PARAFAC Method Research Articles

Impact of ambient conditions on the minor nutrients content in spirulina “Arthrospira platensis” (AP) extracted from Lake Chad revealed by fluorescence spectroscopy coupled with chemometric methods

In this study, the technique of fluorescence spectroscopy coupled with chemometric methods is used to analyse samples of Lake Chad Spirulina “Arthrospira platensis” (AP), either harvested and conditioned by using the traditional method at different seasons or industrially processed. The content of minor fluorescent nutrients is investigated. To this end, fluorescence excitation-emission matrices (EEMs) of 46 AP samples are recorded in aqueous solution. Synchronous fluorescence (SF) spectra are extracted from these EEMs and their important features are compared to those of PARAFAC methods. Synchronous fluorescence scanning allows different AP samples to be characterized in a single scan. The SF and PARAFAC methods yielded two groups of fluorescent compounds; the first group, consisting of vitamin-like molecules, shows excitation/emission (ex/em) peaks at 340/460, 390/462, 370/440 and 450/526 nm, attributed to caffeic acid, vitamin K, E and riboflavins respectively, while the second group, consisting of pigments, shows ex/em peaks at 610/654, 590/630 and 570/644 nm, attributed to phycocyanins, C-phycocyanin and allophycocyanin. Our fluorescence data showed that while both vitamins and pigments are present in AP during the rainy season, only fluorescent components of vitamin-like compounds are present during the dry season. PCA methods allowed classifying different AP samples according to their geographic origin and harvesting season. Fluorescence spectroscopy therefore appears to be a powerful technique for rapidly assessing the chemical composition of AP.

Spectral Analysis, Biocompounds, and Physiological Assessment of Cork Oak Leaves: Unveiling the Interaction with Phytophthora cinnamomi and Beyond

The cork oak tree (Quercus suber L.) symbolizes the Montado landscape in Portugal and is a central element in the country’s social and economic history. In recent decades, the loss of thousands of cork oaks has been reported, revealing the ongoing decline of these agroforestry ecosystems. This emblematic tree of the Mediterranean Basin is host to the soil-born root pathogen Phytophthora cinnamomi, an active cork oak decline driver. In this framework, the early diagnosis of trees infected by the oomycete by non-invasive methods should contribute to the sustainable management of cork oak ecosystems, which motivated this work. Gas exchange and visible/near-infrared (400–1100 nm) reflectance spectroscopy measurements were conducted on leaves of both control and P. cinnamomi inoculated plants. These measurements were taken at 63, 78, 91, 126, and 248 days after inoculation. Additionally, at the end of the experiment, biochemical assays of pigments, sugars, and starch were performed. The spectroscopic measurements proved effective in distinguishing between control and inoculated plants, while the standard gas exchange and biochemistry data did not exhibit clear differences between the groups. The spectral data were examined both daily and globally, utilizing the PARAFAC method applied to a three-way array of samples × wavelengths × days. The separation of the two plant groups was attributed to variations in water content (4v (O−H)); shifts in the spectra red edge; and structural modifications in the epidermal layer and leaves’ mesophyll. These spectral signatures can assist in the field identification of cork oaks that are interacting with P. cinnamomi.

Popularity Prediction of Social Media Post Using Tensor Factorization

The traditional method of doing business has been disrupted by social media. In order to develop the enterprise, it is essential to forecast the level of interaction that a new post would receive from social media users. It is possible for the user’s interest in any one social media post to be impacted by external factors or to dwindle as a result of changes in his behaviour. The popularity detection strategies that are user-based or population-based are unable to keep up with these shifts, which leads to inaccurate forecasts. This work makes a prediction about how popular the post will be and addresses any anomalies caused by factors outside of the study. A novel improved PARAFAC (A-PARAFAC) method that is tensor factorization-based has been presented in order to cope with the user criteria that will be used in the future to rate any project. We consolidated the information on the historically popular content, and we accelerated the computation by choosing the top contents that were most like each other. The tensor is factorised with the application of the Adam optimization. It has been modified such that the bias is now included in the gradient function of A-PARAFAC, and the value of the bias is updated after each iteration. The prediction accuracy is improved by 32.25% with this strategy compared to other state of the art methods.

