Three-way Array Research Articles

Revealing static and dynamic biomarkers from postprandial metabolomics data through coupled matrix and tensor factorizations.

Longitudinal metabolomics data from a meal challenge test contains both fasting and dynamic signals, that may be related to metabolic health and diseases. Recent work has explored the multiway structure of time-resolved metabolomics data by arranging it as a three-way array with modes: subjects, metabolites, and time. The analysis of such dynamic data (where the fasting data is subtracted from postprandial states) reveals dynamic markers of various phenotypes, and differences between fasting and dynamic states. However, there is still limited success in terms of extracting static and dynamic biomarkers for the same subject stratifications. Through joint analysis of fasting and dynamic metabolomics data, our goal is to capture static and dynamic biomarkers of a phenotype for the same subject stratifications providing a complete picture, that will be more effective for precision health. We jointly analyze fasting and dynamic metabolomics data collected during a meal challenge test from the COPSAC cohort using coupled matrix and tensor factorizations (CMTF), where the dynamic data (subjects by metabolites by time) is coupled with the fasting data (subjects by metabolites) in the subjects mode. The proposed data fusion approach extracts shared subject stratifications in terms of BMI (body mass index) from fasting and dynamic signals as well as the static and dynamic metabolic biomarker patterns corresponding to those stratifications. Specifically, we observe a subject stratification showing the positive association with all fasting VLDLs and higher BMI. For the same subject stratification, a subset of dynamic VLDLs (mainly the smaller sizes) correlates negatively with higher BMI. Higher correlations of the subject quantifications with the phenotype of interest are observed using such a data fusion approach compared to individual analyses of the fasting and postprandial state. The CMTF-based approach provides a complete picture of static and dynamic biomarkers for the same subject stratifications-when markers are present in both fasting and dynamic states.

Sparse Tensor Decomposition of Multi-Sensory Data for Fault Localization in Rotating Machinery Health Monitoring

Multi-sensor monitoring is prevalent in modern structural health monitoring (SHM) practice. As the number of sensors and sampling requirements increase, a monitoring sensor network can generate substantial data which are high-volume and high-dimensional, especially for large structure and machinery. In condition-based monitoring (CBM) of rotating machines, e.g., gas turbine, rotor fault diagnosis serves a significant role in the system reliability, safety, and efficiency, which helps reduce potential damage to the on-rotor structures and avoid catastrophic failures. For multi-channel vibration measurement, classical rotor diagnosis approaches typically involve data fusion techniques based on cross spectral analysis for identifying spectral correlation, e.g., cross power spectral density, and/or matrix analysis tool for dimensionality reduction, e.g., principal component analysis. The operation on vectors or matrices may limit their effectiveness for higher-order array or tensor data. To circumvent this limitation, the third order vibrational spectral tensor is generated by representing the multi-channel acceleration spectra as the three-way array. Through nonnegative Tucker decomposition (NTD), the spectral tensor is decomposed into multiple principal factors: the spectral factor, segmental factor, channel-wise factor, and the dense tensor core. The correlations across the characteristic spectral contents and the sensor channels are revealed by the factorization, which enables the diagnosis of rotor fault and facilitates fault localization by identifying the dominant channels of the characteristic spectral factor. The method is validated on an experimental rotor testbed where the faulty channel with crack, rub, or misalignment fault, is effectively localized via 4-channel vibration measurements, which presents a promising approach for multi-sensor fusion and fault diagnosis in rotating machinery health monitoring.

Characterizing human postprandial metabolic response using multiway data analysis

IntroductionAnalysis of time-resolved postprandial metabolomics data can improve our understanding of the human metabolism by revealing similarities and differences in postprandial responses of individuals. Traditional data analysis methods often rely on data summaries or univariate approaches focusing on one metabolite at a time.ObjectivesOur goal is to provide a comprehensive picture in terms of the changes in the human metabolism in response to a meal challenge test, by revealing static and dynamic markers of phenotypes, i.e., subject stratifications, related clusters of metabolites, and their temporal profiles.MethodsWe analyze Nuclear Magnetic Resonance (NMR) spectroscopy measurements of plasma samples collected during a meal challenge test from 299 individuals from the COPSAC2000 cohort using a Nightingale NMR panel at the fasting and postprandial states (15, 30, 60, 90, 120, 150, 240 min). We investigate the postprandial dynamics of the metabolism as reflected in the dynamic behaviour of the measured metabolites. The data is arranged as a three-way array: subjects by metabolites by time. We analyze the fasting state data to reveal static patterns of subject group differences using principal component analysis (PCA), and fasting state-corrected postprandial data using the CANDECOMP/PARAFAC (CP) tensor factorization to reveal dynamic markers of group differences.ResultsOur analysis reveals dynamic markers consisting of certain metabolite groups and their temporal profiles showing differences among males according to their body mass index (BMI) in response to the meal challenge. We also show that certain lipoproteins relate to the group difference differently in the fasting vs. dynamic state. Furthermore, while similar dynamic patterns are observed in males and females, the BMI-related group difference is observed only in males in the dynamic state.ConclusionThe CP model is an effective approach to analyze time-resolved postprandial metabolomics data, and provides a compact but a comprehensive summary of the postprandial data revealing replicable and interpretable dynamic markers crucial to advance our understanding of changes in the metabolism in response to a meal challenge.

