PARAFAC Model Research Articles

Joint interpretation of the hydrochemistry of two neighbouring basins by N-way multivariate methods

The objective of this work is to compare the chemical composition and the spatial and temporal variabilities of groundwater in two basins, the Langueyu and Del Azul creeks basins, located in the Pampean plain, Buenos Aires province (Argentina). The Pampean plain is the most productive region in Argentina, agriculture and livestock being the main economic activities. Groundwater is the principal water resource in the region, with a strong and growing demand for human supply and for agriculture and industrial activities. Several sampling campaigns were carried out on shallow wells of the two studied basins along a period of 3 years (2010–2013) to identify seasonal variations. Electrical conductivity, pH, bicarbonate, chloride, sulphate, nitrate, calcium, magnesium, sodium and potassium were determined following standard methods. For hydrochemical interpretation, descriptive statistical analyses, matrix augmentation principal component analysis, MA-PCA, and multidimensional principal component analysis, N-PCA (Parafac and Tucker3 models), were applied to the hydrochemical datasets from both basins. Three main hydrochemical processes have been identified in both basins: saline enrichment in the groundwater flow direction caused by dissolution of carbonates; exchange of calcium and magnesium by sodium in the same direction, and located areas of nitrate pollution. The paper shows that N-PCA is a good tool to deepen in the understanding of the hydrochemical features of groundwater from two neighbour basins, simplifying the analysis of large amounts of data, as well as establishing relations between the compared basins. Therefore the work is considered an interesting contribution to the study of groundwater resources with a regional scope. This knowledge is essential in basins with high socio-economic interests it causes a direct impact on resources management.

Advanced Residuals Analysis for Determining the Number of PARAFAC Components in Dissolved Organic Matter.

Parallel factor analysis (PARAFAC) has facilitated an explosion in research connecting the fluorescence properties of dissolved organic matter (DOM) to its functions and biogeochemical cycling in natural and engineered systems. However, the validation of robust PARAFAC models using split-half analysis requires an oft unrealistically large number (hundreds to thousands) of excitation-emission matrices (EEMs), and models with too few components may not adequately describe differences between DOM. This study used self-organizing maps (SOM) and comparing changes in residuals with the effects of adding components to estimate the number of PARAFAC components in DOM from two data sets: MS (110 EEMs from nine leaf leachates and headwaters) and LR (64 EEMs from the Lena River). Clustering by SOM demonstrated that peaks clearly persisted in model residuals after validation by split-half analysis. Plotting the changes to residuals was an effective method for visualizing the removal of fluorophore-like fluorescence caused by increasing the number of PARAFAC components. Extracting additional PARAFAC components via residuals analysis increased the proportion of correctly identified size-fractionated leaf leachates from 56.0 ± 0.8 to 75.2 ± 0.9%, and from 51.7 ± 1.4 to 92.9 ± 0.0% for whole leachates. Model overfitting was assessed by considering the correlations between components, and their distributions amongst samples. Advanced residuals analysis improved the ability of PARAFAC to resolve the variation in DOM fluorescence, and presents an enhanced validation approach for assessing the number of components that can be used to supplement the potentially misleading results of split-half analysis.

Strong linkages between DOM optical properties and main clades of aquatic bacteria

