Multi-way Data Research Articles

Multiway Graph Signal Processing on Tensors: Integrative analysis of irregular geometries.

Graph signal processing (GSP) is an important methodology for studying data residing on irregular structures. As acquired data is increasingly taking the form of multi-way tensors, new signal processing tools are needed to maximally utilize the multi-way structure within the data. In this paper, we review modern signal processing frameworks generalizing GSP to multi-way data, starting from graph signals coupled to familiar regular axes such as time in sensor networks, and then extending to general graphs across all tensor modes. This widely applicable paradigm motivates reformulating and improving upon classical problems and approaches to creatively address the challenges in tensor-based data. We synthesize common themes arising from current efforts to combine GSP with tensor analysis and highlight future directions in extending GSP to the multi-way paradigm.

Exploring the potential of combining chemometric approaches to model non-linear multi-way data with quantitative purposes – A case study

Second-order based calibration methods have been widely investigated capitalizing on the inherent benefits of the data structure and the decomposition models, demonstrating that second-order advantage is a property that conspires to a high likelihood success in the resolution of systems of varying complexity. This work aims to demonstrate the applicability of a combined chemometric strategy to solve non-linear multivariate calibration systems in the presence of non-multilinear multi-way data. The determination of histamine by differential pulse voltammetry at different pH is presented as case study. The experimental system has the outstanding difficulty arisen from the large displacement along the potential axis by the pH, which was successfully overcome by implementation of the presented combined strategy. For data modeling, MCR-ALS, U-PLS/RBL and U-PCA/RBL-RBF were used. MCR-ALS allowed unraveling the non-linear behavior between the signal and the concentration, and extracting the underlying profiles of the constituent. Quantitative analysis was performed through the three models, and a comparative evaluation of the predictive performance was done. The best results were achieved with U-PCA/RBL-RBF (mean recovery = 101%) whereas, MCR-ALS yield the lowest mean recovery for all samples (70%)

Simultaneous Determination of Medicinal Drugs with Overlapping Profiles Contained in Low Chromatographic Resolution Data using HPLC-DAD and Multivariate Curve Resolution

Background: The increasing demand of effective pharmaceutical products directed to fight against malaria lead to the combination of at least two antimalarial drugs. This combination aims to minimize the Plasmodium falciparum resistance which is found when the most frequently used drugs are taken individually. Within this context, proguanil hydrochloride and chloroquine phosphate which have independent modes of action are taken together to prevent malaria. This paper aims to develop a fast and powerful analytical method for the simultaneous determination of proguanil hydrochloride and chloroquine phosphate in the commercial Paludrine/Avloclor dosage forms using a multi-way chromatographic calibration based on high-performance liquid chromatography with diode array detection (HPLC-DAD) and multivariate curve resolution – alternating leastsquares (MCR-ALS). Methods: A rapid and powerful analytical method based on HPLC-DAD and MCR-ALS was developed for the simultaneous quantification of proguanil hydrochloride and chloroquine phosphate in the commercial Paludrine/Avloclor antimalarial drugs. An isocratic mobile phase composed by 0.2 M ammonium acetate, acetonitrile, and methanol (40:25:35) and a flow rate of 1.2 mL min-1 were employed in the chromatographic runs with an elution time about 5 min. Results: This approach demonstrates that chromatographic analysis may become considerably simpler and economical in terms of time, cost, and organic solvent consumption when coupled to multiway calibration models such as MCR-ALS. In fact, this multi-way chromatographic calibration based on second-order HPLC-DAD data matrices (with extremely low chromatographic resolution) and MCR-ALS allows the development of greener analytical methods for complex samples. The proposed analytical method allowed the simultaneous quantification of two antimalarial APIs present in the commercial Paludrine/Avloclor drugs with low REP values below 8% for the simultaneous determination of proguanil hydrochloride and chloroquine phosphate. Conclusion: The proposed multi-way chromatographic strategy can be used for routine control of pharmaceutical dosage forms. It should be highlighted that MCR-ALS allowed to: (a) achieve the second- order advantage and the quantification of analytes in the presence of uncalibrated compounds such as coeluted profile measured in different magnitude of the signal in each successive chromatographic run and significant overlapping profiles and (b) separate the contribution of several components from chromatographic runs with extremely low separation of peaks through the deconvolution of the signal obtained, performing the so-called mathematical chromatography.

