基于支持向量机和模糊系统的机器学习方法及其应用研究
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摘要
机器学习是现代智能技术中十分重要的一个方面,主要研究如何从观测数据出发得出目前尚不能通过原理分析得到的规律,利用这些规律去分析客观现象,对未来数据或无法观测的数据进行预测。简单地说就是用计算机来模拟人的学习能力,以达到自动获取知识的目的。有三类基本的机器学习问题,它们分别是模式识别、函数逼近以及概率密度估计。在过去十几年的时间里,基于支持向量机和模糊系统的机器学习方法及其应用得到了广泛研究和迅猛发展,并在信号处理、智能控制、模式识别、系统辨识、生物信息学、医疗和行为科学及商业等领域取得了丰硕的成果。
本文针对基于支持向量机和模糊系统的机器学习方法中的几个问题进行了研究,包括支持向量机的学习速度、支持向量机的泛化能力、一型和二型模糊系统的可解释性以及一型和二型模糊系统跟概率理论和微积分理论的结合。本文的创造性研究成果主要有:
(1)利用微分逼近理论提出了一种改进核函数的方法:将原来的核函数乘以一个融入数据信息的正定函数。改进后的核函数增大了分类间隔,改善了支持向量机的性能。实验结果表明改进后的核函数不但提高了分类精度,而且具有很少的支持向量,从而加快了学习速度。
(2)利用再生核理论提出了一类新的核函数——再生核核函数。这类新核函数是再生核空间的再生核。由于再生核核函数兼具多项式核函数和高斯核函数的优点,所以它不仅具有良好的全局性质,而且还具有很强的内推能力。
(3)利用超椭球面坐标变换公式构造出了一类核函数——同维映射超椭球坐标变换核。由于是同维映射,而且增大了分类间隔,所以支持向量机的性能得到了很大改善。对人工数据集和标准数据集的实验结果表明,使用坐标变换核的支持向量机学习速度快,推广能力强。
(4) Epanechnikov混合模型和Mamdani-Larsen(abbr. ML)模糊系统之间的对应关系被建立:任何一个Epanechnikov混合模型都唯一对应着一个Mamdani-Larsen模糊系统,在一定条件下,Epanechnikov混合模型的条件均值和Mamdani-Larsen模糊模型的输出是等价的。因此可用概率的方法来设计Mamdani-Larsen模糊系统,如期望最大化算法等。将设计的模糊系统应用于时间序列预测,仿真结果表明:利用概率方法设计的Mamdani-Larsen模糊系统精度高,抗噪性强。
(5)基于广义Epanechnikov混合模型提出了一种新的模糊系统——具有多维隶属度函数的规则中心化模糊系统。Epanechnikov混合模型的条件期望输出恰好是规则中心化模糊系统的去模糊化输出。因此可以从概率的角度解释模糊系统,反之亦然,即在概率和模糊之间搭建起了一座桥梁。模糊系统的规则后件恰好是其输出在规则中心的一阶Taylor级数展开式,所以模糊系统具有高度解释性且将模糊理论和微分理论相结合;采用的多维隶属度函数考虑了数据间的相关性,因此模糊系统更符合实际。将该系统应用于时间序列和动态系统预测,不但精度高、速度快,而且有很好的鲁棒性。
(6)建立了不确定的高斯混合模型和具有可加性的二型Takagi-Sugeno-Kang (TSK)模糊系统之间的对应关系:任何一个不确定的高斯混合模型都唯一对应着一个二型模糊系统,不确定高斯混合模型的条件均值和二型模糊系统的去模糊化输出是等价的。因此可以从概率的角度解释、研究和训练二型模糊系统。这种新颖的训练方法提高了二型模糊系统的精度和抗噪性。不同类型数据的实验结果表明了该模型的优良性能。
(7)针对模糊系统缺乏优化结构的辨识方法的问题,提出了一种新颖的基于再生核空间理论的模糊系统。该模糊系统的结构辨识算法由再生核函数的的逼近性质给出,简单易处理;并且参数学习只涉及前件的学习,降低了系统的复杂度。仿真结果验证了这种模糊系统及其结构辨识算法的有效性。
Machine learning is one of important tasks of modern intelligence technoloy, it studies mainly how to summarize the laws which is drived from the observed data but can’t be obtained by analyzing the principles at present. And use these laws to analyse objective phenomenon, predict further data or the data which can’t be obseved. To be short, in order to gain the knowledge automatically, it uses computers to simulate the human learning ability. There are there kinds of basic machine learning problems, including pattern recognition, function approximation and probability density estimation.
In the past decades, machine learning methods and applications which are based on Support Vector Machine (abbr. SVM) and fuzzy system are studied widely and develop rapidly. And it also achieves fruitful achievements in the fields of signal processing, intelligent control, pattern recognition, system identification, bioinformatics, medical treatment, behavioral science and business.
