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  • 1
    Electronic Resource
    Electronic Resource
    New York, NY : Wiley-Blackwell
    Journal of Chemometrics 8 (1994), S. 147-154 
    ISSN: 0886-9383
    Keywords: RAFA ; GRAM ; Eigenvalue problem ; Complex solution ; Degenerate solution ; Chemistry ; Analytical Chemistry and Spectroscopy
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology
    Notes: Rank annihilation factor analysis (RAFA) is a method for multicomponent calibration using two data matrices simultaneously, one for the unknown and one for the calibration sample. In its most general form, the generalized rank annihilation method (GRAM), an eigenvalue problem has to be solved. In this first paper different formulations of GRAM are compared and a slightly different eigenvalue problem will be derived. The eigenvectors of this specific eigenvalue problem constitute the transformation matrix that rotates the abstract factors from principal component analysis (PCA) into their physical counterparts. This reformulation of GRAM facilitates a comparison with other PCA-based methods for curve resolution and calibration. Furthermore, we will discuss two characteristics common to all formulations of GRAM, i.e. the distinct possibility of a complex and degenerate solution. It will be shown that a complex solution-contrary to degeneracy-should not arise for components present in both samples for model data.
    Type of Medium: Electronic Resource
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  • 2
    Electronic Resource
    Electronic Resource
    New York, NY : Wiley-Blackwell
    Journal of Chemometrics 8 (1994), S. 273-285 
    ISSN: 0886-9383
    Keywords: GRAM ; Least-squares problem ; Eigenvalue problem ; NIPALS ; Performance index ; Condition number ; Chemistry ; Analytical Chemistry and Spectroscopy
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology
    Notes: In this paper we discuss the practical implementation of the generalized rank annihilation method (GRAM). The practical implementation comes down to developing a computer program where two critical steps can be distinguished: the construction of the factor space and the oblique rotation of the factors. The construction of the factor space is a least-squares (LS) problem solved by singular value decomposition (SVD), whereas the rotation of the factors is brought about by solving an eigenvalue problem. In the past several formulations for GRAM have been published. The differences essentially come down to solving either a standard eigenvalue problem or a generalized eigenvalue problem. The first objective of this paper is to discuss the numerical stability of the algorithms resulting from these formulations. It is found that the generalized eigenvalue problem is only to be preferred if the construction of the factor space is not performed with maximum precision. This is demonstrated for the case where the dominant factors are calculated by the non-linear iterative partial least-squares (NIPALS) algorithm. Several performance measures are proposed to investigate the numerical accuracy of the computed solution. The previously derived bias and variance are proposed to estimate the number of physically significant digits in the computed solution. The second objective of this paper is to discuss the relevance of theoretical considerations for application of GRAM in the presence of model errors.
    Additional Material: 1 Ill.
    Type of Medium: Electronic Resource
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  • 3
    ISSN: 0886-9383
    Keywords: RAFA ; GRAM ; Eigenvalues ; Bias ; Variance ; Chemistry ; Analytical Chemistry and Spectroscopy
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology
    Notes: Rank annihilation factor analysis (RAFA) is a method for multicomponent calibration using two data matrices simultaneously, one for the unknown and one for the calibration sample. In its most general form, the generalized rank annihilation method (GRAM), an eigenvalue problem has to be solved. In this second paper expressions are derived for predicting the bias and variance in the eigenvalues of GRAM. These expressions are built on the analogies between a reformulation of the eigenvalue problem and the prediction equations of univariate and multivariate calibration. The error analysis will also be performed for Lorber's formulation of RAFA. It will be demonstrated that, depending on the size of the eigenvalue, large differences in performance must be expected. A bias correction technique is proposed that effectively eliminates the bias if the error in the bias estimate is not too large. The derived expressions are evaluated by Monte Carlo simulations. It is shown that the predictions are satisfactory up to the limit of detection. The results are not sensitive to an incorrect choice of the dimension of the factor space.
    Additional Material: 3 Ill.
    Type of Medium: Electronic Resource
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  • 4
    Electronic Resource
    Electronic Resource
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 87 (1987), S. 2263-2270 
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: The pair correlation function in a homogeneous hard sphere fluid at various densities has been measured in a large system, using the Monte Carlo method. The corresponding direct correlation function C2(r) has been determined directly from these measurements, and will be given here in closed form. We conclude that density functional models that neglect the effect of C3 and higher order direct correlation functions, that are defined in bulk fluids, are not able to describe an inhomogeneous hard sphere fluid accurately.
    Type of Medium: Electronic Resource
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  • 5
    ISSN: 0886-9383
    Keywords: Standard errors ; Eigenvalues ; PCA ; MLR ; GRAM ; Rank estimation ; Chemistry ; Analytical Chemistry and Spectroscopy
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology
    Notes: New 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.
    Additional Material: 8 Ill.
    Type of Medium: Electronic Resource
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