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  • 1
    Electronic Resource
    Electronic Resource
    Standard errors in the eigenvalues of a cross-product matrix: Theory and applications (1993)
    New York, NY : Wiley-Blackwell
    Journal of Chemometrics 7 (1993), S. 495-526 
    Details
    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
    URL: http://dx.doi.org/10.1002/cem.1180070605
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  • 2
    Electronic Resource
    Electronic Resource
    Generalized rank annihilation method. II: Bias and variance in the estimated eigenvalues (1994)
    New York, NY : Wiley-Blackwell
    Journal of Chemometrics 8 (1994), S. 181-203 
    Details
    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
    URL: http://dx.doi.org/10.1002/cem.1180080303
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  • 3
    Electronic Resource
    Electronic Resource
    Generalized rank annihilation method. III: Practical implementation (1994)
    New York, NY : Wiley-Blackwell
    Journal of Chemometrics 8 (1994), S. 273-285 
    Details
    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
    URL: http://dx.doi.org/10.1002/cem.1180080406
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  • 4
    Electronic Resource
    Electronic Resource
    Evaluation of curve resolution and iterative target transformation factor analysis in quantitative analysis by liquid chromatography (1987)
    New York, NY : Wiley-Blackwell
    Journal of Chemometrics 1 (1987), S. 57-71 
    Details
    ISSN: 0886-9383
    Keywords: Curve resolution ; Liquid chromatography ; Accuracy ; Chemistry ; Analytical Chemistry and Spectroscopy
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology
    Notes: The performance of curve resolution and iterative target transformation curve resolution in quantitative liquid chromatography with a diode array detector is evaluated. Quantitative accuracy of the elution profiles obtained for two- and three-component mixtures has been evaluated as a function of spectral similarity, chromatographic resolution and ratio of the peak areas, using a number of simulations analysed by an analysis of variance. The results obtained by simulation are compared and validated with the results obtained for the separation of phenylene diamines in reversed phase liquid chromatography and proteins in gel permeation liquid chromatography. The results that were obtained are summarized and the chromatographic implications are discussed.
    Additional Material: 3 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/cem.1180010108
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  • 5
    Electronic Resource
    Electronic Resource
    Program for processing of historical laboratory data based on fuzzy set theory as a tool for decision support (1989)
    New York, NY : Wiley-Blackwell
    Journal of Chemometrics 3 (1989), S. 81-95 
    Details
    ISSN: 0886-9383
    Keywords: Fuzzy set theory ; Classification ; Data reduction ; Historical laboratory data ; Chemistry ; Analytical Chemistry and Spectroscopy
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology
    Notes: A computer program (FEA) is presented for processing historical laboratory data. It performs on a list of sample entries stored in a laboratory information management system. Using an algorithm which is based on fuzzy set theory, FEA classifies the entries into a limited number of clusters called sample types. The classification is fully user-defined. The program transforms the historical data into a representation which is more suitable for studying the performance of the laboratory or which can be used as preparation for a simulation project.
    Additional Material: 8 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/cem.1180030109
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  • 6
    Electronic Resource
    Electronic Resource
    The performance of least squares and robust regression in the calibration of analytical methods under non-normal noise distributions (1989)
    New York, NY : Wiley-Blackwell
    Journal of Chemometrics 3 (1989), S. 329-342 
    Details
    ISSN: 0886-9383
    Keywords: Calibration ; Robust regression ; Iteratively reweighted least squares ; M-estimator ; Monte Carlo simulations ; Chemistry ; Analytical Chemistry and Spectroscopy
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology
    Notes: By means of Monte Carlo simulations a comparison has been made between ordinary least squares regression and robust regression. The robust regression procedure is based on the Huber estimate and is computed by means of the iteratively reweighted least squares algorithm. The performance of both procedures has been evaluated for estimation of the parameters of a calibration function and for determination of the concentration of unknown samples. The influence of the distributional characteristics skewness and kurtosis has been studied, and the number of measurements used for constructing the calibration curve has also been taken into account. Under certain conditions robust regression offers an advantage over least squares regression.
    Additional Material: 6 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/cem.1180030203
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  • 7
    Electronic Resource
    Electronic Resource
    Expert system for the characterization of laboratory sample input signals: AUTOCORR (1989)
    New York, NY : Wiley-Blackwell
    Journal of Chemometrics 3 (1989), S. 511-527 
    Details
    ISSN: 0886-9383
    Keywords: Expert system ; Time series analysis ; Autocorrelation ; Laboratory input signals ; Chemistry ; Analytical Chemistry and Spectroscopy
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology
    Notes: An expert system is presented for automated time series analysis of laboratory sample input signals. The system, AUTOCORR, builds a model of the time series by identifying the processes that are present. These are an uncorrelated random process and, underlying this, possibly one or more of the following: a first-order autoregressive process, a trend and a periodic process. AUTOCORR has a knowledge base of 44 rules and 41 facts for this purpose. The employed shell, INFER, allows the use of algorithmic procedures. Elaborate tests with simulated signals show that AUTOCORR has a very low false positive score and is successful in describing time series for laboratory simulation models.
    Additional Material: 11 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/cem.1180030308
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  • 8
    Electronic Resource
    Electronic Resource
    Generalized rank annihilation method. I: Derivation of eigenvalue problems (1994)
    New York, NY : Wiley-Blackwell
    Journal of Chemometrics 8 (1994), S. 147-154 
    Details
    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
    URL: http://dx.doi.org/10.1002/cem.1180080206
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    Paper (German National Licenses)
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  • 9
    Electronic Resource
    Electronic Resource
    The construction of simultaneous optimal experimental designs for several polynomials in the calibration of analytical methods (1990)
    New York, NY : Wiley-Blackwell
    Journal of Chemometrics 4 (1990), S. 171-185 
    Details
    ISSN: 0886-9383
    Keywords: Calibration ; Polynomial regression ; D-optimality ; Experimental design ; Model discrimination ; Lack of fit ; Least squares regression ; Chemistry ; Analytical Chemistry and Spectroscopy
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology
    Notes: In this paper a criterion is described for the construction of experimental designs for the evaluation of calibration models in analytical chemistry. The proposed criterion seeks a compromise between the D-optimal designs for estimating the parameters of different polynomial models. A computer algorithm is presented for a sequential construction of experimental designs using the optimality criterion. The performance of the optimality criterion and the computer algorithm is elaborated for the problem of discrimination between a first- to a third-degree polynomial for the calibration of analytical methods. An experimental design consisting of replicate measurements at five distinct levels equally spaced over the calibration range proved a good solution.
    Additional Material: 6 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/cem.1180040208
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    Paper (German National Licenses)
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