Curve Resolution Methods Research Articles

Independent components analysis as a means to have initial estimates for multivariate curve resolution-alternating least squares

Multivariate Curve Resolution with Alternating Least Squares (MCR-ALS) is a curve resolution method based on a bilinear model which assumes that the observed spectra are a linear combination of the spectra of the pure components in the system. The algorithm steps include the determination of the number of components by rank analysis methods, initial estimates for the concentrations and/or spectra and an iterative optimization. Sometimes, suitable results may not be achieved when MCR-ALS is applied. One reason for this is the importance of the initial estimates of the spectral profiles. In that case, the MCR-ALS algorithm may reach a local minimum instead of a global minimum and this can result in ineffective curve resolution. The most popular algorithm used to find the initial estimates (PURE derived from SIMPLISMA) suffers from an essential drawback, which is the necessity to have “pure” variables related to a single spectral component, which cannot be expected in all cases because of the strong signal overlapping as in the Ultraviolet–Visible (UV–Vis) spectroscopy. This work summarizes this problem, presenting a case study based on UV–Vis spectroscopy of heated olive oil. To solve the problems of the need for “pure” variables and to avoid local minima with MCR-ALS, Independent Components Analysis (ICA) was used to calculate initial estimates for MCR-ALS. The results from this study suggest that this use of ICA prior to MCR-ALS improves the resolution for UV–Vis data and provides acceptable resolution results when compared to the most used method, PURE.

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.

Peak group analysis for the extraction of pure component spectra

Structure elucidation for the reactive or catalytic specie s of a chemical reaction system can significantly be supporte d by spectroscopic measurements. If the spectroscopic data contains isolated signals or groups of partially separated peaks, then the identification of correlations between thes e peaks can help to determine the pure components by their functional groups. A computational method is presented which constructs from a certain frequency window, which contains a single peak or a peak group, an associated pure component spectrum on the full frequency range. This global spectrum reproduces the spectrum in the local frequency window or, at least, reproduces the contribution from the dominant component in the local window. The method is called the Peak Group Analysis (PGA). The methodological background of the PGA are a multivariate curve resolution method and the solution of a minimization problem with weighted soft constraints. The method is tested for two experimental FT-IR data sets from investigations into equilibria of hydroformylation catalysts based on rhodium and iridium. An implementation of the PGA is presented as a part of the FACPACK software.

“Parallel factor analysis of multi-excitation ultraviolet resonance Raman spectra for protein secondary structure determination”

Protein secondary structural analysis is important for understanding the relationship between protein structure and function, or more importantly how changes in structure relate to loss of function. The structurally sensitive protein vibrational modes (amide I, II, III and S) in deep-ultraviolet resonance Raman (DUVRR) spectra resulting from the backbone C–O and N–H vibrations make DUVRR a potentially powerful tool for studying secondary structure changes. Experimental studies reveal that the position and intensity of the four amide modes in DUVRR spectra of proteins are largely correlated with the varying fractions of α-helix, β-sheet and disordered structural content of proteins. Employing multivariate calibration methods and DUVRR spectra of globular proteins with varying structural compositions, the secondary structure of a protein with unknown structure can be predicted. A disadvantage of multivariate calibration methods is the requirement of known concentration or spectral profiles. Second-order curve resolution methods, such as parallel factor analysis (PARAFAC), do not have such a requirement due to the “second-order advantage.” An exceptional feature of DUVRR spectroscopy is that DUVRR spectra are linearly dependent on both excitation wavelength and secondary structure composition. Thus, higher order data can be created by combining protein DUVRR spectra of several proteins collected at multiple excitation wavelengths to give multi-excitation ultraviolet resonance Raman data (ME-UVRR). PARAFAC has been used to analyze ME-UVRR data of nine proteins to resolve the pure spectral, excitation and compositional profiles. A three factor model with non-negativity constraints produced three unique factors that were correlated with the relative abundance of helical, β-sheet and poly-proline II dihedral angles. This is the first empirical evidence that the typically resolved “disordered” spectrum represents the better defined poly-proline II type structure.