DOD and DOA Estimation From Incomplete Data Based on PARAFAC and Atomic Norm Minimization Method

In this article, we propose an efficient direction of departure (DOD) and direction of arrival (DOA) estimation method for bistatic multiple-input multiple-output (MIMO) radar with faulty arrays. A third-order tensor model is built, and the measurement 3-D structure can be better utilized than the traditional matrix model. Subsequently, the atomic norm minimization (ANM) technique is used to further improve the angle estimation performance. Furthermore, we found in the research process that when the faulty arrays still maintain the symmetry property, the measurement tensor can be converted to the real-valued domain by the forward–backward averaging technique and the unitary transform technique. The new algorithm we proposed exploits the multidimensional structure of the signal without estimating the signal subspace. Comparing with traditional matrix completion (MC) methods, it has a better performance in terms of robustness and resolving correlated targets. Also, the algorithm proposed in this article does not require angle pairing. Simulation results verify the effectiveness of the proposed algorithm.

Study on the Migration Law of Dissolved Organic Matter in Mine Water Treatment Station

With the improvement of coal-mining mechanizations and the intensification of human activities, the organic matter pollution of mine water is becoming severe. In this study, the chemical compositions of the influents and effluents from 15 mine water treatment stations in the mining area bordering Mongolia and Shaanxi were measured. The occurrence of DOM (dissolved organic matter) in the effluent from the mine water treatment stations in this area was determined by the EEM (excitation emission matrix), combined with the PARAFAC (parallel factor analysis) method. The DOM removal from the mine water treatment station in the Caojiatan coal mine is specifically discussed here, although trends are similar across the 15 mines. The treatment capacity of this treatment process for different types of pollutants is also evaluated, and a mine water treatment process suitable for the current coal-mining mode is suggested. The results show that the DOM of the mine water treatment stations in this area mainly has four components: a fulvic-acid-like substance (C1/C3), a protein tryptophan-like substance (C2), and a protein tyrosine-like substance (C4). The coagulation, filtration, and disinfection process has a removal efficiency of more than 90% for the protein-like tryptophan components, COD (chemical oxygen demand), and NO2−, and an efficiency of ~50% for TOC (total organic carbon), <30% for Cu2+ and F−, and almost no removal effect for protein-like tyrosine components, EC (electrical conductivity), TDS (total dissolved solids), and NH4+. These conclusions show that aliphatic hydrocarbons, such as alkanes and cycloalkanes, in mine water are removed by the treatment process, whereas macromolecular aromatic hydrocarbons and other groups are not removed by the treatment process. Based on this, an ozone-demulsification process for the special removal of protein tyrosine-like pollutants in mine water is proposed. This conclusion can provide theoretical support for research on the source and fate of the carbon trajectory in the water-cycle process and provides technical guidance for the removal of DOM from mine water.

Application of Parallel Factor Analysis (PARAFAC) to the Regional Characterisation of Vineyard Blocks Using Remote Sensing Time Series

Monitoring wine-growing regions and maximising the value of production based on their region/local specificities requires accurate spatial and temporal monitoring. The increasing amount and variability of information from remote sensing data is a potential tool to assess this challenge for the grape and wine industry. This article provides a first insight into the capacity of a multiway analysis method applied to Sentinel-2 time series to assess the value of simultaneously considering spectral and temporal information to highlight site-specific canopy evolution in relation to environmental factors and management practices, which present a large diversity at this regional scale. Parallel Factor Analysis (PARAFAC) was used as an unsupervised technique to recover pure spectra and temporal signatures from multi-way spectral imagery of vineyards in the Languedoc-Roussillon region in the south of France. The model was developed using a time series of Sentinel-2 satellite imagery collected over 4978 vineyard blocks between May 2019 and August 2020. From the Sentinel-2 (spectral and temporal) signal, the PARAFAC analysis allowed the identification of spectral and temporal profiles in the form of pure components, which corresponded to vegetation and soil. The PARAFAC analysis also identified that two of the pure spectra were strongly related to characteristics and dynamics of vineyard cultivation at a regional scale. A conceptual framework was proposed in order to simultaneously consider both vegetation and soil profiles and to summarise the mass of data accordingly. This methodology allowed the computation of a concentration index that characterised how close a field was to a vegetation or a soil profile over the season. The concentration indices were validated for the vegetation and the soil over two growing seasons (2019 and 2020) with geostatistical analysis. A non-random distribution of the concentration index at the regional scale was assumed to highlight a strongly spatially organised phenomenon related to spatially organised environmental factors (soil, climate, training system, etc.). In a second step, spatial patterns of indices were subjected to the expertise of a panel of advisors of the wine industry in order to validate them in relation to vine-growing conditions. Results showed that the introduction of the PARAFAC method opened up the possibility to identify relevant spectro-temporal profiles for vine monitoring purposes.

