State Estimation in Chemometrics

The Kalman Filter and Beyond

Pierre C. Thijssen, Sillas Hadjiloucas

• Publisher: Woodhead Publishing
• Year: 2020
• ISBN:
  9780081026229
• Edition: 2
• Topic: Sciences physiques
• Subtopic: Chimie analytique

This unique text blends together state estimation and chemometrics for the application of advanced data-processing techniques. State Estimation in Chemometrics, second edition describes the basic methods for chemical analysis—the multicomponent, calibration and titration systems—from a new perspective. It succinctly reviews the history of state estimation and chemometrics and provides examples of its many applications, including classical estimation, state estimation, nonlinear estimation, the multicomponent, calibration and titration systems and the Kalman filter. The concepts are introduced in a logical way and built up systematically to appeal to specialist post-graduates working in this area as well as professionals in other areas of chemistry and engineering. This new edition covers the latest research in chemometrics, appealing to readers in bio-engineering, food science, pharmacy, and the life sciences fostering cross-disciplinary research.

• Features a new chapter surveying the most up-to-date scientific literature on chemometrics, highlighting developments that have occurred since the first edition published
• Includes a new chapter devoted to new applications for state estimation in chemometrics
• Covers a new chapter entirely devoted to subspace identification methods
• Provides several new real-life examples of methods such as multiple modeling, principal component analysis, iterative target transformation factor analysis, and the generalized standard addition method

Chapter 1: Introduction

Abstract

Short reviews on the history of state estimation and chemometrics are given. The manual, instrumental, and automated stage in analytical chemistry and chemometrics are described. Chemometrics should be considered as a new dimension added to analytical chemistry. Chemical analysis depends exclusively on the multicomponent, calibration, and titration methods or combinations hereof. From the viewpoint of system theory, the application of state estimation in chemometrics evolves a modular framework, which enables the development of intelligent analyzers.

Keywords

History; State estimation; Analytical chemistry; Chemometrics; System theory; Intelligent analyzers

Chapter outline

• 1.1 History
• 1.2 Chemometrics
• 1.3 System view

1.1: History

The development of data processing techniques can be reviewed briefly in a historical perspective. At the beginning of the 19th century, Gauss (1809) developed the method of least squares and employed it in a simple orbit measurement system. During the next hundred years, several others made contributions to the field of estimation. A breakthrough came when Fisher (1910), working with probability density functions, reinvented the approach of the maximum likelihood. Much of this work has been employed thereafter in the broad area of statistics. A major change of viewpoint occurred when Kolmogorov (1941) and Wiener (1942) operated on random processes in the frequency domain. This approach describes the estimation problem in terms of correlation functions and the filter impulse response. It was limited to stationary processes and ensures only optimal estimates in the steady-state regime. Over the next 20 years, this work was extended in an often-cumbersome way to include nonstationary and multiple sensor systems. In the early 1960s, Kalman et al. (1960) advanced estimation with the concept of the state space model in the time domain and set the foundation of modern state estimation.

State estimation is concerned with the extraction of noise from measurements about some quantities that are essential to a system. A state is a minimal set of values sufficient to describe the behavior of a system. Three types of estimation are of interest: prediction is concerned with extrapolation of the state into the future, filtering recovers the state using measurements up to the current point, and smoothing involves interpolation of the state backward in the past. The Kalman filter is probably the most common estimation technique used in practice. Here, prediction and filtering are combined for an optimal performance of the estimation procedure. This approach is based on the online or recursive, rather than the batch processing of the measurements. It is ideally suited for computer implementation in automated systems and meets a broad application range: from ship navigation, image enhancement, process control, satellite orbit tracking, aircraft autopilot, earthquake forecasting to water resource planning.

The present book governs particularly applications of state estimation in the field of chemometrics. A lot of problems may arise when state estimation is applied in the practice of chemometrics. State estimation, for example, does not solve either the problem of modeling, how to acquire the noise statistics, how to select an optimal measurement schedule, or how to deal with computational errors and so on. Other design criteria, in addition to those used to derive the estimation algorithms, must be imposed to resolve such requirements.

Therefore, the blending together of state estimation and chemometrics is shown to be fruitful for both approaches.

At a first glance, the investigated systems in chemometrics often behave nonlinearly and/or nonstationary, and the modeling problem first has to be tackled.

In this book, mainly discrete linear state space models and preset noise statistics are involved for state estimation. In addition, some extensions are made toward the application of nonlinear models and the adaptation of the noise variances.

1.2: Chemometrics

Chemical analysis is referred to as the qualitative and/or quantitative determination of unknown constituents in samples. Here, analytical chemistry is devoted to the use and development of methods to enable chemical analysis. In less than a century, analytical chemistry has been developed from a mystic art to a reliable science. Nowadays, chemical analysis offers an important contribution to many organizations in society. Applications can be found for example in the petrochemical industry, clinical health survey, food quality assurance, and environmental pollution control.

