Rotational Ambiguity Research Articles

Exploring the effects of sparsity constraint on the ranges of feasible solutions for resolution of GC-MS data

Many practical pattern recognition problems require non-negativity constraints. For example, pixels in digital images and chemical concentrations are non-negative. Sparse non-negative matrix factorizations (SNMFs) are useful when some degrees of sparseness exist in original data, intrinsically. The present contribution is about the implementation of sparsity constraint in multivariate curve resolution-alternating least square (MCR-ALS) technique for analysis of GC-MS/LC-MS data. The GC-MS and LC-MS data are sparse in mass dimension, and implementation of SNMF techniques would be useful for analyzing such two-way chromatographic data. In this work, L1-regularization paradigm has been implemented in each iteration of the MCR-ALS algorithm in order to force the algorithm to return more sparse mass spectra. L1-regularization has been applied by using the least absolute shrinkage and selection operator (Lasso) instead of the ordinary least square. A comprehensive comparison has been made between MCR-ALS and Lasso-MCR-ALS algorithms for resolution of the simulated and real GC-MS data. The comparison has been made by calculation of the values of sum of square errors (SSE) for 5000 times repetition of both algorithms using the random mass spectra and concentration profiles as initial estimates. The results revealed that regularization of L1-norm in mass dimension prevents occurrence of overfitting in ALS algorithm and this increases the probability of finding “true solution” after the resolution procedure. Moreover, the effect of this “sparsity constraint” has been explored on the area of feasible solutions in MCR methods. The results in this work revealed that implementation of this constraint reduces the extent of rotational ambiguity in MCR solutions and can be helpful for resolution of GC-MS data with high degrees of overlapping in mass spectra and concentration profiles.

Closure constraint in multivariate curve resolution

AbstractMultivariate curve resolution techniques try to estimate physically and/or chemically meaningful profiles underlying a set of chemical or related measurements. However, the estimation of profiles is not generally unique and it is often complicated by intensity and rotational ambiguities. Constraints as further information of chemical entities can be imposed to reduce the extent of ambiguities. Not only a long list of constraints has been introduced but also some of them can be applied in different ways. Either investigating constraint effects on the extent of rotational ambiguity or how they can be applied during curve resolution can shed light on curve resolution studies. The motivation behind this contribution is to pave the way to a clarification about the closure constraint. Considering simulated equilibrium and kinetic spectrophotometric data sets, different approaches to closure implementation were applied to demonstrate the geometrical explanation of closure constraint and its effect on multivariate curve resolution‐alternating least squares results. Besides, the closure constraint is compared with normalization and it is proved that the closure constraint is a Borgen norm and has the same effect as other Borgen norms in multivariate curve resolution. Finally, to further examine the closure constraint, a real data set was investigated.

Accurate anisotropy recovery from fluorophore mixtures using Multivariate Curve Resolution (MCR)

Anisotropy resolved multidimensional emission spectroscopy (ARMES) provides valuable insights into multi-fluorophore systems like proteins that have complex overlapping emission bands. The method combines multidimensional fluorescence, anisotropy, and chemometrics to facilitate the differentiation of fluorophores with very similar emission properties. Here, we address the critical issue of standardizing the chemometric methods required to accurately extract spectral and anisotropy information from fluorophore mixtures using two standard sample sets: perylene in glycerol, and a mixture of Erythrosin B and Phloxine B with overlapping emission but different anisotropies. We show for the first time how to accurately model component anisotropy using Multivariate Curve Resolution (MCR) from data collected using total synchronous fluorescence scan (TSFS) and Excitation Emission Matrix (EEM) measurement methods.These datasets were selected to avoid the presence of inner filter effects (IFE) or Förster resonance energy transfer (FRET) that would depolarize fluorescence emission or reduce data tri-linearity. This allowed the non-trilinear TSFS data to yield accurate component anisotropy data once modelled using the correct data augmentation strategy, however, the EEM data proved to be more accurate once optimal constraints (non-negativity and correspondence among species) were employed. For perylene (S2) and Phloxine B which both have very weak anisotropy (<0.06), while the spectral recovery was excellent, the modelled anisotropy values were reasonably accurate (±20% of the real value) because of large relative noise contributions. However, for perylene (S1) and Erythrosin B which have large (>0.2) anisotropies, bilinear and trilinear EEM models built using a total tri-linearity constraint, yielded solutions without any rotational ambiguities and very accurate (±4% of real value) anisotropy values. These sample systems thus provide simple and robust test systems for validating the spectral measurement and chemometric data analysis elements of ARMES.

