Fast kernel Fisher discriminant analysis via approximating the kernel principal component analysis

Kernel Fisher discriminant analysis (KFDA) extracts a nonlinear feature from a sample by calculating as many kernel functions as the training samples. Thus, its computational efficiency is inversely proportional to the size of the training sample set. In this paper we propose a more approach to efficient nonlinear feature extraction, FKFDA (fast KFDA). This FKFDA consists of two parts. First, we select a portion of training samples based on two criteria produced by approximating the kernel principal component analysis (AKPCA) in the kernel feature space. Then, referring to the selected training samples as nodes, we formulate FKFDA to improve the efficiency of nonlinear feature extraction. In FKFDA, the discriminant vectors are expressed as linear combinations of nodes in the kernel feature space, and the extraction of a feature from a sample only requires calculating as many kernel functions as the nodes. Therefore, the proposed FKFDA has a much faster feature extraction procedure compared with the naive kernel-based methods. Experimental results on face recognition and benchmark datasets classification suggest that the proposed FKFDA can generate well classified features.     

一种基于空间变换的核Fisher鉴别分析

引入空间变换的思相想，提出了一种基于空间变换的核Fisher鉴别分析，与KFDA不同的是，该方法只需在一个较低维的空间内执行，从而较大幅度地降低了求解最优鉴别矢量集的计算量，提高了计算速度，在ORL标准人脸库上的试验结果验证了所提方法的有效性。     

Face recognition using a kernel fractional-step discriminant analysis algorithm

Feature extraction is among the most important problems in face recognition systems. In this paper, we propose an enhanced kernel discriminant analysis (KDA) algorithm called kernel fractional-step discriminant analysis (KFDA) for nonlinear feature extraction and dimensionality reduction. Not only can this new algorithm, like other kernel methods, deal with nonlinearity required for many face recognition tasks, it can also outperform traditional KDA algorithms in resisting the adverse effects due to outlier classes. Moreover, to further strengthen the overall performance of KDA algorithms for face recognition, we propose two new kernel functions: cosine fractional-power polynomial kernel and non-normal Gaussian RBF kernel. We perform extensive comparative studies based on the YaleB and FERET face databases. Experimental results show that our KFDA algorithm outperforms traditional kernel principal component analysis (KPCA) and KDA algorithms. Moreover, further improvement can be obtained when the two new kernel functions are used.     

基于核Fisher判别分析的蛋白质氧链糖基化位点的预测

以各种窗口长度的蛋白质样本序列为研究对象,实验样本用稀疏编码方式编码,使用核Fisher判别分析(KFDA)的方法来预测蛋白质氧链糖基化位点。首先通过非线性映射(由核函数隐含定义)将样本映射到特征空间,然后在特征空间中用Fisher判别分析进行分类。进一步,用多数投票策略对各种窗口下的分类器进行组合以综合多个窗口的优势。实验结果表明,使用组合KFDA的方法预测的效果优于FDA和PCA以及单个KFDA分类器的预测效果,预测准确率为86.5%。     

基于低密度分割几何距离的半监督KFDA算法

提出了一种基于低密度分割几何距离的半监督KFDA（kernel Fisher discriminant analysis）算法（semisupervised KFDA based on low density separation geometry distance，简称SemiGKFDA）.该算法以低密度分割几何距离作为相似性度量，通过大量无标签样本，提高KFDA算法的泛化能力.首先，利用核函数将原始空间样本数据映射到高维特征空间中；然后，通过有标签样本和无标签样本构建低密度分割几何距离测度上的内蕴结构一致性假设，使其作为正则化项整合到费舍尔判别分析的目标函数中；最后，通过求解最小化目标函数获得最优投影矩阵.人工数据集和UCI数据集上的实验表明，该算法与KFDA及其改进算法相比，在分类性能上有显著提高.此外，将该算法与其他算法应用到人脸识别问题中进行对比，实验结果表明，该算法具有更高的识别精度.     

