Rapid estimation of the number of coeluting components in liquid chromatography by factor analysis

Shen, Chengbo; Vickers, Thomas J.; Mann, Charles K.

doi:10.1002/cem.1180050503

Cited by 12 publications

(3 citation statements)

References 21 publications

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“…The number of signi® cant abstract factors was determined by using the power spectra method described by Mann and co-workers. 8 Three signi® cant abstract factors were identi® ed (Fig. 4).…”

Section: Resultsmentioning

confidence: 99%

Use of Multivariate Analysis to Determine Temperature from Low-Resolution Infrared Spectra of Carbon Dioxide

Szczepanski

Fountain

2000

Appl Spectrosc

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The remote optical monitoring of gaseous contaminants is important for both military and industrial applications. An important parameter for quantifying chemical species and for predicting plume dynamics is the temperature. While in some industrial monitoring situations it may be practical to independently measure the temperature of stack emissions, for compliance monitoring and military chemical reconnaissance a remote optical means of estimating gas plume temperature is required. It was noticed that the band shape of low-resolution spectra of carbon dioxide in equilibrium with an exhaust plume was very sensitive to temperature. Spectra of carbon dioxide were acquired under controlled laboratory conditions in 5° increments from 20 to 200 °C. Various multivariate models were used to predict the temperature. It was found that partial least-squares (PLS) was unable to effectively model the simultaneous changes in amplitude and bandwidth with temperature. However, principal component regression (PCR) was found to be well correlated with temperature and allowed cross-validated prediction within 4% error.

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Section: Resultsmentioning

confidence: 99%

Use of Multivariate Analysis to Determine Temperature from Low-Resolution Infrared Spectra of Carbon Dioxide

Szczepanski

Fountain

2000

Appl Spectrosc

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“…The human eye is seen to be an excellent pattern recognizer. The application of this method is, however, restricted to ordered data [48]. In the light of the above results it is reasonable to demand that the significance tests yield certainly three significant PCs for EXPl and EXP2 and possibly two for EXP3.…”

Section: Simulated Datamentioning

confidence: 99%

Aspects of pseudorank estimation methods based on an estimate of the size of the measurement error

Faber

Buydens

Kateman

1994

Analytica Chimica Acta

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The estimation of the pseudorank of a matrix, i.e., the rank of a matrix in the absence of measurement error, is a major problem in multivariate data analysis. In the practice of analytical chemistry it is often even the only problem. An important example is the determination, of the purity of a chromatographic peak. In this paper we discuss three pseudorank estimation methods that make use of prior knowledge about the size of the measurement error. The first method (Method A) is based on the standard errors in the diagonal elements of the row-echelon form of the matrix, the second method (Method Bl is based on the eigenvalues of principal component analysis @CA) and the third method (a t-test) is based on the singular values. Methods A and B are modifications of methods that are well known in analytical chemistry. However, these methods cannot provide significance levels for the estimated pseudorank. This holds for the original methods as well as the present modifications. The main reason for introducing these modifications is that in this way relationships are established between the t-test and methods that are already known. The aspects that are covered in this paper include the sampling distribution of the test statistic, the number of degrees of freedom to be used in the test, the adequacy of theoretical predictions and the bias that results from random measurement noise. The object of this paper is to demonstrate that using prior knowledge about the size of the measurement error may yield powerful pseudorank estimation methods. This is illustrated by comparing the significance levels obtained by the r-test and Malinowski's F-test. The r-test yields sharper significance levels for experimental data obtained from the literature as well as simulated data. This can be satisfactorily explained by the larger number of degrees of freedom that is employed in this test. The viability of the new t-test is supported by a thorough evaluation of the test data by a large number of conventional methods. As a remarkable by-product of the present investigation we find that a plot of the singular values yields a promising graphical pseudorank estimation method. Graphical methods have proved their use in the past in the case that the size of the measurement error is unknown. This new graphical method therefore provides a natural complement to the t-test.

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“…Visual inspection of the eigenvectors proves to be a very sensitive method for determining the number of significant factors for ordered data. 49 We have also executed these simulations with uniform noise. The results are summarized in Table 8.…”

Section: Standard Errors In the Eigenvalues Of Pcamentioning

confidence: 99%

Standard errors in the eigenvalues of a cross‐product matrix: Theory and applications

Faber

Buydens

Kateman

1993

Journal of Chemometrics

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SUMMARYNew expressions are derived for the standard errors in the eigenvalues of a cross-product matrix by the method of error propagation. Cross-product matrices frequently arise in multivariate data analysis, especially in principal component analysis (PCA). The derived standard errors account for the variability in the data as a result of measurement noise and are therefore essentially different from the standard errors developed in multivariate statistics. Those standard errors were derived in order to account for the finite number of observations on a fixed number of variables, the so-called sampling error. They can be used for making inferences about the population eigenvalues. Making inferences about the population eigenvalues is often not the purposes of PCA in physical sciences. This is particularly true if the measurements are performed on an analytical instrument that produces two-dimensional arrays for one chemical sample: the rows and columns of such a data matrix cannot be identified with observations on variables at all. However, PCA can still be used as a general data reduction technique, but now the effect of measurement noise on the standard errors in the eigenvalues has to be considered. The consequences for significance testing of the eigenvalues as well as the usefulness for error estimates for scores and loadings of PCA, multiple linear regression (MLR) and the generalized rank annihilation method (GRAM) are discussed. The adequacy of the derived expressions is tested by Monte Carlo simulations.

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Rapid estimation of the number of coeluting components in liquid chromatography by factor analysis

Cited by 12 publications

References 21 publications

Use of Multivariate Analysis to Determine Temperature from Low-Resolution Infrared Spectra of Carbon Dioxide

Use of Multivariate Analysis to Determine Temperature from Low-Resolution Infrared Spectra of Carbon Dioxide

Aspects of pseudorank estimation methods based on an estimate of the size of the measurement error

Standard errors in the eigenvalues of a cross‐product matrix: Theory and applications

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