Three-way Data Sets Research Articles

Visualization and Analysis of Three-Way Data Using Accumulated Concept Graphs

This paper introduces the Accumulated Concept Graph (ACG), a visualization tool based on the quantile method designed to analyze three-way data, including distributional data. Such data often have complex structures that make it difficult to identify patterns using conventional visualization techniques. The ACG represents each object with one or more monotonic line graphs. As a result, the three-way data are visualized as a set of parallel monotonic line graphs that never intersect. This non-intersecting property allows for the easy identification of both macroscopic and microscopic patterns within the data. We demonstrate the usefulness of ACGs and principal component analysis in the analysis of real three-way datasets.

An extension of PARAFAC to analyze multi-group three-way data

This paper introduces a novel methodology for analyzing three-way array data with a multi-group structure. Three-way arrays are commonly observed in various domains, including image analysis, chemometrics, and real-world applications. In this paper, we use a practical case study of process modeling in additive manufacturing, where batches are structured according to multiple groups. Vast volumes of data for multiple variables and process stages are recorded by sensors installed on the production line for each batch. For these three-way arrays, the link between the final product and the observations creates a grouping structure in the observations. This grouping may hamper gaining insight into the process if only some of the groups dominate the controlled variability of the products. In this study, we develop an extension of the PARAFAC model that takes into account the grouping structure of three-way data sets. With this extension, it is possible to estimate a model that is representative of all the groups simultaneously by finding their common structure. The proposed model has been applied to three simulation data sets and a real manufacturing case study. The capability to find the common structure of the groups is compared to PARAFAC and the insights into the importance of variables delivered by the models are discussed.

Consensus and fuzzy partition of dendrograms from a three-way dissimilarity array

In cluster analysis, a problem often addressed is finding a consensus on a set of hierarchical classifications of the same set of objects, named primary hierarchies (dendrograms). A unique consensus of the primary hierarchies, called a secondary hierarchy, is sufficient to synthesize the relevant clustering information only when the primary dendrograms are similar to one another. In contrast, when several differences are observed, more than one consensus secondary hierarchy is required to clearly synthesize the different primary hierarchies.Our methodology, PARoDENo3WD (PARtition of DENdrograms of a 3-Way Data array), aims to obtain a secondary fuzzy partition of the primary hierarchies, where hierarchies belonging to the same class are perceived as similar. Each class is associated with a consensus hierarchy.The fuzzy approach allows each primary hierarchy to contribute to the definition of all classes of the secondary partition, according to different membership degrees. In this way, the ‘clustering uncertainty’ is taken into account. Indeed, in several realistic applications, primary dendrograms belonging to different classes of the secondary partition may share some similar features.The performance of PARoDENo3WD is evaluated by an extended simulation study, generating 1800 three-way datasets. An application of PARoDENo3WD on a real data set is given.

Multilinear Slicing for curve resolution of fluorescence imaging with sequential illumination.

Fluorescence microscopy is an extremely powerful technique that allows to distinguish multiple labels based on their emission color or other properties, such as their photobleaching and fluorescence recovery kinetics. These kinetics are ideally assumed to be mono-exponential in nature, where the time constants intrinsic to each fluorophore can be used to quantify their presence in the sample. However, these time constants also depend on the specifics of the illumination and sample conditions, meaning that identifying the different contributions in a mixture using a single-channel detection may not be straightforward. In this work, we propose a factor analysis approach called Slicing to identify the different contributions in a multiplexed fluorescence microscopy image exploiting a single measurement channel. With Slicing, a two-way dataset is rearranged into a three-way dataset, which allows the application of a trilinear decomposition model to derive individual profiles for all the model components. We demonstrate this method on bleaching - recovery fluorescence microscopy imaging data of U2OS cells, allowing us to determine the spatial distribution of the dyes and their associated characteristic relaxation traces, without relying on a parametric fitting. By requiring little a priori knowledge and efficiently handling perturbation factors, our method represents a general approach for the recovery of multiple mono-exponential profiles from single-channel microscopy data.