一种用于人脸识别的非线性鉴别特征融合方法

最近，在人脸等图像识别领域，用于抽取非线性特征的核方法如核Fisher鉴别分析(KFDA)已经取得成功并得到了广泛应用，但现有的核方法都存在这样的问题，即构造特征空间中的核矩阵所耗费的计算量非常大．而且，抽取得到的单类特征往往不能获得到令人满意的识别结果．提出了一种用于人脸识别的非线性鉴别特征融合方法，即首先利用小波变换和奇异值分解对原始输入样本进行降雏变换，抽取同一样本空间的两类特征，然后利用复向量将这两类特征组合在一起，构成一复特征向量空间，最后在该空间中进行最优鉴别特征抽取．在ORL标准人脸库上的试验结果表明所提方法不仅在识别性能上优于现有的核Fisher鉴别分析方法，而且，在ORL人脸库上的特征抽取速度提高了近8倍．     

Generalized nonlinear discriminant analysis and its small sample size problems

This paper develops a generalized nonlinear discriminant analysis (GNDA) method and deals with its small sample size (SSS) problems. GNDA is a nonlinear extension of linear discriminant analysis (LDA), while kernel Fisher discriminant analysis (KFDA) can be regarded as a special case of GNDA. In LDA, an under sample problem or a small sample size problem occurs when the sample size is less than the sample dimensionality, which will result in the singularity of the within-class scatter matrix. Due to a high-dimensional nonlinear mapping in GNDA, small sample size problems arise rather frequently. To tackle this issue, this research presents five different schemes for GNDA to solve the SSS problems. Experimental results on real-world data sets show that these schemes for GNDA are very effective in tackling small sample size problems.     

Nonlinear feature extraction and classification of multivariate data in kernel feature space

Batch processes have played an essential role in the production of high value-added product of chemical, pharmaceutical, food, bio-chemical, and semi-conductor industries. For productivity and quality improvement, several multivariate statistical techniques such as principal component analysis (PCA) and Fisher discriminant analysis (FDA) have been developed to solve a fault diagnosis problem of batch processes. Fisher discriminant analysis, as a traditional statistical technique for feature extraction and classification, has been shown to be a good linear technique for fault diagnosis and outperform PCA based diagnosis methods. This paper proposes a more efficient nonlinear diagnosis method for batch processes using a kernel version of Fisher discriminant analysis (KFDA). A case study on two batch processes has been conducted. In addition, the diagnosis performance of the proposed method was compared with that of an existing diagnosis method based on linear FDA. The diagnosis results showed that the proposed KFDA based diagnosis method outperforms the linear FDA based method.     

局部Fisher判别分析算法及仿真研究*

复杂化工过程常被多种类型的故障损坏,正常的训练数据无法建立准确的操作模型。为了提高复杂化工过程中故障的检测和分类能力,传统无监督Fisher判别分析(Fisher Discriminant Analysis,FDA)算法无法在多模态故障数据中的应用,本文提出基于局部Fisher判别分析(Local Fisher Discriminant Analysis,LFDA)的故障诊断方法。首先计算训练数据的局部类内和类间离散度矩阵,寻找LFDA的投影方向;其次把训练数据和测试数据向投影向量上投影,提取特征向量;最后计算特征向量间的欧氏距离,运用KNN分类器进行分类。把提出的LFDA方法应用到Tennessee Eastman(TE)过程,监控结果表明,LFDA的效果好于FDA和核Fisher判别分析(Kernel Fisher Discriminant Analysis,KFDA),说明LFDA方法在分类及检测不同类的故障方面具有高准确性及高灵敏度的优势。     

Fault diagnosis of nonlinear and large-scale processes using novel modified kernel Fisher discriminant analysis approach

