Orthonormal Subspace Research Articles

Monitoring dynamic process with orthonormal subspace analysis

AbstractTraditional multivariate statistics‐based process monitoring (MSPM) methods are static algorithms, and the “time lag shift” method (TLSM) is the most commonly used approach to handle the dynamic issue. This paper proves in theory that two drawbacks exist in TLSM‐based dynamic approaches: information unrelated to the real‐time data is also analyzed, and information that can be predicted by historical data is counted repeatedly in both real‐time and historical data. This paper adopts orthonormal subspace analysis (OSA) to handle these issues. OSA can successfully separate real‐time data into information that can be predicted by historical data (the dynamic component) and cannot be predicted for process monitoring (the static component), so the detection result is not influenced by redundant information and is more sensitive to process faults than TLSM‐based dynamic methods.

SFENOSA: A Novel KPI-Related Process Monitoring Method by Slow Feature Extraction and Elastic Net Orthonormal Subspace Analysis

SFENOSA: A Novel KPI-Related Process Monitoring Method by Slow Feature Extraction and Elastic Net Orthonormal Subspace Analysis

A Novel Dynamic Process Monitoring Algorithm: Dynamic Orthonormal Subspace Analysis

Orthonormal subspace analysis (OSA) is proposed for handling the subspace decomposition issue and the principal component selection issue in traditional key performance indicator (KPI)-related process monitoring methods such as partial least squares (PLS) and canonical correlation analysis (CCA). However, it is not appropriate to apply the static OSA algorithm to a dynamic process since OSA pays no attention to the auto-correlation relationships in variables. Therefore, a novel dynamic OSA (DOSA) algorithm is proposed to capture the auto-correlative behavior of process variables on the basis of monitoring KPIs accurately. This study also discusses whether it is necessary to expand the dimension of both the process variables matrix and the KPI matrix in DOSA. The test results in a mathematical model and the Tennessee Eastman (TE) process show that DOSA can address the dynamic issue and retain the advantages of OSA.

Fault detection and separation of hybrid electric vehicles based on kernel orthogonal subspace analysis

Driving quality and vehicles safety of hybrid electric vehicles (HEVs) are two hot-topic issues in automobile technology. Nowadays, research focuses to more intelligent and convenient HEVs fault detection methods. This paper will focus on the fault detection of HEV powertrain system with a data-driven algorithm. Orthonormal subspace analysis (OSA) is a newly proposed data-driven method which adds the ability of fault separation. Nonetheless, the linear OSA algorithm cannot effectively detect powertrain system faults, since these faults present complex nonlinear characteristics. A new kernel OSA (KOSA) method is proposed to transform the nonlinear problem into a linear problem through the mapping of kernel function and the dimensionality reduction technique of OSA. Testing results on a nonlinear model and real samples of XMQ6127AGCHEVN61 HEV show that KOSA address the nonlinear problems and it performs better than OSA and kernel principal component analysis (KPCA)

A novel multivariate statistical process monitoring algorithm: Orthonormal subspace analysis

Partial least squares (PLS) and canonical correlation analysis (CCA) are two most popular key performance indicators (KPI) monitoring algorithms, which have shortcomings in dealing with the KPI-related information leakage problem, the model identification problem, the component selection problem, and the process hypothesis problem. To overcome these shortcomings, this study proposes a new multivariate statistics-based process monitoring (MSPM) method called the orthonormal subspace analysis (OSA) method. OSA divides process data and KPI data into three orthonormal subspaces through an analytic solution. Hence, OSA not only can detect a fault but also can judge whether the fault is KPI-related or KPI-unrelated. OSA is always effective irrespective of the availability of the KPI variables in the online monitoring stage. In OSA, the cumulative percent variance (CPV) method is adopted for principal component selection. In brief, OSA is a more effective method than PLS and CCA, and it achieves superior performance in fault detection and classification.

Numerical study of POD-Galerkin-DEIM reduced order modeling of cardiac monodomain formulation

The three-dimensional cardiac monodomain model with inhomogeneous and anisotropic conductivity characterizes a complicated system that contains spatial and temporal approximation coefficients along with a nonlinear ionic current term. These complexities make its numerical modeling computationally challenging, and therefore, the formation of an efficient computational approximation is important for studying cardiac propagation. In this paper, a reduced order modeling approach has been developed for the simplified cardiac monodomain model, which yields a significant reduction of the full order dynamics of the cardiac tissue, reducing the required computational resources. Additionally, the discrete empirical interpolation technique has been implemented to accurately estimate the nonlinearity of the ionic current of the cardiac monodomain scheme. The proper orthogonal decomposition technique has been utilized, which transforms a given dataset called ‘snapshots’ to a new coordinate system. The snapshots are computed first from the original system, and they encapsulate all the information observed over both time and parameter variations. Next, the proper orthogonal decomposition provides a reduced order basis for projecting the original solution onto a low-dimensional orthonormal subspace. Finally, a reduced set of unknowns of the forward problem is obtained for which the solution involves significant computational savings compared to that for the original system of unknowns. The efficiency of the model order reduction technique for finite difference solution of cardiac electrophysiology is examined concerning simulation time, error potential, activation time, maximum temporal derivative, and conduction velocity. Numerical results for the monodomain show that its solution time can be reduced by a significant factor, with only 0.474 mV RMS error between the full order and reduced dimensions solution.

