Generalized Rank Annihilation Method Research Articles

Analysis of Tryptophan and Tyrosine in the Presence of Other Bioactive Molecules Using Generalized Rank Annihilation Method on Excitation-emission Fluorescence Spectroscopic Data Sets

Excitation-emission matrix fluorescence (EEMF) spectroscopy can describe the fluorescence activity of a molecular mixture in a single three dimensional plot describing the variation of excitation and emission spectra as function of increasing emission and excitation wavelength, respectively. Generalized rank annihilation method (GRAM) is a chemometric technique that is known to have second order advantage i.e.. it can be easily used to analyze the analyte of interest even in the presence of the unknown interferences. The present work proposes a simple and sensitive analytical procedure by exploring the advantages associated with EEMF and GRAM technique to analyze the two amino acids, tyrosine and tryptophan, in the presence of complex fluorescence background originating from other bioactive molecules. The obtained results of GRAM analysis on EEMF was found to make precise quantification of tryptophan and tyrosine in different unknown samples.

Outlier detection for the Generalized Rank Annihilation Method in HPLC-DAD analysis

The Generalized Rank Annihilation Method (GRAM) is a second-order calibration method that is used in chromatography to quantify analytes that coelute with interferences. For a correct quantification, the peak of the analyte in the standard and in the test sample must be aligned and have the same shape (i.e., have a trilinear structure). Variations in retention time and shape between the two peaks may cause the test sample to behave as an outlier and produce an incorrect prediction. This situation cannot be detected by checking the coincidence of the recovered spectrum with the known spectrum of the analyte because the spectral domain is not affected. It cannot be detected either by checking if the recovered profile is correct (i.e., unimodal and positive). Several plots are presented to detect such outliers. The first plot compares the particular elution profiles in the standard and in the test sample that are recovered by least-squares regression from the spectra estimated by GRAM. The calculated elution profiles from both peaks should coincide. A second plot uses the elution profiles and spectra calculated by GRAM to define the vector space spanned by the interferences. The measured peaks in the standard and in the test sample are projected onto the space that is orthogonal to the space spanned by the interferences. These projections are proportional (up to the noise) if data are trilinear. The proportionality is checked graphically from the first singular vector of the projected peaks, or from the plot of the orthogonal signal versus the net sensitivity. The use of these graphs is shown for simulated data and for the determination of 4-nitrophenol in river water samples with liquid chromatography/UV–Vis detection.

GRAM satisfies Kruskal's condition

AbstractIt is well known that least‐squares (LS) methods uniquely identify the parameters of the CANDECOMP/PARAFAC model if Kruskal's condition is satisfied. By contrast, a stricter sufficient condition applies to eigenvalue‐based methods like the generalized rank annihilation method (GRAM). This discrepancy suggests that LS methods can solve problems for which GRAM must fail. However, GRAM has been specifically introduced for the special case of a three‐way array with two frontal slices only (i.e. K = 2). Here, it is shown that the two conditions are equivalent for this special case. Copyright © 2008 John Wiley & Sons, Ltd.

Modulation-induced error in comprehensive two-dimensional gas chromatographic separations