This paper is aimed at several issues based on SVM and fuzzy system, including the learning speed and generalization ability of SVM, the interpretability of type-1 and type-2 fuzzy system, the combination of type-1 and type-2 fuzzy system and probability and calculus theory. In this paper, the creative research results are:
(1) A novel method of improving the performance of a support vector machine classifier is presented by modifying kernel function. The method based on the differential approximation of metric merges data information into dynamic kernel by a positive scalar function. The separability is increased by enlarging margin around the separating hyper-plane. Example is given specifically for modifying Gaussian Radial Basis Function kernel. Simulation results for both artificial and real data show remarkable improvement of generalization ability and computational cost.
(2) A kind of novel kernel functions are obtained from the reproducing kernels of Hilbert spaces associated with special inner product. The kernel functions with the advantages of both polynomial kernel and Gauss kernel, not only have a good global property, but also have a strong ability to interpolate. SVM with the proposed kernel functions only need less support vectors to construct two-class hyperplane than that with Gaussian kernel functions, so the proposed kernel functions have the better generalization capability. SVM applied to Wisconsin breast cancer data and artificial data using the proposed kernel functions, and demonstrate that it provides remarkable improvement of support vectors and training time compared with that of SVM with the Gaussian kernels. Especially, the proposed kernel functions become more and more efficient with the increase of orders of the space.
(3) The conventional support vector machine classifier is a nonlinear classifier by mapping a low-dimensional data space into its high-dimensional feature space where it may become linearly separable using kernel functions. The SVM solution (an optimal hyper-plane) is obtained through maximizing the margin between the separating hyper-plane and data in the feature space. The performance of SVM depends heavily on the kernel functions. A class of new kernel functions is proposed, which are named as hyper-ellipsoid coordinate transform kernels using hyper-ellipsoid coordinate transformation formula. The performance of SVM with hyper-ellipsoid kernels may be enhanced as a result of the same dimensional mapping between input space and the feature space and the enlarged spatial resolution. Another advantage is much less support vectors to be required, resulting in faster learning and better generalization capability than the conventional SVM with Gaussian kernels. Experimental results for both artificial and real data confirm the effectiveness of the proposed hyper-ellipsoid support vector machine classifier.
(4) Epanechnikov Mixture Model (abbr. EMM) can be translated to Mamdani-Larsen fuzzy system. The mathematical equivalence is proved between the conditional mean of an EMM, and the defuzzified output of a Mamdani-Larsen fuzzy system. The result provides a study of the new perspective of Mamdani-Larsen fuzzy system by interpreting it from a probabilistic viewpoint. Instead of estimating the parameters of the fuzzy rules directly, the parameters of an Epanechnikov mixture model can be firstly estimated using any popular probability density estimation algorithm, such as Expectation Maximization (abbr. EM). Simulation results show Mamdani-Larsen fuzzy system trained by the new way has high accuracy and strong anti-noise ability.
(5) Generalized Epanechnikov mixture model can be translated into a rule- centered generalized fuzzy system (abbr. RCGFS) with multidimensional membership functions. The conditional mean of a generalized EMM is the defuzzified output of a RCGFS. A bridge is built between probability model and fuzzy system. The distinctive advantage of the proposed fuzzy system induced by a generalized EMM is easy to manipulate and highly interpretable, due to the fact that the coefficients in the consequent polynomials of fuzzy rules can be exactly interpreted as Taylor series coefficients. Moreover, the proposed system is also rooted at multidimensional membership functions that take into account the correlation among data components, which results in a more effective partition of the input space. The power of the proposed system is experimentally demonstrated by means of three benchmark examples: piecewise function, Mackey-Glass chaotic time series and a nonlinear dynamic system.
(6) Uncertain Gaussian mixture model (abbr. UGMM) can be translated to an additive type-2 Takagi-Sugeno-Kang (abbr. TSK) fuzzy logic system. The mathematical equivalence is proved between the conditional mean of a UGMM, and the defuzzified output of a type-2 TSK fuzzy system (abbr. T2-TSK-FS). The results provide a study of the new perspective of type-2 fuzzy systems by interpreting them from a probabilistic viewpoint. T2-TSK-FS can be obtained from UGMM using any popular density estimation algorithm, such as EM. The new way of training a fuzzy model has two advantages, one is diversification of estimating parameters, and another is that T2-TSK-FS has high accuracy and stronger anti-noise ability. After comparing the simulation results with the ones obtained from other system modeling tools, it can be claimed successful results are achieved.