Combining real-time fluorescence spectroscopy and flow cytometry to reveal new insights in DOC and cell characterization of drinking water

Sudden changes in drinking water quality can cause harmful consequences for end users. Thus, real-time monitoring of drinking water quality can allow early warning and provide crucial gains for securing safe water distribution. This study investigated the advantages of simultaneous real-time measuring of flow cytometry and fluorescence spectroscopy. A contamination event was investigated in a laboratory-scale analysis by spiking drinking water samples with organic nutrients. Flow cytometric data were analyzed by creating fingerprints based on differentiation into high and low nucleic acid cells (HNA/LNA). The detailed characterization of these data showed that an increase in HNA cells indicated an increase in the bacterial growth potential even before actual TCC increases. The fluorescence data was decomposed via the PARAFAC method to reveal seven fluorescent components. Three aromatic protein-like components were associated with the microbiological condition of the drinking water cells; namely, Components 4 (λEx = 279 nm, λEm = 351 nm), 6 (λEx = 279 nm, λEm = 332 nm), and 7 (λEx = 276 nm, λEm = 302 nm). Component 6 was identified as a possible organic variable for appropriate monitoring of TCC, whereas Components 4 and 7 were identified as organic compounds representing nutrients for organisms present in drinking water. Overall, combining both methods for real-time monitoring can be a powerful tool to guarantee drinking water quality.

Multi-Sensor Fusion by CWT-PARAFAC-IPSO-SVM for Intelligent Mechanical Fault Diagnosis.

A new method of multi-sensor signal analysis for fault diagnosis of centrifugal pump based on parallel factor analysis (PARAFAC) and support vector machine (SVM) is proposed. The single-channel vibration signal is analyzed by Continuous Wavelet Transform (CWT) to construct the time–frequency representation. The multiple time–frequency data are used to construct the three-dimension data matrix. The 3-level PARAFAC method is proposed to decompose the data matrix to obtain the six features, which are the time domain signal (mode 3) and frequency domain signal (mode 2) of each level within the three-level PARAFAC. The eighteen features from three direction vibration signals are used to test the data processing capability of the algorithm models by the comparison among the CWT-PARAFAC-IPSO-SVM, WPA-PSO-SVM, WPA-IPSO-SVM, and CWT-PARAFAC-PSO-SVM. The results show that the multi-channel three-level data decomposition with PARAFAC has better performance than WPT. The improved particle swarm optimization (IPSO) has a great improvement in the complexity of the optimization structure and running time compared to the conventional particle swarm optimization (PSO.) It verifies that the proposed CWT-PARAFAC-IPSO-SVM is the most optimal hybrid algorithm. Further, it is characteristic of its robust and reliable superiority to process the multiple sources of big data in continuous condition monitoring in the large-scale mechanical system.

Multiway resolution of spectrochromatographic measurements for the quantification of echinuline in marine-derived fungi Aspergillus chevalieri using parallel factor analysis