Multivariate curve resolution of multiway data using the multilinearity constraint

AbstractThe extension of Multivariate Curve Resolution‐Alternating Least Squares (MCR‐ALS) to the analysis of multiway data using the multilinearity constraint is described in detail as one step forward of previous implementations of the trilinearity and quadrilinearity constraints for the analysis of three‐ and four‐way data sets, respectively. As in previous cases, the implementation of the multilinear model for multiway data sets is done algorithmically, within the frame of the alternating least squares (ALS) optimization in the MCR‐ALS method. This implementation is tested using multiway data sets of different complexity, and the obtained results have confirmed the adequacy of the proposed approach. Special advantages of the proposed methodology are that it allows for the implementation of the constraint separately for the different components in their different modes and that it also allows for the introduction of different levels of complexity of the multilinear model, including mixed multilinear models. These two features are especially relevant because they are not present in most of the most used multiway data analysis methods at present.

CLEAR: A Consistent Lifting, Embedding, and Alignment Rectification Algorithm for Multiview Data Association

Many robotics applications require alignment and fusion of observations obtained at multiple views to form a global model of the environment. Multiway data association methods provide a mechanism to improve alignment accuracy of pairwise associations and ensure their consistency. However, existing methods that solve this computationally challenging problem are often too slow for real-time applications. Furthermore, some of the existing techniques can violate the cycle consistency principle, thus drastically reducing the fusion accuracy. This article presents the consistent lifting, embedding, and alignment rectification (CLEAR) algorithm to address these issues. By leveraging insights from the multiway matching and spectral graph clustering literature, CLEAR provides cycle-consistent and accurate solutions in a computationally efficient manner. Numerical experiments on both synthetic and real datasets are carried out to demonstrate the scalability and superior performance of our algorithm in real-world problems. This algorithmic framework can provide significant improvement in the accuracy and efficiency of existing discrete assignment problems, which traditionally use pairwise (but potentially inconsistent) correspondences. An implementation of CLEAR is made publicly available online.

Randomized CP tensor decomposition

The CANDECOMP/PARAFAC (CP) tensor decomposition is a popular dimensionality-reduction method for multiway data. Dimensionality reduction is often sought after since many high-dimensional tensors have low intrinsic rank relative to the dimension of the ambient measurement space. However, the emergence of ‘big data’ poses significant computational challenges for computing this fundamental tensor decomposition. By leveraging modern randomized algorithms, we demonstrate that coherent structures can be learned from a smaller representation of the tensor in a fraction of the time. Thus, this simple but powerful algorithm enables one to compute the approximate CP decomposition even for massive tensors. The approximation error can thereby be controlled via oversampling and the computation of power iterations. In addition to theoretical results, several empirical results demonstrate the performance of the proposed algorithm.

Improved robust tensor principal component analysis for accelerating dynamic MR imaging reconstruction.

Dynamic magnetic resonance imaging (dMRI) strikes a balance between reconstruction speed and image accuracy in medical imaging field. In this paper, an improved robust tensor principal component analysis (RTPCA) method is proposed to reconstruct the dynamic magnetic resonance imaging (MRI) from highly under-sampled K-space data. The MR reconstruction problem is formulated as a high-order low-rank tenor plus sparse tensor recovery problem, which is solved by robust tensor principal component analysis (RTPCA) with a new tensor nuclear norm (TNN). To further exploit the low-rank structures in multi-way data, the core matrix nuclear norm, extracted from the diagonal elements of the core tensor under tensor singular value decomposition (t-SVD) framework, is also integrated into TNN for enforcing the low-rank structure in MRI datasets. The experimental results show that the proposed method outperforms state-of-the-art methods in terms of both MR image reconstruction accuracy and computational efficiency on 3D and 4D experiment datasets, especially for 4D MR image reconstruction. Graphical abstract The flowchart of the proposed method to reconstruct the dynamic magnetic resonance imaging (MRI) from highly under-sampled K-space data in the kth iteration. To further exploit the low-rank structures in multi-way data, the core matrix nuclear norm, extracted from the diagonal elements of the core tensor under tensor singular value decomposition (t-SVD) framework, is also integrated into tensor nuclear norm (TNN) for enforcing the low-rank structure in MRI datasets. In each iteration, the first step is to get low-rank tensor ℓk - 1 by using soft thresholding on the singular values of ℓk - 1 = χk - 1 - ξk - 1, and an improved tensor nuclear norm method is proposed to process the low-rank tensor ℓk - 1 firstly. Then, the shrinkage operator is applied to ξk - 1 = χk - 1 - ℓk - 1 for sparse part ξk - 1. The final reconstructed d-MRI χk is obtained by enforcing data consistency that the residual in K-space is subtracted by the sum of the reconstructed low-rank tensor and sparse tensor.

