Second‐Order Calibration and Higher

Abstract

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).

Alternating penalty trilinear decomposition algorithm for second‐order calibration with application to interference‐free analysis of excitation–emission matrix fluorescence data

“…Moreover, the proposed algorithm can overcome the slow convergence brought about by random initialization or high multicollinearity to some extent. In other words, the APTLD algorithm has a much higher convergence rate compared with the traditional PARAFAC algorithm [37][38]. The results of treating one simulated and one real excitation-emission spectral data set showed that the proposed algorithm performs well.…”

High-Order Calibration and Data Analysis in Chromatography

“…In addition, selecting diagonal elements makes ATLD retain trilinearity property indeed and be insensitive to the excessive estimation of component numbers. The advantages have been reviewed by Fleming and Kowalski [16]. Based on the above advantages, it is very suitable to handle second-order data obtained from hyphenated instruments, such as HPLC-DAD, LC-MS, and GC-MS.…”

A study on the differential strategy of some iterative trilinear decomposition algorithms: PARAFAC‐ALS, ATLD, SWATLD, and APTLD

“…The parallel factor analysis (PARAFAC) /canonical decomposition (CANDECOMP) model is a classical three‐way component model in chemometrics . With the development of second‐order calibration based on PARAFAC/CANDECOMP model, scientists found that the model can provide excellent selectivity even in the case when unknown interference exists . This feature has been recognized as “second‐order advantage” .…”