A multiway resolution of incomplete chromatographic separation was presented for spectrochromatographic quantification of echinuline in marine-derived fungi Aspergillus chevalieri. Two-dimensional spectrochromatographic maps of calibration, validation and real samples were recorded as a function of time and wavelength using UPLC-PDA instrument under non-optimized chromatographic conditions, which gave rise to co-elution of echinuline and the constituents of sample matrix. A three-way array was obtained by concatenating the data matrices of the spectrochromatographic maps. Then, parallel factor analysis was applied to the multiway array to extract the individual contribution of echinuline in three modes (time, wavelength and sample). While time and wavelength profiles were used for the characterization of echinuline, the sample profile was used for its quantitative determination of the analyte in validation set and in real samples. Validity of the analytical method was evaluated by analyzing the validation set, which consist of test samples, standard addition samples, intra-day and inter-day samples. The proposed multiway analysis method was then applied to marine-derived fungi extracts and echinuline content was found to be 31.9 µg/g based on the average of ten assay results. The assay results provided by PARAFAC model were statistically compared with those obtained by a newly developed classical UPLC method, which ensured the complete separation of echinuline in a run time of nine minutes. The assay results were found to be comparable due to the fact that there was no significant difference between the analysis results (F = 1.63, Fcrit = 3.17; t = 0.69, tcrit = 2.11) at the significance level of 95%). Consequently, the PARAFAC method permitted the accurate determination of echinuline in fungal extracts despite the partial chromatographic separation with a run time of only three minutes.

A New Application of PARAFAC Model to UPLC Dataset for the Quantitative Resolution of a Tri-Component Drug Mixture.

In the presented work, a three-way analysis of ultra-performance liquid chromatography-photodiode array (UPLC-PDA) dataset was performed by parallel factor analysis (PARAFAC) for quantitatively resolving a ternary mixture containing paracetamol and methocarbamol with indapamide selected as an internal standard in their co-eluted chromatographic conditions. Paracetamol and methocarbamol were quantified in the working range between 3-24 and 5-50μg/mL by applying PARAFAC decomposition to UPLC-PDA data array obtained under unresolved chromatographic peak conditions. To compare the experimental results provided by co-eluted UPLC-PARAFAC method, an ordinary UPLC method was developed ensuring proper separation of the peaks. The performance of both PARAFAC and ordinary UPLC methods were assessed by quantifying independent test samples, intra- and inter-day samples and spiked samples of pharmaceutical preparations. Then, both methods were applied for quantitative estimation of the related drugs in a commercial pharmaceutical preparation. In this study, PARAFAC method was proved to be a very powerful alternative for the quality control of pharmaceutical preparations containing paracetamol and methocarbamol even in their co-eluted chromatograms with high precision and accuracy in a short chromatographic runtime of 1.2min.

The NMR and spectral study on the structure of molecular size-fractionated lignite humic acid

In this study, high pure humic acid (HA) extracted from lignite was fractionated into four fractions according to their molecular weights by the ultrafiltration method. The structural characteristics of various HA fractions were investigated by UV–Vis,FTIR, and 3D fluorescence spectra and 13 C/ 1 H-NMR tests. The experiment results showed that the HA fractions with the molecular weight ＞50 KDa are major, containing 70.89%. The active hydrogen proportion declined from 43.96% to 21.27% with the increase of HA molecular weight, meaning the decrease of the large HA molecules’ acidic group content. On the other hand, 80.93% of the aromatic nucleus content and 12.5% of the alkyl carbon (aliphatic structure) were observed in the large HA molecules that are much higher than those in the small HA molecules. The NMR and spectra studies indicated that the small HA molecules have more acidic groups such as hydroxy, carboxyl. Meanwhile, the large HA molecules have more aromatic and aliphatic structures. The above research indicated that the HA fractions with low molecular weight could provide more active sites for the interaction between HA and iron minerals. Moreover, the carbon chain and aromatic nucleus in the high molecular weight HA fractions could keep the space structure of HA stable. • High pure lignite humic acid was fractionated successfully by ultrafiltration. • The differences of composition and structure of various HA fractions were investigated. • The structure proportion of HA was quantized by integral and PARAFAC method. • The smaller HA molecular had more abundant oxygen-bearing groups. • The larger HA molecular had more aromatic structures.

Feasibility study for the analysis of coconut water using fluorescence spectroscopy coupled with PARAFAC and SVM methods

PurposeParallel factor analysis (PARAFAC) coupled with support-vector machine (SVM) was carried out to identify and discriminate between the fluorescence spectroscopies of coconut water brands.Design/methodology/approachPARAFAC was applied to reduce three-dimensional data of excitation emission matrix (EEM) to two-dimensional data. SVM was applied to discriminate between six commercial coconut water brands in this study. The three largest variation data from fluorescence spectroscopy were extracted using the PARAFAC method as the input data of SVM classifiers.FindingsThe discrimination results of the six commercial coconut water brands were achieved by three SVM methods (Ga-SVM, PSO-SVM and Grid-SVM). The best classification accuracies were 100.00%, 96.43% and 94.64% for the training set, test set and CV accuracy.Originality/valueThe above results indicate that fluorescence spectroscopy combined with PARAFAC and SVM methods proved to be a simple and rapid detection method for coconut water and perhaps other beverages.

