Search

This paper describes how data from two instrumental techniques - near infrared spectroscopy (NIR) and 13C nuclear magnetic resonance ( 13C-NMR) - are combined by means of multivariate curve resolution-alternating least squares (MCR-ALS) to obtain concentration and spectral profiles for the reaction between phenylglycidylether and aniline. The reaction, in stoichiometric proportions, was monitored by both spectroscopic techniques at 95°C. The concentration values obtained by 13C-NMR were used as an equality constraint during the multivariate curve resolution of the near infrared data. The results obtained were recovered without ambiguities: that is to say, there was a unique solution. The goodness of the results was tested by comparing the recovered concentration profiles with the values obtained by high performance liquid chromatography (HPLC) as a reference technique. The statistical tests showed that there were no significant differences between the results of both methods (α = 5%). Also, the recovered spectra were compared with the experimentally recorded spectra for the reagents (i.e. phenylglycidylether and aniline) and the final product and the correlation coefficients were, in all cases, higher than 0.998. Fil: Garrido, Mariano Enrique. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca; Argentina. Universitat Rovira I Virgili; España Fil: Larrechi, M. S.. Universitat Rovira I Virgili; España Fil: Rius, F. X.. Universitat Rovira I Virgili; España

We study a weighted low-rank approximation that is inspired by a problem of constrained low-rank approximation of matrices as initiated by the work of Golub, Hoffman, and Stewart [Linear Algebra Appl., 88/89 (1987), pp. 317--327]. Our results reduce to that of Golub, Hoffman, and Stewart in the limiting cases. We also propose an algorithm based on the alternating direction method to solve our weighted low-rank approximation problem and compare it with the state-of-art general algorithms such as the weighted total alternating least squares algorithm and the expectation maximization algorithm.

In this paper, we propose a low-rank tensor deconvolution problem which seeks multiway replicative patterns and corresponding activating tensors of rank-1. An alternating least squares (ALS) algorithm has been derived for the model to sequentially update loading components and the patterns. In addition, together with a good initialisation method using tensor diagonalization, the update rules have been implemented with a low cost using fast inversion of block Toeplitz matrices as well as an efficient update strategy. Experiments show that the proposed model and the algorithm are promising in feature extraction and clustering.

An alternating asymmetric trilinear decomposition for three-way data arrays analysis (AATLD) method was introduced. The new proposed algorithm combines the merit of Three-way Alternating Least Squares (Tri-ALS) and Alternating Trilinear Decomposition (ATLD). It retains the second-order advantage of quantification for analyte(s) of interest even in the presence of potentially unknown interferents. As an asymmetric trilinear decomposition, AATLD can perform well when three-way data arrays possess serious collinearity problem. Simulated and real high-performance liquid chromatography data arrays were used to demonstrate these advantages of the algorithm. In contrast with traditional PARAFAC, ATLD and Tri-ALS, the new proposed algorithm performs better when the data are high collinear, e.g., the large condition number of the loading matrices A, B and C. Even with heavily collinear simulated data set, it was also found that the AATLD algorithm is faster than others on obtaining solutions with chemical meaning.

This paper demonstrates a novel implementation of an alternating least squares (ALS) algorithm for resolving three- and four-way data. Computer-simulated multi-way data are studied as well as the multi-way data obtained from typical kinetic experiments observed using liquid chromatography with diode array detection (LC-DAD) and UV–visible spectroscopy. Each data set is analyzed using this new multi-way ALS algorithm, not only providing estimates of the spectral profiles (and retention profiles in the case of LC-DAD measurements) for each of the components involved, but also simultaneously estimating the rate constants for the reaction steps at different experimental conditions using a global kinetic analysis. However, when the reaction conditions do not require that all the rate constants are identical for each experiment, as is the case when the reactions are observed at different temperatures, the data analysis still benefits from the common information present in the data, such as spectral and retention profiles, as well as a common reaction mechanism.

In this paper, we report on a method for quantifying clavulanic acid and amoxicillin simultaneously in pharmaceuticals using sequential injection analysis (SIA) with a diode-array spectrophotometric detector and multivariate curve resolution with alternating least squares (MCR-ALS). We optimized the experimental parameters so that the analytical sequence could distinguish the concentrations and spectrum profiles of the species of interest with optimum resolution quality. After establishing the optimum conditions, we quantified clavulanic acid and amoxicillin in four pharmaceuticals. In most cases our results were slightly higher than those in the prospectus of the pharmaceutical. The relative standard deviations were below 5% for amoxicillin and below 7% for clavulanic acid. These results are acceptable because, to prevent degradation due to bacteriostatic activity, the concentration of the main reactant is usually higher.

