Bilinear Decomposition Based Alignment of Chromatographic Profiles

Article pubs.acs.org/ac

Bilinear Decomposition Based Alignment of Chromatographic Profiles Christophe Tistaert and Yvan Vander Heyden* Department of Analytical Chemistry and Pharmaceutical Technology, Center for Pharmaceutical Research (CePhaR), Vrije Universiteit Brussel (VUB), Laarbeeklaan 103, B-1090 Brussels, Belgium S Supporting Information *

ABSTRACT: A novel alignment procedure for chromatographic signals with photodiode array detection is presented. At first, the complexity of the chromatographic signals is reduced by chemometric resolution of the pure constituents. For this, the application of multivariate curve resolution− alternating least squares leads to the decomposition of the multiway data block into a chemically meaningful bilinear model representing the chromatographic profiles and their spectral signatures. The flexible implementation of a spectral selectivity constraint allows the background to be differentiated from the constituent spectra. Hereby, the pure concentration profiles are obtained which are consequently individually aligned by correlation optimized warping. In its final step, the procedure reconstitutes the original data with the aligned chromatographic profiles and their corresponding spectra. The alignment is evaluated for two sets of chromatographic signals. The new procedure improves the original application of correlation optimized warping minimizing the risks of aligning noncorresponding chromatographic information. ithin the field of chromatography, the characterization of samples is an important part of today’s routine analyses. The chromatographic profiles are able to describe the chemical composition of a sample and provide a unique identification tool, the chromatographic fingerprints.1 By uncovering the variability between the fingerprints, several chemometric methods proved to be useful in the identification, classification, and discrimination of samples.2−4 However, inconsiderate use should be avoided. The variation introduced by instrumental instability may mask the chemical differences between the samples, leading to unacceptable results. A common source of instrumental variation is the retention time shifting from one chromatogram to another. This phenomenon is originating from factors, such as column aging, minor changes in mobile phase composition, and instrumental drift. For consistency of the data, such retention time shifts should be corrected for prior to chemometric analysis.5 Over the past decades, several alignment techniques have been proposed in historic correspondence with the evolving complexity of the chromatographic data. Initially, positional information of selected landmark peaks was used for the alignment of low-complexity profiles.6,7 With the innovations in chromatographic science and the increase in data complexity, such methods became obsolete and alignment techniques which do not require any preliminary information were developed. To date, these techniques remain frequently applied and examples include dynamic time warping,8,9 parametric time warping,10 fuzzy warping,11 and correlation optimized warping.12 The latter is most commonly integrated by chromatographers and chemometricians.

W

© 2012 American Chemical Society

With the current interest in the analysis of biological samples for proteomic and metabolomic studies, the complexity of the data increases even more.3,13,14 The largely varying concentration profiles and patient-to-patient sample compositions complicate the alignment and may result in the mismatching of noncorresponding peaks. Moreover, the popular optimization of the correlation between the chromatograms is greatly influenced by the major peaks. Hence, the maximal calculated correlation coefficient does not necessarily implicate optimal alignment.15 To overcome these emerging problems, the available spectroscopic information from photodiode array detection (PDA) and mass spectrometry (MS) is ideally included to avoid the mismatching of noncorresponding information. Several novel alignment procedures have been proposed,15−20 but are either computationally extremely demanding15 or include only a limited fraction of the available data.18 In this study, a new alignment procedure for chromatographic signals with PDA detection is presented. At first, the complexity of the signals is reduced by chemometric resolution of the pure compounds. For this, the application of bilinear decomposition methods is most beneficial as they are insensitive to the retention time shifts because of the high reproducibility of the spectral mode. With multivariate curve resolution−alternating least squares (MCR-ALS) the multiway data block is decomposed into a chemically meaningful bilinear Received: March 26, 2012 Accepted: June 6, 2012 Published: June 6, 2012 5653

