Loading Matrices Research Articles

Localization of spatially distributed brain sources after a tensor-based preprocessing of interictal epileptic EEG data.

This paper addresses the localization of spatially distributed sources from interictal epileptic electroencephalographic data after a tensor-based preprocessing. Justifying the Canonical Polyadic (CP) model of the space-time-frequency and space-time-wave-vector tensors is not an easy task when two or more extended sources have to be localized. On the other hand, the occurrence of several amplitude modulated spikes originating from the same epileptic region can be used to build a space-time-spike tensor from the EEG data. While the CP model of this tensor appears more justified, the exact computation of its loading matrices can be limited by the presence of highly correlated sources or/and a strong background noise. An efficient extended source localization scheme after the tensor-based preprocessing has then to be set up. Different strategies are thus investigated and compared on realistic simulated data: the "disk algorithm" using a precomputed dictionary of circular patches, a standardized Tikhonov regularization and a fused LASSO scheme.

Determination of polycyclic aromatic hydrocarbons by four-way parallel factor analysis in presence of humic acid

There is not effective method to solve the quenching effect of quencher in fluorescence spectra measurement and recognition of polycyclic aromatic hydrocarbons in aquatic environment. In this work, a four-way dataset combined with four-way parallel factor analysis is used to identify and quantify polycyclic aromatic hydrocarbons in the presence of humic acid, a fluorescent quencher and an ubiquitous substance in aquatic system, through modeling the quenching effect of humic acid by decomposing the four-way dataset into four loading matrices corresponding to relative concentration, excitation spectra, emission spectra and fluorescence quantum yield, respectively. It is found that Phenanthrene, pyrene, anthracene and fluorene can be recognized simultaneously with the similarities all above 0.980 between resolved spectra and reference spectra. Moreover, the concentrations of them ranging from 0 to 8μgL−1 in the test samples prepared with river water could also be predicted successfully with recovery rate of each polycyclic aromatic hydrocarbon between 100% and 120%, which were higher than those of three-way PARAFAC. These results demonstrate that the combination of four-way dataset with four-way parallel factor analysis could be a promising method to recognize the fluorescence spectra of polycyclic aromatic hydrocarbons in the presence of fluorescent quencher from both qualitative and quantitative perspective.

A modified extended finite element method approach for design sensitivity analysis

SummaryThis paper describes a modified extended finite element method (XFEM) approach, which is designed to ease the challenge of an analytical design sensitivity analysis in the framework of structural optimisation. This novel formulation, furthermore labelled YFEM, combines the well‐known XFEM enhancement functions with a local sub‐meshing strategy using standard finite elements. It deviates slightly from the XFEM path only at one significant point but thus allows to use already derived residual vectors as well as stiffness and pseudo load matrices to assemble the desired information on cut elements without tedious and error‐prone re‐work of already performed derivations and implementations. The strategy is applied to sensitivity analysis of interface problems combining areas with different linear elastic material properties. Copyright © 2015 John Wiley & Sons, Ltd.

A regularized nonnegative canonical polyadic decomposition algorithm with preprocessing for 3D fluorescence spectroscopy

We consider blind source separation in chemical analysis focussing on the 3D fluorescence spectroscopy framework. We present an alternative method to process the Fluorescence Excitation‐Emission Matrices (FEEM): first, a preprocessing is applied to eliminate the Raman and Rayleigh scattering peaks that clutter the FEEM. To improve its robustness versus possible improper settings, we suggest to associate the classical Zepp's method with a morphological image filtering technique. Then, in the second stage, the Canonical Polyadic (CP or Candecomp/Parafac) decomposition of a nonnegative three‐way array has to be computed. In the fluorescence spectroscopy context, the constituent vectors of the loading matrices should be nonnegative (since standing for spectra and concentrations). Thus, we suggest a new nonnegative third order CP decomposition algorithm (NNCP) based on a nonlinear conjugate gradient optimization algorithm with regularization terms and periodic restarts. Computer simulations performed on real experimental data are provided to enlighten the effectiveness and robustness of the whole processing chain and to validate the approach. Copyright © 2015 John Wiley & Sons, Ltd.