Multivariate curve resolution (MCR) combined with alternating least squares (ALS) has been applied to dynamic infrared linear dichroism (DIRLD) spectra to provide relative quantification and relate this information to the thermo‐mechanical properties of a poly(ester urethane). MCR‐ALS analysis reduces the DIRLD data matrix to submatrices of factors and scores. The factors represent spectra‐like variables that relate to the DIRLD spectra of the major components constituting the individual DIRLD spectra in the data matrix. The scores are related to the concentration of these factors, and serve as a method to semiquantify the components that represent the primary molecular‐level mechanisms in the dynamic mechanical temperature scan. This paper was by special invitation as a contribution to a special issue of the journal entitled “Quantitative Vibrational Spectrometry in the 21st Century.” This special issue was organized by Professor Miguel de la Guardia, Professor of Analytical Chemistry at Valenci...

Alcoholic fermentation runs under Saccharomyces cerevisiae yeasts were conducted on culture medium batches containing glucose as carbon source. Spectral changes during the process were monitored in-line with a near infrared (NIR) immersion probe. Data were analysed by using a multivariate curve resolution–alternating least-squares (MCR–ALS) method. Different regions of the NIR spectrum were examined in order to ensure optimum application of the ALS algorithm and elucidation of the chemical rank for the system. The ambiguity inherent in the ALS algorithm was resolved by using various combinations of inequality and equality constraints. Some combinations were found to perform quite well in terms of explained variance and lack of fit, even in the absence of information in the form of equality constraints. The resulting model exposed a highly significant relationship between the ALS response and the reference concentration. Application of the model to alcoholic fermentation runs performed under similar conditions resulted in also similar analytical figures of merit. This allows the MCR–ALS method to be used to obtain the analyte profiles as a function of time and the spectral profiles in evolving alcoholic fermentations.

10 pages, 6 figures, 1 table., A novel application of Multivariate Curve Resolution with Alternating Least Squares (MCR-ALS) method for the background correction and for the resolution and quantitation of coeluted analyte mixtures in liquid chromatography with voltammetric detection is presented. The selected voltammetric technique for analyte detection was Linear Sweep Voltammetry (LSV), which provides richer multivariate information than the more widespread employed univariate amperometric detection. Use of other voltammetric detection techniques was also attempted but they gave much poorer results because of instrumental and software limitations concerning data acquisition times. The method is tested in the resolution and quantitation of mixtures of pyrocatechol, ascorbic acid and epinephrine coeluting mixtures. In spite of the extremely large background current contributions observed using this voltammetric technique, MCR-ALS was shown to be able to model this background contribution satisfactory as well as to resolve and give good quantitative estimations of these three analyte in their mixtures. The proposed approach is recommended for resolution and quantitation of complex coeluted mixtures using liquid chromatography with voltammetric detection and also for fast initial screening of optimal potential when amperometric detection is preferred., Financial support from the Ministerio de Ciencia y TecnologÌa of Spain (Projects PB96-0379-CO3 and BQU2000-0642-CO3-01) and from Generalitat of Catalonia (Projects 1997SGR-00385 and 1999SGR-00048).

Analysis of synthetically generated and real Raman imaging data sets were used to show the significance of modified alternating least squares (MALS) regression as a superior method of analysis compared with several other well established mathematical algorithms. The performance of MALS was compared with that of ordinary alternating least squares regression (ALS) and fast non-negative least squares (FNNLS) regression in applications of spectroscopic image analysis and self-modeling curve resolution (SMCR). The MALS algorithm is shown to be superior in terms of computational speed, stability, and component resolution ability in the analysis of both real and synthetic data sets. Results of the analysis show that MALS is significantly faster than FNNLS and generally produces equivalent or superior results. This work also shows that MALS is superior to ordinary ALS in all performance aspects. A detailed description of the regression equations is given along with a discussion of the application of MALS to the general spectroscopic image analysis problem.