Extraction of category orthonormal subspace for multi-class classification

Extraction of pattern class associated discriminative subspace is critical to many pattern classification problems. Traditionally, pattern class labels are regarded as indicators to discriminate between pattern classes. In this work, a novel indicator model is proposed to extract discriminant subspace by projecting samples onto a space where the projected categories are mutually orthogonal and in-category normalized. Category orthonormal property and its connections to discriminative subspace extraction are derived. It is shown that the proposed method has a strong connection with the existing Fukunaga-Koontz Transformation but extends the category number from two to multiple. For applications with a large dimension size but limited number of samples, an analytic least-norm solver is developed for calculating the projection function. A discriminative subspace extraction method for multiple classes is proposed and is evaluated by a combination with classifiers. Experiments demonstrate a promising result of using the extracted category orthonormal subspace for multi-class subspace extraction when sample number is small.

Incoherent Noise Suppression of Seismic Data Based on Robust Low-Rank Approximation

Incoherent noise is one of the most common noise widely distributed in seismic data. To improve the interpretation accuracy of the underground structure, incoherent noise needs to be adequately suppressed before the final imaging. We propose a novel method for suppressing seismic incoherent noise based on the robust low-rank approximation. After the Hankelization, seismic data will show strong low-rank features. Our goal is to obtain the stable and accurate low-rank approximation of the Hankel matrix and then reconstruct the denoised data. We construct a mixed model of the nuclear norm and the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$l_{1}$ </tex-math></inline-formula> norm to express the low-rank approximation of the Hankel matrix constructed in the frequency domain. Essentially, the adopted model is an optimization for the subspace similar to the online subspace tracking method, thus avoiding the time-consuming singular value decomposition (SVD). We introduce the orthonormal subspace learning to convert the nuclear norm to the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$l_{1}$ </tex-math></inline-formula> norm to optimize the orthonormal subspace and the corresponding coefficient. Finally, two optimization strategies—the alternating direction method and the block coordinate descent method—are applied to obtain the optimized orthonormal subspace and the corresponding coefficient for representing the low-rank approximation of the Hankel matrix. We perform incoherent noise attenuation tests on synthetic and real seismic data. Compared with other denoising methods, the proposed method produces small signal errors while effectively suppressing the seismic incoherent noise and has a high computational efficiency.

GPR Clutter Reduction by Robust Orthonormal Subspace Learning

The clutter severely decreases the target visibility, thus the detection rates in ground penetrating radar (GPR) systems. Recently proposed robust principal component analysis (RPCA) based clutter removal method decomposes the GPR image into its low rank and sparse parts corresponding to clutter and target components. Motivated by its encouraging results, many lower complexity low rank and sparse decomposition (LRSD) methods such as go decomposition (GoDec) or robust non-negative matrix factorization (RNMF) have been applied to GPR. This paper proposes a new clutter reduction method using robust orthonormal subspace learning (ROSL). The raw GPR image is decomposed into its clutter and target parts via ROSL. The proposed method is faster than the popular RPCA. Although it has similar complexity, and similar performance with GoDec and RNMF for fine tuned parameters of these methods, the proposed method does not require any presetting of the algorithm parameters. Its performance remains independent for a broad range parameter value. Results demonstrate that the proposed method achieves 14 - 48% higher performance in terms of PSNR values than the state-of-the-art LRSD methods for an arbitrary parameter choice.

A motion and lightness saliency approach for forest smoke segmentation and detection

Most video-based detection systems rely on extracting visual features directly from divided original frames or detected motion regions achieved using conventional background subtraction. However, these approaches are not effective for smoke detection because conventional motion detection methods are sensitive to non-salient motion regions like waves, shaking tree leaves, camera jitter an so on. This tiny motion regions can be easily misclassified as smoke. Second, in the case of light smoke the background is visible with or without motion detection, such that it will deteriorate the feature quality. These regions usually occur far from smoke centers and present unstable and non-salient characteristics. This paper proposes an approach to separate the smoke region based on motion and lightness saliency detection. A low-rank and structured sparse decomposition method is used to extract the foreground regions. Saliency of smoke is then computed for further separation. These aforementioned measures ensure a reliable smoke component extraction. We propose a saliency measurement for group-sparse robust orthonormal subspace and learning (ROSL) in virtue of adaptive parameters. Experiments on challenging data sets demonstrate that the proposed method can work well on a wide range of smoke videos and give better smoke detection results.