Generalized Non-Orthogonal Joint Diagonalization With LU Decomposition and Successive Rotations

Non-orthogonal joint diagonalization (NJD) free of prewhitening has been widely studied in the context of blind source separation (BSS) and array signal processing, etc. However, NJD is used to retrieve the jointly diagonalizable structure for a single set of target matrices which are mostly formulized with a single dataset, and thus is insufficient to handle multiple datasets with inter-set dependences, a scenario often encountered in joint BSS (J-BSS) applications. As such, we present a generalized NJD (GNJD) algorithm to simultaneously perform asymmetric NJD upon multiple sets of target matrices with mutually linked loading matrices, by using LU decomposition and successive rotations, to enable J-BSS over multiple datasets with indication/exploitation of their mutual dependences. Experiments with synthetic and real-world datasets are provided to illustrate the performance of the proposed algorithm.

質的データに対する非計量三相主成分分析法

TUCKER3 (Tucker, 1966) is one of the methods of principal component analysis (PCA) for “quantitative” three-mode data. In this paper,we extend TUCKER3 to “nonmetric” three-mode PCA by deriving quantification scores. We also present an alternating least squares algorithm for finding optimal solution of nonmetric three-mode PCA by updating parameters such as quantification scores, loading matrices and core matrix successively. Calculating quantification scores of qualitative variables allows us to exclude arbitrariness which is included in questionnaire, such as the coding of nominal variables and the equality of intervals of ordinal variables. Two real data examples are given to illustrate the effectiveness of the proposed procedure.

Canonical polyadic decomposition of third-order semi-nonnegative semi-symmetric tensors using LU and QR matrix factorizations

Abstract Semi-symmetric three-way arrays are essential tools in blind source separation (BSS) particularly in independent component analysis (ICA). These arrays can be built by resorting to higher order statistics of the data. The canonical polyadic (CP) decomposition of such semi-symmetric three-way arrays allows us to identify the so-called mixing matrix, which contains the information about the intensities of some latent source signals present in the observation channels. In addition, in many applications, such as the magnetic resonance spectroscopy (MRS), the columns of the mixing matrix are viewed as relative concentrations of the spectra of the chemical components. Therefore, the two loading matrices of the three-way array, which are equal to the mixing matrix, are nonnegative. Most existing CP algorithms handle the symmetry and the nonnegativity separately. Up to now, very few of them consider both the semi-nonnegativity and the semi-symmetry structure of the three-way array. Nevertheless, like all the methods based on line search, trust region strategies, and alternating optimization, they appear to be dependent on initialization, requiring in practice a multi-initialization procedure. In order to overcome this drawback, we propose two new methods, called JD LU + and JD QR + , to solve the problem of CP decomposition of semi-nonnegative semi-symmetric three-way arrays. Firstly, we rewrite the constrained optimization problem as an unconstrained one. In fact, the nonnegativity constraint of the two symmetric modes is ensured by means of a square change of variable. Secondly, a Jacobi-like optimization procedure is adopted because of its good convergence property. More precisely, the two new methods use LU and QR matrix factorizations, respectively, which consist in formulating high-dimensional optimization problems into several sequential polynomial and rational subproblems. By using both LU and QR matrix factorizations, we aim at studying the influence of the used matrix factorization. Numerical experiments on simulated arrays emphasize the advantages of the proposed methods especially the one based on LU factorization, in the presence of high-variance model error and of degeneracies such as bottlenecks. A BSS application on MRS data confirms the validity and improvement of the proposed methods.

The Cross-Cultural Validity of the MMPI-2-RF Higher-Order Scales in a Sample of North Korean Female Refugees.

We investigated the cross-cultural factorial validity of the three Higher-Order (H-O) scales in the Minnesota Multiphasic Personality Inventory-2 Restructured Form (MMPI-2-RF) among a sample of North Korean female refugees (N = 2,732). Given the importance of the H-O scales in the overall structure of the MMPI-2-RF scales and in interpretation, we were interested in exploring their cross-cultural validity. We conducted an exploratory factor analysis (EFA) on the nine Restructured Clinical (RC) scale raw scores and fitted and compared one- to three-factor models. The three-factor model, akin to the model in Tellegen and Ben-Porath, demonstrated the best fit to the data. Furthermore, the pattern matrices of loadings across the current sample and the U.S. samples were comparable despite some differences, such as the RC2 scale's salient, negative loading on a factor analogous to the Behavioral/Externalizing Dysfunction scale. We also investigated the unique psychological characteristics of the refugees, possibly resulting from the arduous, perilous journeys out of North Korea taken by this group of female refugees and discussed the results of EFA in light of those singular psychological traits and experiences. Overall, the three H-O scales of the Korean MMPI-2-RF evidenced reasonable cross-cultural factorial validity among the sample of North Korean female refugees.