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.

Marginal-Preserving Imputation of Three-Way Array Data in Nested Structures, with Application to Small Areal Units

Geospatial population data are typically organized into nested hierarchies of areal units, in which each unit is a union of units at the next lower level. There is increasing interest in analyses at fine geographic detail, but these lowest rungs of the areal unit hierarchy are often incompletely tabulated because of cost, privacy, or other considerations. Here, the authors introduce a novel algorithm to impute crosstabs of up to three dimensions (e.g., race, ethnicity, and gender) from marginal data combined with data at higher levels of aggregation. This method exactly preserves the observed fine-grained marginals, while approximating higher-order correlations observed in more complete higher level data. The authors show how this approach can be used with U.S. census data via a case study involving differences in exposure to crime across demographic groups, showing that the imputation process introduces very little error into downstream analysis, while depicting social process at the more fine-grained level.

Aqueous two-phase systems coupled with chemometrics-enhanced HPLC-DAD for simultaneous extraction and determination of flavonoids in honey.

In this study, an accurate, rapid, green, and environment friendly method for the extraction and quantitative analysis of flavonoids in honey was established by using the aqueous two-phase extraction combined with the chemometrics-assisted HPLC-DAD. The first purpose of this study was to extract seven flavonoids in five different types of honey using alcohol/salt aqueous two-phase system (ATPS). The system with 2.82mL sodium citrate (30%), 1.58mL water, and 3.10mL isopropanol, showed the highest flavonoids extraction yields in the top phase (87.66-101.50%). Additionally, the three-way array of honey samples based on HPLC-DAD was decomposed mathematically by the alternating trilinear decomposition (ATLD) algorithm to obtain reasonable chromatograms, spectra, and concentration profiles for each analyte. Compared with the traditional solid-phase extraction method, the ATPS-ATLD-based method showed satisfactory spiked recoveries, lower limit of detection, and higher sensitivity, further verifying its accuracy and stability.

A multiway spectral analysis model to monitor the kinetic chlorination reaction of caffeine and determine its content in commercial liquid and solid drinks.

A multiway data analysis model, namely parallel factor analysis (PARAFAC) was proposed to decompose a three-way array of second-order kinetic UV measurements, for the chlorination reaction of caffeine with NaOCl, into a set of the spectra, time, and concentration matrices. The multiway resolution provided the simultaneous estimation of spectral, kinetic, and quantitative analysis of caffeine. The ability of the PARAFAC tool was checked by analyzing the validation samples in the presence of interferences. The added recovery and relative standard deviations for caffeine in the spiked samples were calculated as 99.1%-99.5% and 0.52%-1.34% for Iced Coffee Black liquid coffee (ICB), 99.5%-103.0% and 0.42%-1.03% for Jacobs Monarch Gold 100% Instant Coffee (JMG) and 99.5%-101.4% and 0.11%-0.13% for Çaykur Black Filter (Süzen) Bag Tea (BTB). Caffeine in commercial drinks was analyzed using the concentration matrices of the PARAFAC application. The PARAFAC results were statistically compared to those obtained by the developed UPLC method.

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.