Several recently proposed algorithms for fitting the PARAFAC model are investigated and compared to more established alternatives. Alternating least squares (ALS), direct trilinear decomposition (DTLD), alternating trilinear decomposition (ATLD), self-weighted alternating trilinear decomposition (SWATLD), pseudo alternating least squares (PALS), alternating coupled vectors resolution (ACOVER), alternating slice-wise diagonalization (ASD) and alternating coupled matrices resolution (ACOMAR) are compared on both simulated and real data. For the recent algorithms, only unconstrained three-way models can be fitted. In contrast, for example, ALS allows modeling of higher-order data, as well as incorporating constraints on the parameters and handling of missing data. Nevertheless, for three-way data, the newer algorithms are interesting alternatives. It is found that the ALS estimated models are generally of a better quality than any of the alternatives even when overfactoring the model, but it is also found that ALS is significantly slower. Based on the results (in particular the poor performance of DTLD), it is advised that (a slightly modified) ASD may be a good alternative to ALS when a faster algorithm is desired.

We consider factoring low-rank tensors in the presence of outlying slabs. This problem is important in practice, because data collected in many real-world applications, such as speech, fluorescence, and some social network data, fit this paradigm. Prior work tackles this problem by iteratively selecting a fixed number of slabs and fitting, a procedure which may not converge. We formulate this problem from a group-sparsity promoting point of view, and propose an alternating optimization framework to handle the corresponding $\ell_p$ ($0

Extra virgin olive oil, virgin olive oil and mixed oil were analyzed by UV-Vis spectroscopy and Multivariate Curve Resolution with Alternating Least Squares (MCR-ALS). When the oils were heated at high temperatures (frying), oxidation products were formed which were harmful to human health and antioxidants present were degraded. The aim of this study was to evaluate tocopherol (an antioxidant present in oils) and the behaviour of oxidation products in different olive oil classes. MCR-ALS results showed that degradation started at 70 ºC for extra virgin olive oil, while tocopherol concentration decreased and oxidation products increased starting at 40 ºC in virgin olive oil. In mixed oil, commercialized as a mixture of 85% soybean oil and 15% olive oil, tocopherol concentration started to decrease and oxidation products increase at 30 ºC. The results suggested that extra virgin olive oil offered more resistance to increasing temperatures, while virgin olive oil and mixed oil were less resistant.

Clustering and Disjoint Principal Component Analysis (CDP CA) is a constrained principal component analysis recently proposed for clustering of objects and partitioning of variables, simultaneously, which we have implemented in R language. In this paper, we deal in detail with the alternating least-squares algorithm for CDPCA and highlight its algebraic features for constructing both interpretable principal components and clusters of objects. Two applications are given to illustrate the capabilities of this new methodology.

The comprehensive analysis of untargeted metabolomics data acquired using LC-MS is still a major challenge. Different data analysis tools have been developed in recent years such as XCMS (various forms (X) of chromatography mass spectrometry) and multivariate curve resolution alternating least squares (MCR-ALS)-based strategies. In this work, metabolites extracted from rice tissues cultivated in an environmental test chamber were subjected to untargeted full-scan LC-MS analysis, and the obtained data sets were analyzed using XCMS and MCR-ALS. These approaches were compared in the investigation of the effects of copper and cadmium exposure on rice tissue (roots and aerial parts) samples. Both methods give, as a result of their application, the whole set of resolved elution and spectra profiles of the extracted metabolites in control and metal-treated samples, as well as the values of their corresponding chromatographic peak areas. The effects caused by the two considered metals on rice samples were assessed by further chemometric analysis and statistical evaluation of these peak area values. Results showed that there was a statistically significant interaction between the considered factors (type of metal of treatment and tissue). Also, the discrimination of the samples according to both factors was possible. A tentative identification of the most discriminant metabolites (biomarkers) was assessed. It is finally concluded that both XCMS- and MCR-ALS-based strategies provided similar results in all the considered cases despite the completely different approaches used by these two methods in the chromatographic peak resolution and detection strategies. Finally, advantages and disadvantages of using these two methods are discussed. [Figure not available: see fulltext.], The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n. 320737. Also, recognition from the Catalan government (grant 2014 SGR 1106) is acknowledged. JJ acknowledges a CSIC JAE-Doc contract cofounded by the FSE and AGR thanks CONICET for a fellowship.