G-Tric: generating three-way synthetic datasets with triclustering solutions

BackgroundThree-way data started to gain popularity due to their increasing capacity to describe inherently multivariate and temporal events, such as biological responses, social interactions along time, urban dynamics, or complex geophysical phenomena. Triclustering, subspace clustering of three-way data, enables the discovery of patterns corresponding to data subspaces (triclusters) with values correlated across the three dimensions (observations times features times contexts). With increasing number of algorithms being proposed, effectively comparing them with state-of-the-art algorithms is paramount. These comparisons are usually performed using real data, without a known ground-truth, thus limiting the assessments. In this context, we propose a synthetic data generator, G-Tric, allowing the creation of synthetic datasets with configurable properties and the possibility to plant triclusters. The generator is prepared to create datasets resembling real 3-way data from biomedical and social data domains, with the additional advantage of further providing the ground truth (triclustering solution) as output.ResultsG-Tric can replicate real-world datasets and create new ones that match researchers needs across several properties, including data type (numeric or symbolic), dimensions, and background distribution. Users can tune the patterns and structure that characterize the planted triclusters (subspaces) and how they interact (overlapping). Data quality can also be controlled, by defining the amount of missing, noise or errors. Furthermore, a benchmark of datasets resembling real data is made available, together with the corresponding triclustering solutions (planted triclusters) and generating parameters.ConclusionsTriclustering evaluation using G-Tric provides the possibility to combine both intrinsic and extrinsic metrics to compare solutions that produce more reliable analyses. A set of predefined datasets, mimicking widely used three-way data and exploring crucial properties was generated and made available, highlighting G-Tric’s potential to advance triclustering state-of-the-art by easing the process of evaluating the quality of new triclustering approaches.

Nonlinear Fault Detection of Batch Processes Using Functional Local Kernel Principal Component Analysis

In order to guarantee and improve the product quality, the data-driven fault detection technique has been widely used in industry. For three-way datasets of batch process in industry process (i.e., batch × variable × time), a novel method named functional local kernel principal component analysis (FLKPCA) is proposed. Since the variables' trajectories often show functional nature and can be considered as smooth functions rather than just vectors. Firstly, the variables' trajectory is expressed as the combination of smooth basis functions using functional data analysis (FDA), which means that the datasets of batches process would be transformed from the three-ways array into two-ways function matrix. Then, kernel locality preserving projections (LKPCA) is used to perform dimensionality reduction on two-way function matrix directly. Different from kernel principal component analysis (KPCA). LKPCA aims at preserving the both local and global structure of the data in a new optimization objective. Consequently, FLKPCA could more effectively seek the potential information that hidden in the three-ways datasets. Lastly, the effectiveness of the proposed approach is illustrated by the benchmark of fed-batch penicillin fermentation process and the hot strip rolling process.

Bilinear and trilinear modelling of three-way data obtained in two factor designed metabolomics studies

Metabolomic studies of biological samples using experimentally designed experiments at different levels produce large multivariate datasets which can be arranged in three-way data structures and modelled using bilinear and trilinear factor decomposition methods. The goal of these studies is the discovery of the hidden sources of data variability to facilitate their biochemical interpretation. In this paper, the relationship between the effects of the experimental design factors, the structure of the generated three-way datasets and their more appropriate modelling (bilinear or trilinear) are investigated. As example of study, the effects of the dose of a chemical drug on the changes over time in the concentration of lipids in multiple samples of a biological organism are investigated in detail. Different scenarios are considered depending on the type of effects and interactions between the experimental factors. The optimal data modelling results are obtained in case of having reproducible multiplicative effects between the experimental design factors, because in this case the data decomposition can be performed using a trilinear model and the correct lipid profiles are recovered. In the other data scenarios, even in the presence of only additive effects and no interaction between design factors, the correct recovery of the different lipid profiles describing the behavior of the system is not guaranteed and the subsequent rotation ambiguities associated to the bilinear model decompositions can still be present.

Determination of three polycyclic aromatic hydrocarbons in tea using four-way fluorescence data coupled with third-order calibration method

The content of polycyclic aromatic hydrocarbons (PAHs) in tea has attracted widespread attention. In this study, the strategy of four-way fluorescence data coupled with third-order calibration method based on four way parallel factor analysis (four-way PARAFAC) was used to determine phenanthrene, naphthalene and acenaphthene in tea. A new method, introducing an additional temperature mode into the three-way fluorescence data set, was proposed to acquire a four-way data array. The feasibility of the method was validated by experiments. The four-way PARAFAC could give satisfactory analytical results for the determination of three PAHs in tea. The average spiked recoveries were (100.9 ± 7.8)% for phenanthrene, (94.6 ± 8.6)% for naphthalene, (96.0 ± 9.8)% for acenaphthene. The root-mean-square error of prediction (RMSEP) values were 0.495 μg L−1 for phenanthrene, 0.841 μg L−1 for naphthalene, 0.577 μg L−1 for acenaphthene. The average relative predictive errors (ARPE) values for three PAHs were <2.35%. The limits of detection (LOD) of three PAHs were between 0.026 and 0.14 μg L−1. Generally, these results obtained by four-way PARAFAC were better than that of three-way PARAFAC. In addition, analytical figures of merit, such as Sensitivity, Selectivity and Analytical sensitivity, were calculated for three PAHs to evaluate the methods. The results demonstrated that the four-way PARAFAC was more suitable for the determination for three target analytes in this study.