Vehicle Layover Removal in Circular SAR Images via ROSL

Circular synthetic aperture radar (CSAR) has raised interest in both wide-aspect-angle 2-D imaging and 3-D feature reconstruction. As for CSAR 2-D imaging with a 360° aperture, vehicles spotlighted in the scene are depicted with multiview layover, which makes the imagery intuitively less comprehensible. In addition, the shape of the layover bulge depends on the elevation of the radar platform. Thus, otherwise identical vehicle targets appear differently in the image when the elevation deviates from a constant, which makes target discrimination more difficult. In this letter, subaperture images are vectorized and stacked to build a composite matrix. Decomposing the composite matrix via robust orthonormal subspace learning results in a low-rank matrix and a sparse matrix. The layover belongs to the sparse matrix and thus can be get rid of. The performance of the proposed method has been verified on synthesized and real CSAR data sets. Experimental results show that the multiview layover of vehicles is eliminated effectively. Moreover, the CSAR images become insensitive to elevation variation after layover removal, which benefits target discrimination.

Reduced multiple empirical kernel learning machine.

Multiple kernel learning (MKL) is demonstrated to be flexible and effective in depicting heterogeneous data sources since MKL can introduce multiple kernels rather than a single fixed kernel into applications. However, MKL would get a high time and space complexity in contrast to single kernel learning, which is not expected in real-world applications. Meanwhile, it is known that the kernel mapping ways of MKL generally have two forms including implicit kernel mapping and empirical kernel mapping (EKM), where the latter is less attracted. In this paper, we focus on the MKL with the EKM, and propose a reduced multiple empirical kernel learning machine named RMEKLM for short. To the best of our knowledge, it is the first to reduce both time and space complexity of the MKL with EKM. Different from the existing MKL, the proposed RMEKLM adopts the Gauss Elimination technique to extract a set of feature vectors, which is validated that doing so does not lose much information of the original feature space. Then RMEKLM adopts the extracted feature vectors to span a reduced orthonormal subspace of the feature space, which is visualized in terms of the geometry structure. It can be demonstrated that the spanned subspace is isomorphic to the original feature space, which means that the dot product of two vectors in the original feature space is equal to that of the two corresponding vectors in the generated orthonormal subspace. More importantly, the proposed RMEKLM brings a simpler computation and meanwhile needs a less storage space, especially in the processing of testing. Finally, the experimental results show that RMEKLM owns a much efficient and effective performance in terms of both complexity and classification. The contributions of this paper can be given as follows: (1) by mapping the input space into an orthonormal subspace, the geometry of the generated subspace is visualized; (2) this paper first reduces both the time and space complexity of the EKM-based MKL; (3) this paper adopts the Gauss Elimination, one of the on-the-shelf techniques, to generate a basis of the original feature space, which is stable and efficient.

Multiorder Arnoldi Approach for Model Order Reduction of PEEC Models With Retardation

This paper presents an efficient algorithm to create reduced-order models of large linear networks that contain delay elements. The proposed algorithm is based on a multiorder Arnoldi algorithm used to implicitly calculate the moments with respect to frequency. This procedure generates reduced-order models that preserve the structure of the original system, without having to introduce any extra state variables to calculate the moments. In addition, it is shown that the orthonormal subspace of the system, built by introducing extra state variables, is embedded in the subspace constructed by the multiorder Arnoldi approach. Numerical examples of distributed interconnects modeled by partial element equivalent circuits that include retardation effects are described to illustrate the validity of the proposed reduction technique.

A multivariate approach for integrating genome-wide expression data and biological knowledge

Several statistical methods that combine analysis of differential gene expression with biological knowledge databases have been proposed for a more rapid interpretation of expression data. However, most such methods are based on a series of univariate statistical tests and do not properly account for the complex structure of gene interactions. We present a simple yet effective multivariate statistical procedure for assessing the correlation between a subspace defined by a group of genes and a binary phenotype. A subspace is deemed significant if the samples corresponding to different phenotypes are well separated in that subspace. The separation is measured using Hotelling's T(2) statistic, which captures the covariance structure of the subspace. When the dimension of the subspace is larger than that of the sample space, we project the original data to a smaller orthonormal subspace. We use this method to search through functional pathway subspaces defined by Reactome, KEGG, BioCarta and Gene Ontology. To demonstrate its performance, we apply this method to the data from two published studies, and visualize the results in the principal component space.