A multiresolution approach for the convergence acceleration of multivariate curve resolution methods

Modern computerized spectroscopic instrumentation can result in high volumes of spectroscopic data. Such accurate measurements rise special computational challenges for multivariate curve resolution techniques since pure component factorizations are often solved via constrained minimization problems. The computational costs for these calculations rapidly grow with an increased time or frequency resolution of the spectral measurements.The key idea of this paper is to define for the given high-dimensional spectroscopic data a sequence of coarsened subproblems with reduced resolutions. The multiresolution algorithm first computes a pure component factorization for the coarsest problem with the lowest resolution. Then the factorization results are used as initial values for the next problem with a higher resolution. Good initial values result in a fast solution on the next refined level. This procedure is repeated and finally a factorization is determined for the highest level of resolution. The described multiresolution approach allows a considerable convergence acceleration. The computational procedure is analyzed and is tested for experimental spectroscopic data from the rhodium-catalyzed hydroformylation together with various soft and hard models.

Local rank-based spatial information for improvement of remote sensing hyperspectral imaging resolution

This paper shows the effect of using local rank and selectivity constraints based on spatial information of spectroscopic images to increase the performance of Multivariate Curve Resolution (MCR) methods and to decrease the ambiguity of final results. Fixed Size Image Window-Evolving Factor Analysis (FSIW-EFA) is applied to discover which pixels are more suitable for the application of local rank constraints. An automated method to help in setting appropriate threshold values for the application of FSIW-EFA, based on global and local use of Singular Value Decomposition (SVD) is proposed. Additional use of correlation coefficients between selected reference spectra and pixel spectra of the image is shown to provide an alternative way for the application of the selectivity constraint in spectroscopic images for the first time. This alternative method resulted to be satisfactory when pure pixels exist.

Pushing back the limits of Raman imaging by coupling super-resolution and chemometrics for aerosols characterization.

The increasing interest in nanoscience in many research fields like physics, chemistry, and biology, including the environmental fate of the produced nano-objects, requires instrumental improvements to address the sub-micrometric analysis challenges. The originality of our approach is to use both the super-resolution concept and multivariate curve resolution (MCR-ALS) algorithm in confocal Raman imaging to surmount its instrumental limits and to characterize chemical components of atmospheric aerosols at the level of the individual particles. We demonstrate the possibility to go beyond the diffraction limit with this algorithmic approach. Indeed, the spatial resolution is improved by 65% to achieve 200 nm for the considered far-field spectrophotometer. A multivariate curve resolution method is then coupled with super-resolution in order to explore the heterogeneous structure of submicron particles for describing physical and chemical processes that may occur in the atmosphere. The proposed methodology provides new tools for sub-micron characterization of heterogeneous samples using far-field (i.e. conventional) Raman imaging spectrometer.

Application of Multivariate Curve Resolution Approaches to Improve Analytical Separation of Iranian Rice Volatiles by GC-MS

Abstract Multivariate curve resolution (MCR) methods are proposed to improve the analysis of volatile components of an Iranian rice sample (Domsiah) by headspace sampling coupled to gas chromatography-mass spectrometry. A total of 31 components was identified using similarity searches between mass spectra and mass spectrometry database. Then extended multivariate curve resolution-alternating least squares (MCR-ALS) by applying proper constraints is used to obtain pure elution and mass spectral profiles for the components present in each peak cluster. The resolved components were identified by similarity searches using the NIST mass database and their percentage were calculated for qualitative and quantitative analysis, respectively. The number of identified components was extended to 82 by applying the MCR-ALS techniques. The main identified volatile chemicals that were found in an Iranian rice sample (Domsiah) were; nonanal (18.22%), hexanal (14.33%), 2-benzoyl-6,7-dimethoxy-4-methylidene-2H-1,3-benzothiazine (7.81%), epilophodione (7.023%), and 1-methyl-4-(1-methylethenyl)cyclohexene (6.61%).

Data-driven signal-resolving approaches of infrared spectra to explore the macroscopic and microscopic spatial distribution of organic and inorganic compounds in plant.

The nondestructive and label-free infrared (IR) spectroscopy is a direct tool to characterize the spatial distribution of organic and inorganic compounds in plant. Since plant samples are usually complex mixtures, signal-resolving methods are necessary to find the spectral features of compounds of interest in the signal-overlapped IR spectra. In this research, two approaches using existing data-driven signal-resolving methods are proposed to interpret the IR spectra of plant samples. If the number of spectra is small, "tri-step identification" can enhance the spectral resolution to separate and identify the overlapped bands. First, the envelope bands of the original spectrum are interpreted according to the spectra-structure correlations. Then the spectrum is differentiated to resolve the underlying peaks in each envelope band. Finally, two-dimensional correlation spectroscopy is used to enhance the spectral resolution further. For a large number of spectra, "tri-step decomposition" can resolve the spectra by multivariate methods to obtain the structural and semi-quantitative information about the chemical components. Principal component analysis is used first to explore the existing signal types without any prior knowledge. Then the spectra are decomposed by self-modeling curve resolution methods to estimate the spectra and contents of significant chemical components. At last, targeted methods such as partial least squares target can explore the content profiles of specific components sensitively. As an example, the macroscopic and microscopic distribution of eugenol and calcium oxalate in the bud of clove is studied.

