For the newcomer, unfamiliarity with the specialist vocabulary of scientific disciplines like physics and chemistry can act as an obstacle to learning those disciplines. What can be even more challenging is that science employs many words such as force and energy that are used in various ways in everyday language. The science of measurement, in particular, has many terms, such as error, uncertainty and accuracy, that also occur in day-to-day use in contexts far removed from measurement. In this chapter we consider terms used in measurement, including those with an everyday or popular meaning such as error, and we clarify their meaning when used in the context of measurement.

Measurement and related terms

Measurement

Measurement is a process by which a value of a particular quantity such as the temperature of a water bath or the pH of a solution is obtained. In the case of length measurement, this might involve measuring the atomic-scale topography of a surface using an instrument such as an atomic-force microscope (AFM), or measuring the length of a pendulum using a metre rule. Values obtained through measurement form the foundation upon which we are able to

• test both new and established scientific theories;

• decide whether a component, such as a resistor, is within specification;

• compare values obtained by workers around the world of a particular quantity, such as the thickness of the ozone layer of the atmosphere;

• quantify the amount of a particular chemical species, such as the amount of steroid in a sample of urine taken from an athlete; and

• establish the proficiency of laboratories involved with the testing and calibration of equipment.

Random errors arise from uncontrollable small changes in the measurand, instrumentation or environment. These changes are evident as variations in the values obtained when we carry out repeat measurements. In this chapter we shall consider methods of quantifying these variations: that is, describing them numerically using statistical methods. Some basic statistical concepts will therefore be introduced and discussed.

Sampling from a population

In statistics, the term population refers to the number of possible, but not necessarily actual, measured values. In some situations a population consists of an infinite number of values. In practice, we can measure only a sample drawn from a population, since time and resources are always limited. We hope and expect that the sample is representative of the population. In almost every case of measurement we sample a population, and the quantities of interest obtained from the sample (sometimes called sample statistics) should reliably represent corresponding parameters in the population (the population parameters). An example of such a quantity of interest, which quantifies the amount of scatter in values, is the standard deviation of the values.

There are cases where a sample may, in fact, be the entire population. Thus the examination results of a class of 30 students can be analysed statistically in order to determine, for example, the mean mark and the range of marks, with no attempt at generalising. The teacher of the class may be interested simply in that particular class.

Random errors, evaluated using statistical methods, create a Type A uncertainty. A known systematic error in a measured value should be corrected for, and after the correction has been made, the uncertainty in the correction contributes to the uncertainty in that value. The uncertainty in the correction, and hence in the value, may be Type A or Type B, depending on how the uncertainty is evaluated. The finally reported uncertainty of a measurand, called the combined uncertainty, is likely to have both Type A and Type B components, but becomes wholly Type B when subsequent use is made of it.

In this chapter we consider how to evaluate the combined uncertainty of a measurand. The procedure to be described makes no distinction between Type A and Type B uncertainties. It may appear then as if we have gone to unnecessary trouble in assigning types to uncertainties, but this classification is desirable since it emphasises the different methods by which they are evaluated. It is also useful as a reminder that, whereas an ‘error’ can be random or systematic, ‘uncertainty’ is a separate concept whose two types are distinguished from each other by different names, ‘Type A’ and ‘Type B’. However, once uncertainties have been classified, Type A and Type B uncertainties are treated identically thereafter.

In this chapter we describe how consistency and clarity may be brought to the calculation and expression of uncertainty in measurement.

The goal of any measurement is to establish a numerical value for the measurand. Depending on the accuracy that we wish to claim for the numerical value, the procedure that gives us the value may be relatively simple and direct, involving no more than a tape-measure, for example. In other situations the process may be more complicated, with several intermediate stages requiring the resources of a well-equipped laboratory. Thus, if the measurand is the width of a table, the tapemeasure is all that is needed. On the other hand, if the measurand is the accurate mass of an object, we need to know the value of the buoyancy correction (since the weight of the object is less by an amount equal to the weight of the volume of air that it displaces). This in turn requires knowledge of the volume of the object and of the density of air (which is a function of temperature, pressure and composition) at the time of measurement.

There are three components of a measurement: the measurand itself; the measuring instrument (which can be a stand-alone instrument such as a thermometer, or a complex system that occupies a whole laboratory); and the environment (which includes the human operator). The environment will, in general, affect both the measurand and the measuring instrument.