This paper describes an improved three-way alternating least-squares multivariate curve resolution algorithm that makes use of the recently introduced multi-dimensional arrays of MATLAB®. Multi-dimensional arrays allow for a convenient way to apply chemically sound constraints, such as closure, in the third dimension. The program is designed for kinetic studies on liquid chromatography with diode array detection but can be used for other three-way data analysis. The program is tested with a large number of synthetic data sets and its flexibility is demonstrated, especially when non-trilinear data sets are fit. In this case, the algorithm finds a solution with a better fit than direct trilinear decomposition (DTD). When trilinear data are used, the optimal fit is not as good as when a direct decomposition method is used. Most real data sets, however, have some degree of non-trilinearity. This makes this method a better choice to analyze non-trilinear, three-way data than direct trilinear decomposition.

In this work, a methodology has been proposed and evaluated for identification the muscle synergy patterns during human grasping.The proposed approach is based on decomposition analysis of involved muscle activation profiles utilizing the Hierarchical Alternating Least Squares (HALS) algorithm. The surface EMG signals of Flexor Digital Superfacialis and Flexor Pollicis Longus muscles were recorded during grasping an cylindrical object. EMG signals were full-wave rectified and smoothed through a low pass filter. Then the HALS algorithm was utilized for decomposition of muscle activation profiles. The HALS algorithm can be efficiently used instead of NNMF (non-negative matrix factorization) method. The HALS method not only provides a very good convergence property but also there is not the non-negativity constraint for the decomposed factors. The results of evaluations are interesting and promising.

This paper presents a discussion of principal component analysis by alternating least squares optimal scaling (PRINCIPALS) and its application to measure development.

The aim of this study was to develop a methodology using Raman hyperspectral imaging and chemometric methods for identification of pre- and post-blast explosive residues on banknote surfaces. The explosives studied were of military, commercial and propellant uses. After the acquisition of the hyperspectral imaging, independent component analysis (ICA) was applied to extract the pure spectra and the distribution of the corresponding image constituents. The performance of the methodology was evaluated by the explained variance and the lack of fit of the models, by comparing the ICA recovered spectra with the reference spectra using correlation coefficients and by the presence of rotational ambiguity in the ICA solutions. The methodology was applied to forensic samples to solve an automated teller machine explosion case. Independent component analysis proved to be a suitable method of resolving curves, achieving equivalent performance with the multivariate curve resolution with alternating least squares (MCR-ALS) method. At low concentrations, MCR-ALS presents some limitations, as it did not provide the correct solution. The detection limit of the methodology presented in this study was 50 μg cm −2 .

The optimisation of the resolution in high-performance liquid chromatography is traditionally performed attending only to the time information. However, even in the optimal conditions, some peak pairs may remain unresolved. Such incomplete resolution can be still accomplished by deconvolution, which can be carried out with more guarantees of success by including spectral information. In this work, two-way chromatographic objective functions (COFs) that incorporate both time and spectral information were tested, based on the peak purity (analyte peak fraction free of overlapping) and the multivariate selectivity (figure of merit derived from the net analyte signal) concepts. These COFs are sensitive to situations where the components that coelute in a mixture show some spectral differences. Therefore, they are useful to find out experimental conditions where the spectrochromatograms can be recovered by deconvolution. Two-way multivariate selectivity yielded the best performance and was applied to the separation using diode-array detection of a mixture of 25 phenolic compounds, which remained unresolved in the chromatographic order using linear and multi-linear gradients of acetonitrile–water. Peak deconvolution was carried out using the combination of orthogonal projection approach and alternating least squares.

The body-site variation of skin autofluorescence was monitored in vivo in order to characterize individual contributions to the overall signal. Body sites having a range of chronic solar exposure were selected. The site-to-site spectral data sets were acquired from thirteen subjects. Incident light of 325-nm wavelength was used to excite fluorescence, observed between 360 nm and 570 nm. Principal factor analysis coupled with alternating least-squares optimization of within-subject body-site dependence revealed that two factors accounted for all of the body-site spectral variability. The factors were virtually the same for all subjects. The spectra were interpreted in terms of collagen cross-links and NADH. The body-site variation of component scores (relative concentrations) exhibited a consistent pattern between subjects. The factor assigned to collagen had highest scores at protected body sites; in contrast, NADH had highest scores at more exposed sites.