Kernelizing: A way to increase accuracy in trilinear decomposition analysis of multiexponential signals

The unmixing of multiexponential decay signals into monoexponential components using soft modelling approaches is a challenging task due to the strong correlation and complete window overlap of the profiles. To solve this problem, slicing methodologies, such as PowerSlicing, tensorize the original data matrix into a three-way data array that can be decomposed based on trilinear models providing unique solutions. Satisfactory results have been reported for different types of data, e.g., nuclear magnetic resonance or time-resolved fluorescence spectra. However, when decay signals are described by only a few sampling (time) points, a significant degradation of the results can be observed in terms of accuracy and precision of the recovered profiles.In this work, we propose a methodology called Kernelizing that provides a more efficient way to tensorize data matrices of multiexponential decays. Kernelizing relies on the invariance of exponential decays, i.e., when convolving a monoexponential decaying function with any positive function of finite width (hereafter called “kernel”), the shape of the decay (determined by the characteristic decay constant) remains unchanged and only the preexponential factor varies. The way preexponential factors are affected across the sample and time modes is linear, and it only depends on the kernel used. Thus, using kernels of different shapes, a set of convolved curves can be obtained for every sample, and a three-way data array generated, for which the modes are sample, time and kernelizing effect. This three-way array can be afterwards analyzed by a trilinear decomposition method, such as PARAFAC-ALS, to resolve the underlying monoexponential profiles. To validate this new approach and assess its performance, we applied Kernelizing to simulated datasets, real time-resolved fluorescence spectra collected on mixtures of fluorophores and fluorescence-lifetime imaging microscopy data. When the measured multiexponential decays feature few sampling points (down to fifteen), more accurate trilinear model estimates are obtained than when using slicing methodologies.

Photoluminescence Probes in Data-Enabled Sensing.

This review summarizes the current status of development in photoluminescent probes, multidimensional photoluminescence detection, and multivariate data analysis methods. It then highlights reports featuring multivariate analysis of multidimensional measurements of photoluminescent probes published between June 2015 and June 2022, emphasizing work in the last 5 years. Important trends include the development of probe arrays, which provide fingerprint responses to the analyte(s) of interest and facilitate the analysis of complex samples; the application of neural networks and deep learning to pattern recognition and feature selection in photoluminescence images; and the application of multiway multivariate analysis to mining matrices, three-way arrays, and higher-order measurements, including hyperspectral intensity and lifetime images. These examples illustrate the increase in information extraction provided by the combination of multidimensional measurements and multivariate analysis.

Treatment effect identification using two-level designs with partially ignorable missing data

Missingness is often present in treatment effect inference studies with response datasets. However, the existing methods could not deal with the partially ignorable missingness (PIM) of response data from many time points. These data often have two characteristics: (a) missing in the framework of counterfactual outcomes; (b) missing responses arising from dropouts. One challenge is to identify the treatment effect in three-way arrays of outcomes with missing data.Thus, this paper proposes a novel method, PIM estimator (PIME), for inferring the differential effect of treatment. This method aims to estimate the average treatment effect using the three-way outcome array of a two-level design. This method can identify the selection bias and quantify it with the difference in outcome means between complete and incomplete cases. With the sensitivity analysis of dropout fractions, the results suggest there exists a minimum point of the upper bound of the difference estimation with the specified models. The results with real applications show that the proposed method is efficient for the treatment effect estimation. The proposed method was compared with the related methods (with dropouts imputed or removed). The results indicate that patients with higher blood glucose levels are more likely to drop out than the others.

A Note on Weaker Conditions for Identifying Restricted Latent Class Models for Binary Responses.

Restricted latent class models (RLCMs) are an important class of methods that provide researchers and practitioners in the educational, psychological, and behavioral sciences with fine-grained diagnostic information to guide interventions. Recent research established sufficient conditions for identifying RLCM parameters. A current challenge that limits widespread application of RLCMs is that existing identifiability conditions may be too restrictive for some practical settings. In this paper we establish a weaker condition for identifying RLCM parameters for multivariate binary data. Although the new results weaken identifiability conditions for general RLCMs, the new results do not relax existing necessary and sufficient conditions for the simpler DINA/DINO models. Theoretically, we introduce a new form of latent structure completeness, referred to as dyad-completeness, and prove identification by applying Kruskal's Theorem for the uniqueness of three-way arrays. The new condition is more likely satisfied in applied research, and the results provide researchers and test-developers with guidance for designing diagnostic instruments.

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.