dx.doi.org/10.1021/ac300735a | Anal. Chem. 2012, 84, 5653−5660

Analytical Chemistry

Article

points and the spectral dimension consists of experimentally derived spectral signatures of 126 scan points. Chemometric Resolution of the Pure Contributions. Chemometric resolution aims to describe the underlying contributions to a data set. The second-order data from chromatographic systems coupled to multichannel detectors can be resolved by bilinear decomposition models describing the chromatographic data as the pure concentration profiles of the constituents, and the spectral data as their corresponding spectral signatures.24 The applied procedure is based on MCRALS with a flexible implementation of mathematical constraints, resulting in a chemically meaningful bilinear model. Data Organization. Prior to resolution, the three-way chromatographic data block X(JxKxL) where J is the number of time points for K samples and L is the number of wavelengths, is reshaped into a two-dimensional structure X(JKxL) representing the chromatographic information for all samples in its first dimension and the spectral information in its second. The inclusion of all K samples into a single data block aids in the chemometric resolution as it simplifies the development of calibration relationships, and helps to improve the resolved profiles by averaging the information across samples, improving the S/N of the results.25 Initial Estimate. Iterative resolution methods require an initial estimate of the concentration profiles or spectral signatures to start the optimization process. Numerous possibilities to generate chemically meaningful estimates are available. Our implementation is based on iterative key set factor analysis (IKSFA),26,27 a preferred subset selection method that seeks to find the key set of N spectra under the assumption that the purest spectra are more dissimilar from one another than the mixture spectra. IKSFA results in a singular value decomposition of which the normalized rows of the left singular vectors ur̃ are used to determine the key rows. The first key row uk̃ ey is selected based on the largest absolute value of ũr,1 for all rows, after which the following key rows are selected based on the maximum determinant found with the first key row until N key rows are identified. Once the initial key set is selected, the first key row is replaced with the first normalized left singular vector and the determinant is calculated. If the value of the determinant for the new key set is less than that of the initial key set, the initial key remains unchanged and the calculations are repeated with the remaining vectors. The procedure cycles through the entire set of key rows and reaches convergence when no changes in the key set occur during an entire cycle. The initial input parameter for IKSFA, that is, the number of pure spectra N, is determined based on a SCREE plot of the singular values of X(JKxL). For the simulated data, seven pure spectral components were defined as shown in Figure 2b. Multivariate Curve Resolution−Alternating Least Squares. MCR-ALS results in the bilinear decomposition of X(JKxL) into three matrices (eq 1):21−24

model representing the chromatographic profiles and their spectral signatures.21,22 The flexible implementation of a spectral selectivity constraint allows the background to be differentiated from the constituent spectra.23 Hereby, the pure concentration profiles are obtained, which are consequently individually aligned by correlation optimized warping (COW).12 In its final step, the procedure, referred to as MCR-COW, reconstitutes the original data with the aligned chromatographic profiles and their spectral signatures, resulting in chromatographic signals aligned over their entire spectral range.

■

THEORY The flow chart in Figure 1 is a detailed representation of the proposed alignment procedure. In the following sections, the

Figure 1. Flow chart of the three-step alignment procedure: (I) chemometric resolution, (II) correlation optimized warping, and (III) data reconstitution.

three individual steps of the procedure are discussed and illustrated on simulated HPLC-DAD data (Figure 2). Ten samples of four spectrally unique peaks with varying peak widths, large dynamic ranges, and retention time shifts were generated on an experimentally derived three-component background. An overlap between peaks three and four was simulated. Peak five spectrally rank deficient to peak one and in overlap with peak two was introduced in the last five samples. Peak three was left out of samples two and three (Figure 2a). The chromatographic dimension was simulated with 150 scan

X = C ·ST + E

where C describes the resolved chromatographic profiles with dimensions JK × N, S the resolved spectral signatures with dimensions L × N, and E the residuals with dimensions JK × L. On the basis of the initial estimate for So, an iterative leastsquares optimization of the chromatographic and spectral matrices is performed with implementation of mathematical constraints of chemical significance.22,28 While all the chromatographic profiles are constrained by non-negativity, 5654