Multivariate Analysis of Mixed Lipid Aggregate Phase Transitions Monitored Using Raman Spectroscopy.

The phase behavior of aqueous 1,2-dimyristoyl-sn-glycero-3-phosphorylcholine (DMPC)/1,2-dihexanoyl-sn-glycero-3-phosphocholine (DHPC) mixtures between 8.0 ℃ and 41.0 ℃ were monitored using Raman spectroscopy. Temperature-dependent Raman matrices were assembled from series of spectra and subjected to multivariate analysis. The consensus of pseudo-rank estimation results is that seven to eight components account for the temperature-dependent changes observed in the spectra. The spectra and temperature response profiles of the mixture components were resolved by applying a variant of the non-negative matrix factorization (NMF) algorithm described by Lee and Seung (1999). The rotational ambiguity of the data matrix was reduced by augmenting the original temperature-dependent spectral matrix with its cumulative counterpart, i.e., the matrix formed by successive integration of the spectra across the temperature index (columns). Successive rounds of constrained NMF were used to isolate component spectra from a significant fluorescence background. Five major components exhibiting varying degrees of gel and liquid crystalline lipid character were resolved. Hydrogen-bonded water networks exhibiting varying degrees of organization are associated with the lipid components. Spectral parameters were computed to compare the chain conformation, packing, and hydration indicated by the resolved spectra. Based on spectral features and relative amounts of the components observed, four components reflect long chain lipid response. The fifth component could reflect the response of the short chain lipid, DHPC, but there were no definitive spectral features confirming this assignment. A minor component of uncertain assignment that exhibits a striking response to the DMPC pre-transition and chain melting transition also was recovered. While none of the spectra resolved exhibit features unequivocally attributable to a specific aggregate morphology or step in the gelation process, the results are consistent with the evolution of mixed phase bicelles (nanodisks) and small amounts of worm-like DMPC/DHPC aggregates, and perhaps DHPC micelles, at low temperature to suspensions of branched and entangled worm-like aggregates above the DMPC gel phase transition and perforated multi-lamellar aggregates at high temperature.

On rotational ambiguity in parallel profiles with linear dependencies

Several curve resolution techniques have been developed to analyze higher‐order data structures, among which one can name parallel profiles with linear dependencies (PARALIND). The PARALIND is a variant of the Parallel Factor Analysis (PARAFAC) to cope with linear dependencies. Applying a PARAFAC model with different random initializations to a noisy three‐way data with linearly dependent loadings always converges to a best‐fitted noisy solution. Since PARAFAC maximally fits the noise part of the data, rank‐deficiency is broken, and it causes a noise‐induced unique decomposition. The uniqueness due to the presence of noise in rank‐deficient cases is an artificial phenomenon called “surface uniqueness.” The PARALIND model causes non‐uniqueness to be revealed, ie, it leads to a different set of profiles in each run. However, it might not give us all the possible solutions. Therefore, there is still a demand for determining all and every possible solution. The aim of this paper is 3‐fold: (1) A method is developed for determining feasible regions using the PARALIND model. (2) The surface uniqueness phenomenon is clarified. (3) The effect of interaction matrices in PARALIND as an active constraint is emphasized and discussed in detail using simulated noisy and noise‐free two‐ and three‐component data arrays.

The effect of data matrix augmentation and constraints in extended multivariate curve resolution–alternating least squares

The reliability of results obtained by multivariate curve resolution (MCR) methods is strongly dependent on the absence or presence of a small degree of rotational ambiguity associated to them. In this work, the effect of rotational ambiguities on the profiles resolved by MCR methods is examined in detail for cases of interest to analytical chemistry, where a number of calibration samples are usually prepared containing analyte standards, while test samples may contain additional uncalibrated constituents. These multiple chemical data sets having common constituents are simultaneously analyzed using matrix augmentation strategies. In these cases, conditions for better resolution and improved profiles are more easily achieved. To evaluate the extension of rotational ambiguities and to quantify their reduction after matrix augmentation, we applied the MCR‐BANDS procedure. Results obtained by the application of this procedure confirmed that the simultaneous analysis of multiple data sets decreased considerably the extension of rotational ambiguities compared with those obtained when only a single data set is analyzed. Simulated and experimental data sets of interest to second‐order analytical calibration are discussed.

Local Rank Deficiency Caused Problems in Analyzing Chemical Data.