Near and mid infrared spectroscopy and multivariate data analysis in studies of oxidation of edible oils

Infrared spectroscopic techniques and chemometric methods were used to study oxidation of olive, sunflower and rapeseed oils. Accelerated oxidative degradation of oils at 60°C was monitored using peroxide values and FT-MIR ATR and FT-NIR transmittance spectroscopy. Principal component analysis (PCA) facilitated visualization and interpretation of spectral changes occurring during oxidation. Multivariate curve resolution (MCR) method found three spectral components in the NIR and MIR spectral matrix, corresponding to the oxidation products, and saturated and unsaturated structures. Good quantitative relation was found between peroxide value and contribution of oxidation products evaluated using MCR – based on NIR (R2=0.890), MIR (R2=0.707) and combined NIR and MIR (R2=0.747) data. Calibration models for prediction peroxide value established using partial least squares (PLS) regression were characterized for MIR (R2=0.701, RPD=1.7), NIR (R2=0.970, RPD=5.3), and combined NIR and MIR data (R2=0.954, RPD=3.1).

Thermodynamic model and Raman spectra of CaO–P2O5 glasses

Raman spectra of xZnO (100−x)P2O5 (x = 35, 40, 45, 50, 55) glasses were described by the thermodynamic model of Shakhmatkin and Vedishcheva using the method of Malfait. The glasses were considered as the ideal solution of unreacted oxides and four compounds (ZnP2O6, Zn2P2O7, Zn3P2O8, and ZnP2O11). Due to missing thermodynamic data, thermodynamic model was based on the estimates of reaction Gibbs energies of formation of considered compounds obtained by fitting the 31P MAS NMR Q-distribution in ZnO–MoO3–P2O5 glasses. Simultaneously, the set of Raman spectra was analyzed by multivariate curve resolution method. The very good coincidence between the results of spectral decomposition based on the thermodynamic model on one side and on the results of MCR method on the other can be considered as plausible validation of the thermodynamic data used, i.e., as a general guideline for the situations where the thermodynamic data needed for the construction of Shakhmatkin and Vedishcheva thermodynamic model are missing but the detailed structural information is available.

Spectral Resolving of Deep Raman Spectra of PMMA–trans‐Stilbene Bilayer with Generalized 2D Correlation Spectroscopy#

Spatially offset Raman spectroscopy of bilayer analytes results in complicated spectra from each component. To separate the respective spectrum, we applied generalized two‐dimensional correlation spectroscopy (2DCOS) as a curve resolution method in comparison with principal component analysis and the multiple curve resolution–alternating least square method. Our results show that composite Raman spectra of analytes are resolved more clearly using 2DCOS than the other methods. We propose that 2DCOS improves the resolving capability using the out‐of‐phase information of spectral intensity variation as a function of offset.

Elimination of chromatographic and mass spectrometric problems in GC-MS analysis of Lavender essential oil by multivariate curve resolution techniques: Improving the peak purity assessment by variable size moving window-evolving factor analysis.

In analysis of complex natural matrices by gas chromatography-mass spectrometry (GC-MS), many disturbing factors such as baseline drift, spectral background, homoscedastic and heteroscedastic noise, peak shape deformation (non-Gaussian peaks), low S/N ratio and co-elution (overlapped and/or embedded peaks) lead the researchers to handle them to serve time, money and experimental efforts. This study aimed to improve the GC-MS analysis of complex natural matrices utilizing multivariate curve resolution (MCR) methods. In addition, to assess the peak purity of the two-dimensional data, a method called variable size moving window-evolving factor analysis (VSMW-EFA) is introduced and examined. The proposed methodology was applied to the GC-MS analysis of Iranian Lavender essential oil, which resulted in extending the number of identified constituents from 56 to 143 components. It was found that the most abundant constituents of the Iranian Lavender essential oil are α-pinene (16.51%), camphor (10.20%), 1,8-cineole (9.50%), bornyl acetate (8.11%) and camphene (6.50%). This indicates that the Iranian type Lavender contains a relatively high percentage of α-pinene. Comparison of different types of Lavender essential oils showed the composition similarity between Iranian and Italian (Sardinia Island) Lavenders.