Monitoring Framework Based on Generalized Tensor PCA for Three-Dimensional Batch Process Data

Unfolding is a pretreating operation in most existing batch process modeling methods, but it would destroy the essential structure of the raw data. Also, the number of estimated parameters in the unfolding model is prohibitively large. In this work, a generalized tensor PCA (GTPCA) method is proposed for batch process monitoring. It can be performed on the 3-D batch process data directly to avoid the “curse of dimensionality” problem and potential information loss caused by data unfolding. Also, the uneven-length problem can be solved naturally without batch trajectory synchronization to prevent distorting data, which guarantees better quality models and monitoring performances. Moreover, within-batch detection can be easily performed using the tensor-based modeling scheme. The advantages and effectiveness of the proposed method are illustrated through a numerical case and an injection molding process in comparison with the traditional MPCA, PARAFAC, and Tucker3 methods.

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).

Exploiting second-order advantage for simultaneous and interference-free determination of orthophosphate and arsenate in water samples

ABSTRACT A novel approach for the simultaneous determination of orthophosphate and arsenate as a vital parameter for environmental controlling was developed using the molybdenum blue method. The second-order chemometrics analysis was employed to overcome significant spectral overlapping among analytes and interferences. Two powerful second-order calibration methods including, Bilinear least squares/residual bilinearization (BLLS/RBL) and Parallel factor analysis (PARAFAC) methodologies, were used for analysing kinetic-spectroscopic second-order data and the decomposition of spectra and quantification of analytes. These methodologies are capable of accurate and reliable discernment of the target analyte signals, even in the presence of unknown and uncalibrated interferences, which are inherently capable of providing the second-order advantage. No sample pretreatment or separation steps were required. The proposed method was performed on the water samples, and the accuracies of both methods were successfully validated by the analytical figures of merit (FOM). The limits of detection (LOD) were 0.28 and 0.29 µmol L−1 using BLLS/RBL method and 0.14 and 0.16 µmol L−1 using PARAFAC method for orthophosphate and arsenate, respectively. Orthophosphate and arsenate recoveries in real water samples were determined between 97.0 and 100.5% using both methods.

Low-Level Fusion of Fourier Transform Ion Cyclotron Resonance Mass Spectrometry Data Sets for the Characterization of Nitrogen and Sulfur Compounds in Vacuum Gas Oils.

A total of 18 vacuum gas oils have been analyzed by Fourier transform ion cyclotron resonance mass spectrometry considering six replicates in three different ionization modes (electrospray ionization (ESI)(+), ESI(-), and atmospheric pressure photoionization (APPI)(+)) to characterize the nitrogen and sulfur compounds contained in these samples. Classical data analysis has been first performed on generated data sets using double bond equivalents (DBE) versus number of carbon atoms (#C) plots in order to observe similarities and differences within the nitrogen and sulfur-containing molecular classes from samples produced by different industrial processes. In a second step, three-way arrays have been generated for each ionization mode considering three dimensions: DBE related to aromaticity, number of carbon atoms related to alkylation, and sample. These three-way arrays have then be concatenated using low-level data fusion strategy to obtain a new tensor with three new modes: aromaticity, alkylation, and sample. The PARAFAC method has then been applied for the first time to this three-way data structure. A two components decomposition has allowed us to highlight unique samples with unexpected reactivity behaviors throughout hydrotreatment. The obtained loadings led to the identification of the variables responsible for this specific character. This original strategy has provided a fast visualization tool able to highlight simultaneously the impact of the three ionization modes in order to explain the differences between the samples and compare them.

