Rank Deficiency Research Articles

On Hidden Rank Deficiency in MCR Problems

ABSTRACTPure component decomposition problems in chemometrics can be classified into rank‐regular and rank‐deficient problems. Rank‐deficient problems are characterized by a spectral data matrix that has a lower rank than the number of chemical species. However, it is possible that there exists rank‐regular factorization of the spectral data matrix, but none of these solutions can be interpreted chemically, and only a solution of the MCR problem with rank deficiency is chemically meaningful. Then we say that the underlying problem suffers from a hidden rank deficiency. In this paper, MCR problems with hidden rank deficiency are introduced and analyzed with several examples for problems of rank 2 and rank 3. The area of feasible solutions is determined with the help of additional constraints on the solution.

Adaptive current decoupling control scheme based on online multi-parameter identification for high-speed permanent magnet synchronous motor in fuel cell

The precise control of centrifugal hydrogen compressor (CHC) driven by high-speed permanent magnet synchronous motor (PMSM) is the key to the stable and efficient operation of hydrogen fuel cell (HFC). In this paper, an adaptive current deviation decoupling control (CDDC) strategy based on variable step size affine projection algorithm (VSS-APAs) is proposed to solve the high-speed PMSM control problem caused by current cross-coupling and parameter perturbation under complex operating conditions. Firstly, the step size is dynamically adjusted by VSS-APA through a normalized gradient descent to respond to system fluctuations, achieving rapid parameter identification during dynamic changes and accurate tracking in steady states. Secondly, stator inductance, magnetic flux linkage, stator resistance, and disturbance torque are identified in two-time-scale based on the characteristics of parameter changes, overcoming the rank deficiency in motor mathematical equations. Finally, utilizing online-identified motor parameters, the gains of the CDDC are adjusted, and compensation is applied for back electromotive force (BEMF) and disturbance torque, thereby ensuring the control accuracy of the motor under parameter perturbations and system disturbances. Experimental results demonstrate that the proposed method enhances the responsiveness and accuracy of the control strategy through efficient parameters identification. Consequently, it guarantees high-efficiency and stable performance of the CHC across different working conditions, markedly advancing the overall functionality and dependability of hydrogen energy system.

Kaczmarz-type methods for solving matrix equations

In this paper, several Kaczmarz-type numerical methods for solving the matrix equations AX = B and XA = C are proposed, where the coefficient matrix A may be full rank or deficient rank. These methods are iterative methods without matrix multiplication. Theoretically, the convergence of these methods is proved. The numerical results show that these methods are more efficient than iterative methods involving matrix multiplication for high-dimensional matrices.

Adjusting effective multiplicity ( ) for family-wise error rate in functional near-infrared spectroscopy data with a small sample size.

The advancement of multichannel functional near-infrared spectroscopy (fNIRS) has enabled measurements across a wide range of brain regions. This increase in multiplicity necessitates the control of family-wise errors in statistical hypothesis testing. To address this issue, the effective multiplicity ( ) method designed for channel-wise analysis, which considers the correlation between fNIRS channels, was developed. However, this method loses reliability when the sample size is smaller than the number of channels, leading to a rank deficiency in the eigenvalues of the correlation matrix and hindering the accuracy of calculations. We aimed to reevaluate the effectiveness of the method for fNIRS data with a small sample size. In experiment 1, we used resampling simulations to explore the relationship between sample size and values. Based on these results, experiment 2 employed a typical exponential model to investigate whether valid could be predicted from a small sample size. Experiment 1 revealed that the values were underestimated when the sample size was smaller than the number of channels. However, an exponential pattern was observed. Subsequently, in experiment 2, we found that valid values can be derived from sample sizes of 30 to 40 in datasets with 44 and 52 channels using a typical exponential model. The findings from these two experiments indicate the potential for the effective application of correction in fNIRS studies with sample sizes smaller than the number of channels.