Determination of6Li and7Li abundances by inductively coupled plasma-optical emission spectrometry combined with derivative spectroscopy, multivariate curve resolution and multivariate calibration methods

The development of a rapid method for the determination of6Li and7Li abundances by ICP-OES coupled with computational methods.

Definition and detection of data-based uniqueness in evaluating bilinear (two-way) chemical measurements

Multivariate curve resolution methods, frequently used in analyzing bilinear data sets, result in ambiguous decomposition in general. Implementing the adequate constraints may lead to reduce the so-called rotational ambiguity drastically, and in the most favorable cases to the unique solution. However, in some special cases, non-negativity constraint as minimal information of the system is a sufficient condition to resolve profiles uniquely. Although, several studies on exploring the uniqueness of the bilinear non-negatively constrained multivariate curve resolution methods have been made in the literature, it has still remained a mysterious question. In 1995, Manne published his profile-based theorems giving the necessary and sufficient conditions of the unique resolution. In this study, a new term, i.e., data-based uniqueness is defined and investigated in details, and a general procedure is suggested for detection of uniquely recovered profile(s) on the basis of data set structure in the abstract space. Close inspection of Borgen plots of these data sets leads to realize the comprehensive information of local rank, and these argumentations furnish a basis for data-based uniqueness theorem. The reported phenomenon and its exploration is a new stage (it can be said fundament) in understanding and describing the bilinear (matrix-type) chemical data in general.Our proposed detection tool is restricted to three-component systems because of the visual limitations of the Borgen plot, but the theorem is universal for systems with more than three components. A recently published experimental four-component system is used for illustrating this theorem in the case of systems with more than three components.

Phase identification and interfacial transitions in ternary polymer blends by ToF-SIMS

Phase identification and the study of the interphase region in multi-component polymer blends with a chemically similar structure using conventional techniques is a challenge. In this work, the detailed morphological analysis of such systems is examined. A ternary blend comprised of poly butylene succinate (PBS); poly lactic acid (PLA); and polycaprolactone (PCL) with a partial wetting morphology is carefully selected since all three components are polyesters with low interfacial tensions. It will be shown that a novel technique by applying multivariate analysis (MVA) on time-of-flight secondary ion mass spectrometry (ToF-SIMS) data can effectively identify the complex phase structure, especially in blends with chemically similar components. Furthermore, for the first time for such systems, this technique provides detailed information about interfacial thicknesses and transitions. By employing the principal component analysis (PCA) method on the ToF-SIMS data of pure polymers, specific peaks with a certain molecular ion mass related to each polymer are determined. Using overlaid mappings on the surface of the blend by ToF-SIMS and selected ion masses to identify each polymer results in the differentiation of the various phases represented as a morphological image. In a second step, the multivariate curve resolution (MCR) method is used as a “self modeling curve resolution” for the recovery of pure components from a multi-component mixture when little or no prior information is available. Total pseudo-color RGB images of PBS/PLA/PCL show that PLA droplets unambiguously partially wet the PBS and PCL phases. Since each pixel from the analysis in the high lateral resolution image represents a 200 nm diameter, the interfacial transitions can also be studied for both PLA/PBS and PLA/PCL interfaces. The results show the concentration variation of phases across the interfaces. A complete trace line across the two interfaces (PLA/PBS and PLA/PCL) allows for the quantitative determination of interfacial thickness for the first time for such systems.

XRPD to Characterize Disordered and Amorphous Pharmaceutical Samples

X-ray powder diffraction is commonly used to characterize pharmaceutical samples to identify crystalline forms as well as differentiate crystalline vs amorphous materials. Using XRPD for amorphous materials is usually limited to determining if the materials are fully amorphous or a mixture of crystalline and amorphous materials. Changes in the amorphous halos under various conditions (grinding, drying, water sorption, etc) are usually not investigated, but can give information on the system. This presentation will discuss other uses of XRPD in characterizing amorphous and disordered systems, including amorphous solid dispersions. Patterns involving nanocrystalline, amorphous, and defected samples will be presented and examples on using various methods to obtain further information from the data (pair distribution functions, pure curve resolution methods, diffuse scattering, and Rietveld analysis) will be included.