Audio segmentation-by-classification approach based on factor analysis in broadcast news domain

This paper studies a novel audio segmentation-by-classification approach based on factor analysis. The proposed technique compensates the within-class variability by using class-dependent factor loading matrices and obtains the scores by computing the log-likelihood ratio for the class model to a non-class model over fixed-length windows. Afterwards, these scores are smoothed to yield longer contiguous segments of the same class by means of different back-end systems. Unlike previous solutions, our proposal does not make use of specific acoustic features and does not need a hierarchical structure. The proposed method is applied to segment and classify audios coming from TV shows into five different acoustic classes: speech, music, speech with music, speech with noise, and others. The technique is compared to a hierarchical system with specific acoustic features achieving a significant error reduction.

Linear Precoder Design for Doubly Correlated Partially Coherent Fading MIMO Channels

We consider the single-user multiple-input multiple-output (MIMO) precoder design problem for the doubly spatially correlated partially coherent Rayleigh fading channels with discrete inputs. The objective is to design a linear precoder to adapt to the degradation caused by the imperfect channel estimation at the receiver and the transmit-receive antenna correlation. The system is partially coherent so that the MIMO channel coefficients are estimated at the receiver and its error covariance matrix is fed back to the transmitter. We utilized the cutoff rate (CR) expression, an alternative to the mutual information (MI), and propose to use it as a design criterion to design the linear precoders. A linear precoder is obtained by numerically maximizing the CR with respect to the precoder matrix with a given average power constraint. Furthermore, the precoder matrix is decomposed using singular value decomposition (SVD) into the input shaping matrix, power loading matrix, and beamforming matrix. The beamforming matrix is found to coincide with the eigenvectors of the transmit correlation matrix. The power loading and input shaping matrices are solved numerically using the difference of convex (d.c.) functions programming algorithm and optimization under the unitary constraint, respectively. A 2-block alternating optimization (AO) algorithm is proposed to solve the input shaping matrix and power loading matrix iteratively. Precoders are designed to be used in conjunction with two MIMO transmission schemes: the spatial multiplexing (SM) and space-time (ST) block transmission modes. The frame error rate (FER) and average MI are used as the performance metrics to validate the performance of the newly designed CR-precoders in comparison with the conventional no-precoding case and cutoff rate optimized partially coherent constellations (PCCs). Numerical examples show that the performance gains of the designed precoders are significant compared to the CR-PCCs and conventional codewords.

PROJECTED PRINCIPAL COMPONENT ANALYSIS IN FACTOR MODELS.

This paper introduces a Projected Principal Component Analysis (Projected-PCA), which employees principal component analysis to the projected (smoothed) data matrix onto a given linear space spanned by covariates. When it applies to high-dimensional factor analysis, the projection removes noise components. We show that the unobserved latent factors can be more accurately estimated than the conventional PCA if the projection is genuine, or more precisely, when the factor loading matrices are related to the projected linear space. When the dimensionality is large, the factors can be estimated accurately even when the sample size is finite. We propose a flexible semi-parametric factor model, which decomposes the factor loading matrix into the component that can be explained by subject-specific covariates and the orthogonal residual component. The covariates' effects on the factor loadings are further modeled by the additive model via sieve approximations. By using the newly proposed Projected-PCA, the rates of convergence of the smooth factor loading matrices are obtained, which are much faster than those of the conventional factor analysis. The convergence is achieved even when the sample size is finite and is particularly appealing in the high-dimension-low-sample-size situation. This leads us to developing nonparametric tests on whether observed covariates have explaining powers on the loadings and whether they fully explain the loadings. The proposed method is illustrated by both simulated data and the returns of the components of the S&P 500 index.

Blind source separation of underdetermined mixtures of event-related sources

This paper addresses the problem of blind source separation for underdetermined mixtures (i.e., more sources than sensors) of event-related sources that include quasi-periodic sources (e.g., electrocardiogram (ECG)), sources with synchronized trials (e.g., event-related potentials (ERP)), and amplitude-variant sources. The proposed method is based on two steps: (i) tensor decomposition for underdetermined source separation and (ii) signal extraction by Kalman filtering to recover the source dynamics. A tensor is constructed for each source by synchronizing on the “event” period of the corresponding signal and stacking different periods along the second dimension of the tensor. To cope with the interference from other sources that impede on the extraction of weak signals, two robust tensor decomposition methods are proposed and compared. Then, the state parameters used within a nonlinear dynamic model for the extraction of event-related sources from noisy mixtures are estimated from the loading matrices provided by the first step.The influence of different parameters on the robustness to outliers of the proposed method is examined by numerical simulations. Applied to clinical electroencephalogram (EEG), ECG and magnetocardiogram (MCG), the proposed method exhibits a significantly higher performance in terms of expected signal shape than classical source separation methods such as π CA and FastICA.