AbstractDespite considerable research on the linkages between dissolved organic matter (DOM) and bacteria, it is not yet clear how the abundance of the main aquatic clades relates to DOM composition in natural aquatic systems. We evaluated this relation using PARAFAC modeling of excitation–emission fluorescence spectroscopy and spectroscopic indexes to characterize DOM composition, and fluorescence in situ hybridization, to quantify the major bacterial groups in a subtropical lagoon. The DOM exhibited marked temporal variations in concentration, molecular weight, aromaticity, color, degree of humification, and freshness, and proportion of the three different fluorescent components identified. All major bacterial clades (Alphaproteobacteria, Betaproteobacteria, Gammaproteobacteria, and Cytophaga‐Flavobacteria) were significantly linked to DOM concentration and/or composition, being those crucial factors for modeling their abundance in situ. The combination and significance of the factors was specific for each bacterial group, strongly indicating that they behave as coherent and distinctive units. Cytophaga‐Flavobacteria and Betaproteobacteria were the groups which correlated with more DOM properties. Alphaproteobacteria and Gammaproteobacteria abundances were significantly explained by low or high dissolved organic carbon concentrations, respectively. The significant relationships between DOM properties and the main bacterial groups delineated a profile of each group regarding DOM preferences/dislikes, in agreement with evidence derived from genome analysis to single‐cell substrate uptake. These results highlight the specificities of the main bacterial clades, providing support for a functional classification of the bacterioplankton regarding DOM processing at the level of bacterial classes.

Label-free fluorescent catalytic biosensor for highly sensitive and selective detection of the ferrous ion in water samples using a layered molybdenum disulfide nanozyme coupled with an advanced chemometric model.

In this work, we developed a novel layered molybdenum disulfide (MoS2) nanosheet peroxidase mimetic-based fluorescent catalytic biosensor for the sensitive and selective detection of Fe(2+). It was found that Fe(2+) remarkably enhanced the catalytic activity of the MoS2 nanosheet for oxidation of OPD to form a highly fluorescent substance, 2,3-diaminophenazine (DAPN), and the MoS2/OPD/H2O2 biosensor displayed substantial fluorescence enhancement after addition of Fe(2+) in a concentration-dependent manner. The fluorescence intensity was proportional to the concentration of Fe(2+) over a range of 0.005-0.20 μM with a limit of detection of 3.5 nM (signal/noise = 3). When compared with the OPD/H2O2 biosensor, the MoS2/OPD/H2O2 biosensor provided a higher sensitivity and selectivity for Fe(2+), suggesting the validity of the use of the MoS2 nanosheets. To further demonstrate the feasibility of the MoS2/OPD/H2O2 biosensor for Fe(2+) detection in real water samples, we measured the three-dimensional excitation-emission spectra of the real system, and submitted the excitation-emission matrix (EEM) data to an advanced chemometrics model based on parallel factor analysis (PARAFAC). The results showed that the use of the PARAFAC model could further enhance the selectivity of the biosensor and determine Fe(2+) concentration in the presence of unexpected interferents from real water samples. This work opens up new opportunities for the use of the catalytic properties of the MoS2 nanosheets and advanced chemometrics models in the field of biosensors.

Sparse Tucker2 analysis of three-way data subject to a constrained number of zero elements in a core array

Three-way principal component analysis (3WPCA) models have been developed for analyzing a three-way data array of objects × variables × sources. Among the 3WPCA models, the least restrictive is the Tucker2 model, in which an extended core array describes the source-specific relationships between the components underlying objects and those for variables. In contrast to Tucker2 with the core array unconstrained, the Parafac model is highly restrictive in that the core slices in the array are constrained to be diagonal matrices. In this paper, we propose a procedure by which a suitably constrained intermediate model between Tucker2 and Parafac can be found. In the proposed procedure, the Tucker2 loss function is minimized subject to a specified number of core elements being zero with their locations unknown; the optimal locations of zero elements and nonzero parameter values are simultaneously estimated. This technique can be called sparse core Tucker2 (ScTucker2), as the matrices including a number of zeros are said to be sparse. We present an alternating least squares algorithm for ScTucker2 with a procedure for selecting a suitable number of zero elements. This procedure is assessed in a simulation study and illustrated with real data sets.

Multi-set factor analysis by means of Parafac2.