Article

Figure 2. Illustration of alignment procedure: (a) 10 chromatographic samples are simulated based on four experimentally derived constituent spectra and a three-component experimental background; (b) singular value decomposition leads to an estimate of seven unique spectral signatures, which are consequently determined by IKSFA; (c) the chromatographic profiles with their corresponding spectral signatures are resolved by MCRALS; (d) the resolved constituent profiles are aligned by COW; and (e) the aligned chromatographic data is reconstituted.

limited number of data points. Starting at the last segment, the correlation coefficient to the target is calculated for every end point and the solution with the highest correlation coefficient is stored and interpolated to the length of the corresponding section of the target profile. The global warping solution is defined as the highest cumulative sum of the correlation coefficients for all sections. To avoid the effects of outlying observations, such as rank deficient peaks, our implementation utilizes the median of the concentration profiles as the target profile. The algorithm is set to only apply COW to the concentration profiles of the constituents, that is, the constrained components, and ignore the background information. The outcome for the alignment of the four constituent profiles is presented in Figure 2d. Data Reconstitution. With the aligned concentration profiles, the three-way chromatographic data block X(JxKxL) can be reconstituted. The N aligned concentration profiles with dimensions J × K are organized in a J × K × N matrix and reshaped into a two-dimensional JK × N matrix of aligned chromatographic information. The information is multiplied with the spectral signatures of S, resulting in the aligned data block Xaligned which is reshaped to the dimensions of the original X matrix. One should note that the background components can be removed from the data, resulting in chromatographic information solely attributable to the constituents. This is demonstrated for the simulated data for which the three background components are excluded from the reconstitution (Figure 2e).

spectral selectivity and non-negativity are only applied to the constituent spectra. The application of spectral selectivity sets the absorbances of the constituent spectra to zero within a specified wavelength interval. Typically, the constraint is imposed at wavelengths exceeding 440 nm. Chemical constituents usually do not absorb at these higher wavelengths, while the background components are subjected to refractive index changes. This flexible implementation allows for the differentiation between the constituent profiles and the background.23 The chemometric resolution reduces the data complexity and, for the simulated data, results in four constituent profiles with unique spectral signatures and three background components. The overlapped peaks 3 and 4, and the low intensity profile of peak 2 in overlap with peak 5 are resolved. Rank deficiency is also observed in the last five samples where peak 5 presents a spectral signature highly similar to peak 1 and is attributed to the same component (Figure 2c). Alignment. The alignment is performed on the N concentration profiles of the K samples individually, minimizing the risks of aligning noncorresponding information. For this, the JK × N matrix C of chromatographic profiles is reshaped to a three-dimensional matrix of dimensions J × K × N, after which the concentration profiles are aligned by COW.12 COW aligns two signals by means of piecewise linear stretching and compression of the chromatogram in correlation with a target chromatogram. By dividing both profiles into a user-defined number of segments, each section of the profile is individually stretched or compressed by moving the section’s end point by a 5655



■

Article

EXPERIMENTAL SECTION The MCR-COW procedure is evaluated on two experimental case-studies: replicate injections of a standard mixture of five βblockers, and green tea fingerprints. The performance, the required user-intervention, and the computational demands are compared to the standard implementation of COW. The experimental details on the development of both data sets are available in the Supporting Information. The MCR-COW toolbox can be downloaded at http://www.vub.ac.be/fabi/ toolboxes.html.