Exploring dynamic metabolomics data with multiway data analysis: a simulation study

BackgroundAnalysis of dynamic metabolomics data holds the promise to improve our understanding of underlying mechanisms in metabolism. For example, it may detect changes in metabolism due to the onset of a disease. Dynamic or time-resolved metabolomics data can be arranged as a three-way array with entries organized according to a subjects mode, a metabolites mode and a time mode. While such time-evolving multiway data sets are increasingly collected, revealing the underlying mechanisms and their dynamics from such data remains challenging. For such data, one of the complexities is the presence of a superposition of several sources of variation: induced variation (due to experimental conditions or inborn errors), individual variation, and measurement error. Multiway data analysis (also known as tensor factorizations) has been successfully used in data mining to find the underlying patterns in multiway data. To explore the performance of multiway data analysis methods in terms of revealing the underlying mechanisms in dynamic metabolomics data, simulated data with known ground truth can be studied.ResultsWe focus on simulated data arising from different dynamic models of increasing complexity, i.e., a simple linear system, a yeast glycolysis model, and a human cholesterol model. We generate data with induced variation as well as individual variation. Systematic experiments are performed to demonstrate the advantages and limitations of multiway data analysis in analyzing such dynamic metabolomics data and their capacity to disentangle the different sources of variations. We choose to use simulations since we want to understand the capability of multiway data analysis methods which is facilitated by knowing the ground truth.ConclusionOur numerical experiments demonstrate that despite the increasing complexity of the studied dynamic metabolic models, tensor factorization methods CANDECOMP/PARAFAC(CP) and Parallel Profiles with Linear Dependences (Paralind) can disentangle the sources of variations and thereby reveal the underlying mechanisms and their dynamics.

Three-Way Analysis-Based pH-UV-Vis Spectroscopy for Quantifying Allura Red in An Energy Drink and Determining Colorant's pKa

Three-way analysis-based pH-UV-Vis spectroscopy was proposed for quantifying allura red in an energy drink product without the need for chromatographic analysis, and determining the colorant’s pKa without using any titration technique. In this study, UV-Vis spectroscopic data matrices were obtained from absorbance measurements at five different pH levels from pH 8 to pH 12 and arranged as a three-way array (wavelength × sample × pH). In the three-way analysis procedure, parallel factor analysis (PARAFAC) was implemented to decompose the three-way array into a set of trilinear components. Each set of three components relates to spectral, pH and relative concentration profiles of allura red and sample matrix in the energy drink. First, UV-Vis spectra of the colorant’s acid-base pair and sample’s matrix were characterized by using the estimated spectral profile. Then, from the pH profile the pKa value was found to be 11.28 for the related colorant. Finally, allura red in energy drink samples was determined using the estimated concentration curve in the relative concentration profile. In the quantitation procedure, the working concentration range was 0.8–19.2 μg/mL. PARAFAC approach was tested in terms of selectivity, precision, and accuracy of the method. Added recovery results obtained by applying the proposed method to spiked samples were between 101.5% and 103.5%. In the application of the method to the analysis of real samples, successful results were reported. For a comparison, an ultra-performance liquid chromatographic method was developed for the quantitation of the colorant. Compared to the chromatographic method, we observed that PARAFAC model was simple and less expensive without requiring separation.

Resolving isomers of nitrogen containing polycyclic aromatic compounds by travelling wave ion mobility spectrometry – Mass spectrometry and multiway curve resolution

Separation of nitrogen containing heterocyclic aromatic compounds in oil products is challenging due to the high number of structural isomers. Resolving these isomers is a major challenge in chromatography, even when combined with mass spectrometry (MS), because they have similar volatility, polarity, and mass spectra. Ion-mobility spectrometry (IMS) can address this problem by separating molecules according to their “bulkiness”. When used in combination with mass spectrometry on multiple samples, three-way arrays can be generated which can be modelled by a PARAFAC model to further enhance the resolution. In the three-way case, PARAFAC components cannot be uniquely resolved if the compounds they represent have the same profile in one of the modes. The PARALIND model – i.e., PARAFAC with Linear Dependencies, partly copes with this problem by introducing a dependency matrix. PARAFAC and PARALIND were tested to extract the signals of ten nitrogen containing heterocyclic aromatic compounds from IMS-MS data. The mixtures included three pairs of isomers, which could not be fully resolved by optimization of the ion mobility parameters. The results indicate that ion mobility combined with PARAFAC or PARALIND is a powerful complementary analysis platform that could aid the analysis of products containing a mixture of isomeric hetero-atomic compounds.