Adaptive Covariance Hybridization for the Assimilation of SST Observations Within a Coupled Earth System Reanalysis

AbstractEnsemble data assimilation methods, such as the Ensemble Kalman Filter (EnKF), are well suited for climate reanalysis because they feature flow‐dependent covariance. However, because Earth System Models are heavy computationally, the method uses a few tens of members. Sampling error in the covariance matrix can introduce biases in the deep ocean, which may cause a drift in the reanalysis and in the predictions. Here, we assess the potential of the hybrid covariance approach (EnKF‐OI) to counteract sampling error. The EnKF‐OI combines the flow‐dependent covariance computed from a dynamical ensemble with another covariance matrix that is static but less prone to sampling error. We test the method within the Norwegian Climate Prediction Model, which combines the Norwegian Earth System Model and the EnKF. We test the performance of the reanalyzes in an idealized twin experiment, where we assimilate synthetic sea surface temperature observations monthly over 1980–2010. The dynamical and static ensembles consist respectively of 30 members and 315 seasonal members sampled from a pre‐industrial run. We compare the performance of the EnKF to an EnKF‐OI with a global hybrid coefficient, referred to as standard hybrid, and an EnKF‐OI with adaptive hybrid coefficients estimated in space and time. Both hybrid covariance methods cure the bias introduced by the EnKF at intermediate and deep water. The adaptive EnKF‐OI performs best overall by addressing sampling noise and rank deficiencies issues and can sustain low analysis errors by doing smaller updates than the standard hybrid version.

Bayesian inference of vorticity in unbounded flow from limited pressure measurements

We study the instantaneous inference of an unbounded planar flow from sparse noisy pressure measurements. The true flow field comprises one or more regularized point vortices of various strength and size. We interpret the true flow's measurements with a vortex estimator, also consisting of regularized vortices, and attempt to infer the positions and strengths of this estimator assuming little prior knowledge. The problem often has several possible solutions, many due to a variety of symmetries. To deal with this ill posedness and to quantify the uncertainty, we develop the vortex estimator in a Bayesian setting. We use Markov-chain Monte Carlo and a Gaussian mixture model to sample and categorize the probable vortex states in the posterior distribution, tailoring the prior to avoid spurious solutions. Through experiments with one or more true vortices, we reveal many aspects of the vortex inference problem. With fewer sensors than states, the estimator infers a manifold of equally possible states. Using one more sensor than states ensures that no cases of rank deficiency arise. Uncertainty grows rapidly with distance when a vortex lies outside of the vicinity of the sensors. Vortex size cannot be reliably inferred, but the position and strength of a larger vortex can be estimated with a much smaller one. In estimates of multiple vortices their individual signs are discernible because of the nonlinear coupling in the pressure. When the true vortex state is inferred from an estimator of fewer vortices, the estimate approximately aggregates the true vortices where possible.

Design of truss structures with multiple eigenfrequency constraints via rank minimization

Rank deficiency of the dynamic stiffness matrix is an indicator for resonance of a structure at a given frequency. This indicator can be exploited as a heuristic optimization objective to achieve resonance at several frequencies. Log-det heuristic provides a tractable surrogate function for matrix rank in the case of affine dependency of stiffness and mass matrices on design parameters, which applies to truss structures. Reducing the rank of the dynamic stiffness matrix for higher frequencies implies that the matrix is not semi-positive definite. For this case, the log-det heuristic is valid with a combination of interior-point methods and Fazel's semi-definite embedding via linear matrix inequalities. Further constraints on the fundamental frequency and compliance can be easily added within the framework as linear matrix inequalities. Several successful numerical examples illustrate the performance of the approach.

Challenges in determining individual chemical potentials of system elements by Gibbs energy minimization

Accounting for the elusive gas phase in thermodynamic equilibrium calculations is demonstrated to be important for achieving accurate and unique determination of constituent element chemical potentials, when Gibbs energy minimization (GEM) encounter indeterminacy due to rank deficiency in mass-balance equations. The molar GEM of the equilibrium vapor phase offers a robust solution for unambiguously and accurately determining element chemical potentials. To enable this, a novel optimization method for mass-balance equation is presented, paving the way for reliable thermodynamic calculations.

A note on rank deficiency and numerical modeling

AbstractLinearly dependent concentration profiles of a chemical reaction can result in a spectral data matrix with a chemical rank smaller than the number of absorbing chemical species. Such a rank deficiency is problematic for a factor analysis as some information on the pure component spectra cannot be recovered from the mixture data. Matrix augmentation can break rank deficiencies and enable successful pure component recovery. In contrast to this, an artificial breakdown of a rank deficiency can be caused by a numerical finite precision simulation of the underlying kinetic model and can fake a successful MCR analysis. This work discusses the problem and points out some remedies.