There is a fundamental difference between data collected in comprehensive two-dimensional gas chromatographic (GC × GC) separations and data collected by one-dimensional GC techniques (or heart-cut GC techniques). This difference can be ascribed to the fact that GC × GC generates multiple sub-peaks for each analyte, as opposed to other GC techniques that generate only a single chromatographic peak for each analyte. In order to calculate the total signal for the analyte, the most commonly used approach is to consider the cumulative area that results from the integration of each sub-peak. Alternately, the data may be considered using higher order techniques such as the generalized rank annihilation method (GRAM). Regardless of the approach, the potential errors are expected to be greater for trace analytes where the sub-peaks are close to the limit of detection (LOD). This error is also expected to be compounded with phase-induced error, a phenomenon foreign to the measurement of single peaks. Here these sources of error are investigated for the first time using both the traditional integration-based approach and GRAM analysis. The use of simulated data permits the sources of error to be controlled and independently evaluated in a manner not possible with real data. The results of this study show that the error introduced by the modulation process is at worst 1% for analyte signals with a base peak height of 10 × LOD and either approach to quantitation is used. Errors due to phase shifting are shown to be of greater concern, especially for trace analytes with only one or two visible sub-peaks. In this case, the error could be as great as 6.4% for symmetrical peaks when a conventional integration approach is used. This is contrasted by GRAM which provides a much more precise result, at worst 1.8% and 0.6% when the modulation ratio ( M R) is 1.5 or 3.0, respectively for symmetrical peaks. The data show that for analyses demanding high precision, a M R of 3 should be targeted as a minimum, especially if multivariate techniques are to be used so as to maintain data density in the primary dimension. For rapid screening techniques where precision is not as critical lower M R values can be tolerated. When integration is used, if there are 4–5 visible sub-peaks included for a symmetrical peak at M R = 3.0, the data will be reasonably free from phase-shift-induced errors or a negative bias. At M R = 1.5, at least 3 sub-peaks must be included for a symmetrical peak. The proposed guidelines should be equally relevant to LC × LC and other similar techniques.

Second-order standard addition for deconvolution and quantification of fatty acids of fish oil using GC–MS

In the present work two second-order calibration methods, generalized rank annihilation method (GRAM) and multivariate curve resolution-alternating least square (MCR-ALS) have been applied on standard addition data matrices obtained by gas chromatography–mass spectrometry (GC–MS) to characterize and quantify four unsaturated fatty acids cis-9-hexadecenoic acid (C16:1ω7c), cis-9-octadecenoic acid (C18:1ω9c), cis-11-eicosenoic acid (C20:1ω9) and cis-13-docosenoic acid (C22:1ω9) in fish oil considering matrix interferences. With these methods, the area does not need to be directly measured and predictions are more accurate. Because of non-trilinear conditions of GC–MS data matrices, at first MCR-ALS and GRAM have been used on uncorrected data matrices. In comparison to MCR-ALS, biased and imprecise concentrations (%R.S.D. = 27.3) were obtained using GRAM without correcting the retention time-shift. As trilinearity is the essential requirement for implementing GRAM, the data need to be corrected. Multivariate rank alignment objectively corrects the run-to-run retention time variations between sample GC–MS data matrix and a standard addition GC–MS data matrix. Then, two second-order algorithms have been compared with each other. The above algorithms provided similar mean predictions, pure concentrations and spectral profiles. The results validated using standard mass spectra of target compounds. In addition, some of the quantification results were compared with the concentration values obtained using the selected mass chromatograms. As in the case of strong peak-overlap and the matrix effect, the classical univariate method of determination of the area of the peaks of the analytes will fail, the “second-order advantage” has solved this problem successfully.

An absorbance-based micro-fluidic sensor for diffusion coefficient and molar mass determinations

The H-Sensor reported herein is a micro-fluidic device compatible with flow injection analysis (FIA) and high performance liquid chromatography (HPLC). The device detects analytes at two separate off-chip absorbance flow cells, providing two simultaneous absorbance measurements. The ratio of these two absorbance signals contains analyte diffusion coefficient information. A theoretical model for the sensing mechanism is presented. The model relates the signal Ratio to analyte diffusion coefficient. The model is qualitatively evaluated by comparing theoretical and experimental signal Ratio values. Experimental signal Ratios were collected via FIA for a variety of analytes, including sodium azide, benzoic acid, amino acids, peptides, and proteins. Measuring absorbance at multiple wavelengths provides higher order data allowing the analyte signals from mixtures to be deconvolved via classical least squares (CLS). As a result of the H-Sensor providing two simultaneous signals as a function of time for each sample injection, two simulated second-order HPLC chromatograms were generated using experimental H-Sensor data. The chemometric deconvolution method referred to as the generalized rank annihilation method (GRAM) was used to demonstrate chromatographic and spectroscopic deconvolution. GRAM also provides the signal Ratio value, therefore simultaneously obtaining the analyte diffusion coefficient information during deconvolution. The two chromatograms successfully serve as the standard and unknown for the GRAM deconvolution. GRAM was evaluated on chromatograms at various chromatographic resolutions. GRAM was found to function to a chromatographic resolution at and above 0.25 with a percent quantitative error of less then 10%.