With regard to history, a number of major stages can be distinguished:

In the manual stage, the analyst carries out chemical analysis with common laboratory glasswork and tools. The analyst possibly with help of a balance, polarimeter, or densitometer performs the measurement visually. The manual methods allow for easy operations and are often inexpensive. However, in practice, they may become tedious and manpower consuming. Examples of the former are gravimetric analysis, color-indicated titrations, and test tube procedures.

The instrumental stage introduces a great variety of novelties based on chemical or physical effects, which are transformed into an electrical signal. The measurement is performed by means of a recorder, voltmeter, or oscilloscope. The calculation of the required results follows after measurement by the analyst with simple arithmetics and graphics. The first sign of this development can be traced back to the early previous century. The design of instrumental methods grew simultaneously with the progress made in electronics. Contributions can be found in spectroscopy, chromatography, electrochemistry, flow injection analysis, etc.

Recently, the digital computer became available as a new achievement. Chemical analysis can now be exploited in a more efficient way by the capability to store and to process large amount of data. The automated stage introduces the new avenue of chemometrics to achieve, maintain, and improve the quality (or precision, accuracy, time, costs) of the analytical results. Chemometrics investigates strategies in chemical analysis to obtain a maximum of relevant information with minimal means and efforts. Mathematical and statistical methods are applied to design or to select optimal procedures and experiments. Various examples of progress can be given: the description, control, and surveillance of time series; experimental optimization by factorial designs or the simplex method; method selection by measurability and information theory; signal enhancement through estimation techniques; principal component analysis, partial least squares, and curve resolution; classification with pattern recognition; and finally digital simulation of laboratory organizations.

Nowadays, most of the instruments involved in chemical analysis are computer compatible and automated for control purposes, signal registration, data processing, and report generation. Automated instruments exploiting chemometrical techniques and innovations from artificial intelligence are the present state of the art. In the last category, the application of expert systems, neural networks, genetic algorithms, and support vector machines is worth mentioning.

What is the object of this book? Traditionally, the measurements are collected batchwise and computations follow afterward. The advent of today’s computers offers the interactive coupling with an analytical instrument. Now, online data processing schemes such as the Kalman filter can be applied. As soon as a new measurement is available, calculations are updated and its results may be used more effectively. Relatively little attention has been paid to the linking of state estimation and chemometrics. Chemometrics should not be considered as just an outgrowth of but rather as a new dimension added to analytical chemistry. The first important step is to focus on the projection of the great variety of manual, instrumental, and automated methods for chemical analysis to the chemometrical axis. From this viewpoint, chemical analysis depends exclusively on the multicomponent, calibration, and titration methods, or combinations hereof. The application of modern state estimation in chemometrics is therefore demonstrated for these important elementary methods.

Firstly, some aspects of multicomponent analysis as applied in spectroscopy are investigated. Especially, the optimal design problem and the adaptation of the unknown measurement noise variance are of interest. In addition, the extension of the state space model with stochastic drift allows for the compensation of an unknown disturbance spectrum in the measured sample.

Secondly, state estimation is demonstrated in the field of the linear calibration method suffering from drifting parameters. Theoretical considerations particularly on state space modeling, evaluation of unknowns, quality control, optimal design, and variance reduction are investigated and applied in practice. Furthermore, nonlinear estimation is applied when the calibration graph has an exponential shape.

Thirdly, state estimation is employed in the titration method for determining its curve and derivatives. From the estimated state, the setpoint(s) and inflection point(s) can be evaluated offline afterward. The online control to obtain equidistant measurements using variable volume additions in the discrete titration and the online endpoint control in case of continuous titration are of particular interest.

Finally, online processing the measurements for multiple modeling, principal component analysis and the generalized standard addition method is described. Also, iterative target transformation factor analysis evaluated offline is outlined. It further features chapters on subspace identification methods, new applications and recent advances in chemometrics.

In the next section, an attempt is made to develop a modular framework for state estima...

Table of contents

• Cover image
• Title page
• Table of Contents
• Copyright
• Inscription
• About the authors
• Chapter 1: Introduction
• Chapter 2: Classical estimation
• Chapter 3: State estimation
• Chapter 4: Statistics
• Chapter 5: Nonlinear estimation
• Chapter 6: The multicomponent system
• Chapter 7: The calibration system
• Chapter 8: The titration system
• Chapter 9: Miscellaneous
• Chapter 10: Subspace identification methods
• Chapter 11: New applications in chemometrics
• Chapter 12: Recent advances in chemometrics
• Appendix
• Bibliography
• Index