Generalized Co-clustering Analysis via Regularized Alternating Least Squares

Biclustering is an important exploratory analysis tool that simultaneously clusters rows (e.g., samples) and columns (e.g., variables) of a data matrix. Checkerboard-like biclusters reveal intrinsic associations between rows and columns. However, most existing methods rely on Gaussian assumptions and only apply to matrix data. In practice, non-Gaussian and/or multi-way tensor data are frequently encountered. A new CO-clustering method via Regularized Alternating Least Squares (CORALS) is proposed, which generalizes biclustering to non-Gaussian data and multi-way tensor arrays. Non-Gaussian data are modeled with single-parameter exponential family distributions and co-clusters are identified in the natural parameter space via sparse CANDECOMP/PARAFAC tensor decomposition. A regularized alternating (iteratively reweighted) least squares algorithm is devised for model fitting and a deflation procedure is exploited to automatically determine the number of co-clusters. Comprehensive simulation studies and three real data examples demonstrate the efficacy of the proposed method. The data and code are publicly available at https://github.com/reagan0323/CORALS.

The First Identification of the Uniqueness and Authentication of Maltese Extra Virgin Olive Oil Using 3D-Fluorescence Spectroscopy Coupled with Multi-Way Data Analysis.

The potential application of multivariate three-way data analysis techniques, namely parallel factor analysis (PARAFAC) and discriminant multi-way partial least squares regression (DN-PLSR), on three-dimensional excitation emission matrix (3D-EEM) fluorescent data were used to identify the uniqueness and authenticity of Maltese extra virgin olive oil (EVOO). A non-negativity constrained PARAFAC model revealed that a four-component model provided the most appropriate solution. Examination of the extracted components in mode 2 and 3 showed that these belonged to different fluorophores present in extra virgin olive oil. Application of linear discriminate analysis (LDA) and binary logistic regression analysis on the concentration of the four extracted fluorophores, showed that it is possible to discriminate Maltese EVOOs from non-Maltese EVOOs. The application of DN-PLSR provided superior means for discrimination of Maltese EVOOs. Further inspection of the extracted latent variables and their variable importance plots (VIPs) provided strong proof of the existence of four types of fluorophores present in EVOOs and their potential application for the discrimination of Maltese EVOOs.

Spatio-temporal variations and uncertainty in land surface modelling for high latitudes: univariate response analysis