In chemistry, PARAFAC is one of the most widely used algorithms for trilinear decomposition. However, the problem of PARAFAC requiring an accurate estimation of the number of factors in the system under study limits its applications to some extent. This troublesome problem has been tackled by the pseudo alternating least squares (PALS) algorithm designed in this paper. PALS is a unique algorithm which tries to alternately optimize three different objective functions to obtain the solutions for the trilinear decomposition model. It has the outstanding feature of being resistant to the influence of N (the number of factors chosen in calculation), which has been proved mathematically under some mild conditions. Although the optimization procedure of PALS is different from that of PARAFAC, an alternating least squares scheme, and hinders a straightforward analysis of its convergence properties, studies on simulated as well as real data arrays reveal that PALS can often converge to satisfactory results within a reasonable computation time, even if excess factors are used in calculation. Copyright © 2001 John Wiley & Sons, Ltd.

This paper illustrates the possibilities of chemometric methods in the resolution and quantification of various compounds in overlapping peaks from capillary electrophoresis. Ebrotidine and most of its metabolites were efficiently separated by capillary zone electrophoresis (CZE) in a fused-silica capillary. However, the procedure was not suitable for the physical separation of the three less ionizable metabolites, which comigrated and overlapped with the electroosmotic flow signal. Multivariate curve resolution based on an alternating least squares procedure was used for their mathematical resolution. For such a purpose, data obtained in the CZE system with a diode array detector, which consisted of UV spectra registered over time, were analyzed. The ebrotidine metabolites were successfully resolved and quantified in synthetic mixtures and urine samples.

An alternating coupled matrices resolution (ACOMAR) method is developed for decomposition of three-way data arrays. By utilizing alternating least squares algorithm to minimize the proposed coupled matrices resolution error, the intrinsic profiles are found. Moreover, it yields simultaneously a numerically exact solution for all analytes present in the samples. This method retains the second-order advantage of quantization for analyte(s) of interest in the presence of potentially unknown interferents. The performance of a simulated experiment and a real analytical example shows that the proposed method works well when the number of components is chosen to be equal to or greater than the actual model dimensionality. The insensitivity of the ACOMAR method to the estimated component number escapes the difficulty of determining a proper component number for the model, which is hard to handle for the PARAFAC algorithm. Furthermore, this method circumvents the two-factor degeneracy, which is intrinsic in the PARAFAC algorithm.

A method that indicates the numbers of components to use in fitting the three-mode principal components analysis (3MPCA) model is proposed. This method, called DIFFIT, aims to find an optimal balance between the fit of solutions for the 3MPCA model and the numbers of components. The achievement of DIFFIT is compared with that of two other methods, both based on two-way PCAs, by means of a simulation study. It was found that DIFFIT performed considerably better than the other methods in indicating the numbers of components.The 3MPCA model can be estimated by the TUCKALS3 algorithm, which is an alternating least squares algorithm. In a study of how sensitive TUCKALS3 is at hitting local optima, it was found that, if the numbers of components are specified correctly, TUCKALS3 never hits a local optimum. The occurrence of local optima increased as the difference between the numbers of underlying components and the numbers of components as estimated by TUCKALS3 increased. Rationally initiated TUCKALS3 runs hit local optima less often than randomly initiated runs.

An improvement to the Andersson–Bro (A-B) alternating least squares (ALS) algorithm for the Tucker3 three-way model is presented. The published A-B algorithm deals cyclically with the three modes of the problem. In each ALS substep, the whole array is projected onto a different mode. The projections are the dominating workload. In the improved version, each whole-array projection is utilized for two ALS substeps. The same ALS steps are performed as before but the number of full-sized projections is cut to half. This almost doubles the speed of the algorithm without changing its convergence properties. The possibility to utilize each full-sized projection for more than two ALS substeps is discussed.

Multivariate curve resolution–alternating least-squares (MCR–ALS) is introduced specifically for calibration with nontrilinear data of type 1, i.e., data involving changes in constituent profiles along one of the instrumental data modes. This behavior is typical of chromatographic-spectral three-way/second-order data. However, the algorithm is more general and can also be applied to trilinear data. Details are given as to how the MVC2 graphical interface can be employed to implement MCR–ALS and produce three-way/second-order calibrations of analytes and their prediction in test samples containing potential interferents.