Using clustering as pre-processing in the framework of signal unmixing for exhaustive exploration of archaeological artefacts in Raman imaging

Analytical chemistry on archaeological material is an essential part of modern archaeological investigations and from year to year, instrumental improvement has made it possible to generate data at a high spatial and temporal frequency. In particular, Raman spectral imaging can be successfully applied in archaeological research by its simplicity of implementation to study past human societies through the analysis of their material remains. This technique makes it possible to simultaneously obtain spatial and spectral information by preserving sample integrity. However, because of the inherent complexity of the samples in Archaeology (e.g. seniority, fragility, lack or full absence of any information about its composition), chemical interpretation can be difficult at first glance. Indeed, specific problems of spectral selectivity related to unexpected chemical compounds could appear due to their state of conservation. Furthermore, detecting minor compounds becomes challenging as major components impose their contributions in the acquired spectra. Therefore, a relevant chemometric approach has been introduced in this context to characterize distinct spectral sources in a Raman imaging dataset of an archaeological specimen – a mosaic fragment. The fragment was unearthed during the Ruscino archaeological dig on the outskirts of Perpignan, France. It dates back to the oppidum period. The aim is to extract selective spectral information from pixel clustering analysis in order to enhance the initial optimisation step within the Multivariate Curve Resolution and Alternating Least-Squares (MCR-ALS) algorithm, a well-known signal unmixing technique. The underlying principle of the MCR-ALS is that the acquired spectra can be expressed as linear combinations of pure spectra of all individual components present in the chemical system under study. Sometimes it can be difficult to obtain the desired results through the algorithm, particularly if initial estimates of spectral or concentration profiles are inaccurate due to complex signals, noise or lack of selectivity, resulting in rank deficiency (i.e. a poor estimation of the total number of pure signals). For this reason, an innovative threshold-based clustering algorithm, combined with multiple Orthogonal Projection Approaches (OPA), has been developed to improve matrix rank investigation and thus the initialisation step of the MCR-ALS approach before optimisation. The effective analysis of Raman imaging data for an archaeological mosaic played a crucial role in uncovering significant chemical information about a particular biogenic material. This insight sheds light on the origins of mortar manufacture during the oppidum period.

Autonomous navigation method of satellite constellation based on adaptive forgetting factors

To address the problem that model uncertainty and unknown time-varying system noise hinder the filtering accuracy of the autonomous navigation system of satellite constellation, an autonomous navigation method of satellite constellation based on the Unscented Kalman Filter with Adaptive Forgetting Factors (UKF-AFF) is proposed. The process noise covariance matrix is estimated online with the strategy that combines covariance matching and adaptive adjustment of forgetting factors. The adaptive adjustment coefficient based on squared Mahalanobis distance of state residual is employed to achieve online regulation of forgetting factors, equipping this method with more adaptability. The intersatellite direction vector obtained from photographic observations is introduced to determine the constellation satellite orbit together with the distance measurement to avoid rank deficiency issues. Considering that the number of available measurements varies online with intersatellite visibility in practical applications such as time-varying constellation configurations, the smooth covariance matrix of state correction determined by innovation and gain is adopted and constructed recursively. Stability analysis of the proposed method is also conducted. The effectiveness of the proposed method is verified by the Monte Carlo simulation and comparison experiments. The estimation accuracy of constellation position and velocity of UKF-AFF is improved by 30% and 44% respectively compared to those of the extended Kalman filter, and the method proposed is also better than other several adaptive filtering methods in the presence of significant model uncertainty.

Stabilizing nodal integration in dynamic smoothed particle finite element method: A simple and efficient algorithm

The Smoothed Particle Finite Element Method (SPFEM) is a powerful numerical tool for the modelling of large deformation problems in the field of geomechanics. However, the standard SPFEM with nodal integration suffers from the rank deficiency problem due to under-integration. In this study, an hourglass-control based stabilization scheme is presented to overcome the rank deficiency in standard SPFEM with nodal integration. By introducing the hourglass-control based correction forces, the spurious low-energy models in standard SPFEM with nodal integration can be effectively suppressed. The presented stabilization scheme is simple and computationally efficient compared to the existing strain gradient-based stabilization approaches in SPFEM. The proposed hourglass-control based stabilized SPFEM is firstly validated by means of three benchmark examples and the results show that the stabilization scheme based on the hourglass-control can eliminate the rank deficiency instability in standard SPFEM and free of volumetric locking. Finally, two numerical examples related to the cohesive soil flow and the progressive failure of sensitive clay slopes are studied to highlight the robustness of the proposed stabilized SPFEM in solving large deformation geotechnical problems.