Abstract. A range of applications analysing the impact of environmental changes due to climate change, e.g. geographical spread of climate-sensitive infections (CSIs) and agriculture crop modelling, make use of land surface modelling (LSM) to predict future land surface conditions. There are multiple LSMs to choose from that account for land processes in different ways and this may introduce predictive uncertainty when LSM outputs are used as inputs to inform a given application. For useful predictions for a specific application, one must therefore understand the inherent uncertainties in the LSMs and the variations between them, as well as uncertainties arising from variation in the climate data driving the LSMs. This requires methods to analyse multivariate spatio-temporal variations and differences. A methodology is proposed based on multiway data analysis, which extends singular value decomposition (SVD) to multidimensional tables and provides spatio-temporal descriptions of agreements and disagreements between LSMs for both historical simulations and future predictions. The application underlying this paper is prediction of how climate change will affect the spread of CSIs in the Fennoscandian and north-west Russian regions, and the approach is explored by comparing net primary production (NPP) estimates over the period 1998–2013 from versions of leading LSMs (JULES, CLM5 and two versions of ORCHIDEE) that are adapted to high-latitude processes, as well as variations in JULES up to 2100 when driven by 34 global circulation models (GCMs). A single optimal spatio-temporal pattern, with slightly different weights for the four LSMs (up to 14 % maximum difference), provides a good approximation to all their estimates of NPP, capturing between 87 % and 93 % of the variability in the individual models, as well as around 90 % of the variability in the combined LSM dataset. The next best adjustment to this pattern, capturing an extra 4 % of the overall variability, is essentially a spatial correction applied to ORCHIDEE-HLveg that significantly improves the fit to this LSM, with only small improvements for the other LSMs. Subsequent correction terms gradually improve the overall and individual LSM fits but capture at most 1.7 % of the overall variability. Analysis of differences between LSMs provides information on the times and places where the LSMs differ and by how much, but in this case no single spatio-temporal pattern strongly dominates the variability. Hence interpretation of the analysis requires the summation of several such patterns. Nonetheless, the three best principal tensors capture around 76 % of the variability in the LSM differences and to a first approximation successively indicate the times and places where ORCHIDEE-HLveg, CLM5 and ORCHIDEE-MICT differ from the other LSMs. Differences between the climate forcing GCMs had a marginal effect up to 6 % on NPP predictions out to 2100 without specific spatio-temporal GCM interaction.

Exploration advantages of data combination and partition: First chemometric analysis of liquid chromatography–mass spectrometry data in full scan mode with quadruple fragmentor voltages

A novel soft strategy for combination and partition of mass spectra data recorded at different fragmentor voltages in full scan mode of a mass spectrometer was developed to generate abundant multi-way data. It is the first time that non-linear four-way and combined three-way LC-MS data have been obtained simultaneously in a single chromatographic run. This strategy ensures that each analyte can be ionized and detected at the most appropriate MS conditions (ionization modes, fragmentor voltages) and avoids a hard chromatographic segmentation in subsequent chemometric analysis. Two different experimental datasets were analyzed to validate the feasibility and applicability of this strategy. Some simple pretreatments were carried out before LC-MS analysis to prevent potential matrix effects. Proper chemometric tools were used to resolve three-way (partitioned data) and enhanced three-way LC-MS (combined data) data, respectively. The method was assessed by comparing the analytical results obtained from the same chemometric algorithm with both combined and partitioned datasets: (1) the combined data provided the best global overall resolution, higher sensitivity and more reliable results, (2) the partitioned data provided higher selectivity for some specific analytes. The results showed that the developed method could be a soft and ingenious tool to handle the unordered but information-rich raw LC-MS data. Moreover, the proposed strategy could take extra analytical advantages in terms of higher sensitivity and more reliable quantitative results when compared with LC-MS (with single fragmentor voltage) strategy and showed nearly the same capability in analytical quality as classic LC-MS/MS method.

Generalized inverses of Boolean tensors via the Einstein product

Applications of the theory and computations of Boolean matrices are of fundamental importance to study a variety of discrete structural models. But the increasing ability of data collection systems to store huge volumes of multidimensional data, the Boolean matrix representation of data analysis is not enough to represent all the information content of the multiway data in different fields. From this perspective, it is appropriate to develop an infrastructure that supports reasoning about the theory and computations. In this paper, we discuss the generalized inverses of the Boolean tensors with the Einstein product. Further, we elaborate on this theory by producing a few characterizations of different generalized inverses and several equivalence results on Boolean tensors. We explore the space decomposition of the Boolean tensors and present reflexive generalized inverses through it. In addition to this, we address rank and the weight for the Boolean tensor.

Novel Three-Dimensional Resolution of a pH and Ultraviolet-Visible Absorption Spectral Dataset for the Determination of Desloratadine in a Pharmaceutical Product and Its Acid Dissociation Constant