Duality based interpretation of uniqueness in the trilinear decompositions

Application of trilinearity constraint in curve resolution of three-way data sets can play an important role in the triadic decompositions. Matrix Augmented-MCR-ALS is a two-way analysis method while its incorporation with the trilinearity constraint can provide a triadic decomposition like PARAFAC model. Trilinearity is a very strong constraint which can lead unique decomposition under mild conditions. Additionally, duality concept represents a relation between column and row spaces of bilinear data matrices. Thus, in case of a unique solution or uniqueness condition, the duality concept is a general and powerful approach for visualization. Based on the duality concept, it is necessary to define a particular hyper-plane in the dual abstract space in order to have a unique profile. It is discussed and visualized that the trilinearity constraint with the idea of parallel proportional profiles can define a particular hyper-plane in the space causing/leading to uniqueness in the dual space. Several simulated and real data sets were exemplified to show the ability of the duality concept to study and visualize the uniqueness due to the application of trilinearity constraint.

Partial Least-Squares Method for Three-Mode Three-Way Datasets Based on Tucker Model

When analyzing two three-mode three-way datasets (object × variable × condition), the objective is to obtain common factors that show the relationships between the two datasets. The partial least-squares (PLS) method has been applied to such datasets to investigate the common factors. However, the PLS method was proposed for two-mode two-way datasets, such as multivariate datasets. Therefore, this method does not consider the condition when searching for relationships between datasets; that is, it tends to regard the same variable under different conditions as different variables. To address this problem, we extended the PLS method to three-mode three-way datasets by using the Tucker model so that the same variable under different conditions is regarded as the same. Moreover, we can apply the proposed method to three-mode three-way datasets with different dimensions for the conditions and variables, and the output is obtained in the form of three-mode three-way datasets. We show the advantage of the proposed method by applying it to a multicollinearity case as a numerical example.

ANALYSIS OF WELL-BEING IN OECD COUNTRIES THROUGH STATIS METHODOLOGY

This paper presents the main concepts and results of Data Analysis aims to analyze the evolution of some developed countries and also of some emerging countries that are members of the Organisation for Economic Co-operation and Development (OECD) in what concerns some indicators or variables of well-being during the period 2011-2015, through the STATIS (Structuring Three-way data sets in Statistics) methodology. This methodology allows to analyze the presence of a common structure in several data tables obtained over time, to identify the differences and similarities along the period of time under study and according to well-being indicators included in the “Your Better Life Index” of the OECD, and to analyze the trajectories of the countries.

Analytical solution and meaning of feasible regions in two-component three-way arrays

Although many efforts have been directed to the development of approximation methods for determining the extent of feasible regions in two- and three-way data sets; analytical determination (i.e. using only finite-step direct calculation(s) instead of the less exact numerical ones) of feasible regions in three-way arrays has remained unexplored. In this contribution, an analytical solution of trilinear decomposition is introduced which can be considered as a new direct method for the resolution of three-way two-component systems. The proposed analytical calculation method is applied to the full rank three-way data array and arrays with rank overlap (a type of rank deficiency) loadings in a mode. Close inspections of the analytically calculated feasible regions of rank deficient cases help us to make clearer the information gathered from multi-way problems frequently emerged in physics, chemistry, biology, agricultural, environmental and clinical sciences, etc. These examinations can also help to answer, e.g., the following practical question: “Is two-component three-way data with proportional loading in a mode actually a three-way data array?” By the aid of the additional information resulted from the investigated feasible regions of two-component three-way data arrays with proportional profile in a mode, reasons for the inadequacy of the seemingly trilinear data treatment methods published in the literature (e.g., U-PLS/RBL-LD that was used for extraction of quantitative and qualitative information reported by Olivieri et al. (Anal. Chem. 82 (2010) 4510–4519)) could be completely understood.

Phase adaptive RVM model for quality prediction of multiphase batch processes with limited modeling batches

For quality prediction of batch processes under limited modeling batches, the relevance vector machine (RVM) has recently been introduced. By unfolding the three-way dataset through the variable direction, significant nonlinearities are remained in the process data, which in turn explored the nonlinear modeling ability of RVM. For multiphase batch processes, however, different phases may have simultaneous impacts on the final product quality, which should be connected together in the modeling stage. In this paper, a new phase adaptive RVM model is proposed for quality prediction in multiphase batch processes. Based on the information transfer of relevance vectors in each RVM model, different phases are connected one after another, providing simultaneous information for prediction of the final product quality. A detailed industrial case study is given to show the efficiency of the new developed method.