Multivariate Curve Resolution with Alternating Least Squares (MCR-ALS) is a resolution method that has been efficiently applied in many different fields, such as process analysis, environmental data and, more recently, hyperspectral image analysis. When applied to second order data (or to three-way data) arrays, recovery of the underlying basis vectors in both measurement orders (i.e. signal and concentration orders) from the data matrix can be achieved without ambiguities if the trilinear model constraint is considered during the ALS optimization. This work summarizes different protocols of MCR-ALS application, presenting a case study: near-infrared image spectroscopy.

A proof for the point-wise convergence of the factors in the higher-order power method for tensors towards a critical point is given. It is obtained by applying established results from the theory of \L{}ojasiewicz inequalities to the equivalent, unconstrained alternating least squares algorithm for best rank-one tensor approximation.

Avocado and pequi oils were analyzed by UV-Vis spectroscopy and chemometric methods of curve resolution. When vegetable oils are heated at high temperatures, oxidation products are formed, which besides being harmful to human health strongly accelerates the antioxidants degradation. In this sense, monitoring antioxidants and oxidation products can provide information about the best conditions to its consumption. Therefore, this study aimed to evaluate antioxidants and oxidation products behavior in avocado and pequi oils. By applying Independent Component Analysis (ICA) in the avocado oil it was detected that antioxidant concentration decreased after 90°C. To the pequi oil, by applying Multivariate Curve Resolution Alternating Least Squares (MCR37ALS) it was possible to detect that the transformations between antioxidant/oxidation products took place at 70°C. The results suggested that avocado oil is more resistant to high temperatures then pequi oil. Moreover, the methods used in this study suggest that chemomstric tools are feasible to accomplish spectrophotometric techniques in food quality evaluation.

Sensor arrays are used in many applications, where their ability to localize signal sources is essential. One of the sensor applications is the signal processing of multi antennas with multi sensors. In this paper, we present an application of the proposed Canonical Polyadic decomposition (CP decomposition), with isolation of scaling matrix to multiarray multisensor systems. A simple blind receiver based on the enhanced alternating least squares (E-ALS) algorithm is presented. For illustrating this application, computer simulations are provided and demonstrate the good behavior of these algorithm compared to the old ALS algorithm.

The SINDCLUS algorithm for fitting the ADCLUS and INDCLUS models deals with a parameter matrix that occurs twice in the model by considering the two occurrences as independent parameter matrices. This procedure has been justified empirically by the observation that upon convergence of the algorithm to the global optimum, the two independently treated parameter matrices turn out to be equal. In the present paper, results are presented that contradict this finding, and a modification of SINDCLUS is presented which obviates the need for independently treating two occurrences of the same parameter matrix.

The introduction of constraints in multivariate curve resolution methods, such as the Alternating Least Squares (ALS), is commonly used to limit the span of possible solutions, guiding the iterative process to a final result as close as possible to the true situation. In the present work, two modifications of the unimodality constraint and a new constraint for chromatographic concentration profiles related to the prevention of fronting have been checked. Simulated data sets as well as real data have been used to evaluate the effect of these new constraints in the resolution results. The parameters measured to assess the goodness of the constraints are related to the recovery of the concentration profiles and the quality of the data fit.

A general approach for fitting a model to a data matrix by weighted least squares (WLS) is studied. This approach consists of iteratively performing (steps of) existing algorithms for ordinary least squares (OLS) fitting of the same model. The approach is based on minimizing a function that majorizes the WLS loss function. The generality of the approach implies that, for every model for which an OLS fitting algorithm is available, the present approach yields a WLS fitting algorithm. In the special case where the WLS weight matrix is binary, the approach reduces to missing data imputation.

An extension of Generalized Structured Component Analysis (GSCA), called Functional GSCA, is proposed to analyze functional data that are considered to arise from an underlying smooth curve varying over time or other continua. GSCA has been geared for the analysis of multivariate data. Accordingly, it cannot deal with functional data that often involve different measurement occasions across participants and a large number of measurement occasions that exceed the number of participants. Functional GSCA addresses these issues by integrating GSCA with spline basis function expansions that represent infinite-dimensional curves onto a finite-dimensional space. For parameter estimation, functional GSCA minimizes a penalized least squares criterion by using an alternating penalized least squares estimation algorithm. The usefulness of functional GSCA is illustrated with gait data.