Multivariate Calibration in Chemometrics: the Robust Approach

In this paper we show the usage of robust multivariate statistical methods in geostatistics. A usual procedure to estimate the values of variables (e.g. geochemical variables) measured at certain points of a region is to apply geostatistical methods like Krige estimation (based on the estimation of variograms). Here we emphasize robust principal component and factor analysis for the preliminary investigation of the data to reduce the dimension. Geostatistical methods are applied afterwards to the estimated factor scores. The final results show the influence of certain combinations of variables in the considered region. Moreover, the estimated factor scores with the robust procedure indicate outlying observations in a much better way. Copyright © 1999 John Wiley & Sons, Ltd.

Advantages of robust procedures over ordinary least-squares procedures in geochemical data analysis is demonstrated using NURE data from the Hot Springs Quadrangle, South Dakota, U.S.A. Robust principal components analysis with 5% multivariate trimming successfully guarded the analysis against perturbations by outliers and increased the number of interpretable factors. Regression with SINE estimates significantly increased the goodness-of-fit of the regression and improved the correspondence of delineated anomalies with known uranium prospects. Because of the ubiquitous existence of outliers in geochemical data, robust statistical procedures are suggested as routine procedures to replace ordinary least-squares procedures.

Principal component analysis (PCA) is widely used to extract features and reduce dimensionality in various computer vision and image/video processing tasks. Conventional approaches either lack robustness to outliers and corrupted data or are designed for one-dimensional signals. To address this problem, we propose a robust PCA model for two-dimensional images incorporating structured sparse priors, referred to as structured sparse 2D-PCA. This robust model considers the prior of structured and grouped pixel values in two dimensions. As the proposed formulation is jointly nonconvex and nonsmooth, which is difficult to tackle by joint optimization, we develop a two-stage alternating minimization approach to solve the problem. This approach iteratively learns the projection matrices by bidirectional decomposition and utilizes the proximal method to obtain the structured sparse outliers. By considering the structured sparsity prior, the proposed model becomes less sensitive to noisy data and outliers in two dimensions. Moreover, the computational cost indicates that the robust two-dimensional model is capable of processing quarter common intermediate format video in real time, as well as handling large-size images and videos, which is often intractable with other robust PCA approaches that involve image-to-vector conversion. Experimental results on robust face reconstruction, video background subtraction data set, and real-world videos show the effectiveness of the proposed model compared with conventional 2D-PCA and other robust PCA algorithms.

In this study, the observation model of Kalman Filter (KF) is extended to an errors-in-variables (EIV) model because the observations may exist in the design matrix of the observation model. Then, a robust total least squares method (RTLS) is introduced into the KF, and a robust total Kalman filter (RTKF) algorithm is derived. The RTKF is a simple, flexible and effective algorithm. It is simple because its computational formulae are similar to the computational formulae of a standard KF; it is flexible because it can be used in a wide range of applications; it is effective because the influence of outliers on estimated results is weakened. Finally, the simulated example of the indoor location and the empirical example of pseudorange differential positioning are used to demonstrate the performance of the RTKF algorithm. The results prove the validity, robustness, and reliability of the RTKF in dealing with the outliers that exist in both observation vector and design matrix of the EIV model. Furthermore, the results of the empirical example show that the RTKF improves the precision of a pseudorange differential positioning compared with KF and robust Kalman filter (RKF) algorithms regardless the observation model has outliers or not in this empirical example.

Chemical exchange saturation transfer (CEST) MRI requires the acquisition of multiple saturation-weighted images and can last several minutes. Misalignments among these images, which are often due to the inevitable motion of the subject, will corrupt CEST contrast maps and result in large quantification errors. Therefore, the registration of the CEST series is critical. However, registration is challenging since common intensity-based registration algorithms may fail to differentiate CEST signals from motion artifacts. Herein, we studied how different patterns of motion affect CEST quantification and proposed a cascaded two-step registration scheme by utilizing features extracted from the entire Z-spectral image series instead of direct registration to a single image. The proposed approach is conducted in two stages: during the first coarse registration, the Z-spectral image series is decomposed by robust principal component analysis (RPCA) to separate CEST contrast from motion. The recomposed image series using only the low-rank component, which contains minimized motion, are averaged to generate a reference for the alignment of the image series. To further remove residual misalignments, the coarse registration is followed by a refinement stage, which uses PCA iteratively to generate motionless synthetic reference series with the first few principal components (PCs) that correspond to CEST contrast. In the end, the quality check is performed to exclude the images with unsuccessful registration. The proposed registration scheme (RPCA + PCA_R) was assessed by both phantom experiments and in vivo data of tumor-bearing mouse brain, with simulated random rigid motion in different patterns applied to the acquired static Z-spectral image series. For comparison, previous correction schemes using an explicit image [either S0 or Ssat(∆ω)] as registration reference were also performed, named as S0_R and Ssat_R respectively. To illustrate the advantage of combination of RPCA and PCA, registration was also exploited using either only the RPCA-based method (RPCA_R) or only the PCA-based one (PCA_R). Compared with the above four methods, RPCA + PCA_R allowed for more accurate correction of the corrupted Z-spectral images, exhibiting smaller MTRasym(∆ω) error maps and lower residual Z-spectra referring to the static data. Among all the five correction methods, the corrected Z-spectral image series by RPCA + PCA_R and the resulting MTRasym(∆ω) maps achieved the highest correlation coefficients (CC) with respect to the static ones. The registration scheme of RPCA + PCA_R provides robust motion correction between two specific Z-spectral images and among an entire image series, through extraction of the static component from the entire Z-spectra set and inclusion of a PCA-based refinement step. Therefore, this method can help improve CEST acquisition and quantification.