the interpretation of the results, the chromatographic peaks were further divided over two clusters (1.40−3.20 min and 3.20−7.45 min) (Figure 3). The results of the alignment procedure are visualized in Figure 4. As no constituent data was expected between 7.45 and 14.00 min, this zone is ideal for the determination of the background and led to the resolution of two background components (Figure 4a). On the basis of the a priori knowledge of the data, the determination of the two background components and a SCREE plot, the two clusters of chromatographic peaks are resolved. Between 1.40 and 3.20 min, a fivecomponent resolution led to the determination of three constituent profiles and two background profiles (Figure 4b). The constituent profiles are resolved at 1.59, 1.73, and 2.35 min and identified as carteolol, metoprolol, and oxprenolol, respectively, with retention times and spectral signatures highly correlated to the standard injections. In the same way, the peak cluster between 3.20 and 7.45 min was resolved by a fourcomponent model and led to the determination of the two background profiles and the constituent profiles of the partially coeluting compounds carazolol and propranolol at 5.75 and 6.31 min, respectively (Figure 4c). The zone with the injection peak (0−1.40 min) was considered uninformative and therefore discarded for alignment. An exploratory analysis by Principal Component Analysis (PCA) on the raw chromatographic profiles (240 nm), the profiles aligned by COW, and the profiles aligned by MCRCOW resulted in a PC1-PC2 score plot describing 91.93% of the variance (Figure 5). The unprocessed profiles are spread over the score plot (▲) along the PC1-axis (81.97% of the variation), while the profiles aligned by COW (■) are clustered in the PC1 direction and present an increasing spread along PC2 (9.96% of the variation). The best results were achieved by MCR-COW, resulting in a centrally situated, dense cluster of all six profiles (●) along both the PC1 and PC2 axis. Green Tea Fingerprints. HPLC analysis of the green tea samples resulted in 11 min fingerprints with 17 clearly visible peaks within the 3.70 to 9.60 min interval (Figure 6, peaks a− q). The chromatographic time frame between 0 and 3.70 min (injection peak) was considered uninformative, while no compounds eluted between 9.60 and 11.00 min. The fingerprints were subtracted with a blank injection and organized as a 2064 × 4 × 377 three-dimensional matrix. Contrary to the standard mixture, no a priori information on the samples is available. Additionally, the fingerprints represent different samples. The constituents present in the green tea samples are not well-defined, the chromatographic peaks are mostly unidentified, and the peak purity is uncertain. All of this compromises the alignment and may result in a mismatch of noncorresponding peaks. To ensure the alignment of the corresponding constituents, MCR-COW was applied to the fingerprints. The interpretation and visualization of the results, as well as the accommodation of the complexity of such biological data, was facilitated by dividing the chromatographic time frame into six subwindows. As the HPLC analysis involved a gradient elution program, the mobile phase composition is continuously changing. The resolved background components may be subjected to changes within the different timeframes of the elution program. Therefore, their determination was performed for each considered subwindow individually on a blank injection. This led to the resolution of three background components present

■

RESULTS AND DISCUSSION Standard Mixture. HPLC analysis of the standard mixture resulted in a chromatographic profile with four distinguishable peaks. While carteolol and metoprolol remained unresolved at 1.68 min (Rs = 0), oxprenolol was baseline separated at 2.35 min, and carazolol and propranolol were partially coeluting at 5.75 and 6.31 min (Rs = 0.85), respectively (Figure 3). Six

Figure 3. Chromatographic profiles of the standard mixture prior and after alignment. The data was divided into four distinct zones: the injection peak (0−1.40 min), two clusters of chromatographic peaks (1.40−3.20 min and 3.20−7.45 min with zoomplots), and baseline (7.45−14.00 min).

replicate measurements were performed and retention time shifts up to 5 s were observed. A blank injection was subtracted from the chromatographic profiles, which were subsequently organized as a 1314 × 6 × 377 three-dimensional matrix (chromatographic data × number of injections × spectral data). Prior to chemometric resolution and subsequent alignment of the samples, the data was divided into three distinct zones: the injection peak (0−1.40 min), the chromatographic peaks (1.40−7.45 min), and baseline (7.45−14.00 min). To facilitate 5656



Article

Figure 4. MCR-COW applied on the standard mixture profiles. (a) As no constituents were observed between 7.45 and 14.00 min, the zone is ideal for the determination of the background components. (b) Between 1.40 and 3.20 min, a five component resolution led to three constituent and two background profiles. (c) The peak cluster between 3.20 and 7.45 min was resolved by a four-component model with two constituent and two background profiles.