We consider multi-set data consisting of Nk observations, k=1,…, K (e.g., subject scores), on J variables in K different samples. We introduce a factor model for the J×J covariance matrices Σk, k=1,…, K, where the common part is modelled by Parafac2 and the unique variances Uk, k=1,…, K, are diagonal. The Parafac2 model implies a common loadings matrix that is rescaled for each k, and a common factor correlation matrix. We estimate the unique variances Uk by minimum rank factor analysis on Σk for each k. The factors can be chosen orthogonal or oblique. We present a novel algorithm to estimate the Parafac2 part and demonstrate its performance in a simulation study. Also, we fit our model to a data set in the literature. Our model is easy to estimate and interpret. The unique variances, the factor correlation matrix and the communalities are guaranteed to be proper, and a percentage of explained common variance can be computed for each k. Also, the Parafac2 part is rotationally unique under mild conditions.

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.

PARAFAC: A tool for the analysis of phosphor mixture luminescence

Parallel factor analysis (PARAFAC) was demonstrated as a tool for the analysis of the time-resolved emission spectra of mixtures of rare-earth phosphors. The time-resolved luminescence spectra of mixtures of europium(III)-doped Gd2O3, Gd2Ti2O7, and GdVO4 powders were measured in the 570–650nm emission range with time delays varying between 0 and 10ms. Then, PARAFAC was applied to decompose the time-resolved spectra of the mixtures into the emission spectra and emission decay profiles of the individual phosphors. The second-order advantage of this method was demonstrated by calculating the concentrations of the phosphors in the mixtures after calibrating the PARAFAC model by using samples of known concentrations. This method offered 0.1% detection limit and an average prediction error of 0.034%.

Semi-Blind Receivers for Non-Regenerative Cooperative MIMO Communications Based on Nested PARAFAC Modeling

In cooperative communication systems, multiple-input multiple-output (MIMO) amplify-and-forward (AF) relaying is a promising solution for upcoming wireless standards due to its intrinsic benefits in terms of extended coverage and increased spatial diversity. However, the deployment of MIMO AF relays faces some key challenges such as the need of channel state information (CSI) for the partial channels associated with the different communication links, which is required by some transmit optimization techniques. In this paper, we consider a one-way two-hop MIMO AF cooperative scheme employing space-time (ST) codings at the transmitter and relay nodes. First, we formulate a nested PARAFAC model for the tensor of signals received at destination. A theoretical study on the identifiability and uniqueness issues for the nested PARAFAC model is also carried out. Then, two iterative semi-blind receivers that exploit differently the algebraic properties of this model are formulated for a joint channel and symbol estimation at the destination. Uniqueness and least squares identifiability conditions are derived for these receivers, and their computational complexities are analyzed. Simulation results evaluate the bit error rate and the channel estimation accuracy obtained with the proposed semi-blind receivers, corroborating the performance improvements over recent semi-blind and supervised competitors, thanks to the use of a double Khatri-Rao coding associated with the source and relay nodes.

Influence of pH on fluorescent dissolved organic matter photo-degradation

A novel semi-continuous excitation emission matrix (EEM) fluorescence and absorbance monitoring system has been developed. Full EEMs were collected simultaneously with absorbance spectra every 20 min during 24 h solar-simulated irradiation experiments, and the kinetic change of fluorescence of Suwannee River natural organic matter IHSS standard material (SRNOM) at various pH values was investigated. Parallel factor analysis (PARAFAC) was then used to isolate the photo-labile and pH-influenced fluorescent components of SRNOM. Kinetic analysis showed increasing rates of fluorescence loss with increasing pH. This has significant implications for the photo-degradation of dissolved natural organic matter during estuarine mixing, when large increases of pH are common. The influence of pH on fluorescence and photo-degradation kinetics emphasizes the need for pH to be monitored and accurately controlled during laboratory experiments. It is also highly recommended that when constructing PARAFAC models or monitoring changes in fluorescence data between samples of different origins, that the pH be held constant to remove any potential artifacts or misinterpretation of data.

Multilinear Educational Data Analysis for Evaluation of Engineering Education

In this paper, we present a research study of educational evaluation in an engineering undergraduate degree. A multilinear decomposition, called Parallel Factors Analysis (PARAFAC), was used to extract intrinsic information about twokinds of evaluation: Learning Outcomes (LO) and Learning Context (LC). The results obtained allow us to identify common characteristics and similarities in disciplines in the undergraduate degree curriculum, both in terms of students' opinion and students' performance. The PARAFAC model also demonstrated significant potential for extracting data information related latent variables in educational contexts considering the sample multidimensionality.