This paper presents a novel algorithm - robust kernel principal component analysis (robust KPCA), on the basis of the research of kernel principal component analysis (KPCA) and robust principal component analysis (RPCA). First, this algorithm sets the radius of the images of the training samples in the feature space using kernel tricks, then determines whether the samples are outliers or not, and finally analyzes the training samples which have eliminated the outliers using KPCA algorithm. The improved KPCA algorithm not only retains the non-linearity property of KPCA algorithm but also gets better robustness. Because the effects of outliers are eliminated, robust KPCA algorithm gets higher accuracy than KPCA algorithm for data analysis. The simulation experiments show that the robust KPCA algorithm developed is better than the KPCA algorithm.

In this paper, we propose an efficient singing voice extraction method from a music sound consisting of a singing voice and a drum sound. The proposed method is based on robust principal component analysis (RPCA) which is a technique to separate a given matrix into a sparse matrix and a low rank matrix. Dealing with a spectrogram of the music sound as the given matrix, RPCA gives an extracted singing voice spectrogram as a sparse spectrogram. We improve the capability of RPCA method by removing a residual drum sound from the sparse spectrogram. The residual drum sound is estimated as spectral components which repeatedly arise in the sparse spectrogram. Removing the estimated residual drum sound spectrogram from the sparse spectrogram gives an improved singing voice spectrogram. Simulation results show that the proposed method can improve 10dB of GNSDR as compared with the conventional RPCA singing voice extraction method.

Separating vocal and accompaniment signals from a monaural music signal is a challenging task. Recently, robust principal component analysis (RPCA) has been proposed for use in the magnitude spectrogram domain to separate the low-rank and sparse residual matrices, which are assumed to represent the accompaniment and vocal signals, respectively. In this paper, we propose two extended methods based on the RPCA algorithm for more effective vocal separation. First, we extend the conventional RPCA and propose to use in the spectrogram decomposition framework Schatten p- and lp-norms, which are generalized versions of the nuclear norm and l1-norm used in RPCA, respectively. Second, we apply proper scale compression to the magnitude spectrogram, making it a more appropriate representation for the decomposition. Experiments using the MIR-1K dataset show that the proposed methods yield significantly better separation performance than the conventional RPCA.

Robust principal component analysis (PCA) has been proved to be an efficient multivariate statistical method for extraction of a multivariate structure of geochemical data containing outlier values. Furthermore, because of the compositional characteristics of geochemical data, logratio transformation approaches also have to be implemented to transform the data from the simplex space into real space. In order to model the extracted principal component of transformed data, we used sequential Gaussian simulation (SGS) to overcome the drawbacks associated with the traditional kriging method, e.g. bias conditionally relevant to underestimation and overestimation, the smoothing effect and so on. The results of implementing robust PCA on both untransformed and logratio transformed data, which have been acquired from an orogenic gold deposit, indicate that robust PCA analysis of untransformed data is not capable of certifying element association of gold occurrences. However, in robust PCA analysis of transformed data, both varieties of gold mineralization, associated with pyrite mineral and quartz veins, have been revealed. The first variety of gold occurrence is detected in an integrated PC obtained from PC1, PC2 and PC3, which are associated with Pb, As, Hg, Mo, S and Sb elements in quartz–sulphide veins. The second one, PC4, could be correlated with quartz veins. Furthermore, the results of the geostatistical SGS approach are compatible with the detected anomaly in the Alut gold deposit.