Three-way data analysis (or multi-way data analysis) of higher-order records is required to develop new and potential resolution methods to reveal multiple latent information or hidden features from multicomponent systems in presence of unknown interferences. This article presents a new application of parallel factor analysis (PARAFAC), which is one of the multiway analysis techniques, to quantify desloratadine in sirup and to simultaneously determine its acid dissociation constant (pKa) in the presence of spectral interferences without using complex and expensive instrumental methods. The three-dimensional pH-absorbance dataset, which was obtained as a function of wavelength, pH and concentration, was decomposed using the PARAFAC algorithm to obtain the estimated spectral, pH and relative concentration profiles of acid-base pair of desloratadine and sirup excipients. The quantity of the drug in sirup formulation was calculated from the estimated relative concentration profile, which was one of the three modes. At the same time, the acid dissociation constant of desloratadine was determined using the estimated pH profile of the acid and base species in their equilibrium. The applicability of the PARAFAC method was confirmed by the analysis of validation samples. In the practical application of the PARAFAC method to real samples, successful results were reported for the quality control of a sirup formulation containing desloratadine and for the estimation of desloratadine’s pKa value. The PARAFAC assay results were compared to results obtained by ultra-performance liquid chromatography (UPLC).

Multiway data analysis approach toward understanding of photoluminescence and energy transfer in carbon nanodots.

In this study, dilution analysis and anion exchange chromatography (AEC) were employed to provide insights into the photoluminescence (PL) of carbon nanodots (CNDs). A stepwise dilution process revealed that some of the fluorophores with higher energy emission were quenched in the high concentration solution and appeared in the dilute solutions. AEC fractionation led to seven sorts of CND fractions with similar surface charges. The fractionation for this CND mixture showed that excitation wavelength dependence was lower for separated CND particles. The wavelength dependence of excitation spectra could be due to energy exchange between particles that was reduced in diluted solutions and separated fractions. Multivariate analysis of AEC's data demonstrated that there were five distinct fluorophores, which formed the total CND emission. It is interesting that none of these fluorophores had a clear contribution to the surface charge of the CND particles. Further characterization through FTIR spectroscopy and 1 H NMR revealed that optical properties of CNDs did not follow the surface functional groups in CNDs. This situation means that the optical behaviour of particles and their fluorophores differed depending on the surface functional groups.

Stochastic Gradients for Large-Scale Tensor Decomposition

Tensor decomposition is a well-known tool for multiway data analysis. This work proposes using stochastic gradients for efficient generalized canonical polyadic (GCP) tensor decomposition of large-scale tensors. GCP tensor decomposition is a recently proposed version of tensor decomposition that allows for a variety of loss functions such as Bernoulli loss for binary data or Huber loss for robust estimation. The stochastic gradient is formed from randomly sampled elements of the tensor and is efficient because it can be computed using the sparse matricized-tensor times Khatri--Rao product tensor kernel. For dense tensors, we simply use uniform sampling. For sparse tensors, we propose two types of stratified sampling that give precedence to sampling nonzeros. Numerical results demonstrate the advantages of the proposed approach and its scalability to large-scale problems.

Теоретические и прикладные аспекты измерения потенциалов экономического развития регионов России

The paper presents a new direction in methodology of modeling the potentials of regional economic growth developed by the authors in their previous studies. The proposed approach is based on decomposition of the extended production function in the Mankiw – Romer – Weil model and measuring statistically unobserved composition factors characterizing innovative, investment and human potential of regional development. This model enables defining the weight vectors of initial indicators for different groups of spatial entities. Detected weights of statistically observed indicators are the basis for structural analysis of economic growth specifics in different regions. Multiway Data Analysis is the basic method to define these potentials quantitatively. Based on this method of multivariate analysis with statistically observed indicators describing various components of regional economic growth in the period from 2001 to 2015, obtained were estimates of each of the three specified potentials per year per Russian Federation constituent member under consideration. The dynamic analysis in form of such calculations assists in comparing the potentials in individual regions. The information gained also helps in the dynamic analysis of different potentials within regions concerning cities numbering over a million inhabitants. A detailed retrospective study of the regional growth made it possible to carry out an integrated analysis and form identical groups thereafter. The time series obtained can serve as statistical tools for determining the contribution of the three composition factors of growth to each constituent member of the Russian Federation through repeated application of Multiway Data Analysis already to these factors. By way of such calculation the aggregate potential of economic growth for the group of the regions under consideration can be formed

Tensor Low-Rank Representation for Data Recovery and Clustering.