fingerprints by COW. The overall correlation coefficients between the fingerprints ranged from 0.7895 to 0.9827 for the raw fingerprints. After alignment, little difference is observed between both applied procedures with correlation coefficients between 0.9716 and 0.9971 for the COW aligned fingerprints, and 0.9749 and 0.9977 for the MCR-COW aligned fingerprints. Interestingly, when evaluating the correlations of the subwindows (Supporting Information Table S-1), subwindow VI after COW stands out with a correlation as low as 0.5332 compared to 0.9576 for MCR-COW. Therefore, the correlation maps of corresponding fingerprint sections for all three sets of fingerprints were considered (Figure 7). As expected from the overall correlation coefficients, both COW and MCR-COW provided a considerable improvement compared to the raw chromatographic data. Differences between both alignment procedures were however visible. While MCR-COW resulted in high values of the correlation coefficients over the entire fingerprint, COW presented two regions with lower correlations, down to 0.5 (5.10−5.90 and 8.10−9.50 min). These regions concern parts of the data where only small peaks are present. Hence, they have only little influence on the overall correlation coefficients. Discussion. Following the preceding case-studies, some considerations concerning the benefits and drawbacks of MCRCOW are listed. For both the replicate injections of the standard mixture and the green tea fingerprints, the MCR-COW alignment procedure performed better than the standard implementation of COW. More specifically, and as was clearly demonstrated for the green tea data, the alignment of regions with small peaks improved significantly by resolving the data prior to the alignment. This could be explained from the COW procedure

Figure 5. PC1-PC2 score plot of the standard mixture profiles at 240 nm. The unprocessed chromatographic profiles (▲) are spread over the score plot along the PC1-axis, while the profiles aligned by COW (■) are clustered in the PC1 direction but are increasingly spread in PC2. The best results were achieved with the proposed procedure, resulting in a centrally situated dense cluster of all six profiles (●).

throughout the entire data set, with slightly diverging signatures within each subwindow. The subsequent resolution of the constituent profiles revealed the presence of 27 constituents within the 3.70−9.60 min interval, whereas only 17 peaks were clearly visible on the raw fingerprint profiles (Figure 6). The constituent profiles were individually aligned and the fingerprints were reconstituted with subtraction of the background components. Supporting Information Table S-1 provides the reader with an overview of the constituents visualized prior (a−q) and after (1−27) chemometric resolution, as well as the alignment results of the resolved chromatographic profiles and of the raw 5657



Article

Figure 6. HPLC analysis of the green tea samples resulted in fingerprints with 17 clearly visible peaks within the 3.70−9.60 min interval (peaks a−q). The subsequent resolution of the constituent profiles revealed the presence of 27 constituents, which were aligned individually prior to reconstitution of the data. To facilitate the interpretation and visualization of the results, as well as to accommodate the complexity of such biological data, the chromatographic time frame was divided into six subwindows (marked I−VI).

which utilizes the optimal calculated correlation coefficient for the alignment of the chromatographic profiles. The presence of multiple peaks, together with potential changes in the concentration profiles and the dominating effect of the major peaks on the correlation coefficient, implied that the correlation coefficient is not always a reliable parameter to correct for retention time shifts. This resulted in the generation of suboptimal results and could even have led to the mismatching of noncorresponding information. Such problems were avoided by the MCR-COW procedure. The flexible implementation of MCR-ALS resolved the constituents one from another based on the presence of unique spectral signatures. The decomposition decreased the complexity of the data and, if no constituents with highly similar spectra were present, reduced the number of peaks present in each MCR-ALS component to one. Subsequently, the concentration profiles could be individually aligned while assuring the identity of the constituents and thus avoiding the mismatching of noncorresponding peaks. This resulted in chromatographic profiles which did not only present a high overall correlation coefficient, but also had high correlation coefficients between randomly selected subwindows. Moreover, with MCR-COW, the information of the PDA detector was included, aligning the data over its entire spectral range in a single procedure, whereas COW was applied to a single wavelength. Notwithstanding the benefits and the satisfactory results, the procedure presented some drawbacks when it comes to the required user-intervention and computational demands. Besides

