Estimation of prediction error for samples within the calibration range

Karstang, Terje V.; Toft, Jostein; Kvalheim, Olav M.

doi:10.1002/cem.1180060403

Cited by 17 publications

(10 citation statements)

References 12 publications

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“…A literature survey shows that deriving formulas using the method of error propagation has been a major research topic. When employing first-order multivariate data, most publications are concerned with standard (i.e., linear) PLSR [22,23,[82][83][84][85][86][87][88][89][90][91][92][93][94][95][96][97][98][99][100][101] 8 The limit of detection is the analyte level that with sufficiently high probability (1 -β) will lead to a correct positive detection decision. The detection decision amounts to comparing the prediction ĉ with the critical level (L c ).…”

Section: Previously Proposed Methodology In Multivariate Calibrationmentioning

confidence: 99%

Uncertainty estimation and figures of merit for multivariate calibration (IUPAC Technical Report)

Olivieri

Faber²,

Ferré

et al. 2006

Pure and Applied Chemistry

325

166

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This paper gives an introduction to multivariate calibration from a chemometrics perspective and reviews the various proposals to generalize the well-established univariate methodology to the multivariate domain. Univariate calibration leads to relatively simple models with a sound statistical underpinning. The associated uncertainty estimation and figures of merit are thoroughly covered in several official documents. However, univariate model predictions for unknown samples are only reliable if the signal is sufficiently selective for the analyte of interest. By contrast, multivariate calibration methods may produce valid predictions also from highly unselective data. A case in point is quantification from near-infrared (NIR) spectra. With the ever-increasing sophistication of analytical instruments inevitably comes a suite of multivariate calibration methods, each with its own underlying assumptions and statistical properties. As a result, uncertainty estimation and figures of merit for multivariate calibration methods has become a subject of active research, especially in the field of chemometrics.

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Section: Previously Proposed Methodology In Multivariate Calibrationmentioning

confidence: 99%

Uncertainty estimation and figures of merit for multivariate calibration (IUPAC Technical Report)

Olivieri

Faber²,

Ferré

et al. 2006

Pure and Applied Chemistry

325

166

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“…T a ͪs (6) Special notation to emphasize the dependence of ␤ PCR on A is avoided for simplicity. The inversion of S is stabilized by discarding PCs associated with small eigenvalues, i.e.…”

Section: Estimating the Regression Vector By Pcrmentioning

confidence: 99%

“…In this paper the leverage is defined with respect to the mean. Martens and Naes 1 formulate the second term in terms of the score vector of the unknown sample in the calibration space, which is also the notation preferred by Karstang et al 6 …”

Section: Variance In the Predicted Analyte Concentrationmentioning

confidence: 99%

“…This sample-dependent term constitutes a theoretical limit for the prediction variance. Lorber and Kowalski 5 and Karstang et al 6 give expressions for the standard error in the predicted analyte concentration for mean-centered data (see equation (16) in Reference 5 and equation (9) in Reference 6). A comparison with these results is facilitated by working out equation (60) in terms of the unknown sample leverage h u and applying the zeroth-order approximation in the case of PCR and PLS to giveV…”

Section: Variance In the Predicted Analyte Concentrationmentioning

confidence: 99%

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Propagation of measurement errors for the validation of predictions obtained by principal component regression and partial least squares

1997

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SUMMARYMultivariate calibration aims to model the relation between a dependent variable, e.g. analyte concentration, and the measured independent variables, e.g. spectra, for complex mixtures. The model parameters are obtained in the form of a regression vector from calibration data by regression methods such as principal component regression (PCR) or partial least squares (PLS). Subsequently, this regression vector is used to predict the dependent variable for unknown mixtures. The validation of the obtained predictions is a crucial part of the procedure, i.e. together with the point estimate an interval estimate is desired. The associated prediction intervals can be constructed from the covariance matrix of the estimated regression vector. However, currently known expressions for PCR and PLS are derived within the classical regression framework, i.e. they only take the uncertainty in the dependent variable into account. This severely limits their capability for establishing realistic prediction intervals in practical situations. In this paper, expressions are derived using the method of error propagation that also account for the measurement errors in the independent variables. An exact linear relation is assumed between the dependent and independent variables. The obtained expressions are therefore valid for the classical errors-in-variables (EIV) model. In order to make the presentation reasonably self-contained, relevant expressions are reviewed for the classical regression model as well as the classical EIV model, especially for ordinary least squares (OLS). The consequences for the limit of detection, wavelength selection, sample selection and local modeling are discussed. Diagnostics are proposed to determine the adequacy of the approximations used in the derivations. Finally, PCR and PLS are so-called biased regression methods. Compared with OLS, they yield small variance at the expense of increased bias. It follows that bias may be an important ingredient of the obtained predictions. Therefore considerable attention is paid to the quantification of bias and new stopping rules for model selection in PCR and PLS are proposed. The theoretical ideas are illustrated by the analysis of real data taken from the literature (classical regression model) as well as simulated data (classical EIV model).

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“…The expressions of Karstang et al 40 differ from those of Bauer et al 24 by the substitution of the matrix of sensitivities by the score matrix. The use of scores enables the identification of the position of a sample in the predictor space.…”

Section: ( I J + 6 E ) -* U T = Q(g+de)-luti Q ( E -L -D E E -2 ) Imentioning

confidence: 93%

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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Estimation of prediction error for samples within the calibration range

Cited by 17 publications

References 12 publications

Uncertainty estimation and figures of merit for multivariate calibration (IUPAC Technical Report)

Uncertainty estimation and figures of merit for multivariate calibration (IUPAC Technical Report)

Propagation of measurement errors for the validation of predictions obtained by principal component regression and partial least squares

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

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