Multi-way or tensor data analysis has attracted increasing attention recently, with many important applications in practice. This article develops a tensor low-rank representation (TLRR) method, which is the first approach that can exactly recover the clean data of intrinsic low-rank structure and accurately cluster them as well, with provable performance guarantees. In particular, for tensor data with arbitrary sparse corruptions, TLRR can exactly recover the clean data under mild conditions; meanwhile TLRR can exactly verify their true origin tensor subspaces and hence cluster them accurately. TLRR objective function can be optimized via efficient convex programing with convergence guarantees. Besides, we provide two simple yet effective dictionary construction methods, the simple TLRR (S-TLRR) and robust TLRR (R-TLRR), to handle slightly and severely corrupted data respectively. Experimental results on two computer vision data analysis tasks, image/video recovery and face clustering, clearly demonstrate the superior performance, efficiency and robustness of our developed method over state-of-the-arts including the popular LRR and SSC methods.

Genomic data integration by WON-PARAFAC identifies interpretable factors for predicting drug-sensitivity in vivo

Integrative analyses that summarize and link molecular data to treatment sensitivity are crucial to capture the biological complexity which is essential to further precision medicine. We introduce Weighted Orthogonal Nonnegative parallel factor analysis (WON-PARAFAC), a data integration method that identifies sparse and interpretable factors. WON-PARAFAC summarizes the GDSC1000 cell line compendium in 130 factors. We interpret the factors based on their association with recurrent molecular alterations, pathway enrichment, cancer type, and drug-response. Crucially, the cell line derived factors capture the majority of the relevant biological variation in Patient-Derived Xenograft (PDX) models, strongly suggesting our factors capture invariant and generalizable aspects of cancer biology. Furthermore, drug response in cell lines is better and more consistently translated to PDXs using factor-based predictors as compared to raw feature-based predictors. WON-PARAFAC efficiently summarizes and integrates multiway high-dimensional genomic data and enhances translatability of drug response prediction from cell lines to patient-derived xenografts.

Multilevel Features for Sensor-Based Assessment of Motor Fluctuation in Parkinson's Disease Subjects.

Motor fluctuations are a frequent complication in patients with Parkinson's disease (PD) where the response to medication fluctuates between ON states (medication working) and OFF states (medication has worn off). This paper describes a new data analysis approach that can be used along with two wearable IMU (inertial measurement units) sensors to continuously assess motor fluctuations in PD patients while moving in their natural environment. We hypothesized that joint analysis of the sensor data in its spectral, temporal and sensor domain could generate multilevel features that can be used to detect PD-related patterns successfully as the subject's motor state fluctuates between medication ON and OFF states. For this purpose, we utilized time-frequency (TF) representation and multiway data analysis tools (i.e., tensor decomposition) to decompose the TF representation of the two sensors' data into its multilevel structures, which were next used to extract multilevel features representing the PD symptoms in different medication states. The extracted multilevel features were used in a classification model based on support vector machine to detect medication ON and OFF states. For comparison purposes, we implemented a traditional feature extraction method. We also developed a hierarchical feature extraction method based on the combination of those two methods. The performances of the three methods were evaluated using a dataset of 19 PD subjects with a total duration of 17.54 hours. The multilevel features achieved 8.25% improvement in the accuracy over the traditional features, and the hierarchical features resulted in 10.73% improvement indicating that our approach holds great promise to continuously detect medication states from continuous monitoring of the subjects' movement. Such information can be used by the treating physician to tailor the adjustments to each subject's unique impairment(s), thereby improving therapeutic decision-making and patient outcomes.

Low CP Rank and Tucker Rank Tensor Completion for Estimating Missing Components in Image Data

Tensor completion recovers missing components of multi-way data. The existing methods use either the Tucker rank or the CANDECOMP/PARAFAC (CP) rank in low-rank tensor optimization for data completion. In fact, these two kinds of tensor ranks represent different high-dimensional data structures. In this paper, we propose to exploit the two kinds of data structures simultaneously for image recovery through jointly minimizing the CP rank and Tucker rank in the low-rank tensor approximation. We use the alternating direction method of multipliers (ADMM) to reformulate the optimization model with two tensor ranks into its two sub-problems, and each has only one tensor rank optimization. For the two main sub-problems in the ADMM, we apply rank-one tensor updating and weighted sum of matrix nuclear norms minimization methods to solve them, respectively. The numerical experiments on some image and video completion applications demonstrate that the proposed method is superior to the state-of-the-art methods.