Multi-level data fusion strategies for modeling three-way electrophoresis capillary and fluorescence arrays enhancing geographical and grape variety classification of wines

Capillary electrophoresis with diode array detection (CE-DAD) and multidimensional fluorescence spectroscopy (EEM) second-order data were fused and chemometrically processed for geographical and grape variety classification of wines. Multi-levels data fusion strategies on three-way data were evaluated and compared revealing their advantages/disadvantages in the classification context. Straightforward approaches based on a series of data preprocessing and feature extraction steps were developed for each studied level. Partial least square discriminant analysis (PLS-DA) and its multi-way extension (NPLS-DA) were applied to CE-DAD, EEM and fused data matrices structured as two-way and three-way arrays, respectively. Classification results achieved on each model were evaluated through global indices such as average sensitivity non-error rate and average precision. Different degrees of improvement were observed comparing the fused matrix results with those obtained using a single one, clear benefits have been demonstrated when level of data fusion increases, achieving with the high-level strategy the best classification results.

A New Parameter for Measuring the Prediction Uncertainty Produced by Rotational Ambiguity in Second-Order Calibration with Multivariate Curve Resolution.

Multivariate curve resolution-alternating least-squares (MCR-ALS) is the model of choice when dealing with matrix data that cannot be arranged into a trilinear three-way array, that is, mostly from chromatographic origin with spectral detection. A range of feasible solutions may be found in MCR studies, due to the phenomenon of rotational ambiguity associated with bilinear decompositions of matrices. The application of chemically driven constraints is vital to achieving an adequate solution and minimizing the degree of rotational ambiguity present in the system. However, when studying complex samples, it may not be possible to recover unique solutions, even under the application of proper constraints. In such cases, it is important to be able to assess the propagation of rotation uncertainty to the estimated analyte concentrations, which stems from the existence of a finite range of feasible solutions. In this work, we present a new analytical parameter to estimate the potential uncertainty in analyte prediction brought about by rotational ambiguity, in the form of an associated root-mean-square error, named RMSERA. The proposed parameter comes in the form of a range of values, whose limits are δRA/(12)1/2 and δRA/(3)1/2, with δRA being defined as the difference between the maximum and minimum values of the analyte concentration that would be predicted by the MCR model from its concentration profiles lying in the range of feasible solutions, and corresponding to maximum and minimum area, respectively. We support our proposal on extensive simulations for systems of varying composition, and demonstrate its application on experimental data aimed at the determination of four pollutants in environmental water samples.

A novel bound constrained optimization method for three-way chemical data analysis with application to fluorescence excitation-emission matrix spectroscopy

There is a great deal of interests in multivariate approaches which may offer a necessary, and often sufficient, data analysis technique in many fields such as analytical chemistry, biology and environmental chemistry. However, few of these multivariate approaches have paid attention to the nonnegativity constraint in the decomposition process. In this paper, a novel bound constrained optimization method was proposed for three-way chemical data analysis. In this method, nonnegative matrix factorization was introduced to replace the traditional trilinear decomposition to provide the constraints on the nonnegative boundary on excitation-emission matrix spectra data. And the least-squares problem was transformed into a bound constrained optimization problem which can be solved by projected gradient methods. The alternating least squares were applied during each optimization iteration to obtain the individual components. Analysis of simulated three-way arrays indicated that the proposed method has a better performance than parallel factor analysis and alternating trilinear decomposition methods in nonnegativity. Experiments of real excitation-emission matrix spectra data also show that the proposed method is robust with the background interferences in practical applications.

A novel time–frequency–space method with parallel factor theory for big data analysis in condition monitoring of complex system

The spatial information of the signal is neglected by the conventional frequency/time decompositions such as the fast Fourier transformation, principal component analysis, and independent component analysis. Framing of the data being as a three-way array indexed by channel, frequency, and time allows the application of parallel factor analysis, which is known as a unique multi-way decomposition. The parallel factor analysis was used to decompose the wavelet transformed ongoing diagnostic channel–frequency–time signal and each atom is trilinearly decomposed into spatial, spectral, and temporal signature. The time–frequency–space characteristics of the single-source fault signal was extracted from the multi-source dynamic feature recognition of mechanical nonlinear multi-failure mode and the corresponding relationship between the nonlinear variable, multi-fault mode, and multi-source fault features in time, frequency, and space was obtained. In this article, a new method for the multi-fault condition monitoring of slurry pump based on parallel factor analysis and continuous wavelet transform was developed to meet the requirements of automatic monitoring and fault diagnosis of industrial process production lines. The multi-scale parallel factorization theory was studied and a three-dimensional time–frequency–space model reconstruction algorithm for multi-source feature factors that improves the accuracy of mechanical fault detection and intelligent levels was proposed.