A biplot method for multivariate normal populations with unequal covariance matrices

Some previous ideas about non-linear biplots to achieve a joint representation of multivariate normal populations and any parametric function without assumptions about the covariance matrices are extended. Usual restrictions on the covariance matrices (such as homogeneity) are avoided. Variables are represented as curves corresponding to the directions of maximum means variation. To demonstrate the versatility of the method, the representation of variances and covariances as an example of further possible interesting parametric functions have been developed. This method is illustrated with two different data sets, and these results are compared with those obtained using two other distances for the normal multivariate case: the Mahalanobis distance (assuming a common covariance matrix for all populations) and Rao’s distance, assuming a common eigenvector structure for all the covariance matrices. This work is supported by DGICYT grant (Spain), BFM2000-0801 and also 1999SGR00059.     

An EWMA chart for monitoring the covariance matrix of a multivariate process based on dissimilarity index

In this article, we propose an exponentially weighted moving average (EWMA) control chart for monitoring the covariance matrix of a multivariate process based on the dissimilarity index of 2 matrices. The proposed control chart essentially monitors the covariance matrix by comparing the individual eigenvalues of the estimated EWMA covariance matrix with those of the estimated covariance matrix from the in‐control (IC) phase I data. It is different from the conventional EWMA charts for monitoring the covariance matrix, which are either based on comparing the sum or product or both of the eigenvalues of the estimated EWMA covariance matrix with those of the IC covariance matrix. We compare the performance of the proposed chart with that of the best existing chart under the multivariate normal process. Furthermore, to prevent the control limit of the proposed EWMA chart developed using the limited IC phase I data from having extensively excessive false alarms, we use a bootstrap resampling method to adjust the control limit to guarantee that the proposed chart has the actual IC ARL(average run length) not less than the nominal level with a certain probability. Finally, we use an example to demonstrate the applicability and implementation of the proposed EWMA chart.     

Maximum likelihood principal components regression on wavelet-compressed data

Maximum likelihood principal component regression (MLPCR) is an errors-in-variables method used to accommodate measurement error information when building multivariate calibration models. A hindrance of MLPCR has been the substantial demand on computational resources sometimes made by the algorithm, especially for certain types of error structures. Operations on these large matrices are memory intensive and time consuming, especially when techniques such as cross-validation are used. This work describes the use of wavelet transforms (WT) as a data compression method for MLPCR. It is shown that the error covariance matrix in the wavelet and spectral domains are related through a two-dimensional WT. This allows the user to account for any effects of the wavelet transform on spectral and error structures. The wavelet transform can be applied to MLPCR when using either the full error covariance matrix or the smaller pooled error covariance matrix. Simulated and experimental near-infrared data sets are used to demonstrate the benefits of using wavelets with the MLPCR algorithm. In all cases, significant compression can be obtained while maintaining favorable predictive ability. Considerable time savings were also attained, with improvements ranging from a factor of 2 to a factor of 720. Using the WT-compressed data in MLPCR gave a reduction in prediction errors compared to using the raw data in MLPCR. An analogous reduction in prediction errors was not always seen when using PCR.     

基于数据缩减的分频段小波模态参数快速识别

针对利用小波进行模态参数识别效率较低的问题,提出了一种基于数据缩减的分频段小波模态参数快速识别算法。利用奇异值分解对协方差信号在保留数据信息量的情况下进行缩减以减少参与计算的数据量,对正功率谱密度矩阵的奇异值分解确定识别系统的模态阶数及相应的频率范围,利用小波变换对缩减后的数据进行各阶模态逐频段识别。相比原始算法,文中方法减少了小波分析的数据量并避免了一些无用频带的小波分解从而减少计算量。通过对一个3阶线性时不变系统以及一个大桥模型的参数识别验证了文中方法在保持识别精度的情况下大幅度地提升了计算效率。     

Bilinear decomposition based alignment of chromatographic profiles

A novel alignment procedure for chromatographic signals with photodiode array detection is presented. At first, the complexity of the chromatographic signals is reduced by chemometric resolution of the pure constituents. For this, the application of multivariate curve resolution-alternating least squares leads to the decomposition of the multiway data block into a chemically meaningful bilinear model representing the chromatographic profiles and their spectral signatures. The flexible implementation of a spectral selectivity constraint allows the background to be differentiated from the constituent spectra. Hereby, the pure concentration profiles are obtained which are consequently individually aligned by correlation optimized warping. In its final step, the procedure reconstitutes the original data with the aligned chromatographic profiles and their corresponding spectra. The alignment is evaluated for two sets of chromatographic signals. The new procedure improves the original application of correlation optimized warping minimizing the risks of aligning noncorresponding chromatographic information.     