Simultaneous Spectrofluorometric Determination of Piroxicam and Pyridoxine Using Generalized Rank Annihilation Method

The application of generalized rank annihilation method (GRAM) to the analysis of fluorescence excitation-emission matrices of mixtures of piroxicam and pyridoxine is described. The input of GRAM consists of two bilinear data matrices, i.e. one for unknown and one for the calibration sample. The excitation wavelength range was from 290 to 340 nm and the emission was recorded from 370 to 560 nm. Piroxicam and pyridoxine were determined in the concentration ranges 0.33 - 4.00 microg ml(-1) and 0.66 - 8.00 microg ml(-1), respectively. To check the accuracy of the proposed method, several binary synthetic mixtures and one real sample were analyzed successfully. No matrix effect was observed in mixture analysis, so a single external calibration sample was used for each analyte. The ability of GRAM to quantify the studied compounds and the comparability of GRAM results were evaluated by comparing them with those of PLS regression as a standard first-order calibration.

Resolution of overlapping capillary electrophoresis peaks by using chemometric analysis: Improved quantification by using internal standard

Discussed in this paper is the application of internal standard for quantification of compounds in overlapping peaks from capillary electrophoresis diode array detection, which has not often been studied. A strategy is proposed to calculate the calibration coefficient in this case. The resolution of overlapping capillary electrophoresis peaks is achieved by using second order multivariate curve resolution method. Before resolution, the signals are baseline-corrected by automation baseline correction, in which a smoothed signal is used for both baseline recognition and modeling. An evolving routine is used to determine the peak shift and improve the repeatability of migration times. The quantification results are compared with those of generalized rank annihilation method. For multivariate curve resolution method, improved quantification precision results from using internal standard, but further research is expected in the case of generalized rank annihilation method.

The chemometric resolution and quantification of overlapped peaks form comprehensive two-dimensional liquid chromatography

The chemometric resolution and quantification of overlapped peaks from comprehensive two-dimensional (2D) liquid chromatography (LC × LC) data are demonstrated. The LC × LC data is produced from an in-house LC × LC analyzer that couples an anion-exchange column via a multi-port valve with a reversed-phase column connected to a UV absorbance detector. Three test mixtures, each containing a target analyte, are subjected to partial LC × LC separations to simulate likely cases of signal overlap. The resulting unresolved target-analyte signals are then analyzed by the standard-addition method and two chemometric methods. The LC × LC analyses of a test mixture and its corresponding standard-addition mixture results in two data matrices, one for each mixture. The stacking of these two data matrices produces a data structure that can then be analyzed by trilinear chemometric methods. One method, the generalized rank annihilation method (GRAM), uses a non-iterative eigenvalue-based approach to mathematically resolve overlapped trilinear signals. The other method, parallel factor analysis (PARAFAC), uses an iterative approach to resolve trilinear signals by the optimization of initial estimates using alternating least squares and signal constraints. In this paper, GRAM followed by PARAFAC analysis is shown to produce better qualitative and quantitative results than using each method separately. For instance, for all three test mixtures, the GRAM-PARAFAC approach improved quantitative accuracy by at least a factor of 4 and quantitative precision by more than 2 when compared to GRAM alone. This paper also introduces a new means of correcting run-to-run retention time shifts in comprehensive 2D chromatographic data.