Connected Categorical Canonical Covariance Analysis for Three-mode Three-way data Sets Based on Tucker Model

When we work with two three-mode three-way data sets, such as panel data, we often investigate two types of factors: common factors, which represent relationships between the two data sets, and unique factors, which show the uniqueness of each data set relative to the other. We propose a method for investigating common and unique factors simultaneously. Canonical covariance analysis is an existing method that allows the estimation of common and unique factors simultaneously; however, this method was proposed for use with two-mode two-way data, and it is limited to quantitative data. Thus, applying canonical covariance analysis to three-mode three-way data sets or to categorical data sets is not suitable. To overcome this problem, we build on the concept of the Tucker model and the concept of non-metric principal component analysis to develop and propose a method suitable the analysis of categorical three-mode three-way data sets. Moreover, we introduce connector matrices, making it easy to determine which factors are common and allowing the selection of different numbers of dimensions for the factors.

Investigation of kinetics and thermodynamics of DNA hybridization by means of 2-D fluorescence spectroscopy and soft/hard modeling techniques

Reversible hybridization reaction plays a key role in fundamental biological processes, in many laboratory techniques, and also in DNA based sensing devices. Comprehensive investigation of this process is, therefore, essential for the development of more sophisticated applications. Kinetics and thermodynamics of the hybridization reaction, as a second order process, are systematically investigated with the aid of the soft and hard chemometric methods. Labeling two complementary 21 mer DNA single strands with FAM and Texas red fluorophores, enabled recording of the florescence excitation−emission matrices during the experiments which led to three-way data sets. The presence of fluorescence resonance energy transfer in excitation and emission modes and the closure in concentration mode, made the three-way data arrays rank deficient. To acquire primary chemical information, restricted Tucker3 as a soft method was employed. Herein a model-based method, hard restricted trilinear decomposition, is introduced for in depth analysis of rank deficient three-way data sets. By employing proposed hard method, the nonlinear model parameters as well as the correct profiles could be estimated. In addition, a simple constraint is presented to extract chemically reasonable output profiles regarding the core elements of restricted Tucker3 model.

Three-way analysis of the UPLC–PDA dataset for the multicomponent quantitation of hydrochlorothiazide and olmesartan medoxomil in tablets by parallel factor analysis and three-way partial least squares

An application of parallel factor analysis (PARAFAC) and three-way partial least squares (3W-PLS1) regression models to ultra-performance liquid chromatography–photodiode array detection (UPLC–PDA) data with co-eluted peaks in the same wavelength and time regions was described for the multicomponent quantitation of hydrochlorothiazide (HCT) and olmesartan medoxomil (OLM) in tablets. Three-way dataset of HCT and OLM in their binary mixtures containing telmisartan (IS) as an internal standard was recorded with a UPLC–PDA instrument. Firstly, the PARAFAC algorithm was applied for the decomposition of three-way UPLC–PDA data into the chromatographic, spectral and concentration profiles to quantify the concerned compounds. Secondly, 3W-PLS1 approach was subjected to the decomposition of a tensor consisting of three-way UPLC–PDA data into a set of triads to build 3W-PLS1 regression for the analysis of the same compounds in samples. For the proposed three-way analysis methods in the regression and prediction steps, the applicability and validity of PARAFAC and 3W-PLS1 models were checked by analyzing the synthetic mixture samples, inter-day and intra-day samples, and standard addition samples containing HCT and OLM. Two different three-way analysis methods, PARAFAC and 3W-PLS1, were successfully applied to the quantitative estimation of the solid dosage form containing HCT and OLM. Regression and prediction results provided from three-way analysis were compared with those obtained by traditional UPLC method.

Dissecting the space-time structure of tree-ring datasets using the partial triadic analysis.