The phase-field fracture model enriched by interpolation cover functions for brittle fracture problems

In this study, interpolation cover functions were introduced into the phase-field fracture model (PFM) to improve its convergence and efficiency. The traditional PFM has proven to be successful in simulating brittle fracture problems. However, one disadvantage is that it requires a relatively dense mesh. The PFM model enriched by interpolation cover functions (EPFM) was used for numerical fracture analysis under a relatively coarse mesh. The rank deficiency problem caused by the introduction of interpolation cover functions was resolved using the rank deficiency repair technique. Higher gradients were captured by introducing interpolation cover functions without changing the mesh structure. The numerical examples showed the improvement in convergence. The high efficiency of the proposed EPFM was demonstrated through a series of numerical examples. The numerical results of the EPFM and PFM under different length-scale parameters and meshes were also compared. The results showed that the proposed EPFM is an excellent alternative for evaluating engineering fracture problems.

Form-finding of tensegrity structures based on genetic algorithm

We develop a novel form-finding method that utilizes force density and a genetic algorithm. Firstly, the equilibrium equation is derived by using the force density method. Next, the force density matrix is decomposed through QR decomposition. Subsequently, an optimization objective function is introduced, which incorporates information about the force density values. The optimal solution for the objective function is obtained through the use of a genetic algorithm. We determine a suitable set of force density values that satisfy the requirements of the equilibrium matrix rank deficiency, force density rank deficiency, and precision, thereby establishing the equilibrium configuration of the structure. The simulation results verify the reliability of the proposed method. With its advantages of straightforward calculations, rapid convergence, and high precision, this method proves to be well-suited for the form-finding of both regular and irregular tensegrity structures.

This is no "ICA bug": response to the article, "ICA's bug: how ghost ICs emerge from effective rank deficiency caused by EEG electrode interpolation and incorrect re-referencing".

This is no "ICA bug": response to the article, "ICA's bug: how ghost ICs emerge from effective rank deficiency caused by EEG electrode interpolation and incorrect re-referencing".

Improved modal analyses using the novel quadrilateral overlapping elements

In this work, the novel quadrilateral overlapping element incorporated with the harmonic trigonometric functions is presented to reduce the efforts in mesh division for modal analysis. The new partition of unity functions in the overlapping element are cubic polynomials. Therefore, low-order polynomials can be utilized to augment local approximation space without the linear dependence problem. Moreover, the original polynomial approximation spaces are further enriched by the harmonic trigonometric functions without introducing additional rank deficiencies in the system matrices. Thus, the presented enriched overlapping finite element method (EOFEM) provides very precise eigenfrequencies and eigenmodes by using a very coarse mesh. However, the shape functions in the presented EOFEM do not possess the Kronecker delta function property. Hence, a conforming method that modifies the shape functions rather than the variational principle is proposed to impose the essential boundary conditions. Several numerical examples in two dimensions are then investigated to elaborate on the performance of the presented EOFEM in modal analyses. Besides, a block-diagonal lumped mass matrix is presented and further extended to the structure vibration analysis by incorporating the explicit time integration method. It is proven that the enriched overlapping element is computationally more efficient than the quadratic finite element and the original overlapping element.

Reduced-Rank Envelope Vector Autoregressive Model

The standard vector autoregressive (VAR) models suffer from overparameterization which is a serious issue for high-dimensional time series data as it restricts the number of variables and lags that can be incorporated into the model. Several statistical methods, such as the reduced-rank model for multivariate (multiple) time series (Velu, Reinsel, and Wichern; Reinsel and Velu; Reinsel, Velu, and Chen) and the Envelope VAR model (Wang and Ding), provide solutions for achieving dimension reduction of the parameter space of the VAR model. However, these methods can be inefficient in extracting relevant information from complex data, as they fail to distinguish between relevant and irrelevant information, or they are inefficient in addressing the rank deficiency problem. We put together the idea of envelope models into the reduced-rank VAR model to simultaneously tackle these challenges, and propose a new parsimonious version of the classical VAR model called the reduced-rank envelope VAR (REVAR) model. Our proposed REVAR model incorporates the strengths of both reduced-rank VAR and envelope VAR models and leads to significant gains in efficiency and accuracy. The asymptotic properties of the proposed estimators are established under different error assumptions. Simulation studies and real data analysis are conducted to evaluate and illustrate the proposed method.