Multivariate curve resolution (MCR) is a powerful methodology for analyzing chemical data in different application fields such as pharmaceutical analysis, agriculture, food chemistry, environment, and industrial and clinical chemistry. However, MCR results are often complicated by rotational ambiguity, meaning that there is a range of feasible solutions that fulfill the constraints and explain equally well the observed experimental data. Constraints determine the properties of resolved profiles in MCR methods by enforcing different assumptions on data. The applied constraints on chemical data sets should be derived from the physical nature and prior knowledge of the system under study. Therefore, the reliability of the constraints in order to get accurate results is a critical aspect that should be considered by analytical chemists who use MCR methods. Local rank information plays a key role in the curve resolution of multicomponent chemical systems. Applying the local rank constraint can reduce the extent of rotational ambiguity considerably, and in some cases, unique solutions can be achieved. Local rank exploratory methods like Evolving Factor Analysis (EFA) method provide local rank maps in order to obtain the presence pattern of components on the main assumption that the number of components in each window is equal to its rank. It is shown in this work that the local rank is a mathematical concept that may not be in concordance with chemical information. Thus, applying the local rank constraint for restricting the rotational ambiguity in MCR methods can lead to incorrect solutions! This problem is due to "local rank deficiency", which is introduced in this contribution.

Effect of adding variables on rotational ambiguity in positive matrix factorization solutions

A major problem that limit the use of the model-free analysis methods, like positive matrix factorization (PMF), for the resolution of multivariate data is that usually, there is rotational ambiguity in the results. Implementing adequate constraints may reduce such ambiguities. In some special cases, non-negativity constraints may serve as a sufficient condition to uniquely resolve profiles. In this study, effect of adding new variables on the extent of rotational ambiguity is investigated, and profiles with lower rotation have been obtained on the basis of data set structure. The displacement (DISP) method of error estimation (EE) for the factor elements incorporated into the US Environmental Protection Agency's version of positive matrix factorization (PMF) has been applied to capture the uncertainty of PMF analyses due to rotational ambiguity. To explore these considerations, results are presented for synthetic including submicron particle size distributions (12–470nm) measured between January 2008 and December 2010 at the New York State Department of Environmental Conservation (NYS DEC) site in Rochester, NY. Generally, added informative variables can reduce the ambiguity in the profiles. The results of the PMF analysis showed the effect of adding new variables to feasible solutions led to significantly decreased (up to 44%) DISP intervals. More robust results with lower rotational uncertainty became possible and the results were more easily interpreted. This approach can be applied to any other field that utilize quantitative factor analysis methods.

Error propagation along the different regions of multivariate curve resolution feasible solutions

Evaluation of uncertainties due to rotational ambiguities and noise propagation is essential to ascertain the reliability of Multivariate Curve Resolution estimations. When ambiguity is present, every resolved profile can be represented by a band of feasible solutions instead of by a unique profile. In the presence of experimental noise the estimation of this band of feasible solutions is more difficult and uncertain. The aim of this work is to show how experimental noise and profiles overlapping affect the reliability of feasible solutions. For this purpose, feasible solutions of several two and three components data systems with different profiles overlapping and noise levels have been systematically investigated. Results obtained using simulated and experimental data showed that increasing noise levels and profiles overlapping in one mode (e.g. concentration profiles) make more uncertain the estimation of feasible solutions in the other mode (e.g. spectral profiles), and this uncertainty is propagated in a non-uniform and complex way.

Study of Chemical Intermediates by Means of ATR-IR Spectroscopy and Hybrid Hard- and Soft-Modelling Multivariate Curve Resolution-Alternating Least Squares.

3,5-Diamino-1,2,4-triazole (DAT) became a significant energetic materials intermediate, and the study of its reaction mechanism has fundamental significance in chemistry. The aim of this study is to investigate the ability of online attenuated total reflection infrared (ATR-IR) spectroscopy combined with the novel approach of hybrid hard- and soft-modelling multivariate curve resolution-alternating least squares (HS-MCR) analysis to monitor and detect changes in structural properties of compound during 3,5-diamino-1,2,4-triazole (DAT) synthesis processes. The subspace comparison method (SCM) was used to obtain the principal components number, and then the pure IR spectra of each substance were obtained by independent component analysis (ICA) and HS-MCR. The extent of rotation ambiguity was estimated from the band boundaries of feasible solutions calculated using the MCR-BANDS procedure. There were five principal components including two intermediates in the process in the results. The reaction rate constants of DAT formation reaction were also obtained by HS-MCR. HS-MCR was used to analyze spectroscopy data in chemical synthesis process, which not only increase the information domain but also reduce the ambiguities of the obtained results. This study provides the theoretical basis for the optimization of synthesis process and technology of energetic materials and provides a strong technical support of research and development of energy material with extraordinary damage effects.