(7) The existing methods of fuzzy system identification hardly keep good trade-off between precision and fuzzy meaning. One of the main reasons is short of systematic optimal structure identification methods.Thus, a novel fuzzy system whose fuzzy membership functions are the reproducing kernel functions is constructed based on the theorey of the reproducing kernel space. An adaptive learning algorithm for structure identification of the new fuzzy system is proposed using Fuzzy C-Means (abbr. FCM) algorithm and the approximating property of the reproducing kernel functions. Simulation results show the efficiency of the fuzzy system and its structure identification algorithm.
本文针对基于支持向量机和模糊系统的机器学习方法中的几个问题进行了研究,包括支持向量机的学习速度、支持向量机的泛化能力、一型和二型模糊系统的可解释性以及一型和二型模糊系统跟概率理论和微积分理论的结合。本文的创造性研究成果主要有:
(1)利用微分逼近理论提出了一种改进核函数的方法:将原来的核函数乘以一个融入数据信息的正定函数。改进后的核函数增大了分类间隔,改善了支持向量机的性能。实验结果表明改进后的核函数不但提高了分类精度,而且具有很少的支持向量,从而加快了学习速度。
(2)利用再生核理论提出了一类新的核函数——再生核核函数。这类新核函数是再生核空间的再生核。由于再生核核函数兼具多项式核函数和高斯核函数的优点,所以它不仅具有良好的全局性质,而且还具有很强的内推能力。
(3)利用超椭球面坐标变换公式构造出了一类核函数——同维映射超椭球坐标变换核。由于是同维映射,而且增大了分类间隔,所以支持向量机的性能得到了很大改善。对人工数据集和标准数据集的实验结果表明,使用坐标变换核的支持向量机学习速度快,推广能力强。
(4) Epanechnikov混合模型和Mamdani-Larsen(abbr. ML)模糊系统之间的对应关系被建立:任何一个Epanechnikov混合模型都唯一对应着一个Mamdani-Larsen模糊系统,在一定条件下,Epanechnikov混合模型的条件均值和Mamdani-Larsen模糊模型的输出是等价的。因此可用概率的方法来设计Mamdani-Larsen模糊系统,如期望最大化算法等。将设计的模糊系统应用于时间序列预测,仿真结果表明:利用概率方法设计的Mamdani-Larsen模糊系统精度高,抗噪性强。
(5)基于广义Epanechnikov混合模型提出了一种新的模糊系统——具有多维隶属度函数的规则中心化模糊系统。Epanechnikov混合模型的条件期望输出恰好是规则中心化模糊系统的去模糊化输出。因此可以从概率的角度解释模糊系统,反之亦然,即在概率和模糊之间搭建起了一座桥梁。模糊系统的规则后件恰好是其输出在规则中心的一阶Taylor级数展开式,所以模糊系统具有高度解释性且将模糊理论和微分理论相结合;采用的多维隶属度函数考虑了数据间的相关性,因此模糊系统更符合实际。将该系统应用于时间序列和动态系统预测,不但精度高、速度快,而且有很好的鲁棒性。
(6)建立了不确定的高斯混合模型和具有可加性的二型Takagi-Sugeno-Kang (TSK)模糊系统之间的对应关系:任何一个不确定的高斯混合模型都唯一对应着一个二型模糊系统,不确定高斯混合模型的条件均值和二型模糊系统的去模糊化输出是等价的。因此可以从概率的角度解释、研究和训练二型模糊系统。这种新颖的训练方法提高了二型模糊系统的精度和抗噪性。不同类型数据的实验结果表明了该模型的优良性能。
(7)针对模糊系统缺乏优化结构的辨识方法的问题,提出了一种新颖的基于再生核空间理论的模糊系统。该模糊系统的结构辨识算法由再生核函数的的逼近性质给出,简单易处理;并且参数学习只涉及前件的学习,降低了系统的复杂度。仿真结果验证了这种模糊系统及其结构辨识算法的有效性。
Machine learning is one of important tasks of modern intelligence technoloy, it studies mainly how to summarize the laws which is drived from the observed data but can’t be obtained by analyzing the principles at present. And use these laws to analyse objective phenomenon, predict further data or the data which can’t be obseved. To be short, in order to gain the knowledge automatically, it uses computers to simulate the human learning ability. There are there kinds of basic machine learning problems, including pattern recognition, function approximation and probability density estimation.
In the past decades, machine learning methods and applications which are based on Support Vector Machine (abbr. SVM) and fuzzy system are studied widely and develop rapidly. And it also achieves fruitful achievements in the fields of signal processing, intelligent control, pattern recognition, system identification, bioinformatics, medical treatment, behavioral science and business.