Tree-ring datasets are used in a variety of circumstances, including archeology, climatology, forest ecology, and wood technology. These data are based on microdensity profiles and consist of a set of tree-ring descriptors, such as ring width or early/latewood density, measured for a set of individual trees. Because successive rings correspond to successive years, the resulting dataset is a ring variables × trees × time datacube. Multivariate statistical analyses, such as principal component analysis, have been widely used for extracting worthwhile information from ring datasets, but they typically address two-way matrices, such as ring variables × trees or ring variables × time. Here, we explore the potential of the partial triadic analysis (PTA), a multivariate method dedicated to the analysis of three-way datasets, to apprehend the space-time structure of tree-ring datasets. We analyzed a set of 11 tree-ring descriptors measured in 149 georeferenced individuals of European larch (Larix decidua Miller) during the period of 1967–2007. The processing of densitometry profiles led to a set of ring descriptors for each tree and for each year from 1967–2007. The resulting three-way data table was subjected to two distinct analyses in order to explore i) the temporal evolution of spatial structures and ii) the spatial structure of temporal dynamics. We report the presence of a spatial structure common to the different years, highlighting the inter-individual variability of the ring descriptors at the stand scale. We found a temporal trajectory common to the trees that could be separated into a high and low frequency signal, corresponding to inter-annual variations possibly related to defoliation events and a long-term trend possibly related to climate change. We conclude that PTA is a powerful tool to unravel and hierarchize the different sources of variation within tree-ring datasets.

The accuracy of two- and three-way positive matrix factorization models: Applying simulated multisite data sets

The application of three-way data sets (combined multisite data sets) for source apportionment has become common, but its influence on the performance of receptor modeling techniques has not yet been explored systematically. To study the influence of site-to-site correlations of source contributions and the spatial variability of source profiles on two- and three-way positive matrix factorization (PMF), simulated three-way data sets were constructed and modeled by different applications of PMF (PMF2 for each site individually, PMF2 for data sets combining all sites together, and PMF3 for all sites). In addition, the performance of PMF was evaluated under conditions of collinearity and different source categories at two sites. The results indicated that if the sites were contributed by sources with identical profiles, the site-to-site correlations of source contributions would not influence the PMF2, and the three-way blocks could be used by PMF2. However, the PMF2 using three-way data sets was sensitive to the spatial variability of source profiles. For the three-way model, PMF3 could perform well only when all of the sources exhibited strong site-to-site associations among all sites, and at the same time, the spatial variability of source profiles were sufficiently small. It might due to the algorithm that, for each source, PMF3 produces the same source profile and the same temporal variation in daily contributions among all sites.Implications: The application of multisite data sets for source apportionment has become common. However, limited work investigated the accuracy of two- and three-way PMFs when using multisite data sets. If the application of PMFs using multisite data sets were not appropriate, the results would be unreasonable. The unreasonable results would supply confused information for PM control strategies. In this work, simulated multisite data sets were modeled by different applications of PMFs. The effort to assess and compare the performance of two- and three-way PMFs using multisite data sets is very limited. The findings could provide information for multisite source apportionment.

Finding the limit of diverging components in three-way Candecomp/Parafac—A demonstration of its practical merits

Three-way Candecomp/Parafac (CP) is a three-way generalization of principal component analysis (PCA) for matrices. Contrary to PCA, a CP decomposition is rotationally unique under mild conditions. However, a CP analysis may be hampered by the non-existence of a best-fitting CP decomposition with R≥2 components. In this case, fitting CP to a three-way data array results in diverging CP components. Recently, it has been shown that this can be solved by fitting a decomposition with several interaction terms, using initial values obtained from the diverging CP decomposition. The new decomposition is called CPlimit, since it is the limit of the diverging CP decomposition. The practical merits of this procedure are demonstrated for a well-known three-way dataset of TV-ratings. CPlimit finds main components with the same interpretation as Tucker models or when imposing orthogonality in CP. However, CPlimit has higher joint fit of the main components than Tucker models, contains only one small interaction term, and does not impose the unnatural constraint of orthogonality. The uniqueness properties of the CPlimit decomposition are discussed in detail.

Application of three-way data analysis (second-order tensor decomposition) algorithms in analysis of liquid chromatography

Using dropline separation, tangent skimming, and triangulation to estimate the area of an overlapping chromatographic peak might contribute to a large deviation. It is easy, however, to eliminate these errors caused by geometric segmentation using three-way data analysis (second-order tensor decomposition) algorithms. This method of chromatographic analysis has many advantages: automation, anti-interference, high accuracy in the resolution of overlapping chrom- atographic peaks. It even makes the final goal of analytical chemistry achievable without the aid of complicated separation procedures. The core of this method is the process of utilizing useful information and building models through chemometric algorithms. Three-way chromatographic data set can be divided into trilinear dataset and nontrilinear dataset, correspondingly, three-way data analysis (second-order tensor decomposition) algorithms can be divided into trilinear algorithms and nontrilinear algorithms. In this paper, three-way calibration used in liquid chromatography for complex chemical systems in the last decade is reviewed, and focused on sample pretreatment, auxiliary algorithms, the combination and comparison of correction algorithms.