Unveiling multiple solid-state transitions in pharmaceutical solid dosage forms using multi-series hyperspectral imaging and different curve resolution approaches

Solid-state transitions at the surface of pharmaceutical solid dosage forms (SDF) were monitored using multi-series hyperspectral imaging (HSI) along with Multivariate Curve Resolution – Alternating Least Squares (MCR-ALS) and Parallel Factor Analysis (PARAFAC and PARAFAC2). First, the solid-state transformation due to the dehydration of the monohydrate forms of piroxicam and lactose to their respective anhydrate counterparts in tablets were monitored using temperature series NIR-HSI. PARAFAC and MCR-ALS solutions were hampered by the lack of strict trilinearity of the pixels among the unfolded series NIR-images (three-way array) and due to rotational ambiguity (augmented matrix), respectively, while PARAFAC2 resolved satisfactorily the profile of the corresponding compounds in the pixels in the series NIR-images. Next, the amorphous-to-crystalline transitions were monitored in solid dispersion of indomethacin with polyvinylpyrrolidone using time series MIR-HSI. MCR-ALS properly resolved the known solid-state forms of the drug in the pixels of the series MIR-images, while PARAFAC and PARAFAC2 failed to properly resolve all the drug forms in the series MIR-images due to i) strict trilinearity leak in the three-way array and ii) the mandatory constant cross-product AkTAk over the k series MIR-images (A is the loadings of the shift mode), respectively. The highlighting of the advantages and limitation of the corresponding curve resolution methods stressed their potential applicability when handling multi-series HSI to study solid-state transitions in pharmaceutical SDFs.

Assessment of PM2.5 sources and their corresponding level of uncertainty in a coastal urban area using EPA PMF 5.0 enhanced diagnostics

Datasets that include only the PM elemental composition and no other important constituents such as ions and OC, should be treated carefully when used for source apportionment. This work is demonstrating how a source apportionment study utilizing PMF 5.0 enhanced diagnostic tools can achieve an improved solution with documented levels of uncertainty for such a dataset. The uncertainty of the solution is rarely reported in source apportionment studies or it is reported partially. Reporting the uncertainty of the solution is very important especially in the case of small datasets. PM2.5 samples collected in Patras during the year 2011 were used. The concentrations of 22 elements (Z=11–33) were determined using PIXE. Source apportionment analysis revealed that PM2.5 emission sources were biomass burning (11%), sea salt (8%), shipping emissions (10%), vehicle emissions (33%), mineral dust (2%) and secondary sulfates (33%) while unaccounted mass was 3%. Although Patras city center is located in a very close proximity to the city's harbor, the contribution of shipping originating emissions was never before quantified. As rotational stability is hard to be achieved when a small dataset is used the rotational stability of the solution was thoroughly evaluated. A number of constraints were applied to the solution in order to reduce rotational ambiguity.

Multivariate resolution of flattened absorption spectra with weighted least squares; weight selection and rotational ambiguity

Multivariate curve resolution (MCR) of absorption spectra is now a ubiquitously used tool. However, MCR methods, which use ordinary least squares (OLS) approach, assume that the measurement uncertainties are unbiased and homoscedastic. This is not true for absorption measurements, in which uncertainty variance and bias both increase as the true absorbance increases. The bias produces a well‐known flattening/saturation of the peaks at high optical densities, which makes the data nonlinear and unsuitable for OLS‐based MCR analysis. This problem can be reduced by using weighted least squares (WLS).In the present paper, the ability of WLS‐based MCR to handle simulated and real datasets with realistic optical noise and flattening was assessed. Three weighting schemes were tested: OLS (unity weights), weights based on the maximum likelihood principle (MLP) and the physics of absorption measurement, and weights based on empirical cutoff (zero weights for saturated data points). The abilities of MCR to recover the true profiles and to evaluate rotational ambiguity of the solutions were compared for the 3 weighting schemes. MLP‐ and cutoff‐based WLS‐MCR produced better resolution of flattened data than OLS, but the success of the extension to strongly flattened spectra depended on data structure. MLP‐based MCR was general and stable, while cutoff‐based MCR was more sensitive to the data but could recover unbiased profiles. Generally, the use of WLS can expand MCR functionality to the analysis of flattened spectra.The specifics of finding WLS bilinear solutions and approaches to migrate factor‐based MCR methods from OLS to WLS are also discussed.