This paper is aimed at several issues based on SVM and fuzzy system, including the learning speed and generalization ability of SVM, the interpretability of type-1 and type-2 fuzzy system, the combination of type-1 and type-2 fuzzy system and probability and calculus theory. In this paper, the creative research results are:
(1) A novel method of improving the performance of a support vector machine classifier is presented by modifying kernel function. The method based on the differential approximation of metric merges data information into dynamic kernel by a positive scalar function. The separability is increased by enlarging margin around the separating hyper-plane. Example is given specifically for modifying Gaussian Radial Basis Function kernel. Simulation results for both artificial and real data show remarkable improvement of generalization ability and computational cost.
(2) A kind of novel kernel functions are obtained from the reproducing kernels of Hilbert spaces associated with special inner product. The kernel functions with the advantages of both polynomial kernel and Gauss kernel, not only have a good global property, but also have a strong ability to interpolate. SVM with the proposed kernel functions only need less support vectors to construct two-class hyperplane than that with Gaussian kernel functions, so the proposed kernel functions have the better generalization capability. SVM applied to Wisconsin breast cancer data and artificial data using the proposed kernel functions, and demonstrate that it provides remarkable improvement of support vectors and training time compared with that of SVM with the Gaussian kernels. Especially, the proposed kernel functions become more and more efficient with the increase of orders of the space.
(3) The conventional support vector machine classifier is a nonlinear classifier by mapping a low-dimensional data space into its high-dimensional feature space where it may become linearly separable using kernel functions. The SVM solution (an optimal hyper-plane) is obtained through maximizing the margin between the separating hyper-plane and data in the feature space. The performance of SVM depends heavily on the kernel functions. A class of new kernel functions is proposed, which are named as hyper-ellipsoid coordinate transform kernels using hyper-ellipsoid coordinate transformation formula. The performance of SVM with hyper-ellipsoid kernels may be enhanced as a result of the same dimensional mapping between input space and the feature space and the enlarged spatial resolution. Another advantage is much less support vectors to be required, resulting in faster learning and better generalization capability than the conventional SVM with Gaussian kernels. Experimental results for both artificial and real data confirm the effectiveness of the proposed hyper-ellipsoid support vector machine classifier.
(4) Epanechnikov Mixture Model (abbr. EMM) can be translated to Mamdani-Larsen fuzzy system. The mathematical equivalence is proved between the conditional mean of an EMM, and the defuzzified output of a Mamdani-Larsen fuzzy system. The result provides a study of the new perspective of Mamdani-Larsen fuzzy system by interpreting it from a probabilistic viewpoint. Instead of estimating the parameters of the fuzzy rules directly, the parameters of an Epanechnikov mixture model can be firstly estimated using any popular probability density estimation algorithm, such as Expectation Maximization (abbr. EM). Simulation results show Mamdani-Larsen fuzzy system trained by the new way has high accuracy and strong anti-noise ability.
(5) Generalized Epanechnikov mixture model can be translated into a rule- centered generalized fuzzy system (abbr. RCGFS) with multidimensional membership functions. The conditional mean of a generalized EMM is the defuzzified output of a RCGFS. A bridge is built between probability model and fuzzy system. The distinctive advantage of the proposed fuzzy system induced by a generalized EMM is easy to manipulate and highly interpretable, due to the fact that the coefficients in the consequent polynomials of fuzzy rules can be exactly interpreted as Taylor series coefficients. Moreover, the proposed system is also rooted at multidimensional membership functions that take into account the correlation among data components, which results in a more effective partition of the input space. The power of the proposed system is experimentally demonstrated by means of three benchmark examples: piecewise function, Mackey-Glass chaotic time series and a nonlinear dynamic system.
(6) Uncertain Gaussian mixture model (abbr. UGMM) can be translated to an additive type-2 Takagi-Sugeno-Kang (abbr. TSK) fuzzy logic system. The mathematical equivalence is proved between the conditional mean of a UGMM, and the defuzzified output of a type-2 TSK fuzzy system (abbr. T2-TSK-FS). The results provide a study of the new perspective of type-2 fuzzy systems by interpreting them from a probabilistic viewpoint. T2-TSK-FS can be obtained from UGMM using any popular density estimation algorithm, such as EM. The new way of training a fuzzy model has two advantages, one is diversification of estimating parameters, and another is that T2-TSK-FS has high accuracy and stronger anti-noise ability. After comparing the simulation results with the ones obtained from other system modeling tools, it can be claimed successful results are achieved.
(7) The existing methods of fuzzy system identification hardly keep good trade-off between precision and fuzzy meaning. One of the main reasons is short of systematic optimal structure identification methods.Thus, a novel fuzzy system whose fuzzy membership functions are the reproducing kernel functions is constructed based on the theorey of the reproducing kernel space. An adaptive learning algorithm for structure identification of the new fuzzy system is proposed using Fuzzy C-Means (abbr. FCM) algorithm and the approximating property of the reproducing kernel functions. Simulation results show the efficiency of the fuzzy system and its structure identification algorithm.
引文
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