Dimensionality Reduction Based on PARAFAC Model

Abstract In hyperspectral image analysis, dimensionality reduction is a preprocessing step for hyperspectral image (HSI) classification. Principal component analysis (PCA) reduces the spectral dimension and does not utilize the spatial information of an HSI. To solve it, the tensor decompositions have been successfully applied to joint noise reduction in spatial and spectral dimensions of hyperspectral images, such as parallel factor analysis (PARAFAC). However, the PARAFAC method does not reduce the dimension in the spectral dimension. To improve it, two new methods were proposed in this article, that is, combine PCA and PARAFAC to reduce both the dimension in the spectral dimension and the noise in the spatial and spectral dimensions. The experimental results indicate that the new methods improve the classification compared with the PARAFAC method.

Joint transmit-receive B-PARAFAC method for angle estimation in bistatic MIMO radar

In this paper, a joint transmit-receive beamspace parallel factor (B-PARAFAC) method is proposed for angle estimation in bistatic MIMO radar. Combining with the forward-backward averaging technique, the beamspace transformation technique is firstly extended to the tensor case, which utilizes the multidimensional structure of the received data. Specially, the transmit beamspace (TB) and receive beamspace (RB) matrices are obtained through convex optimization, whose matrix structure can be properly designed to construct a real-valued tensor model. Furthermore, different from the traditional discrete Fourier transform (DFT) beamspace filters, the bandwidth and sidelobe level of the proposed beamspace filters can be flexibly controlled as required, which can greatly improve the mainlobe-to-sidelobe ratio (MSR). We prove that the constructed real-valued tensor after beamspace transformation still follows a PARAFAC model. Compared with the unitary PARAFAC (U-PARAFAC) method, the proposed B-PARAFAC method has more attractive performance especially in low signal-to-noise ratio (SNR) and requires less computation. Simulation results demonstrate the effectiveness of the proposed method.

Dimensionality reduction based on parallel factor analysis model and independent component analysis method

In hyperspectral image (HSI) analysis, dimensionality reduction is a preprocessing step for HSI classification. Independent component analysis (ICA) reduces the spectral dimension and does not utilize the spatial information of the HSI. To solve it, tensor decompositions have been successfully applied to joint noise reduction in spatial and spectral dimensions of HSIs, such as parallel factor analysis (PARAFAC). However, the PARAFAC method does not reduce the dimension in the spectral dimension. We proposed a method to improve it, which combines ICA and PARAFAC to reduce both the dimension in the spectral dimension and the noise in the spatial and spectral dimensions. The experimental results indicate that this method improves the classification compared with the previous methods.

Aesthetic compatibility assessment of consolidants for wall paintings by means of multivariate analysis of colorimetric data

Background and methodsWall paintings and architectural surfaces in outdoor environments are exposed to several physical, chemical and biological agents, hence they are often treated with different products to prevent or slow down their deterioration. Among the factors that have to be taken into account in the selection of the most suitable treatment for decorated surfaces, the aesthetic compatibility with the substrate is of great importance in the cultural heritage field; minimizing colour variation after treatment application is a crucial issue in particular for painted surfaces. In the framework of the European Project Nanomatch the color variation induced on wall painting mock-ups by the two innovative consolidants (calcium alkoxides) developed was evaluated using colorimetry in comparison with two traditional products. In this work these innovative consolidants have been also tested in combination with two commercial biocides and the results of colorimetric measurements discussed. Moreover, as the univariate approach didn’t allow to draw clear conclusions on the relation between the different sources of data variability, multivariate analysis was performed on colorimetric data.ResultsPrincipal Component Analysis and multi-way Parallel Factor Analysis (PARAFAC) were successfully applied to colorimetric data to investigate the short-term effects of the application of different consolidants on wall painting surfaces, making it possible to study at the same time the different sources of data variability, i.e. treatments, painting techniques, pigments. Finally, a ranking list of the treatments under study in terms of colour variation induced on the surface was established, in function of the painting technique and pigment, taking also in consideration the combination consolidant/biocide. In particular, given the true multi-way nature of the data, PARAFAC model turned out to be extremely useful in the study of the dependence of colour variation on pigments, a critical issue for painted surfaces, that was not clear using univariate approach.ConclusionsMultivariate approach to colorimetric data and especially 3-way PARAFAC method resulted a powerful technique to evaluate in short-term the color compatibility of consolidants for wall paintings, improving data interpretation and visualization, and thus outperforming the univariate statistical analysis.