A new approach for SSVEP detection using PARAFAC and canonical correlation analysis.

This paper presents a new way for automatic detection of SSVEPs through correlation analysis between tensor models. 3-way EEG tensor of channel × frequency × time is decomposed into constituting factor matrices using PARAFAC model. PARAFAC analysis of EEG tensor enables us to decompose multichannel EEG into constituting temporal, spectral and spatial signatures. SSVEPs characterized with localized spectral and spatial signatures are then detected exploiting a correlation analysis between extracted signatures of the EEG tensor and the corresponding simulated signatures of all target SSVEP signals. The SSVEP that has the highest correlation is selected as the intended target. Two flickers blinking at 8 and 13 Hz were used as visual stimuli and the detection was performed based on data packets of 1 second without overlapping. Five subjects participated in the experiments and the highest classification rate of 83.34% was achieved, leading to the Information Transfer Rate (ITR) of 21.01 bits/min.

Fluorescence fingerprinting of bottled white wines can reveal memories related to sulfur dioxide treatments of the must.

For the first time, Excitation Emission Matrix (EEM) fluorescence spectroscopy was combined with parallel factor statistical analysis (PARAFAC) and applied to a set of 320 dry white wines of the Chardonnay grape variety. A four component PARAFAC model (C1, C2, C3 and C4) best explained the variability of fluorescence signatures of white wines. Subtle changes were observed in EEMs of white wines from two different vintages (2006 and 2007), where different concentrations of sulfur dioxide (0, 4, and 8 g·hL(-1)) were added to the grape must at pressing. PARAFAC results clearly indicated that sulfur dioxide added to the must subsequently influenced white wine chemistry into three distinct sulfur dioxide dose-dependent aging mechanisms. For both vintages, C1 and C2 were the dominant components affected by sulfur dioxide and likely reacting with phenolic compounds associated with some presumably proteinaceous material. Distinct component combinations revealed either SO2 dependent or vintage-dependent signatures, thus, showing the extent of the complex versatile significance underlying such fluorescence spectra, even after several years of bottle aging.

Monitoring of thermal changes in meat by synchronous fluorescence spectroscopy

Here we report the potential of synchronous front-face fluorescence spectroscopy in combination with chemometric tools to detect thermal changes in cooked meat. Samples of bovine meat (Longissimus dorsi muscle) were cooked at 66, 90 & 237°C for 0, 1, 2, 5, 7, and 10min. Synchronous front-face fluorescence spectra were collected on meat samples in the excitation wavelength range of 250–550nm using offsets (Δλ) of 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, 130, 140, 150, and 160nm between excitation and emission wavelengths. The data from the synchronous fluorescence landscape containing 1080 spectra were analyzed using PCA & PARAFAC models. PCAs were applied to determine the potential of synchronous spectra to differentiate these samples as a function of cooking time and to retrieve additional information from different offset values. These results confirmed that the synchronous fluorescence spectra provide information related to the molecular structure of meat, allowing classification of samples as a function of cooking time & temperature. The best PARAFAC model was obtained with 3 components, having 55% core consistency values and 98% of the explained variance. The loading profiles of 1st, 2nd and 3rd components had an optimal Δλ of 60, 30 and 80nm, respectively, allowing the determination of excitation (exc.) and emission (em.) maxima wavelengths of 1st (fluorescence band at about exc.: 291 & em.: 351), 2nd (exc.: 297nm & em.: 327nm) and 3rd (exc.: 468nm & em.: 548nm) components. Loadings and scores of the PARAFAC model developed from the synchronous front-face fluorescence spectra enabled us to get information regarding the changes occurring in meat fluorophores during cooking of meat at 66, 90 & 237°C from 0 to 10min.