On the ambiguity of the reaction rate constants in multivariate curve resolution for reversible first-order reaction systems

If for a chemical reaction with a known reaction mechanism the concentration profiles are accessible only for certain species, e.g. only for the main product, then often the reaction rate constants cannot uniquely be determined from the concentration data. This is a well-known fact which includes the so-called slow-fast ambiguity.This work combines the question of unique or non-unique reaction rate constants with factor analytic methods of chemometrics. The idea is to reduce the rotational ambiguity of pure component factorizations by considering only those concentration factors which are possible solutions of the kinetic equations for a properly adapted set of reaction rate constants. The resulting set of reaction rate constants corresponds to those solutions of the rate equations which appear as feasible factors in a pure component factorization.The new analysis of the ambiguity of reaction rate constants extends recent research activities on the Area of Feasible Solutions (AFS). The consistency with a given chemical reaction scheme is shown to be a valuable tool in order to reduce the AFS. The new methods are applied to model and experimental data.

Investigating the effect of flexible constraints on the accuracy of self‐modeling curve resolution methods in the presence of perturbations

Self‐modeling curve resolution methods have continuously been improved during recent years. Many efforts have been made on curve resolution methods to reduce the rotational ambiguity by means of different types of constraints. Choosing proper constraints and cost functions is critically important for the reduction of the rotational ambiguity because the constraints have a direct influence on the accuracy of the area of feasible solution (AFS).In this work, we introduce a new improved cost function, which serves to apply nonnegativity, unimodality, equality, and closure constraints. We also investigate the reduction of the AFS under hard and soft constraints. Another point of this work is to evaluate the accuracy and precision of the reduced AFS in the presence of noise and perturbations, under hard and soft implementation of nonnegativity, unimodality, equality, and closure constraints. A comparison is given between the reduced AFS with soft constraints (small deviations from constraints are accepted) and the reduced AFS under hard constraints (restrictedly forced constraints). A graphical visualization of this comparison is presented for various model problems. The results show that an AFS computation with soft constraints provides more reliable results, especially in the presence of noise. The test problems substantiate significant advantages of soft constraints over hard constraints because the obtained profiles are closer to the potentially true noisy profiles, which contain small deviations from ideal responses. Using tunable parameters ϵ,γ,ω,δ is one of the advantages of soft constrained cost function that allows the small deviations from ideal responses. Ultimately, soft constraints can help to reduce the lack‐of‐fit, and they are a proper instrument to handle the effect of noise on the AFS. Copyright © 2016 John Wiley &amp; Sons, Ltd.

PERSISTENT ASYMMETRIC STRUCTURE OF SAGITTARIUS A* ON EVENT HORIZON SCALES

ABSTRACT The Galactic Center black hole Sagittarius A* (Sgr A*) is a prime observing target for the Event Horizon Telescope (EHT), which can resolve the 1.3 mm emission from this source on angular scales comparable to that of the general relativistic shadow. Previous EHT observations have used visibility amplitudes to infer the morphology of the millimeter-wavelength emission. Potentially much richer source information is contained in the phases. We report on 1.3 mm phase information on Sgr A* obtained with the EHT on a total of 13 observing nights over four years. Closure phases, which are the sum of visibility phases along a closed triangle of interferometer baselines, are used because they are robust against phase corruptions introduced by instrumentation and the rapidly variable atmosphere. The median closure phase on a triangle including telescopes in California, Hawaii, and Arizona is nonzero. This result conclusively demonstrates that the millimeter emission is asymmetric on scales of a few Schwarzschild radii and can be used to break 180° rotational ambiguities inherent from amplitude data alone. The stability of the sign of the closure phase over most observing nights indicates persistent asymmetry in the image of Sgr A* that is not obscured by refraction due to interstellar electrons along the line of sight.

A systematic study on the effects of multi-set data analysis on the range of feasible solutions

The objective of self modeling curve resolution (SMCR) methods is to decompose a second-order bilinear data matrix into a range of chemically meaningful matrices without any knowledge about the chemical or physical model describing the considered system. In addition, SMCR methods are efficient approaches to deeply investigate data structures by finding not only one of the solutions but all possible ones.Multi-set data analysis can be a powerful tool to decrease the range of feasible solutions in the absence of appropriate conditions for unique resolution. Using SMCR methods, we have investigated the impact of multi-set data analysis on the accuracy of soft modeling results. Interestingly, the feasible regions of individual and simultaneous analysis are compared in a common abstract space. It is demonstrated how such global analysis can result in the reduction of rotational ambiguity in soft modeling analysis. Moreover, as a systematic study, different factors are considered in order to discover the advantages and limitations of multi-set data analysis and lead to a proper design for more accurate results.