基于多尺度高阶奇异谱熵的信号特征提取方法

提出了基于变分模态分解(VMD)的高阶奇异谱熵的特征提取方法,并应用在滚动轴承故障诊断中。首先,使用4阶累积量切片代替奇异谱熵分析(SSEA)的协方差矩阵,引入VMD分解实现方法多尺度化,提出信号多分辨高阶奇异谱熵分析(M-HSSEA)方法;通过信号分析,VMD解决了模态混叠的问题,且能够实现信号滤波,同时该方法提取的熵特征向量增强了相空间重构参数鲁棒性;通过和小波奇异谱提取特征的方法对比,结果表明所提出的方法在克服频率混叠现象,提取的特征点总体离散度小等方面更具优势;最后,结合深度信念网络分类器实现了对故障的分类,实验结果验证了所提方法的有效性和可行性。     

Maximum likelihood principal component analysis with correlated measurement errors: theoretical and practical considerations

Procedures to compensate for correlated measurement errors in multivariate data analysis are described. These procedures are based on the method of maximum likelihood principal component analysis (MLPCA), previously described in the literature. MLPCA is a decomposition method similar to conventional PCA, but it takes into account measurement uncertainty in the decomposition process, placing less emphasis on measurements with large variance. Although the original MLPCA algorithm can accommodate correlated measurement errors, two drawbacks have limited its practical utility in these cases: (1) an inability to handle rank deficient error covariance matrices, and (2) demanding memory and computational requirements. This paper describes two simplifications to the original algorithm that apply when errors are correlated only within the rows of a data matrix and when all of these row covariance matrices are equal. Simulated and experimental data for three-component mixtures are used to test the new methods. It was found that inclusion of error covariance information via MLPCA always gave results which were at least as good and normally better than PCA when the true error covariance matrix was available. However, when the error covariance matrix is estimated from replicates, the relative performance depends on the quality of the estimate and the degree of correlation. For experimental data consisting of mixtures of cobalt, chromium and nickel ions, maximum likelihood principal components regression showed an improvement of up to 50% in the cross-validation error when error covariance information was included.     

Direct fast method for time-limited signal reconstruction

We consider reconstruction of signals by a direct method for the solution of the discrete Fourier system. We note that the reconstruction of a time-limited signal can be simply realized by using only either the real part or the imaginary part of the discrete Fourier transform (DFT) matrix. Therefore, based on the study of the special structure of the real and imaginary parts of the discrete Fourier matrix, we propose a fast direct method for the signal reconstruction problem, which utilizes the numerically truncated singular value decomposition. The method enables us to recover the original signal in a stable way from the frequency information, which may be corrupted by noise and/or some missing data. The classical inverse Fourier transform cannot be applied directly in the latter situation. The pivotal point of the reconstruction is the explicit computation of the singular value decomposition of the real part of the DFT for any order. Numerical experiments for 1D and 2D signal reconstruction and image restoration are given.     

Memory-type multivariate charts with fixed and variable sampling intervals for process mean when covariance matrix is unknown

Memory-type multivariate charts have been widely recognized as a potentially powerful process monitoring tool because of their excellent speed in detecting small-to-moderate shifts in the mean vector of a multivariate normally distributed process, namely, the multivariate EWMA (MEWMA), double MEWMA, Crosier multivariate CUSUM (MCUSUM), and Pignatiello and Runger MCUSUM charts. These multivariate charts are based on the assumption that the covariance matrix is known in advance; but, it may not be known in practice. It is thus not possible to use these multivariate charts unless a large Phase I dataset is available from an in-control process. In this paper, we propose multivariate charts with fixed and variable sampling intervals for the process mean vector when the covariance matrix is estimated from sample. Using the Monte Carlo simulation method, the run length characteristics of the multivariate charts are computed. It is shown that the in-control and out-of-control run length performances of the proposed multivariate charts are robust to the changes in the process covariance matrix, while the existing multivariate charts are not. A real dataset is taken to explain the implementation of the proposed multivariate charts.     