Traditional robust principle component analysis (RPCA) has a high computational cost because RPCA needs to calculate the singular value decomposition of large matrices. To address this issue, this paper proposes a matrix-factorization-based RPCA (MFRPCA) model. MFRPCA has high computation efficiency while improving the robustness and flexibility of traditional RPCA using a non-convex low-rank approximation. Experiment results on challenging datasets demonstrate superior performance of MFRPCA compared with several advanced low-rank reconstruction methods.

RPCA (Robust Principal Component Analysis) recovers sparse and low rank components from the original observation data, RPCA commonly uses ADM (Alternate Direction Method) for solving, the efficiency of algorithm depends on the nuclear norm optimization solution, that is SVD. And the application of RPCA in computer vision, image and video and so on, large amounts of data, such as images and video, make it difficult for large-scale data SVD. In this paper, we used the random matrix algorithm to improve the SVD, respectively the Count Sketch algorithm, the Prototype Randomized k-SVD algorithm and the Faster Randomized k-SVD algorithm, The main idea is to reduce the size of the original large-scale data matrix and sample randomly. Using the random projection algorithm to obtain an approximation of the original matrix, and operate QR decomposition of the approximate matrix, get the unitary matrix corresponding to the approximate matrix, and do the corresponding SVD, finally we can get the results which is similar to the original matrix calculation. Obtaining approximation of the original data matrix. But the cost time and space have been greatly optimized. Simulation experiments based on single image and video foreground detection show that the proposed methods can greatly improve the efficiency of RPCA iterative optimization.

A comparison between two robust PCA algorithms

Principal Component Analysis (PCA) is the main method of dimension reduction and data processing when the dataset is of high dimension. Therefore, PCA is a widely used method in almost all scientific fields. Because PCA is a linear combination of the original variables, the interpretation process of the analysis results is often encountered with some difficulties. The approaches proposed for solving these problems are called to as Sparse Principal Component Analysis (SPCA). Sparse approaches are not robust in existence of outliers in the data set. In this study, the performance of the approach proposed by Croux et al. (2013), which combines the advantageous properties of SPCA and Robust Principal Component Analysis (RPCA), will be examined through one real and three artificial datasets in the situation of full sparseness. In the light of the findings, it is recommended to use robust sparse PCA based on projection pursuit in analyzing the data. Another important finding obtained from the study is that the BIC and TPO criteria used in determining lambda are not much superior to each other. We suggest choosing one of these two criteria that give an optimal result.

Recent advances in optical coherence tomography (OCT) lead to the development of OCT angiography to provide additional helpful information for diagnosis of diseases like basal cell carcinoma. In this paper, we investigate how to extract blood vessels of human skin from full-field OCT (FF-OCT) data using the robust principal component analysis (RPCA) technique. Specifically, we propose a short-time RPCA method that divides the FF-OCT data into segments and decomposes each segment into a low-rank structure representing the relatively static tissues of human skin and a sparse matrix representing the blood vessels. The method mitigates the problem associated with the slow-varying background and is free of the detection error that RPCA may have when dealing with FF-OCT data. Both short-time RPCA and RPCA methods can extract blood vessels from FF-OCT data with heavy speckle noise, but the former takes only half the computation time of the latter. We evaluate the performance of the proposed method by comparing the extracted blood vessels with the ground truth vessels labeled by a dermatologist and show that the proposed method works equally well for FF-OCT volumes of different quality. The average F-measure improvements over the correlation-mapping OCT method, the modified amplitude-decorrelation OCT angiography method, and the RPCA method, respectively, are 0.1835, 0.1032, and 0.0458.

Video background recovery is a very important task in computer vision applications. Recent research offers robust principal component analysis (RPCA) as a promising approach for solving video background recovery. RPCA works by decomposing a data matrix into a low-rank matrix and a sparse matrix. Our previous work shows that when the desired rank of the low-rank matrix is known, fixing the rank in the algorithm called FrALM (fixed-rank ALM) yields more robust and accurate results than existing RPCA algorithms. However, application of RPCA to video background recovery requires that each frame in the video is encoded as a column in the data matrix. This is impractical in real applications because the videos can be easily larger than the amount of memory in a computer. This paper presents an algorithm called iFrALM (incremental fixed-rank ALM) that computes fixed-rank RPCA incrementally by splitting the video frames into an initial batch and an incremental batch. Comprehensive tests show that iFrALM uses less memory and time compared to FrALM. Moreover, the initial batch size and batch quality can be carefully selected to ensure that iFrALM reduces memory and time complexity without sacrificing accuracy.KeywordsBackground recoveryIncremental SVDFixed-rank robust PCA