Resolving of challenging gas chromatography–mass spectrometry peak clusters in fragrance samples using multicomponent factorization approaches based on polygon inflation algorithm

Analysis of fragrance composition is very important for both the fragrance producers and consumers. Unraveling of fragrance formulation is necessary for quality control, competitor and trace analysis. Gas chromatography–mass spectrometry (GC–MS) has been introduced as the most appropriate analytical technique for this type of analysis, which is based on Kovats index and MS database. The most straightforward method to analyze a GC–MS dataset is to integrate those peaks that can be recognized by their mass profiles. But, because of common problems of chromatographic data such as spectral background, baseline offset and specially overlapped peaks, accurate quantitative and qualitative analysis could be failed. Some chemometric modeling techniques such as bilinear multivariate curve resolution (MCR) methods have been introduced to overcome these problems and obtained well resolved chromatographic profiles. The main drawback of these methods is rotational ambiguity or nonunique solution that is represented as area of feasible solutions (AFS).Polygonal inflation algorithm (PIA) is an automatic and simple to use algorithm for numerical computation of AFS. In this study, the extent of rotational ambiguity in curve resolution methods is calculated by MCR-BAND toolbox and the PIA. The ability of the PIA in resolving GC–MS data sets is evaluated by simulated GC–MS data in comparison with other popular curve resolution methods such as multivariate curve resolution alternative least square (MCR-ALS), multivariate curve resolution objective function minimization (MCR-FMIN) by different initial estimation methods and independent component analysis (ICA). In addition, two typical challenging area of total ion chromatogram (TIC) of commercial fragrances with overlapped peaks were analyzed by the PIA to investigate the possibility of peak deconvolution analysis.

Soft constraints for reducing the intrinsic rotational ambiguity of the area of feasible solutions

The reduction of the rotational ambiguity in multivariate curve resolution problems is a central challenge in order to construct an effective chemometric method. Soft modeling is a method of choice to solve this problem.The aim of this paper is to demonstrate the impact of soft constraints on the full set of all feasible, nonnegative solutions. To this end the starting point is the Area of Feasible Solutions (AFS) for a three-component system. Then soft constraints, namely constraints on the unimodality, monotonicity and windowing for certain concentration profiles, are used in order to reduce the AFS. This process extracts chemically meaningful solutions from the set of all feasible nonnegative factors and demonstrates the mode of action of soft constraints. Results are presented for a model problem as well as for FT-IR data for a catalytic subsystem of the rhodium-catalyzed hydroformylation process. Typically, the AFS can significantly be reduced by adding soft constraints.

Implementing constrained multi-time approach with bootstrap analysis in ME-2: An application to PM2.5 data from Florence (Italy)

Advanced receptor models have been recently developed and tested in order to improve the resolution of apportionment problems reducing rotational ambiguity of results and aiming at identifying a larger number of sources. In particular, multi-time model is a factor analysis method able to compute source profiles and contributions using aerosol compositional data with different time resolutions. Unlike traditional factor analysis, each measured value can be inserted into multi-time model with its original time schedule, thus all temporal information can be effectively used in the modelling process. In this work, multi-time model was expanded in order to impose constraints on modelled factors aiming at improving the source identification. Moreover, as far as we know for the first time, a suitable bootstrap technique was implemented in the multi-time scheme to estimate the uncertainty of the final constrained solutions. These implemented approaches were tested on a PM2.5 (particulate matter with aerodynamic diameter <2.5μm) dataset composed of 24-h samples collected during one year and hourly data sampled in parallel for two shorter periods in Florence (Italy). The daily samples were chemically characterised for elements, ions and carbonaceous components while elemental concentrations only were available for high-time resolved samples. The application of the advanced model revealed the major contribution from traffic (accounting for 37% of PM2.5 as annual average) and allowed an accurate characterisation of involved emission processes. In particular, exhaust and non-exhaust emissions were identified. The constraints imposed in the continuation run led to a better description of the factor associated to nitrates and also of biomass burning profile and the bootstrap results gave useful information to assess the reliability of source apportionment solutions. Finally, the comparison with the results computed by ME-2 base model applied to daily and hourly compositional data separately demonstrated the advantages provided by the multi-time approach.