Audio segmentation-by-classification approach based on factor analysis in broadcast news domain

This paper studies a novel audio segmentation-by-classification approach based on factor analysis. The proposed technique compensates the within-class variability by using class-dependent factor loading matrices and obtains the scores by computing the log-likelihood ratio for the class model to a non-class model over fixed-length windows. Afterwards, these scores are smoothed to yield longer contiguous segments of the same class by means of different back-end systems. Unlike previous solutions, our proposal does not make use of specific acoustic features and does not need a hierarchical structure. The proposed method is applied to segment and classify audios coming from TV shows into five different acoustic classes: speech, music, speech with music, speech with noise, and others. The technique is compared to a hierarchical system with specific acoustic features achieving a significant error reduction.

A Tale of Two Matrix Factorizations

In statistical practice, rectangular tables of numeric data are commonplace, and are often analyzed using dimension-reduction methods like the singular value decomposition and its close cousin, principal component analysis (PCA). This analysis produces score and loading matrices representing the rows and the columns of the original table and these matrices may be used for both prediction purposes and to gain structural understanding of the data. In some tables, the data entries are necessarily nonnegative (apart, perhaps, from some small random noise), and so the matrix factors meant to represent them should arguably also contain only nonnegative elements. This thinking, and the desire for parsimony, underlies such techniques as rotating factors in a search for “simple structure.” These attempts to transform score or loading matrices of mixed sign into nonnegative, parsimonious forms are, however, indirect and at best imperfect. The recent development of nonnegative matrix factorization, or NMF, is an attractive alternative. Rather than attempt to transform a loading or score matrix of mixed signs into one with only nonnegative elements, it directly seeks matrix factors containing only nonnegative elements. The resulting factorization often leads to substantial improvements in interpretability of the factors. We illustrate this potential by synthetic examples and a real dataset. The question of exactly when NMF is effective is not fully resolved, but some indicators of its domain of success are given. It is pointed out that the NMF factors can be used in much the same way as those coming from PCA for such tasks as ordination, clustering, and prediction. Supplementary materials for this article are available online.

PRM76 - Exploratory Structural Equation Models: A Simulation Study Exploring Geomin and Target Rotation Techniques on Variations of ESEM Models

This simulation study evaluates the impact of Geomin and Target rotation criteria on factor loading matrices in the recently developed exploratory structural equation models (ESEM), a method that can be considered a strong alternative to the exclusive use of exploratory factor analysis (EFA) and confirmatory factor analysis (CFA) in patient related outcomes measurement. By combining the steps of EFA and CFA in one unified approach, ESEM saves significant time and effort usually invested in separate iterations of EFA and CFA. One hundred replications of ESEM models were carried out for variations of sample size and latent factors. Simulation study 1 examined the behavior of ESEM parameter estimates by changing the values of rotation constant (0.01, 0.001 and 0.0001) in Geomin rotation for a three-factor single group model and for N=300 and 1000. Simulation study 2 evaluated the behavior of ESEM parameter estimates for multi-group models using three- and four-factor models for N=150 and 500 per group. Bias, Mean Square Errors (MSE) and standard errors were used to evaluate accuracy of parameter recovery. Item parameters were generated from 27 items belonging to a pilot graduate creativity instrument. For study 1, Geomin rotated parameter estimates of factor loadings, means and covariances produce higher MSEs than the follow up Target rotations. In study 2, the parameter estimates for ESEM Geomin show small sample size bias for some parameters while the standard errors produced correct coverage for all parameters under Target rotation method for large N=500 per group. Overall, there was accurate recovery of parameter estimates in the smallest sample of 300 especially in the multi-group models specifically when Geomin rotations were employed. This bodes well for analysis of real data and for the study of measurement invariance across groups. Future studies could include the examination of number of items affecting recovery of parameter estimates.