In this chapter we present four case studies based on typical undergraduate experiments, involving the determination of best estimates of measurands, standard uncertainties, expanded uncertainties and coverage intervals. For completeness, we include a brief description of each experiment. The equipment required is inexpensive or can usually be found in an undergraduate science laboratory. The account of each experiment contains data obtained in an actual experiment.

We have not included a detailed introduction to each experiment, nor have we indicated how each might be improved or ‘finessed’. The account of each experiment is biased towards giving details of the data analysis such as the calculations of standard uncertainties and coverage intervals. A more detailed analysis would normally require consideration of the uncertainty in the calibration of instruments used. For many undergraduate experiments such information is not available, and therefore we have not included the contribution of the calibration uncertainty to the combined standard uncertainty. At the end of the account of each experiment we suggest practically based exercises related to the experiment.

Reporting measurement results

An account of an experiment, as presented in a formal report, may contain many sections with headings such as introduction, materials and methods, results, analysis and conclusion. With respect to the analysis of data, best estimates of particular quantities obtained through experiment and by other means should be communicated clearly, concisely, and in a manner that is useful to others.

After measurement, we assign an estimated value to a measurand as well as an accompanying uncertainty. The uncertainty is usually expressed as an interval around the estimated value. With any such interval we associate a probability that the actual or true value of the measurand falls within that interval. Measurands are usually continuous quantities such as temperature, voltage and time. However, when discussing probabilities in the context of measurement it is convenient first to consider ‘experiments’ in which the outcomes are discrete, for example tossing a coin, where the outcome is a head or a tail.

Distribution of scores when tossing coins or dice

A fair coin falls heads up with probability 1/2 and tails up also with probability 1/2. A fair coin is an idealised object (since all real coins have a slight bias towards either heads or tails) and presents the simplest case of a ‘uniform’ probability distribution. When a probability distribution is uniform, the possible outcomes of an experiment (tossing a coin in this case) occur with equal probability. We will show how non-uniform probabilities emerge as soon as two or more fair coins are considered. These non-uniformities tend to a characteristic pattern called a Gaussian (or ‘normal’) probability density distribution. For the sake of brevity we shall usually refer to the ‘Gaussian probability distribution’ as simply the ‘Gaussian distribution’. Likewise we shall usually refer to the ‘uniform probability density distribution’ as the ‘uniform distribution’.

Lord Kelvin, a renowned scientist born in Ireland in the nineteenth century, recognised the importance of measurement and spoke about it in passionate terms:

Measurement is essential to science. Without measurement, scientific models and theories cannot be rigorously tested or challenged. Ambitious scientific and technical endeavours such as the exploration of the surface of Mars, medical diagnosis using magnetic-resonance imaging (MRI) and the evaluation of renewable energy sources would not be possible. Measurement is no less critical in areas such as international trade, with the global economy becoming ever more pervasive.

In this chapter we consider matters key to measurement and the communication of the results of measurement. These include the system of units, scientific notation and significant figures.

The system of units of measurement

To measure the length of a particular object and have the result of that measurement recognised and understood by other people, there must be mutual agreement on a basic unit of length. Over past centuries many units have been adopted as the basis of length measurement in different parts of the world. Some of those units, like the metre, the mile and the fathom, are still in use today.

A systematic error causes a measured value to be consistently greater or less than the true value. The amount by which the value differs from the true value may be a constant. Such a situation would occur, for example, when using a micrometer that has a ‘zero error’: the scale of the micrometer indicating a non-zero value when the jaws of the micrometer are closed. In other circumstances, a systematic error may be proportional to the magnitude of the quantity being measured. For example, if a wooden metre rule has expanded along its whole length as a consequence of absorbing moisture, the size of the systematic error is not constant but increases with the size of the object being measured.

Systematic errors may be revealed in two ways: by means of specific information or when the experimental set-up is changed (whether intentionally in order to identify systematic errors, or for some other reason). In both cases we need a good understanding of the science underlying the measurement. In general, statistical analysis may or may not be involved in assessing the uncertainty associated with a systematic error, so this uncertainty may be Type A or B. When the effect of random errors has been minimised, for example by taking the mean of many values, the influence of systematic errors remains unless they too have been identified and corrected for.

Since a systematic error does not necessarily cause measured values to vary, it often remains hidden (and may be larger than the random errors). Experienced experimenters consistently review their methods in an effort to identify and quantify systematic errors.

We live with uncertainty every day. Will the weather be fine for a barbecue at the weekend? What is the risk to our health posed by a particular item of diet or environmental pollutant? Have we invested our money wisely?