2D correlation spectroscopy and multivariate curve resolution in analyzing pH-dependent evolving systems monitored by FT-IR spectroscopy,a comparative study

Multivariate curve resolution (MCR) and 2D correlation spectroscopy (2D-CoS), including sample-sample correlation, have been applied to the analysis of evolving midinfrared spectroscopic data sets obtained from titrations of organic acids in aqueous solution. In these data sets, well-defined species with significant differences in their spectra are responsible for the spectral variation observed. The two fundamentally different chemometric techniques have been evaluated and discussed on the basis of experimental and supportive simulated data sets. MCR gives information that can be directly related to the chemical species that is of importance from a practical point of view, whereas 2D-CoS results normally require more interpretation. The obtained conclusions are regarded valid for similar evolving data, which are increasingly being encountered in analytical chemistry when multivariate detectors are used to follow dynamic processes, including separations as well as chemical reactions, among others.     

基于卷积盲源分离的噪声鲁棒性语音识别的研究

研究了一种基于卷积盲分离算法与MFCC（Mel-Frequency Cepstral Coefficient）特征相结合的噪声鲁棒语音识别方法。该方法在预处理阶段,首先计算预白化观测数据的多阶自相关协方差矩阵,以获得多时延处理的二阶解相关统计信息。然后利用得到的二阶统计信息构建两个对称正定矩阵,通过Cholesky因式分解等一系列变换获得唯一存在的矩阵,根据此矩阵估算语音信号并提取MFCC特征用于后续识别。实验结果表明,在低信噪比条件下,该方法对于数字语音的识别性能优于基本的MFCC识别器和文献中已有的卷积分离算法。     

延伸奇异值分解包及其在高速列车轮对轴承故障诊断中的应用

结合多分辨奇异值分解包的分解结构和对滚动轴承故障信号的Hankel矩阵的奇异值分布特性研究,提出了延伸奇异值分解包。该算法的核心包括矩阵递推构造和矩阵重构。以分量信号能量为指标,提出了有效分量信号的筛选准则,并基于该准则,进一步提出了延伸奇异值分解包的快速算法。仿真结果表明,延伸奇异值分解包对信号中共振频带分量信号具有很好的分解能力,方法具有强鲁棒性,同时极大地改善了奇异值分解包中出现的模态混叠。应用高速列车轮对轴承试验数据对该方法进行试验验证,结果表明,该方法能有效分离高速列车轮对轴承复合故障信号的不同共振频带信号,对筛选的有效分量信号进行包络分析,可有效提取不同类型的故障特征频率及其谐波,对共振频带的聚集性和故障的表征力相比奇异值分解包均有显著提高。     

一种新的差分奇异值比谱及其在轮对轴承故障诊断中的应用

因滚动体和保持架的随机滑动,轴承故障信号多为伪循环平稳信号。针对这种情况,提出了应用周期截断矩阵的奇异值分解的轮对轴承故障诊断方法。研究了轴承故障伪循环平稳信号的奇异值分布,结合奇异值能量差分和奇异值比,提出了一种新的能量差分奇异值比谱作为周期截断矩阵的嵌入维度计算方法;利用能量差分奇异值比谱计算嵌入维度并利用轮对轴承振动信号构造周期截断矩阵,对矩阵进行奇异值分解,并提出利用差分能量谱确定奇异值有效秩阶次并重构矩阵从而分离出周期信号;对该信号做包络分析以实现轮对轴承的故障诊断。应用轮对实验台的复合故障轴承振动数据对该方法进行验证,结果表明,所提方法能够有效提取轴承外圈、滚动体及保持架的特征频率的基频及其倍频,与传统应用Hankel矩阵进行奇异值分解降噪方法相比,该方法抗干扰能力显著,能够分离同频带的不同故障周期信号,且得到的包络谱谱线清晰,谐波丰富,使故障诊断的可靠性得到了显著提高。     

Three-way alternating least squares using three-dimensional tensors in MATLAB®

This paper describes an improved three-way alternating least-squares multivariate curve resolution algorithm that makes use of the recently introduced multi-dimensional arrays of MATLAB®. Multi-dimensional arrays allow for a convenient way to apply chemically sound constraints, such as closure, in the third dimension. The program is designed for kinetic studies on liquid chromatography with diode array detection but can be used for other three-way data analysis. The program is tested with a large number of synthetic data sets and its flexibility is demonstrated, especially when non-trilinear data sets are fit. In this case, the algorithm finds a solution with a better fit than direct trilinear decomposition (DTD). When trilinear data are used, the optimal fit is not as good as when a direct decomposition method is used. Most real data sets, however, have some degree of non-trilinearity. This makes this method a better choice to analyze non-trilinear, three-way data than direct trilinear decomposition.     