Long-term effect of a simple storage protocol on fluorescence fingerprint of dissolved organic matter

Fitting a pre-established region-specific Parallel Factor Analysis (PARAFAC) model to new dissolved organic matter (DOM) samples has become a popular approach in DOM studies. A key step during the development of the pre-established model is to identify the fluorescence fingerprint, i.e. the number of fluorescent DOM (FDOM) components as well as their excitation and emission spectral features. In practice the samples to use for model development may not be measured immediately after sampling and will instead be stored for a relatively long time, thus raising the question whether the storage would change the intrinsic fingerprints. In this study, two PARAFAC models were separately developed and validated for the same set of surface water DOM samples from an estuary and its river, immediately after sampling and after 9-month storage respectively. The fingerprint did not change after storage, regardless of the change of the intensity of some components. The findings in this study highlighted that DOM samples stored using a simple protocol (i.e. filtration to 0.45 μm pore size without any preservatives and placed at 4 °C in the dark under airtight conditions) for a few months would not change fluorescence fingerprints for PARAFAC and broaden our understanding of the application of PARAFAC to DOM studies.

Assessment of phytoplankton class abundance using fluorescence excitation-emission matrix by parallel factor analysis and nonnegative least squares

The feasibility of using fluorescence excitation-emission matrix (EEM) along with parallel factor analysis (PARAFAC) and nonnegative least squares (NNLS) method for the differentiation of phytoplankton taxonomic groups was investigated. Forty-one phytoplankton species belonging to 28 genera of five divisions were studied. First, the PARAFAC model was applied to EEMs, and 15 fluorescence components were generated. Second, 15 fluorescence components were found to have a strong discriminating capability based on Bayesian discriminant analysis (BDA). Third, all spectra of the fluorescence component compositions for the 41 phytoplankton species were spectrographically sorted into 61 reference spectra using hierarchical cluster analysis (HCA), and then, the reference spectra were used to establish a database. Finally, the phytoplankton taxonomic groups was differentiated by the reference spectra database using the NNLS method. The five phytoplankton groups were differentiated with the correct discrimination ratios (CDRs) of 100% for single-species samples at the division level. The CDRs for the mixtures were above 91% for the dominant phytoplankton species and above 73% for the subdominant phytoplankton species. Sixteen of the 85 field samples collected from the Changjiang River estuary were analyzed by both HPLC-CHEMTAX and the fluorometric technique developed. The results of both methods reveal that Bacillariophyta was the dominant algal group in these 16 samples and that the subdominant algal groups comprised Dinophyta, Chlorophyta and Cryptophyta. The differentiation results by the fluorometric technique were in good agreement with those from HPLC-CHEMTAX. The results indicate that the fluorometric technique could differentiate algal taxonomic groups accurately at the division level.

Parallel factor analysis for monitoring data from a grape harvest in Qualified Designation of Origin Rioja including spatial and temporal variability

The proper ripening of grapes is the key to obtain a high-quality wine. Ripening is a temporal process that is influenced, in addition to uncontrollable climate factors, by the spatial distribution of the vineyard and planted variety. It is a complex process that cannot be characterized by a single parameter; rather, it is a modification of the profile of the compounds of the grape. This paper analyzes the joint evolution of twelve physicochemical parameters determined in red grapes from four different varieties, in sixteen representative (in both geographical and edaphic point of view) plots belonging to the Qualified Designation of Origin (DOC) Rioja. Samples were collected in September 2009 during four consecutive weeks prior to harvest.The data are naturally structured in a three-dimensional cube of data (named tensor) with these profiles: sample (plot/variety), physicochemical, and temporal. For the first time, studies are made for jointly analyzing the effect of the spatial and temporal distribution on the physicochemical profile of the grape. A PARAFAC model shows an internal data structure that accurately reproduces the ripening process with its distinctive features according to plot/variety, evolution of the physicochemical parameter of grapes, and ripening time.PARAFAC analysis reveals the different grape maturity depending on the vineyard's situation (zone and altitude) and the grape variety (garnacha, tempranillo, graciano, mazuelo). The obtained information allows evaluating the attainable quality in the grape related to the potential of the zone and its variety.