Use of second-order calibration for residue screening of some triazines in the presence of coeluting interferences by gas chromatography–selected ion mass spectrometry

The quantities of residues of some triazines such as prometon, propazine, atrazine and simazine in complex matrices of apple samples were determined, using gas chromatography–selected ion mass (GC–SIM) spectrometry. Generalized rank annihilation method (GRAM) as a second-order calibration technique was used for screening, resolving and finally determining the amounts of the residues. Before the GRAM analysis, different steps of data preprocessing such as background correction, de-skewing and standardization for rank alignment was used for every target analyte. The de-skewing and rank alignment algorithms were used for bilinearity and trilinearity corrections, respectively. The two data matrices needed for the GRAM analysis were obtained from the sample solution, which contained different spiked levels of target analytes, and standard addition solution GC–SIM data. By using the GRAM method we were able to obtain the mass chromatograms and the true ion abundance ratio of each target analyte and precisely quantify the residue amounts of triazines in the different sample solutions by using only one standard data matrix. Unambiguous detection of a target analyte by this method was compared with the traditional approach of measuring peak height that fails to accurately detect the target because of severe coeluting interferences. Finally the method was successfully used for the simultaneous determination of two-target analytes of simazine and atrazine in a sample solution with severely convoluted mass chromatographic profiles.

Quantification from highly drifted and overlapped chromatographic peaks using second-order calibration methods

For determining low levels of pesticides and phenolic compounds in river and wastewater samples by high performance liquid chromatography (HPLC), solid phase extraction (SPE) is commonly used before the chromatographic separation. This preconcentration step is not necessarily selective for the analytes of interest and it may retain other compounds of similar characteristics as well. In this case, we present, humic and fulvic acids caused a large baseline drift and overlapped the analytes to be quantified. The inaccurate determinations of the area of the peaks of these analytes made it difficult to quantify them with univariate calibration. Here we compare three second-order calibration algorithms (generalized rank annihilation method (GRAM), parallel factor analysis (PARAFAC) and multivariate curve resolution–alternating least squares (MCR–ALS)) which efficiently solve this problem. These methods use second-order data, i.e., a matrix of responses for each peak, which is easily obtained with a high performance liquid chromatography–diode array detector (HPLC–DAD). With these methods, the area does not need to be directly measured and predictions are more accurate. They also save time and resources because they can quantify analytes even if the peaks are not resolved. GRAM and PARAFAC require trilinear data. Biased and imprecise concentrations (relative standard deviation, %R.S.D.=34) were obtained without correcting the time-shift. Hence, a time-shift correction algorithm to align the peaks was needed to obtain accurate predictions. MCR–ALS was the most robust to the time-shift. All three algorithms provided similar mean predictions, which were comparable to those obtained when sulfite was added to the samples. However, the predictions for the different replicates were more similar for the second-order algorithms (%R.S.D.=3) than the ones obtained by univariate calibration after the sulfite addition (%R.S.D.=13).

Graphical criterion for assessing trilinearity and selecting the optimal number of factors in the generalized rank annihilation method using liquid chromatography–diode array detection data

A weight parameter ( α) was introduced into the generalized rank annihilation method (GRAM) to calculate and reduce the bias in the predicted concentration. Here we show that α can be used as an indicator to determine whether the trilinearity assumptions are met and to select the right number of factors to calculate the model. The procedure is to calculate several GRAM models by varying α and the number of factors. When the experimental data are trilinear and the right number of factors is used, α does not affect the predicted concentration. If the condition of trilinearity is not met or the correct number of factors is not used, the predicted concentration changes when α changes. A graph shows this behavior. Both simulated and real data were checked for trilinearity. Deviations from the ideal mathematical model, such as the time shift or different shape of the chromatographic profiles, were simulated. These parameters have an effect but the greatest effect was produced by the selection of the number of factors.

Second-order bilinear calibration for determining polycyclic aromatic compounds in marine sediments by solvent extraction and liquid chromatography with diode-array detection

A method based on solvent extraction followed by chromatographic separation with diode-array detection has been applied to determine polycyclic aromatic hydrocarbons (PAHs) in marine sediment samples. Because interfering substances coeluted with the compounds studied, second-order calibration using the Generalized Rank Annihilation Method (GRAM) was used for quantification. For calibration, a spiked sample was used instead of a standard solution because the complexity of the samples made us suspect the presence of matrix effect. With this method, the compounds do not need to be completely separated from the interferences and quantification can be carried out with only one spiked sample for calibration. Hence, the quantification process is shorter than the one based on univariate calibration. The ability of the method to quantify the studied compounds was evaluated by comparing the results with those of a univariate standard addition calibration, which involved the previous isolation of the analytes.