It is understandable that we would like to be able to eliminate, or at least reduce, uncertainty. If we can reduce it significantly, we become more confident that a desirable event will happen, or that an undesirable event will not. To this end we seek out accredited professionals, such as weather forecasters, medical researchers and financial advisers.

However, in science and technology uncertainty has a narrower meaning, created by the need for accurate measurement. Accurate measurement, which implies the existence of standards of measurement, and the evaluation of uncertainties in a measurement process are essential to all areas of science and technology. The branch of science concerned with maintaining and increasing the accuracy of measurement, in any field, is known as metrology. It includes the identification, analysis and minimisation of errors, and the calculation and expression of the resulting uncertainties.

Whether or not a measurement is regarded as ‘accurate’ depends on the context. Supermarket scales used for weighing fruit or vegetables need not be better than 1% accurate. By contrast, a state-of-the-art laboratory balance is able to determine the value of an unknown mass of nominal value one kilogram to better than one part in ten million. These figures, 1% in one case and one part in ten million in the other, are numerical measures of the degree of accuracy: low in the first case and high in the second, but each of them fit for its particular purpose.

In this book, we are going to examine various types of numerical problems and develop techniques for solving them. We will gradually build up the necessary tools and use a number of case studies to:

1. illustrate the process of formulating a problem, by which we mean translating from an intuitive idea of the problem by writing it down mathematically,

2. motivate and develop algorithms to solve problems, that is, descriptions of operations that can be implemented in software to take a problem specification and return a solution, and

3. illustrate how to match the formulation of the problem to the capabilities of available algorithms, involving, in some cases, transformation of the problem from its initial formulation.

We illustrate how to think about and describe problems to make them amenable to solution by optimization software. In brief: formulation of problems to facilitate their solution.

In the five parts of this book, we will consider five general problem classes. The first problem class is the solution of linear systems of equations (Part I). Solving linear systems of equations will turn out to be at the center of much of our subsequent work. For example, our second problem class, solving non-linear systems of equations (Part II), will be solved by an algorithm that requires repeated solution of linear systems of equations.

We will then go on to use these two algorithms as building blocks in algorithms for:

• unconstrained optimization (Part III), which will in turn form part of the algorithms for:

• equality-constrained optimization (Part IV), which will again form part of the algorithms for:

• inequality-constrained optimization (Part V).

In this chapter, we apply algorithms from Chapter 10 to the two case studies from Chapter 9. We consider the multi-variate linear regression case study in Section 11.1 and the power system state estimation case study in Section 11.2. Both of our case studies will be transformed into least-squares problems. Unconstrained optimization algorithms that exploit the special characteristics of leastsquares problems are described in [45, section 4.7] [84, chapter 13]; however, we will first apply our basic unconstrained optimization algorithm to these problems because in later chapters we will need to solve more general unconstrained problems.

In practice, a special purpose algorithm for least-squares problems can be expected to yield better performance on least-squares problems compared to the performance of a general purpose algorithm for unconstrained problems. That is, as we have discussed previously in Section 2.3.2, we should always in practice try to find the most specifically applicable algorithm for a problem [84, section 13.1]. We will consider such specific algorithms for least-squares problems using further transformations.

Multi-variate linear regression

In Section 11.1.1 we transform the objective of Problem (9.7) and in Section 11.1.2 we compare the transformed and original problem. In Sections 11.1.3 and 11.1.4 we calculate the derivatives of the transformed objective and present the optimality conditions. In Section 11.1.5, we transform the problem further to avoid numerical ill-conditioning issues. Then, in Section 11.1.6, we relate the optimality conditions to linear regression.

In this chapter we will introduce two case studies:

1. production, at least-cost, of a commodity from machines, while meeting a total demand (Section 12.1), and

2. state estimation in an electric power system where the power injections at some of the buses are known to high accuracy (Section 12.2).

Both problems will turn out to be equality-constrained optimization problems. The first will introduce several new ideas in problem formulation, while the second will build on the state estimation case study from Section 9.2.

Least-cost production

Motivation

Consider a machine that makes a certain product, requiring some costly input to produce. In many industries it is possible to stock-pile the product at low cost from day to day, week to week, or even season to season. In this case, it is natural to try to operate the machine at constant output. Ideally, the constant value of machine output would be matched to either:

• the point of maximum operating efficiency of the machine, or

• some other desirable operating point of the machine.

When demand is lower than production, some of the production goes into the stockpile. When demand is higher than production, the stocks are used to help to meet demand.

However, if stock-piling is costly or inconvenient or if demand for the product varies rapidly, then to avoid over-supplies and shortages we have to vary production to follow variations in demand.