Infrared spectroscopy and multivariate analysis to appraise α-cellulose extracted from wood for stable carbon isotope measurements

Wood is a heterogeneous material mainly constituted of cellulose, hemicelluloses, lignin, and extractives which all have different isotopic signatures. For applications where δ13C measurement of α-cellulose is required, it is important that residues of the other constituents remain below an acceptable level. The laboratory method most widely used for the extraction of α-cellulose consists of several successive treatments: organic solvents and boiled water, acidified sodium chlorite (NaClO2) application, and finally a NaOH treatment. Different variants of this method were tested systematically to optimize the extraction of α-cellulose from wood samples of four tree species: oak, beech, poplar, and pine. Mid-infrared spectroscopy in Attenuated Total Reflection mode (IR-ATR) combined with a curve resolution method (Bayesian Positive Sources Separation statistical analysis) was used to monitor the residues of other wood constituents in extracted α-cellulose. IR-ATR spectroscopy was shown to be sensitive enough to detect residual compounds in α-cellulose extracts below a concentration which does not present a measurable bias for δ13C measurements. For all tree species, a residual concentration of lignin below the bias threshold for δ13C measurements was reached with fewer additions of acidic NaClO2 than usually reported. Further the NaOH treatment step was not necessary to remove the hemicelluloses from oak and beech. Infrared spectroscopy combined with a curve resolution method is appropriate to improve α-cellulose extraction species-specifically for reliable stable carbon isotope δ13C measurements. It allows to check the extracted α-cellulose and to reduce the consumption of chemicals, the extraction time and the loss of α-cellulose.

Multivariate curve resolution of co-eluting vapors from a gas chromatograph with microsensor array detector

The application of a hybrid multivariate curve resolution method, which combines evolving factor analysis (EFA) with alternating least squares (ALS), to the analysis of partially overlapping peaks from vapors measured by a microsensor-array gas chromatograph detector is described. The detector comprised an array of four chemiresistors coated with different sorptive thiolate-monolayer-protected gold nanoparticle (MPN) films. Three pairs of vapors, the members of which had array response pattern correlation coefficients, ρ, ranging from −0.57 to 0.85, were tested at different values of chromatographic resolution, Rs, and relative response ratio, RRR. Composite responses were equivalent to the sums of the responses to the individual components, but differences in peak asymmetry among the sensors in the array led to pattern distortions across the spans of all peaks. With data pre-processing to account for the latter, EFA correctly determined the chemical ranks of the binary composite peaks in 57 of the 63 cases (90%), with most errors observed for the most highly correlated pair. By using calibrated response patterns as inputs for the ALS refinements of EFA-extracted responses, the fidelity of recovered response patterns and elution profiles was sufficiently high to differentiate the composite peak components in 124 of 126 cases (98%) and to quantify them to within ±30% of actual values in 95 of 126 (75%) cases. Without such inputs, the corresponding rates were 112 of 126 (89%) and 68 of 126 (54%), respectively. In general, the RRR value was a more important determinant of performance than was the Rs value. The methodology and performance of EFA-ALS in this application are critically assessed.

A fast polygon inflation algorithm to compute the area of feasible solutions for three‐component systems. II: Theoretical foundation, inverse polygon inflation, and FAC‐PACK implementation

The area of feasible solutions (AFS) of a multivariate curve resolution method is the continuum of feasible solutions under the given constraints. In the current paper, the AFS is computed only on the condition of nonnegative solutions. This work is a continuation of a paper (J. Chemometrics 28:106–116, 2013) on the polygon inflation algorithm for AFS computations. In this second part, various properties of the AFS are analyzed. First, its boundedness is proved, which is a necessary condition for its numerical computation. Second, it is shown that the origin is never contained in the area of feasible solutions. This fact is the basis for the inverse polygon inflation algorithm, which allows to compute specific types of an AFS containing a hole.The numerical computation of the AFS is a complicated and computationally expensive process. The construction of proper objective functions for the AFS optimization problem appears to be decisive. The paper contains a comparative analysis of two objective functions and describes the ideas of the new FAC‐PACK toolbox for MATLAB. This freely available toolbox contains a numerical implementation of the polygon inflation and of the inverse polygon inflation algorithm. Copyright © 2014 John Wiley & Sons, Ltd.