Practical Three-Way Calibration

Alejandro Olivieri, Graciela M. Escandar

• Publisher: Elsevier
• Year: 2014
• ISBN:
  9780124104549
• Topic: Physical Sciences
• Subtopic: Analytic Chemistry

Practical Three-Way Calibration is an introductory-level guide to the complex field of analytical calibration with three-way instrumental data. With minimal use of mathematical/statistical expressions, it walks the reader through the analytical methodologies with helpful images and step-by-step explanations. Unlike other books on the subject, there is no need for prior programming experience and no need to learn programming languages. Easy-to-use graphical interfaces and intuitive descriptions of mathematical and statistical concepts make three-way calibration methodologies accessible to analytical chemists and scientists in a wide range of disciplines in industry and academia.

• Numerous detailed examples of slowly increasing complexity
• Exposure to several different data sets and techniques through figures and diagrams
• Computer program screenshots for easy learning without prior knowledge of programming languages
• Minimal use of mathematical/statistical expressions

Chapter 1

Calibration Scenarios

Abstract

An introduction is provided to various calibration approaches: univariate, multivariate, and multiway, based on the complexity of the collected instrumental data. A basic nomenclature for the different data is introduced, using the complementary concepts of way and order. The advantages in going from classical univariate calibration to multiway calibration methodologies are discussed.

Keywords

Multivariate and multiway data; Multiway calibration advantages; Univariate

Do it your way, but do it multiway!

1.1. Calibration

According to the International Union of Pure and Applied Chemistry (IUPAC), calibration is, in a general sense, “an operation that relates an output quantity to an input quantity for a measuring system under given conditions”.¹ The input quantities of our primary interest, i.e., in analytical calibration, are the concentrations of a sample constituent of interest (the analyte), while the output quantities are analytical signals or responses delivered by analytical instruments (a spectrometer, chromatograph, voltammeter, etc.). Therefore, in the present context calibration will mean the operation of relating instrumental signals to analyte concentrations.

1.2. Univariate calibration

A specific case of the general calibration process is the one relating the content of a single analyte in a sample to a single value of an instrumental signal, and is called univariate calibration. In analytical chemistry, univariate calibration employs a calibration curve as a general method for the determination of the concentration of a constituent in an unknown sample.¹ The calibration curve is a plot of the instrumental response as a function of the concentration of the analyte. Usually, the analyst prepares a series of standards across a range of concentrations near the expected concentration of the analyte in the unknown sample. The concentrations of the standards lie within the working range of the employed technique (Figure 1.1). Analyzing each of these standards using the chosen technique will produce a series of measurements. For most analyses, a plot of instrumental response vs analyte concentration will show a linear relationship. After measuring the response for the unknown sample, interpolation in the calibration curve allows one to find the concentration of the analyte (Figure 1.1).

1.3. Multivariate calibration

A more general calibration process involves the relationship between the concentrations of various constituents in a test sample and multiple measured responses, i.e., multivariate instead of univariate. In contrast to univariate calibration, which works with a single instrumental response measured for each experimental sample, multivariate calibration works with many different signals for each sample.

Depending on the instrumental setup, the delivered data for a single sample may have different degrees of complexity. The simplest multivariate data are those produced in vector form, i.e., as a series of responses, which can be placed one on top of each other to generate a mathematical object known as a column vector. This object is also referred as having a single “mode” or “direction”. Spectra and chromatograms are prime examples of vector data.

FIGURE 1.1 (A) Univariate calibration curve showing a linear relationship between instrumental response (vertical axis) and analyte concentration (horizontal axis). Calibration standards are shown as filled circles, with the solid line implying the best fit to the calibration data. (B) Analyte prediction in an unknown sample: the dashed arrows show the process, as the measured response (open square) is interpolated in the calibration curve, leading to the estimated analyte concentration

Multivariate calibration using vectorial spectral data, particularly those measured through sample absorption in the near infrared (NIR) range, has given rise to a highly fruitful analytical field, which today is routine in many industrial laboratories and process control units.² The literature on this field is abundant. Multiple analytes can be determined simultaneously in the presence of other, possibly unknown constituents, provided they have been properly taken into account during the calibration phase.^3,4 The latter statement means that for determining an analyte in complex specimens (foodstuff, industrial materials, biological fluids, etc.) a relatively large number of calibration samples is required, in order to be able to model the effect of the interferents in new samples. The field has undoubtedly been triggered by the useful properties of NIR instruments, i.e., the ability to probe samples in a noninvasive manner.²

More complex data arrays than vectors are possible. The increasing complexity derives from increasingly complex instrumental setups, which generate data that can be arranged into mathematical objects with additional modes, such as a matrix or a three-dimensional array (Figure 1.2). In general, all calibration procedures developed using multiple data measured for each sample are known as multivariate. A proper classification of these calibrations depends on how the instrumental data are classified, as detailed in the next section.

FIGURE 1.2 Different Objects, Which can be Built with Data Measured for a Single Sample, Indicating Their Different Modes

1.4. Nomenclature for data and calibrations

Figure 1.2 shows data objects of growing complexity. In algebraic jargon, a scalar is a zeroth-order object, a vector is first order, a matrix is second order, etc. A nomenclature exists for the different calibrations, based on the measurement of data of various orders for a single sample: zeroth-order calibration is equivalent to univariate calibration, first-order multivariate calibration is equivalent to calibration with vectorial data per sample, second-order multivariate calibration is equivalent to calibration with matrix data per sample, and third-order multivariate calibration is equivalent to calibration with three-dimensional data arrays per sample. The list may continue with data arrays with additional modes per sample.⁵

On the other hand, with data for a group of samples it is possible to create an array having an additional mode, the sample mode. For example, univariate measurements for several samples can be grouped to form a vector, first-order data can be placed adjacent to each other to create a matrix, etc. On this basis, an alternative nomenclature has been developed, in which calibrations are named according to the number of ways (modes) of an array for a sample set. Thus zeroth-order calibration is also one-way calibration, first order is two-way, second order is three-way, etc., and it is customary to name all calibration methodologies involving second- and higher-order data (i.e., three-way and beyond) as multiway calibration, which is thus a subdivision of multivariate calibration.

Figure 1.3 provides a compact view of the hierarchy of data and calibrations. Throughout this book, we will employ both the order-based and way-based nomenclatures for calibration.

1.5. Nomenclature for constituents and samples

In order to...

Table of contents

• Cover image
• Title page
• Table of Contents
• Copyright
• Dedication
• Preface
• Foreword
• Acknowledgments
• Chapter 1. Calibration Scenarios
• Chapter 2. Data Properties
• Chapter 3. Experimental Three-way/Second-order Data
• Chapter 4. The MVC2 Software
• Chapter 5. Parallel Factor Analysis: Trilinear Data
• Chapter 6. Analytical Figures of Merit
• Chapter 7. Parallel Factor Analysis: Nontrilinear Data of Type 1
• Chapter 8. Multivariate Curve Resolution–Alternating Least-Squares
• Chapter 9. Partial Least-Squares with Residual Bilinearization
• Chapter 10. Three-way/Second-order Standard Addition
• Chapter 11. Third-order/Four-way Calibration and Beyond
• Chapter 12. Application Example: PARAFAC
• Chapter 13. Application Example: MCR–ALS
• Chapter 14. Application Example: U–PLS/RBL
• Chapter 15. Solutions to Homework
• Index