Non-invasive imaging of neural activity with magnetic detection electrical impedance tomography (MDEIT): a modelling study

Objectives. (1) Develop a computational pipeline for three-dimensional fast neural magnetic detection electrical impedance tomography (MDEIT), (2) determine whether constant current or constant voltage is preferable for MDEIT, (3) perform reconstructions of simulated neural activity in a human head model with realistic noise and compare MDEIT to EIT and (4) perform a two-dimensional study in a saline tank for MDEIT with optically pumped magnetometers (OPMs) and compare reconstruction algorithms. Approach. Forward modelling and image reconstruction were performed with a realistic model of a human head in three dimensions and at three noise levels for four perturbations representing neural activity. Images were compared using the error in the position and size of the reconstructed perturbations. Two-dimensional MDEIT was performed in a saline tank with a resistive perturbation and one OPM. Six reconstruction algorithms were compared using the error in the position and size of the reconstructed perturbations. Main results. A computational pipeline was developed in COMSOL Multiphysics, reducing the Jacobian calculation time from months to days. MDEIT reconstructed images with a lower reconstruction error than EIT with a mean difference of 7.0%, 5.5% and 11% for three noise cases representing current noise, reduced current source noise and reduced current source and magnetometer noise. A rank analysis concluded that the MDEIT Jacobian was less rank-deficient than the EIT Jacobian. Reconstructions of a phantom in a saline tank had a best reconstruction error of 13%, achieved using 0th-order Tikhonov regularisation with simulated noise-based correction. Significance. This study demonstrated that three-dimensional MDEIT for neural imaging is feasible and that MDEIT reconstructed superior images to EIT, which can be explained by the lesser rank deficiency of the MDEIT Jacobian. Reconstructions of a perturbation in a saline tank demonstrated a proof of principle for two-dimensional MDEIT with OPMs and identified the best reconstruction algorithm.

Data-driven distributed collaborative fault detection and isolation for large-scale dynamic processes in simultaneous-fault cases

This paper proposes a data-driven distributed collaborative fault detection and isolation scheme for large-scale dynamic processes, interconnected by heterogeneous subsystems with both actuator and sensor faults. The residual generator in each subsystem is directly constructed with both local and neighboring process data based on subspace identification. Utilizing the rank deficiency property, the healthy mode residual covariance matrix is estimated with noisy data, and thus the test statistic and detection threshold are obtained. Then, a collaborative simultaneous fault isolation method is developed, where the local residual is decoupled with neighboring actuator fault’s effects and a series of robust residual generators are constructed by solving corresponding optimization problems. In each subsystem, both local fault isolation and neighboring sensor fault isolation are realized. Thus, sensor fault detection and isolation for the whole process can be achieved with a reduced number of monitoring subsystems. Finally, case studies on the Tennessee Eastman (TE) process and a real hot strip mill process (HSMP) verify the feasibility and good performance of the proposed scheme.

Automated measurement and hybrid adaptive identification method for kinematic calibration of hybrid machine tools

Kinematic calibration is an effective method for ensuring the accuracy of hybrid machine tools. Traditional calibration methods are limited by cumbersome operations and poor algorithm robustness. In this study, measurement methods and identification algorithms are studied, and a new calibration method is proposed. First, the configuration of the hybrid machine tool is described, and an error model is constructed. Then, an automated measurement method of tool center point position error for multiple test arbors is proposed using the R-test device. Subsequently, a hybrid adaptive identification algorithm is proposed by sequentially solving the unit inhomogeneity of the identification model, the rank deficiency produced by incomplete measurements, and the multicollinearity between error parameters. Finally, the efficacy of the proposed calibration method is verified through simulations and experiments. The proposed method significantly improves the calibration efficiency and accuracy and proposes a new solution for automated calibration.