Improved method for calibrating a Stokes polarimeter

We present a method for calibrating a polarization state analyzer that uses a set of well- characterized reference polarization states and makes no assumptions about the optics contained in the polarimeter other than their linearity. The method requires that a matrix be constructed that contains the data acquired for each of the reference polarization states and that this matrix be pseudoinverted. Since this matrix is usually singular, we improve the method by performing the pseudoinversion by singular value decomposition, keeping only the four largest singular values. We demonstrate the calibration technique using an imaging polarimeter based upon liquid crystal variable retarders and with light emitting diode (LED) illumination centered at 472 nm, 525 nm, and 630 nm. We generate the reference polarization states by using an unpolarized source, a single polarizer, and a Fresnel rhomb. This method is particularly useful when calibrations are performed on field-grade instruments at a centrally maintained facility and when a traceability chain needs to be maintained.     

基于EMD的奇异值分解技术在滚动轴承故障诊断中的应用

针对滚动轴承故障振动信号的非平稳特征,提出了一种基于经验模态分解(EmpiricalModeDecomposition,简称EMD)和奇异值分解技术的滚动轴承故障诊断方法。该方法首先采用EMD方法将滚动轴承振动信号分解为多个平稳的内禀分量(IntrinsicModefunction,简称IMF)之和,并形成初始特征向量矩阵。然后对初始特征向量矩阵进行奇异值分解得到矩阵的奇异值,将其作为滚动轴承振动信号的故障特征向量,并输入神经网络来识别滚动轴承的工作状态和故障类型。实验分析结果表明,本文方法能有效地应用于滚动轴承故障诊断。     

基于检测指数判决的子空间方位估计方法

针对子空间高分辨方位估计方法稳健性差的问题,根据协方差矩阵稳定估计所需累积次数和各子空间强度谱检测指数差异性,提出一种基于检测指数判决的子空间方位估计方法.该方法将协方差矩阵频域求取过程转换为经相参补偿的时域求取,降低空间数据稳定性对协方差矩阵估计的影响;依据各子空间强度谱检测指数差异,提取各子空间强度谱判决统计量;根...     

一种非高斯随机振动过程数值模拟方法

目的 考虑真实随机振动的非高斯特性,提出一种根据已知信息生成与其相符的非高斯随机振动过程的数值模拟方法。方法 基于均值、方差、偏斜度、峭度及功率谱密度函数（或自相关函数）等约束条件,对非高斯随机振动进行模拟。根据功率谱获取非高斯过程的自相关矩阵;通过Hermite多项式的正交性质和多项式混沌展开方法推导出的公式,构造满足标准正态分布随机过程的协方差矩阵,并对其进行谱分解和主成分分析;最后,利用Karhunen-Loeve展开和多项式混沌展开来表示所模拟的非高斯振动过程。结果 随着采样点个数的增加,实测数据与模拟数据之间的误差越来越小,该方法具有较好的模拟精度。结论 应用多项式混沌展开、Karhunen-Loeve展开以及蒙特卡洛等方法,可生成非高斯随机振动过程,并得到准确有效的各项统计参数模拟值。     

Parametric identification of two-port models in the frequencydomain

The authors treat the problem of parametric estimation of linear time-invariant dynamic two-port models (e.g. the short-circuit admittance matrix) from experimental data. A multivariate frequency-domain Gaussian maximum likelihood estimator is proposed to estimate the unknown coefficients occurring in the rational two-port model. It takes the perturbing noise of all the measured voltages and currents into account. The covariance matrix of the noise is assumed to be known, e.g. from measurements. The estimates and their covariance matrix are obtained as the result of an optimization procedure. The value of the minimized loss function and the covariance matrix of the estimates can be used to determine the model structure. The ability of the estimator to handle real measurement problems is demonstrated by means of experimental results. Using the estimated two-part parameters of an unloaded band-pass filter, it was possible to predict the transfer function of the loaded filter within an error of ±0.01 dB on the magnitude and ±0.1° on the phase