Tensorial calibration: I. First‐order calibration

Sanchez, Eugenio; Kowalski, Bruce R.

doi:10.1002/cem.1180020404

Cited by 111 publications

(53 citation statements)

References 40 publications

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“…We here discuss the prediction phase from a relatively general point of view, without specifying the calibration method used, but assuming that calibration has been carried out, resulting in estimated profiles b A and b B of model (1). A new specimen of unknown diagonal concentration matrix C 0 , with concentration vector c 0 along the diagonal, yields a response matrix Z 0 , which is supposed to follow the same bilinear model (1) as the calibration samples, but with unknown c 0 :…”

Section: Prediction Methodsmentioning

confidence: 99%

“…Sanchez and Kowalski [1] coined the terms zero-, first-and second-order calibration. Zero-order calibration is the univariate situation, represented by for example a monochromatic photometer and a single constituent to be quantified.…”

Section: Second-order Calibrationmentioning

confidence: 99%

“…The data arrays of hyphenated instruments are even richer, and one calibration specimen is in principle enough to determine several constituents in a new specimen (GRAM [2]). Also, in second-order calibration it is possible not only to detect but also to correct for the presence of interfering species [1]. This so-called second-order advantage is occasionally even found as part of the definition of second-order calibration.…”

Section: Second-order Calibrationmentioning

confidence: 99%

“…However, this would obscure the special role of the concentrations in a calibration±prediction situation, so we will keep to the form (1).…”

mentioning

confidence: 99%

“…However, as long as we do not need identifiability, we will use this model representation (1). When it comes to precision of estimation of the model parameters, we will need unique parameters.…”

mentioning

confidence: 99%

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Precision of prediction in second‐order calibration, with focus on bilinear regression methods

Linder

Sundberg

2002

Journal of Chemometrics

103

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We consider calibration of hyphenated instruments with particular focus on determination of the unknown concentrations of new specimens. A hyphenated instrument generates for each specimen a two-way array of data. These are assumed to depend on the concentrations through a bilinear regression model, where each constituent is characterized by a pair of profiles to be determined in the calibration. We discuss the problem of predicting the unknown concentrations in a new specimen, after calibration. We formulate three different predictor construction methods, a`naive' method, a least squares method, and a refined version of the latter that takes account of the calibration uncertainty. We give formulae for the uncertainty of the predictors under white noise, when calibration can be seen as precise. We refine these formulae to allow for calibration uncertainty, in particular when calibration is carried out by the bilinear least squares (BLLS) method or the singular value decomposition (SVD) method proposed by Linder and Sundberg (Chemometrics Intell. Lab. Syst. 1998; 42: 159±178). By error propagation formulae and previous results on the precision of bA and b B we can obtain approximate standard errors for the predicted concentrations, according to each of the two estimation methods. The performance of the predictors and the precision formulae is illustrated on both real (fluorescence) and simulated data.

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Section: Prediction Methodsmentioning

confidence: 99%

Section: Second-order Calibrationmentioning

confidence: 99%

Section: Second-order Calibrationmentioning

confidence: 99%

“…However, this would obscure the special role of the concentrations in a calibration±prediction situation, so we will keep to the form (1).…”

mentioning

confidence: 99%

mentioning

confidence: 99%

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Precision of prediction in second‐order calibration, with focus on bilinear regression methods

Linder

Sundberg

2002

Journal of Chemometrics

103

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Second‐Order Calibration and Higher

Fleming

Kowalski

2000

Encyclopedia of Analytical Chemistry

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Most analytical chemists recognize calibration as the quantitation of analytes by relating the analyte signal and analyte concentration, although it is also possible to obtain information about other characteristics of the sample through calibration. In the calibration process, a number of samples that have known values of the characteristic to be determined, e.g. concentration, are measured. These are reference measurements, and are known as calibration standards. Traditionally, the measurements have consisted of a single number (a univariate or scalar measurement); they are plotted against the known concentration, to generate a “calibration curve”. (An example of a scalar measurement is a pH value or the area under the curve of a chromatographic peak.) When unknown (test) samples are measured under the same experimental conditions as the standards, their concentrations may be found by regressing their measured signals against the standard calibration curve. However, a significant disadvantage of univariate calibration is that a sensor must be fully selective for the analyte of interest, since any interfering species that may be present cannot be detected when measurements are scalar. This has led to the development of “first‐order” calibration methods. These techniques use the relationship between an array of measurements (such as a chromatogram or spectrum) and the analyte concentration to develop a calibration model from which analyte concentrations in test samples can be determined. The array of measurements is a vector, and is known as a first‐order tensor, while the calibration model is equivalent to the calibration curve. Since first‐order calibration uses many measurements per sample to generate a model, rather than just a single measurement as in the univariate case, it has a number of advantages. For example, multianalyte analysis is made possible as long as there is a standard available for each analyte present in a sample, and outlier detection is also feasible. These types of methods have become popular in recent years, and include partial least squares (PLS) and principal component regression (PCR). It makes sense that when more data are available per sample, more information may be extracted, and this is the source of the success of first‐order methods. It therefore follows that, if an analyte signal consists of a matrix of data (also known as a second‐order tensor), even more advantages exist. [Examples of instruments that produce matrices of data for each sample analyzed include liquid chromatography/ultraviolet detection (LC/UV) and gas chromatography/mass spectrometry (GC/MS).] This situation allows powerful calibration techniques, known as second‐order methods, to be used. The primary advantage of these methods is known as the “second‐order advantage”; it allows analytes to be quantitated even in the presence of unmodeled interferents. In other words, the calibration standards do not have to contain any information about the interfering species in the sample. This is of particular value in situations where complex samples are being analyzed, as they do not have to be fully characterized for quantitation of a single analyte. The disadvantage of these methods, however, is their complexity: there are often decisions to be made, such as the values to assign to various parameters, and each method has analysis situations which are more favorable than others. The most common second‐order analysis methods are alternating least‐squares (ALS) methods, such as parallel factor analysis (PARAFAC) and multivariate curve resolution (MCR), and eigenvalue–eigenvector based methods, such as the generalized rank annihilation method (GRAM) and direct trilinear decomposition (DTD).

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Chemometrics in Bioanalytical Chemistry

Yin

Xie

et al. 2016

Encyclopedia of Analytical Chemistry

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Chemometrics, as a young science in chemical discipline, has been widely applied in many fields, such as environmental, clinical and forensic, food, industrial, biological, theoretical, and physical sciences. With the advent of postgenomic era, a powerful tool was urgently needed to extract useful chemical information from huge biological data, which results in the development and application of chemometrics in bioanalytical chemistry. At present, the application of chemometrics in bioanalytical chemistry mainly involves in quantitative bioanalysis and ‘‐omics’. This article focuses on multivariate and multiway calibration in quantitative bioanalysis, which covers fundamentals, current situations, innovative applications, and research trends of these chemometric methods.

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Tensorial calibration: I. First‐order calibration

Cited by 111 publications

References 40 publications

Precision of prediction in second‐order calibration, with focus on bilinear regression methods

Precision of prediction in second‐order calibration, with focus on bilinear regression methods

Second‐Order Calibration and Higher

Chemometrics in Bioanalytical Chemistry

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