It is pretty significant for fault diagnosis timely and accurately to improve the dependability of industrial processes. In this study, fault diagnosis of nonlinear and large-scale processes by variable-weighted kernel Fisher discriminant analysis (KFDA) based on improved biogeography-based optimisation (IBBO) is proposed, referred to as IBBO-KFDA, where IBBO is used to determine the parameters of variable-weighted KFDA, and variable-weighted KFDA is used to solve the multi-classification overlapping problem. The main contributions of this work are four-fold to further improve the performance of KFDA for fault diagnosis. First, a nonlinear fault diagnosis approach with variable-weighted KFDA is developed for maximising separation between the overlapping fault samples. Second, kernel parameters and features selection of variable-weighted KFDA are simultaneously optimised using IBBO. Finally, a single fitness function that combines erroneous diagnosis rate with feature cost is created, a novel mixed kernel function is introduced to improve the classification capability in the feature space and diagnosis accuracy of the IBBO-KFDA, and serves as the target function in the optimisation problem. Moreover, an IBBO approach is developed to obtain the better quality of solution and faster convergence speed. On the one hand, the proposed IBBO-KFDA method is first used on Tennessee Eastman process benchmark data sets to validate the feasibility and efficiency. On the other hand, IBBO-KFDA is applied to diagnose faults of automation gauge control system. Simulation results demonstrate that IBBO-KFDA can obtain better kernel parameters and feature vectors with a lower computing cost, higher diagnosis accuracy and a better real-time capacity.     

一种面向不平衡数据集的核Fisher线性判别分析方法

实际应用中,很多分类问题是面向不平衡数据的分类,而不平衡数据集会导致许多分类器的性能下降。文中介绍核Fisher线性判别分析的分类机制,分析不平衡数据导致核Fisher线性判别分析失效的原因,进而提出一种加权核Fisher线性判别分析方法。该方法通过调整两类样本的核协方差矩阵对核类内离散度矩阵的贡献, 可克服不平衡数据对分类性能的影响。为进一步测试该方法, 对UCI数据集进行实验测试,实验结果表明该方法可有效改进分类器的分类性能。     

Block-wise 2D kernel PCA/LDA for face recognition

Direct extension of (2D) matrix-based linear subspace algorithms to kernel-induced feature space is computationally intractable and also fails to exploit local characteristics of input data. In this letter, we develop a 2D generalized framework which integrates the concept of kernel machines with 2D principal component analysis (PCA) and 2D linear discriminant analysis (LDA). In order to remedy the mentioned drawbacks, we propose a block-wise approach based on the assumption that data is multi-modally distributed in so-called block manifolds. Proposed methods, namely block-wise 2D kernel PCA (B2D-KPCA) and block-wise 2D generalized discriminant analysis (B2D-GDA), attempt to find local nonlinear subspace projections in each block manifold or alternatively search for linear subspace projections in kernel space associated with each blockset. Experimental results on ORL face database attests to the reliability of the proposed block-wise approach compared with related published methods.     

新的非线性鉴别特征抽取方法及人脸识别

在非线性空间中采用新的最大散度差鉴别准则,提出了一种新的核最大散度差鉴别分析方法.该方法不仅有效地抽取了人脸图像的非线性鉴别特征,而且从根本上避免了以往核Fisher鉴别分析中训练样本总数较多时,通常存在的核散布矩阵奇异的问题,计算复杂度大大降低,识别速度有了明显的提高.在ORL人脸数据库上的实验结果验证了该算法的有效性.     

极小化类内散布度的大间距非线性鉴别分析

提出了一种新的非线性鉴别分析算法——极小化类内散布的大间距非线性鉴别分析。该算法的主要思想是将原始样本映射到更高维的空间中,利用核技术对传统的大间距分类算法进行改进,在新的高维空间中利用再生核技术寻找核鉴别矢量,使得在这个新的空间中核类内散度尽可能的小。在ORL人脸数据库上进行实验,分析了识别率及识别时间,结果表明该方法具有一定优势。     

Kernel clustering-based discriminant analysis

In this paper, a kernelized version of clustering-based discriminant analysis is proposed that we name KCDA. The main idea is to first map the original data into another high-dimensional space, and then to perform clustering-based discriminant analysis in the feature space. Kernel fuzzy c-means algorithm is used to do clustering for each class. A group of tests on two UCI standard benchmarks have been carried out that prove our proposed method is very promising.     