the predefinition of the parameters for COW and the inherent risks of peak deformation,12,29 the procedure required the implementation of a manually optimized MCR-based bilinear decomposition of the chromatographic subwindows. The required interventions on the determination of the number of components as well as the implementation of the constraints were based on visual inspection of the results. However, the interpretation of the data and the subsequent decisions were fairly straightforward, did not require a great deal of expertise, and could rather instantly be decided upon. Additionally, and while the currently available hardware allows alignment algorithms, such as COW, to be calculated within 1 min, the computational demands of the MCR-based implementation rose significantly with the increasing complexity of the data (i.e., number of components), the size of the chromatographic window, the number of samples, and the spectral dimensions. Depending on the interaction between the above-described factors, the user may have to redefine the subwindows to decrease the computational time to an acceptable level. A simulated data set mimicking a fivecomponent chromatographic subwindow of 4 min (acquisition rate of 1.5 Hz) with five replicate injections and spectral data points recorded each 4 nm within the 200−700 nm interval resulted in a 360 × 5 × 125 data matrix. The three-step alignment procedure was completed within 9 s. A 5-fold increase of the chromatographic subwindow and the number of samples, together with a duplication of the spectral data density and the data complexity, resulted in a 40-fold increase of the analysis time. 5658



Article

sets. The bilinear decomposition of the data resulted in peak purity assessment and spectral correlation of the resolved constituents. Hereby, a solid base for the alignment of the corresponding constituent profiles was created. With the implementation of a flexible spectral selectivity constraint, the algorithm allowed differentiating the background components from the constituents, which could subsequently be removed from the data. Moreover, the alignment of the individual profiles prior to the reconstitution of the data led to chromatographic signals aligned over the entire spectral range. For the replicate injections of the standard mixture, an exploratory analysis was performed on the raw chromatographic profiles, the profiles aligned by COW, and the profiles aligned by MCR-COW. Optimal results were achieved with the proposed procedure, outperforming the standard implementation of COW. Similarly, for the green tea fingerprints, MCRCOW resulted in high values of the correlation coefficients over the entire fingerprint, while low correlation was observed for COW in two regions with only small peaks. One should note that the procedure makes use of a manually optimized MCR-based bilinear decomposition method. However, the interpretation of the data and the subsequent decisions were fairly straightforward, did not require a great deal of expertise, and could rather instantly be decided upon. Moreover, the rising computational demands with increasing complexity of the data could easily be addressed by redefining the subwindows.

■

ASSOCIATED CONTENT

S Supporting Information *

Additional information as noted in text. This material is available free of charge via the Internet at http://pubs.acs.org.

■

AUTHOR INFORMATION

Corresponding Author

*Tel.: +32 2 477 47 34. Fax: +32 2 477 47 35. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The authors would like to thank the research group of Professor Sarah C. Rutan (Virginia Commonwealth University) for the contributions to the code.

Figure 7. Correlation maps of corresponding sections from the raw chromatographic fingerprints, and from the fingerprints aligned by COW and MCR-COW.

■

■

REFERENCES

(1) IUPAC Compendium of Chemical Terminology; Royal Society of Chemistry; Cambridge, 1997. (2) Dumarey, M.; van Nederkassel, A. M.; Stanimirova, I.; Daszykowski, M.; Bensaid, F.; Lees, M.; Martin, G. J.; Desmurs, J. R.; Smeyers-Verbeke, J.; Vander Heyden, Y. Anal. Chim. Acta 2008, 655, 43−51. (3) Tistaert, C.; Dejaegher, B.; Vander Heyden, Y. Anal. Chim. Acta 2011, 690, 148−161. (4) Daszykowski, M.; Walczak, B. Trends Anal. Chem. 2006, 25, 1081−1096. (5) Malmquist, G.; Danielsson, R. J. Chromatogr., A 1994, 687, 71− 88. (6) Round, A. J.; Aguilar, M. I.; Hearn, M. T. W. J. Chromatogr., A 1994, 661, 61−75. (7) Grung, B.; Kvalheim, O. M. Anal. Chim. Acta 1995, 304, 57−66. (8) Itakura, F. IEEE Trans. Acoust., Speech, Signal Process. 1975, 23, 67−72.