Chemometric approach for the resolution and quantification of unresolved peaks in gas chromatography–selected-ion mass spectrometry data

A semiautomated and integrated chemometric approach is presented for the resolution and quantification of unresolved target-analyte signals in gas chromatography-selected-ion monitoring (GC-SIM) data collected using scanning mass spectrometers. The chemometric approach utilizes an unskewing algorithm and two multivariate chemometric methods known as rank alignment and the generalized rank annihilation method (GRAM). The unskewing algorithm corrects the retention-time differences within a single GC-SIM data matrix caused by using a scanning mass spectrometer. Rank alignment objectively corrects the run-to-run retention-time difference between a sample GC-SIM data matrix and a standard addition GC-SIM data matrix. GRAM analysis uses the sample and standard addition data matrices to mathematically resolve and quantify the target-analyte signal(s). The resolution and quantification of severely unresolved target-analyte signals are demonstrated using GC-SIM data obtained from conventional heart-cut two-dimensional gas chromatography with mass spectrometric detection. In addition, the GC-SIM data is used to demonstrate the result of chemometric analysis when the absence of a target-analyte signal is obscured by interference. Chemometric analysis is shown to unambiguously detect an analyte based on its resolved mass chromatograms in situations where the traditional approach of measuring peak height fails to positively detect it. The predicted analyte concentrations are within 8% of the reference concentrations.

Chemometric analysis of comprehensive two-dimensional gas chromatography data using cryogenic modulation

The two-dimensional (2D) data structure generated under a high resolution GC×GC system with a small number of samplings taken across the first dimension is evaluated for the purpose of the application of chemometric deconvolution methods. Chemometric techniques such as generalized rank annihilation method (GRAM) place high demands on the reproducibility of chromatographic experiments. For GRAM to be employed for GC×GC data interpretation, it is critical that the separation method provides data with a bilinear structure; the peak-shape and retention times on both columns must be reproducible. With a limited number of samplings across a 1 D (first dimension) peak (e.g. four to six samplings) repeatability of the pattern of the modulated peaks (controlled by the modulation phase) becomes important in producing a bilinear data structure. Reproducibility of modulation phase can be affected by both reliability of the modulation period and reproducibility of the retention time of the peak on the first column (which arises from oven temperature and carrier flow rate stability). Evaluation of within-run and run-to-run retention time reproducibility (retention time uncertainty) on both columns, and modulation phase reproducibility using a modulated cryogenic system for a pair of overlapping components (fatty acid methyl esters) was undertaken. An investigation of the quality of data to permit quantification of each component by using GRAM deconvolution, was also conducted. Less than 4% run-to-run retention time uncertainty was obtained on column 1 and less than 9% run-to-run and within-run retention time uncertainty was obtained on column 2, where these R.S.D. measures are reported normalised to peak widths on each respective dimension. The R.S.D. of duplicate quantification results by GRAM ranged from 2 to 26% although the average quantification error using GRAM was less than 5%.

Parallel column liquid chromatography with a single multi-wavelength absorbance detector for enhanced selectivity using chemometric analysis

The selectivity of high performance liquid chromatography (HPLC) separations is increased using a parallel column configuration. In this system, an injected sample is first split between two HPLC columns that provide complementary separations. The effluent from the two columns is recombined prior to detection with a single multiwavelength absorbance detector. Complementary stationary phases are used so that each chemical component produces a detected concentration profile consisting of two peaks. A parallel column configuration, when coupled with multivariate detection, provides increased chemical selectivity relative to a single column configuration with the same multivariate detection. This enhanced selectivity is achieved by doubling the number of peaks in the chromatographic dimension while keeping the run time constant. Unlike traditional single column separation methodology, the parallel column system sacrifices chromatographic resolution while actually increasing the chemical selectivity, thus allowing chemometric data analysis methods to mathematically resolve the multivariate chromatographic data. The parallel column system can be used to reduce analysis times for partially resolved peaks and simplify initial method development as well as provide a more robust methodology if and when subsequent changes in the sample matrix occur (such as when new interferences show up in subsequent samples). Here, a mixture of common aromatic compounds were separated with this system and analyzed using the generalized rank annihilation method (GRAM). Analytes that were significantly overlapped on both stationary phases applied, ZirChrom PBD and CARB phases, when used in traditional single column format, were successfully quantified with a R.S.D.% of typically 2% when the same stationary phases were used in the parallel column format. These results indicate that a parallel column system should substantially improve the chemical selectivity and quantitative precision of the analysis relative to a single-column instrument.

