Matrix Linear Discriminant Analysis

Abstract

We propose a novel linear discriminant analysis (LDA) approach for the classification of high-dimensional matrix-valued data that commonly arises from imaging studies. Motivated by the equivalence of the conventional LDA and the ordinary least squares, we consider an efficient nuclear norm penalized regression that encourages a low-rank structure. Theoretical properties including a nonasymptotic risk bound and a rank consistency result are established. Simulation studies and an application to electroencephalography data show the superior performance of the proposed method over the existing approaches.     

ULDA-based heuristic feature selection method for proteomic profile analysis and biomarker discovery

Uncorrelated linear discriminant analysis (ULDA)-based heuristic feature selection (ULDA-HFS) method was proposed for sample classification and feature extraction for SELDI-TOF MS ovarian cancer data. The ULDA-HFS method includes 4 steps: (1) noise reduction and normalization; (2) selection of discriminatory bins with CHI2 method; (3) peak detection and alignment for each selected bins; and (4) selection of several peaks as potential biomarkers by means of ULDA. As a result, 7 m/z locations were selected in this study; they were 245.3, 559.4, 565.6, 704.2, 717.2, 2667 and 4074.4. To evaluate the classification impression, PCA, PLS-DA and ULDA were performed for discriminant analysis and ULDA obtained the perfect separation. Finally, the 7 selected potential biomarkers were evaluated by ULDA, both sensitivity and specificity were 100%. The 7 m/z values obtained may provide clues for ovarian cancer biomarker discovery. Once the proteins were identified at these m/z locations, it can be used as specific protein for early detection and diagnosis for ovarian cancer.     

Efficient Integration of Sufficient Dimension Reduction and Prediction in Discriminant Analysis

Sufficient dimension reduction (SDR) methods are popular model-free tools for preprocessing and data visualization in regression problems where the number of variables is large. Unfortunately, reduce-and-classify approaches in discriminant analysis usually cannot guarantee improvement in classification accuracy, mainly due to the different nature of the two stages. On the other hand, envelope methods construct targeted dimension reduction subspaces that achieve dimension reduction and improve parameter estimation efficiency at the same time. However, little is known about how to construct envelopes in discriminant analysis models. In this article, we introduce the notion of the envelope discriminant subspace (ENDS) as a natural inferential and estimative object in discriminant analysis that incorporates these considerations. We develop the ENDS estimators that simultaneously achieve sufficient dimension reduction and classification. Consistency and asymptotic normality of the ENDS estimators are established, where we carefully examine the asymptotic efficiency gain under the classical linear and quadratic discriminant analysis models. Simulations and real data examples show superb performance of the proposed method. Supplementary materials for this article are available online.     

Classification of detectors for ion chromatography using principal components regression and linear discriminant analysis

Principal components regression (PCR) and linear discriminant analysis (LDA) have been applied to the classification of ion chromatographic detectors using information about the sample and other IC method conditions (19 attributes in total), a training set of 12 693 cases and a randomly-chosen test set of 1410 cases. Missing data was entered as a separate ‘unknown' code. When the value of each attribute was coded in a simple cardinal series (e.g., column=1, 2, 3, etc.), PCR correctly predicted the detector in 27% of the training set and 28% of the test set. By creating a variable (taking a value between 0 (absent) and 1 (present)) for each value of each attribute, the PCR prediction for both sets increased to 60%. LDA was more successful, predicting 69% of the detectors of each set, using a prior probability of the frequency of a given detector in the database, but this included zero hits for detectors that were poorly represented in the database. If equal prior probabilities were chosen the overall success rate dropped to 33% but now the classification of less frequently used detectors was improved. The ability of these numerically-oriented methods to classify discrete, non-numerical data, is surprisingly good and compares with induction methods, neural networks and expert systems reported previously.     

基于形状特征的线状地物提取方法研究

遥感影像上线状地物的提取是影像处理的主要研究内容之一。通过对线状地物形状特征的分析,将其抽象为长矩形,设定长宽比阈值,进行线状地物提取,取得不错实验结果。     

Fault diagnosis based on variable-weighted kernel Fisher discriminant analysis

In this paper, a new fault diagnosis approach with variable-weighted kernel Fisher discriminant analysis (VW-KFDA) is proposed. The approach incorporates the variable weighting into KFDA. The variable weighting finds out the weight vector of each fault by maximizing separation between the normal and each fault data. With continuous non-negative values, each element of the weight vector represents the corresponding variable's contribution to a special fault. After all fault data are weighted by the corresponding weight vectors, KFDA is performed on these weighted fault data. These weight vectors offer important supplemental classification information to KFDA and effectively improve its multi-classification performance. The proposed approach is applied to the Tennessee Eastman process (TEP). The results show superior capability for fault diagnosis to KFDA and FDA.     