CONCLUSIONS To overcome the emerging problems of mismatching noncorresponding information during the alignment of complex biological fingerprints, there is a need for new developments in the field. In this study, a novel alignment procedure for complex chromatographic signals with photodiode array detection is presented. On the basis of a flexible implementation of MCR-ALS, the multiway chromatographic data is decomposed into its pure underlying contributions. Subsequently, the individual constituent profiles are aligned by COW, after which the data is reconstituted with subtraction of the background components. The procedure, abbreviated as MCR-COW, is illustrated on simulated data closely mimicking an experimental environment, and its applicability is demonstrated on two experimental data 5659



Article

(9) Kassidas, A.; MacGregor, J.; Taylor, P. AIChE J. 1998, 44, 864− 875. (10) Eilers, P. H. C. Anal. Chem. 2004, 76, 404−411. (11) Walczak, B.; Wu, W. Chemom. Intell. Lab. Syst. 2005, 77, 173− 180. (12) Nielsen, N.-P. V.; Carstensen, J. M.; Smedsgaard, J. J. Chromatogr., A 1998, 805, 17−35. (13) Danielsson, R.; Allard, E.; Sjöberg, P. J. R.; Bergquist, J. Chemom. Intell. Lab. Syst. 2011, 108, 33−48. (14) Roux, A.; Lison, D.; Junot, C.; Heilier, J.-F. Clin. Biochem. 2011, 44, 119−135. (15) Gong, F.; Liang, Y.-Z.; Fung, Y.-S.; Chau, F.-T. J. Chromatogr. A 2004, 1029, 173−183. (16) Clifford, D.; Stone, G.; Montoliu, I.; Rezzi, S.; Martin, F.-P.; Guy, P.; Bruce, S.; Kochhar, S. Anal. Chem. 2009, 81, 1000−1007. (17) Daszykowski, M.; Vander Heyden, Y.; Boucon, C.; Walczak, B. J. Chromatrogr., A 2010, 1217, 6127−6133. (18) Kirchner, M.; Saussen, B.; Steen, H.; Steen, J. A. J.; Hamprecht, F. A. J. Stat. Soft. 2007, 18, 1−12. (19) Bloemberg, T. G.; Gerretzen, J.; Wouters, H. J. P.; Gloerich, J.; van Dael, M.; Wessels, H. J. C. T.; van den Heuvel, L. P.; Eilers, P. H. C.; Buydens, L. M. C.; Wehrens, R. Chemom. Intell. Lab. Syst. 2010, 104, 65−74. (20) Comas, E.; Gimeno, R. A.; Ferré, J.; Marcé, R. M.; Borrull, F.; Rius, F. X. Anal. Chim. Acta 2002, 470, 163−173. (21) Tauler, R.; Barcelo, D. Trends Anal. Chem. 1993, 12, 319−327. (22) de Juan, A.; Rutan, S. C.; Tauler, R. In Comprehensive Chemometrics; Brown, S., Tauler, R., Walczak, B. Eds.; Elsevier: Amsterdam, 2009; pp 325−344. (23) Bailey, H. P.; Rutan, S. C. Chemom. Intell. Lab. Syst. 2011, 106, 131−141. (24) Rutan, S. C.; de Juan, A.; Tauler, R. In Comprehensive Chemometrics; Brown, S., Tauler, R., Walczak, B. Eds.; Elsevier: Amsterdam, 2009; pp 249−259. (25) Stoll, D. R.; Li, X.; Wang, X.; Carr, P. W.; Porter, S. E. G.; Rutan, S. C. J. Chromatogr., A 2007, 1168, 3−43. (26) Schostack, K. J.; Malinowski, E. R. Chemom. Intell. Lab. Syst. 1989, 6, 21−29. (27) Bogomolov, A.; McBrien, M. Anal. Chim. Acta 2003, 490, 41− 58. (28) de Juan, A.; Vander Heyden, Y.; Tauler, R.; Massart, D. L. Anal. Chim. Acta 1997, 346, 307−318. (29) van Nederkassel, A. M.; Daszykowski, M.; Eilers, P. H. C.; Vander Heyden, Y. J. Chromatogr., A 2006, 1118, 199−210.

5660


Bilinear Decomposition Based Alignment of Chromatographic Profiles

Recommend Documents