An efficient renovation on kernel Fisher discriminant analysis and face recognition experiments

A reformative kernel algorithm, which can deal with two-class problems as well as those with more than two classes, on Fisher discriminant analysis is proposed. In the novel algorithm the supposition that in feature space discriminant vector can be approximated by some linear combination of a part of training samples, called “significant nodes”, is made. If the “significant nodes” are found out, the novel algorithm on kernel Fisher discriminant analysis will be superior to the naive one in classification efficiency. In this paper, a recursive algorithm for selecting “significant nodes”, is developed in detail. Experiments show that the novel algorithm is effective and much efficient in classifying.     

Distance-Based Classifier via the Kernel Trick

For classifying large data sets, we propose a discriminant kernel that introduces a nonlinear mapping from the joint space of input data and output label to a discriminant space. Our method differs from traditional ones, which correspond to map nonlinearly from the input space to a feature space. The induced distance of our discriminant kernel is Eu- clidean and Fisher separable, as it is defined based on distance vectors of the feature space to distance vectors on the discriminant space. Unlike the support vector machines or the kernel Fisher discriminant analysis, the classifier does not need to solve a quadric program- ming problem or eigen-decomposition problems. Therefore, it is especially appropriate to the problems of processing large data sets. The classifier can be applied to face recognition, shape comparison and image classification benchmark data sets. The method is significantly faster than other methods and yet it can deliver comparable classification accuracy.     

A fast kernel-based nonlinear discriminant analysis for multi-class problems

Nonlinear discriminant analysis may be transformed into the form of kernel-based discriminant analysis. Thus, the corresponding discriminant direction can be solved by linear equations. From the view of feature space, the nonlinear discriminant analysis is still a linear method, and it is provable that in feature space the method is equivalent to Fisher discriminant analysis. We consider that one linear combination of parts of training samples, called “significant nodes”, can replace the total training samples to express the corresponding discriminant vector in feature space to some extent. In this paper, an efficient algorithm is proposed to determine “significant nodes” one by one. The principle of determining “significant nodes” is simple and reasonable, and the consequent algorithm can be carried out with acceptable computation cost. Depending on the kernel functions between test samples and all “significant nodes”, classification can be implemented. The proposed method is called fast kernel-based nonlinear method (FKNM). It is noticeable that the number of “significant nodes” may be much smaller than that of the total training samples. As a result, for two-class classification problems, the FKNM will be much more efficient than the naive kernel-based nonlinear method (NKNM). The FKNM can be also applied to multi-class via two approaches: one-against-the-rest and one-against-one. Although there is a view that one-against-one is superior to one-against-the-rest in classification efficiency, it seems that for the FKNM one-against-the-rest is more efficient than one-against-one. Experiments on benchmark and real datasets illustrate that, for two-class and multi-class classifications, the FKNM is effective, feasible and much efficient.     

Fault diagnosis and classification framework using multi-scale classification based on kernel Fisher discriminant analysis for chemical process system

Fault detection and diagnosis (FDD) in chemical process systems is an important tool for effective process monitoring to ensure the safety of a process. Multi-scale classification offers various advantages for monitoring chemical processes generally driven by events in different time and frequency domains. However, there are issues when dealing with highly interrelated, complex, and noisy databases with large dimensionality. Therefore, a new method for the FDD framework is proposed based on wavelet analysis, kernel Fisher discriminant analysis (KFDA), and support vector machine (SVM) classifiers. The main objective of this work was to combine the advantages of these tools to enhance the performance of the diagnosis on a chemical process system. Initially, a discrete wavelet transform (DWT) was applied to extract the dynamics of the process at different scales. The wavelet coefficients obtained during the analysis were reconstructed using the inverse discrete wavelet transform (IDWT) method, which were then fed into the KFDA to produce discriminant vectors. Finally, the discriminant vectors were used as inputs for the SVM classification task. The SVM classifiers were utilized to classify the feature sets extracted by the proposed method. The performance of the proposed multi-scale KFDA-SVM method for fault classification and diagnosis was analysed and compared using a simulated Tennessee Eastman process as a benchmark. The results showed the improvements of the proposed multiscale KFDA-SVM framework with an average 96.79% of classification accuracy over the multi-scale KFDA-GMM (84.94%), and the established independent component analysis-SVM method (95.78%) of the faults in the Tennessee Eastman process.