Multiscale entropy analysis of resting-state magnetoencephalogram with tensor factorisations in Alzheimer's disease.

Tensor factorisations have proven useful to model amplitude and spectral information of brain recordings. Here, we assess the usefulness of tensor factorisations in the multiway analysis of other brain signal features in the context of complexity measures recently proposed to inspect multiscale dynamics. We consider the "refined composite multiscale entropy" (rcMSE), which computes entropy "profiles" showing levels of physiological complexity over temporal scales for individual signals. We compute the rcMSE of resting-state magnetoencephalogram (MEG) recordings from 36 patients with Alzheimer's disease and 26 control subjects. Instead of traditional simple visual examinations, we organise the entropy profiles as a three-way tensor to inspect relationships across temporal and spatial scales and subjects with multiway data analysis techniques based on PARAFAC and PARAFAC2 factorisations. A PARAFAC2 model with two factors was appropriate to account for the interactions in the entropy tensor between temporal scales and MEG channels for all subjects. Moreover, the PARAFAC2 factors had information related to the subjects' diagnosis, achieving a cross-validated area under the ROC curve of 0.77. This confirms the suitability of tensor factorisations to represent electrophysiological brain data efficiently despite the unsupervised nature of these techniques. This article is part of a Special Issue entitled 'Neural data analysis'.

Utilization of PARAFAC-Modeled Excitation-Emission Matrix (EEM) Fluorescence Spectroscopy to Identify Biogeochemical Processing of Dissolved Organic Matter in a Northern Peatland.

In this study, we contrast the fluorescent properties of dissolved organic matter (DOM) in fens and bogs in a Northern Minnesota peatland using excitation emission matrix fluorescence spectroscopy with parallel factor analysis (EEM-PARAFAC). EEM-PARAFAC identified four humic-like components and one protein-like component and the dynamics of each were evaluated based on their distribution with depth as well as across sites differing in hydrology and major biological species. The PARAFAC-EEM experiments were supported by dissolved organic carbon measurements (DOC), optical spectroscopy (UV-Vis), and compositional characterization by ultrahigh resolution Fourier transform ion cyclotron resonance mass spectroscopy (FT-ICR MS). The FT-ICR MS data indicate that metabolism in peatlands reduces the molecular weights of individual components of DOM, and oxygen-rich less aromatic molecules are selectively biodegraded. Our data suggest that different hydrologic and biological conditions within the larger peat ecosystem drive molecular changes in DOM, resulting in distinctly different chemical compositions and unique fluorescent fingerprints. PARAFAC modeling of EEM data coupled with ultrahigh resolution FT-ICR MS has the potential to provide significant molecular-based information on DOM composition that will support efforts to better understand the composition, sources, and diagenetic status of DOM from different terrestrial and aquatic systems.

PARAFAC models of fluorescence data with scattering: A comparative study

Excitation-Emission Matrix (EEM) fluorescence spectroscopy combined with second order decomposition algorithms such as PARAFAC provides interesting opportunities in analytical chemistry. However, the intrinsic presence of scattering effects in the EEM measurements poses a practical problem. Appropriate handling of the scatter is necessary to avoid detrimental influence on the models. The two most common methods of handling scatter are evaluated in this paper: replacing the scattering area with missing elements or with interpolated values. Both were assessed in terms of stability of the models and quality of predictions. The methods were compared using four different datasets showing diverse scattering effects that was also partly overlapping key chemical areas. The results suggested that the use of missing values lead to more problems related to local minima, which also considerably increases the time of analysis. Only the best fitting models obtained with missing values showed similar spectral recoveries and good predictions in comparison with those ones obtained with interpolated data. Furthermore, the use of missing values in the PARAFAC models hindered the correct decomposition of very complex samples, which showed to be more robust by means of using interpolated data.