Using second-order calibration to identify and quantify aromatic sulfonates in water by high-performance liquid chromatography in the presence of coeluting interferences

We used the Generalized Rank Annihilation Method (GRAM), a second-order calibration method, to quantify aromatic sulfonates in water with high-performance liquid chromatography (HPLC) when interferences coeluted with the analytes of interest. With GRAM, we can quantify in only two chromatographic analyses, one for a calibration sample and one for the unknown sample. The calculated concentrations were not statistically different to those obtained when the chromatographic separation of the unknown sample was modified in order to completely separate the analyte from the interferences before univariate calibration. With GRAM, the concentrations are determined much more quickly because a complete resolution is not required.

Time shift correction in second-order liquid chromatographic data with iterative target transformation factor analysis

When the generalized rank annihilation method (GRAM) is applied to liquid chromatographic data with diode-array detection, an important problem is the time shift of the peak of the analyte in the test sample. This problem leads to erroneous predictions. This time shift can be corrected if a time window is selected so that the chromatographic profile of the analyte in the test sample is trilinear with the peak of the analyte in the calibration sample. In this paper we present a new method to determine when this condition is met. This method is based on the curve resolution with iterative target transformation factor analysis (ITTFA). The calibration and test matrices are independently decomposed into profiles and spectra, and aligned before GRAM is applied. Here we study two situations: first, when the calibration matrix has one analyte and second, when it has two analytes. When the calibration matrix has two analytes, we selectively determine the time window for the analyte to be quantified. There were considerably fewer prediction errors after correction.

Second-order bilinear calibration: the effects of vectorising the data matrices of the calibration set

In a groundbreaking paper, Linder and Sundberg [Chemometr. Intell. Lab. Syst. 42 (1998) 159] developed a statistical framework for the calibration of second-order bilinear data. Within this framework, they formulated three different predictor construction methods [J. Chemom. 16 (2002) 12], namely the so-called naı̈ve method, the bilinear least squares (BLLS) method, and a refined version of the latter that takes account of the calibration uncertainty. Elsewhere [J. Chemom. 15 (2001) 743], a close relationship is established between the naı̈ve method and the generalized rank annihilation method (GRAM) by comparing expressions for prediction variance. Here it is proved that the BLLS method can be interpreted to work with vectorised data matrices, which establishes an algebraic relationship with so-called unfold partial least squares (PLS) and unfold principal component regression (PCR). It is detailed how these results enable quantifying the effects of vectorising bilinear second-order data matrices on analytical figures of merit and variance inflation factors.

Towards a rehabilitation of the generalized rank annihilation method (GRAM).

The trilinear PARAFAC model occupies a central place in multiway analysis, because the components of a data array can often be uniquely resolved. This paper compares the resolution for a large variety of methods, namely the generalized rank annihilation method (GRAM), alternating least squares (ALS), alternating trilinear decomposition (ATLD), alternating coupled vectors resolution (ACOVER), alternating slice-wise diagonalization (ASD), alternating coupled matrices resolution (ACOMAR), self-weighted alternating trilinear decomposition (SWATLD), and pseudo alternating least squares (PALS). The comparison was conducted using Monte Carlo simulations. It was shown that GRAM performs well for moderately and highly overlapped data. These results argue strongly against the previously claimed superiority of the alternatives listed above.