization error, and subset cross-validation on training accuracy are considered to be objective functions in the evolutionary process. The experimental results show that the accuracy of multi-scale RBF kernels is better than that of a single RBF kernel. Moreover, the subset cross-validation on training accuracy is more suitable and it yields the good results on benchmark datasets.     

基于SVM分类问题的核函数选择仿真研究

首先讨论支持向量机(SVM)的基本思想和实现过程,随后着重对SVM核函数进行探讨,从理论上研究常用核函数的选择优化问题。采用UCI数据库中的玻璃识别数据、菖蒲植物数据以及汽车评估数据分别对选择不同的核函数情况进行实验仿真分类和比较。仿真结果表明,同类数据选择不同核函数会产生不同的分类效果,选取合适的核函数对分类效果有很大的影响。     

最小二乘Littlewood-Paley小波支持向量机

基于小波分解理论和支持向量机核函数的条件，提出了一种多维允许支持向量核函数——Littlewood-Paley小波核函数．该核函数不仅具有平移正交性，而且可以以其正交性逼近二次可积空间上的任意曲线，从而提升了支持向量机的泛化性能．在Littlewood-Paley小波函数作为支持向量核函数的基础上，提出了最小二乘Littlewood-Paley小波支持向量机(LS-LPWSVM)．实验结果表明，LS-LPWSVM在同等条件下比最小二乘支持向量机的学习精度要高，因而更适用于复杂函数的学习问题．     

Integration of Dynamic Equation of Spring-Mass-Damper Systems via Chebyshev Polynomials

A numerical scheme based on Chebyshev polynomials for the determination of the response of spring-mass-damper systems is presented. The state vector of the differential equation of the spring-mass-damper system is expanded in terms of Chebyshev polynomials. This expansion reduces the original differential equations to a set of linear algebraic equations where the unknowns are the coefficient of Chebyshev polynomials. A formal procedure to generate the coefficient matrix and the right-hand side vector of this system of algebraic equations is discussed. The numerical efficiency of the proposed method is compared with that of Runge-Kutta method. It is shown that this scheme is accurate and is computationally efficient.     

一种基于SVM新的学习方法及应用

该文提出一种基于支持向量机的组合核函数的学习方法,它首先由遗传算法作为新的学习方法得到训练,组合核函数的权值在学习过程中被确定,并在决策模型的分类阶段用来作为参数。这种学习方法被应用在两个关于癌症诊断的公用数据集中,从而获得分类最优超平面。通过实验,这种学习方法显示出比用单一核函数具有较好的性能。     

基于插值的核函数构造

近年来，统计学习(SLT)和支持向量机(SVM)理论的研究日益受到当前国际机器学习领域的重视．有关核函数的研究则一直是研究的重点．这是因为不同的核函数会导致SVM的泛化能力有很大的不同．如何根据所给数据选择合适的核函数成为人们所关注的核心问题．该文首先指出满足Mercer条件的核函数的具体表达式并非问题关键，在此基础上，该文进一步提出利用散乱数据插值的办法确定特征空间中感兴趣点的内积值以代替传统核函数的一般表达式所起的作用．实验表明该方法不仅能够有效改善支持向量机的设计训练过程中的不确定性，而且泛化能力要优于绝大部分的基于传统核函数的支持向量机．     

半监督图核降维方法

基于图结构的数据表示和分析,在机器学习领域正得到越来越广泛的关注。以往研究主要集中在为图数据定义一个度量其相似性关系的核函数即图核,一旦定义出图核,就可以用标准的支持向量机(SVM)来对图数据进行分类。将图核方法进行扩充,先利用核主成分分析(kPCA)对图核诱导的高维特征空间中的数据进行降维,得到与原始图数据相对应的低维向量表示的数据,然后对这些新得到的数据用传统机器学习方法进行分析;通过在kPCA中利用图数据中的成对约束形式的监督信息,得到基于图核的半监督降维方法。在MUTAG和PTC等标准图数据集上的实验结果验证了所提方法的有效性。     

基于KL散度的支持向量机方法及应用研究

针对ICA提取的说话人语音特征,导出以库尔贝克—莱布勒（KL）散度作为距离测度的KL核函数用来设计支持向量机,实现了一个高分辨率的ICA/SVM说话人确认系统.说话人确认的仿真实验结果表明,使用ICA特征基函数系数比直接使用语音数据训练SVM得到的分类间隔大,支持向量少,而且使用KL核函数的ICA/SVM系统确认的等差率也低于其它传统SVM方法,证明了基于KL散度的支持向量机方法在实现分类和判决上具有高效性能.     

基于原型超平面的多类最接近支持向量机

基于广义特征值的最接近支持向量机（proximal support vector machine via generalized eigenvalues，GEPSVM）摒弃了传统意义下支持向量机典型平面的平行约束，代之以通过优化使每类原型平面尽可能接近本类样本，同时尽可能远离它类样本的准则来解析获得原型平面；从而避免了SVM的二次规划，其分类性能达到甚至超过了SVM．但GEPSVM仍存在如下不足：①仅对两分类问题而提出，无法直接求解多分类问题；②存在正则化因子的选择问题；③求解原型平面的广义特征值问题中所涉及的矩阵一般仅为半正定，容易导致奇异性问题．通过定义新的准则，构建了一个能直接求解多个原型超平面的多分类方法，称之为基于原型超平面的多类最接近支持向量机，较之GEPSVM，该方法优势在于：①无正则化因子选择的困扰；②可同时求解多个超平面，对两分类问题，分类性能达到甚至优于GEPSVM；③超平面的选择问题转化为简单特征值而非广义特征值求解问题；④原型平面的选择只依赖于本类样本，故不必考虑多分类情形时的数据不平衡问题．