Interval Arithmetic and Condence Intervals from the Metrolo- Metrolo-gist's Point of View

We have seen that ordinary interval arithmetic uses only information about the support of the distribution of the random variables involved. An alternative method is to use interval arithmetic to combinecondence intervals. (The standard method can thus be considered to combine 100% condence intervals.) Dierent condence intervals should then give further information about the distribution of the nal result.

This idea has been considered by metrologists in the context of the theory of errors and the propagation of uncertainties arising from physical measurements. We will now examine what they have had to say on this topic. Metrologists have consid-ered the use of condence intervals because of a shortcoming of the standard method of the statement and propagation of experimental uncertainty. (Note that the term

\experimental error" is out of favour nowadays: \the uncertainty, in former times frequently called `error' :::" 846, p.83].) The standard method is to state uncer-tainties as standard deviations and to use the \general law of error propagation"

820]to propagate the errors through subsequent calculations. (A good recent re-view can be found in 22].) This general law is simply the linearisation of nonlinear functions by truncated (rst order) Taylor series expansions about expected values (see section 2.2.2). Sometimes higher order expansions 69]or the more complicated expressions taking account of the covariances 78,352,858]are used. The shortcom-ing of this method arises in the determination of appropriate values of uncertainty to use for subjectively estimated \systematic errors," and the diculty in converting a nal uncertainty statement in terms of standard deviations into an interval result.

An interval statement is often required for calibration or legal purposes. We shall consider these two diculties in turn.

Subjective Interval Estimates of \Systematic Error" and Standard Deviations In recent years the distinction between \random" and \systematic" errors (rst explicitly proposed by D. Bernoulli in 1780 746, p.290]although implicitly adopted by Newton as early as 1676 744, p.222]) has been called into question 302,604, 846,879]. The point is that there has never been an entirely satisfactory criterion for deciding which category to use in any particular instance. (Muller 604, p.375]

quotes Vigoureux: \One has to remember that some errors are random for one person and systematic for another.") Similarly, \what is a systematic error for one experiment may not be for another:::. Much of the skill in experimental work comes from eliminating sources of systematic error" 49, p.72]. The choice is important because the standard (\orthodox") theory propagates random and systematic errors through calculations dierently. Systematic errors are added arithmetically,whereas random errors are added in quadrature. Indeed, as Muller has pointed out, this is precisely the source of the diculty in classication: \the traditional classication of uncertainties depends upon the further use we intend to make of them, and in general this can not be known in advance" 604, p.376].

The new \randomatic" theory avoids the problem of categorization by consid-ering all errors to be random. However it does distinguish between \`objective' statistical estimates" and \`subjective' guesstimates" 302, p.625]. The subjective uncertainties are often given in terms of an interval. This has to be converted into a standard deviation in order to propagate it through any subsequent calculations.

A number of (admittedly somewhatad hoc) methods for doing this can be found in 878]. The subjective uncertainties are usually considered to be independent 878, p.83], although there seems to be no good a priori reason for this to be so. It could well be argued though that any error arising from an ad hoc handling of subjec-tive uncertainties should usually be negligible compared with the variability due to the \objective statistical estimates." Arguments against the randomatic theory of errors and proposals for an improved orthodox theory based on a more careful dis-tinction between the two classes of errors can be found in the closely argued paper of Colclough 158].

Determination of a Con dence Interval for the Final Result

A more serious and older problem is the conversion of a nal uncertainty state-ment in terms of a standard deviation into an interval statestate-ment. We have already observed (section 2.2.2) that the Chebyshev inequality or its generalisations 212, 317,318,537]can be used to determine a condence interval in terms of means and standard deviations. However, in general these will be very loose (pessimistic) in-tervals. Alternative approaches which give tighter, less pessimistic, intervals require assumptions about the underlying distributions.

The oldest method is the assumption of normality. We have already discussed this in section 2.2.1. It has never found universal acceptance and had strong critics even some 60 years ago: \I reject, then, the Gaussian theory of error, without qualication and with the utmost possible emphasis and with it go all theoretical

grounds for adopting the rules that are based on it" 129, p.162]. We need add nothing further to this.

An alternative, which is similar in its general approach to the lower and upper bounds on the distributions we consider in chapter 3, has been considered inde-pendently by Kuznetsov 488]and Weise 857]. Assuming that the true \error"

distribution is symmetrical, unimodal and has nite support, Kuznetsov 488]cal-culated a condence interval in terms of a variance by using the \mean distribution"

derived from lower and upper distribution functions satisfying the distributional as-sumptions. Weise 857], somewhat more ambitiously, considered a whole class of distributions ^D. He then used a distribution which was a mean of all possible dis-tributions over ^D in order to determine a condence interval in terms of a given variance. (Obviously the choice of ^D, and the fact that the mean distribution is used, will aect the result dierent choices will give dierent results.) Nevertheless this approach seems to be a promising way of investigating the eects of dierent distributional assumptions and of handling dierent degrees of optimism.

Direct and Exclusive Use of Con dence Intervals

Although they are preferable to assuming normality, the methods of Kuznetsov and Weise are still not entirely satisfying. An alternative is to avoid completelythe use of standard deviations as a measure of uncertainty and to use condence intervals only from the outset. Muller 603]and others have argued against this idea on the grounds of diculties in combining condence intervals. The general consensus amongst metrologists seems to be described by the DIN standard 1319 part 4 \Basic Concepts of Measurements: Treatment of Uncertainties in the Evaluation of Measurements"

858](a summary appears in 859]. This suggests a careful but fairly straight forward application of the general law for error propagation (in termsof standard deviations).

No distinction is made between \random" and \systematic" errors apart from the methods of initially estimating the numerical value to be attributed. There are still diculties in using standard deviations when repeated measurements are not independent 18], but these seem to be manageable given some idea of the spectrum of the error sequence.

Recently Rowe 692]has presented some very interesting results which could lead to a partial combination of the two techniques (intervals and standard deviations).

Using two separate approaches, Rowe has determined lower and upper bounds for Y and Y in terms of X, X, X, X or limited order statistic information, for a class of transformations Y = g(X). Rowe's approach is preferable to the simple rst order Taylor series approximations because lower and upper bounds for g(X)

and g(X) are given. These allow a more rigorous propagation of uncertainty. Also, when X and X or other distributional information is available, the tighter bounds can be obtained.

Con dence Curves and Fuzzy Numbers as a Generalisation of Interval Arith-metic

Finally, to conclude this somewhat discursive exploration of the metrological uses and signicance of interval arithmetic, we can mention the idea of fuzzy arithmetic 432]. This has been suggested as a natural generalisation of interval arithmetic and error propagation techniques 240]because under the standard sup-min combination rules (see chapter 4) fuzzy numbers can be combined in terms of interval arithmetic on the level sets of their membership functions. We examine fuzzy arithmetic, and point out some similarities to the idea of condence curves introduced by Cox 169]

and developed by Birnbaum and Mau 76,77,553]in chapter 4. These condence curves are made up of nested sets of condence intervals at dierent condence levels and they may provide a useful generalisation for the purposes of the theory and propagation of errors.