Structural Panel VARs

The paper proposes a structural approach to VAR analysis in panels, which takes into account responses to both idiosyncratic and common structural shocks, while permitting full cross member heterogeneity of the response dynamics. In the context of this structural approach, estimation of the loading matrices for the decomposition into idiosyncratic versus common shocks is straightforward and transparent. The method appears to do remarkably well at uncovering the properties of the sample distribution of the underlying structural dynamics, even when the panels are relatively short, as illustrated in Monte Carlo simulations. Finally, these simulations also illustrate that the SVAR panel method can be used to improve inference, not only for properties of the sample distribution, but also for dynamics of individual members of the panel that lack adequate data for a conventional time series SVAR analysis. This is accomplished by using fitted cross sectional regressions of the sample of estimated panel responses to correlated static measures, and using these to interpolate the member-specific dynamics.

Soft-Transition Sub-PCA Fault Monitoring of Batch Processes

Inaccurate substage division problems often emerge when multiway principal component analysis is applied in fault monitoring of multistage batch processes. A new two-step stage division method based on support vector data description (SVDD) is proposed in order to avoid the hard-division and misclassification problems. The loading matrices of the MPCA model are modified using the idea of combining the mechanism knowledge with field data in the rough division step. The model differences are increased by introducing the sampling time to loading matrices, which can avoid division mistakes caused by the fault data. Detailed stage separation is realized here based on the SVDD hypersphere distance to divide the process strictly into steady or transition stages. Then a soft-transition sub-PCA model is given based on the hypersphere distance. The method is applied to monitoring a penicillin fermentation process online. Simulation results show that the proposed method can describe transition stage information in m...

Generalized joint Procrustes analysis

In this paper, we propose a generalized version of Adachi's (Psychometrika 74:667---683, 2009) Joint Procrustes Analysis (GJPA), in order to transform the principal component score and loading matrices obtained for multiple data sets of the same size. The transformation is made so that multiple score and loading matrices are optimally matched to two unknown target matrices, respectively, without affecting the fit of the score and loading matrices to data sets, and without any constraint imposed on the transformation, except for its being nonsingular. The resulting transformed score and loading matrices can reasonably be compared across data sets. A simulation study is performed for assessing an alternate least-squares algorithm for GJPA. Additional procedures for interpreting GJPA solutions are also presented and they are illustrated with a real data example. Finally, GJPA is reconsidered in the contexts of three-way data analysis and canonical correlation analysis.

Bayesian Factor Analysis for Obtaining Simplimax Solutions with an Unspecified Number of Zeros

A Bayesian factor analysis procedure is proposed for obtaining a simple loading matrix based on prior information in which the loading matrix approximates a sparse target matrix including zero elements. A feature of this procedure is that the target matrix is unknown and estimated; the number and locations of zero elements are not specified. We derive the estimation of the full conditional distributions of parameters using Gibbs sampling. A simulation study and real data examples demonstrate that the proposed procedure can provide simpler loading matrices than existing rotation methods.

Determination of fecal contamination origin in reclaimed water open-air ponds using biochemical fingerprinting of enterococci and fecal coliforms

Low levels of fecal indicator bacteria (FIB) were recently detected in two reclaimed water open-air ponds used to irrigate a golf course located in Northeastern Spain. The aim of this study was to evaluate the feasibility of a biochemical fingerprinting method to track the origin of fecal contamination in water with low FIB levels, as in the aforementioned ponds. We also aimed to determine whether FIB presence was due to regrowth of the reclaimed water populations or to a contribution of fecal matter whose source was in the golf facility. Three hundred and fifty enterococcal strains and 308 fecal coliform strains were isolated from the ponds and reclamation plant, and they were biochemically phenotyped. In addition, the inactivation of several microbial fecal pollution indicators (fecal coliforms, total bifidobacteria, sorbitol-fermenting bifidobacteria, somatic bacteriophages, and bacteriophages infecting Bacteroides thetaiotaomicron) was studied using a mesocosm in situ in order to obtain information about their decay rate. Although FIB concentration was low, the biochemical fingerprinting provided evidence that the origin of the fecal contamination in the ponds was not related to the reclaimed water. Biochemical fingerprinting thus proved to be a successful approach, since other microbial source-tracking methods perform poorly when dealing with low fecal load matrices. Furthermore, the mesocosm assays indicated that none of the microbial fecal indicators was able to regrow in the ponds. Finally, the study highlights the fact that reclaimed water may be recontaminated in open-air reservoirs, and therefore, its microbial quality should be monitored throughout its use.