一种基于正交非负矩阵分解的多通道线性预测语音去混响方法

在相对封闭的声学环境中,由于受到混响的影响,麦克风阵列采集到的信号清晰度降低、甚至混淆不清。为了解决这一问题,文章在多通道线性预测(Multi-Channel Linear Prediction, MCLP)语音去混响的基础上,提出了一种改进的多通道线性预测(Multi-ChannelLinearPrediction,MCLP)方法即正交非负矩阵线性预测(Orthogonal Non-negative Matrix Factorization Multi-Channel Linear Prediction, ONMFMCLP)方法。该方法利用纯净语音的短时谱域的稀疏性,构建了基于正交的非负矩阵分解(Non-negative Matrix Factorization, NMF)的Kullback-Leibler(KL)问题,通过对矩阵求迹、利用梯度下降法给出迭代规则,进而改进了MCLP中目标信号矩阵的协方差估计。实验结果表明,相对于其他方法,ONMFMCLP方法具有更好的去混响效果。     

Uncorrelated linear discriminant analysis (ULDA): A powerful tool for exploration of metabolomics data

The theory together with an algorithm for uncorrelated linear discriminant analysis (ULDA) is introduced and applied to explore metabolomics data. ULDA is a supervised method for feature extraction (FE), discriminant analysis (DA) and biomarker screening based on the Fisher criterion function. While principal component analysis (PCA) searches for directions of maximum variance in the data, ULDA seeks linearly combined variables called uncorrelated discriminant vectors (UDVs). The UDVs maximize the separation among different classes in terms of the Fisher criterion. The performance of ULDA is evaluated and compared with PCA, partial least squares discriminant analysis (PLS-DA) and target projection discriminant analysis (TP-DA) for two datasets, one simulated and one real from a metabolomic study. ULDA showed better discriminatory ability than PCA, PLS-DA and TP-DA. The shortcomings of PCA, PLS-DA and TP-DA are attributed to interference from linear correlations in data. PLS-DA and TP-DA performed successfully for the simulated data, but PLS-DA was slightly inferior to ULDA for the real data. ULDA successfully extracted optimal features for discriminant analysis and revealed potential biomarkers. Furthermore, by means of cross-validation, the classification model obtained by ULDA showed better predictive ability than PCA, PLS-DA and TP-DA. In conclusion, ULDA is a powerful tool for revealing discriminatory information in metabolomics data.     

On the direct estimation of creep and relaxation functions

Two alternative approaches for estimating linear viscoelastic material functions from a single experiment under random excitation are derived and analyzed. First, Boltzmann’s superposition integral is discretized into a system of linear equations. Due to the ill-posedness of the resulting matrix equation, Tikhonov’s regularization is introduced. Second, the integral is transformed into a recursive formula, using a Prony series representation of viscoelastic material functions, in which gradient-based optimization is applied. Numerical results are provided to compare and verify the applicability of the presented numerical procedures.     

线阵CCD高速A/D接口技术及开发系统

本文介绍了一种新的不用DMA芯片的高速。A/D数据采集和处理系统,该系统以8098单片机为核心,采用单地址总线,双数据总线的结构。高速A/D采集时,数据流与指令流相分离,A/D结果直接存入相应的RAM区,可达到2MC的采样速率。本文还介绍了由它组成的线阵CCD应用开发系统的特点。此系统可工作在逐位A/D采样测量方式和二值化脉冲测量方式,还可以和286、386微机组成主-从式光信息测控系统。     

Estimation of the number of true null hypotheses when conducting a multiple testing

Application of the standard hypothesis test with no adjustment for the multiple testing leads to a large number of false discoveries. The most convenient error measure in multiple testing is False Discovery Rate, FDR. However, calculation of FDR requires good estimation of the number of true null hypotheses, n₀, (or equivalently, of π₀ = n₀/n, where n denotes the number of all tested hypotheses). Estimation of π₀ is a non-trivial problem and it can be done under several assumptions about input data. In our study, several approaches to the estimation of π₀ values are compared for the different data models (independent features, block-correlated data and mixture models). The presented results give evidence that general dependence of the features leads to a very doubtful estimation of their significance.     

Use of non-negative constraint in Tikhonov regularization for particle sizing based on forward light scattering

Abstract

The set of linear equations in the inversion of particle size distribution (PSD) based on forward light scattering is an ill-posed problem. In order to solve the inverse problem of this kind, a number of inversion algorithms have been proposed. The regularization algorithm can reconstruct the PSD, but in usual case, the solution may contain negative values and is strongly oscillating. Owing to the natural reason, the solution should be non-negative and smooth. In this paper, a simple non-negative constraint (NNC) is used with a combination of the Tikhonov regularization. Simulations and experiments show that the regularization with NNC can achieve more reasonable results.     

Multi-parameter regularization method for particle sizing of forward light scattering

The matrix equation in the inversion of particle sizing based on forward light scattering is an ill-posed problem. To solve such an inversion problem, a number of algorithms have been proposed. The single parameter regularization is effective for retrieving the particle size distribution, but the solution is usually oscillatory in the presence of measurement errors. In this work, a multi-parameter regularization is presented to diminish the oscillations of the solution, which is verified with simulations and experiments.     

Estimating the Expected Probability of Misclassification for a Rule Based on the Linear Discriminant Function: Univariate Normal Case

The problem is to estimate the average probability of misclassifying an observation from a given population in the context of the two group classification problem when populations are univariate normal with unknown means and common known variance, and the rule is based on the linear discriminant function. Several estimators are compared with respect to asymptotic MSE and with respect to the distribution of the absolute error between estimator and parameter, and conclusions drawn about the best estimators.     

Estimating the Expected and the Optimal Probabilities of Misclassification

The problem is to estimate the expected and the optimal probabilities of misclassification in the context of the two group p-dimensional normal classification problem with means and common covariance matrix unknown and a rule based on the linear discriminant function. Performance of several estimators is compared by means of a computer sampling study. For larger p (p = 20) certain estimators are definitely superior for each of the probabilities, while for small p there is less differentiation in performance.