Publisher Summary

This chapter highlights the short reviews on the history of state estimation and chemometrics. From the viewpoint of system theory, state estimation applied to chemical analysis evolves a modular framework, which permits the development of simple intelligent analyzers. 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 problems are of interest: (1) filtering recovers the required information using the measurements up to the current point; (2) prediction is concerned with extrapolation into the future; and (3) smoothing investigates interpolation back in the past. Chemical analysis is referred to as the qualitative and/or quantitative determination of unknown constituents in samples. The analytical chemistry is devoted to the use and development of methods to enable chemical analysis. 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.

1.1 History

The development of data processing techniques can be reviewed briefly in a historical perspective. At the beginning of the 19-th 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 term 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 twenty years, this work was extended in an often-cumbersome way to include nonstationary and multiple sensor systems. In the early sixties of the previous century Kalman and others (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 behaviour of a system. Three types of estimation problems are of interest: filtering recovers the required information using the measurements up to the current point; prediction is concerned with extrapolation into the future; and smoothing investigates interpolation back 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. The 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 qualitative or quantitative chemical analysis. State estimation, for example, does not solve the problem of modeling, how to acquire the noise statistics, how to establish 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 for chemical analysis 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 towards 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. Today, chemical analysis offers an important contribution to many organizations in society. Applications can be found for example in the petrochemical industry, the clinical health survey, the food quality assurance and the environmental pollution control.

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

In the manual stage the analyst with common laboratory glasswork and tools carries out chemical analysis. 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 testtube 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 the innovations from artificial intelligence are the present state of the art. In the last category it is worth mentioning the application of expert systems; neural networks; genetic algorithms and support vector machines.

What is the object of this book? Traditionally, the measurements are collected batchwise and computations follow afterwards. 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. Relative 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 a new dimension added to analytical chemistry. A first important step is to focus on the projection of the great variety of manual, instrumental and automated methods on the chemometrical axis. From this viewpoint quantitative procedures depend strongly 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 classes.

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 particular interest. In addition, the extension of the state space model with stochastic drift allows for the correction of an unknown disturbance spectrum...