Leonid Reznik

School of Communications and Informatics

Victoria University

P.O. Box 14428 MCMC MELBOURNE VIC 8001 AUSTRALIA

E-mail: Leon.Reznik@vu.edu.au

It is a common knowledge that any measurement result is characterised with some uncertainty. The uncertainty of the measurement result reflects the lack of an exact knowledge of its value. In order to standardise the terminology and methods of uncertainty expression and evaluation the Guide to the Expression of Uncertainty in Measurement was developed and introduced by the International Organisation for Standardisation [1]. The guide defines uncertainty (section D.5.2) as an expression of the fact that, for a given measurand and a given result of measurement of it, there is not one value but an infinite number of values dispersed about the result that are consistent with all of the observations and data and one’s knowledge of the physical world, and with varying degrees of credibility can be attributed to the measurand. Historically, the probabilistic models were applied for expressing uncertainty components and statistical methods were used to evaluate it. However, the Guide [1] recognises the limits of statistical models and methods and allows for developing alternative techniques.

An important aspect of uncertainty evaluation formalised and made explicit in the ISO Guide is the categorisation of measurement uncertainty components into two groups according to the method of evaluation. The type A category refers to the application of direct statistical processing of the measurement results of repeated physical experiments received under apparently (and nominally) identical conditions upon which statistical measures of central tendency (mean) and dispersion (variance) are well defined and can be readily estimated. The type B category refers to the evaluation of uncertainty components by any other means, generally by the application of personal expertise to the estimation of those components of the uncertainty, which can not be directly treated by statistical process. This paper proposes the application of emerging intelligent methods such as expert systems, fuzzy logic and neural networks for expression and evaluation of some of the type B components of uncertainty.

As the Guide states (section F.2.1) if a measurement laboratory had limitless time and resources, it could conduct a comprehensive statistical investigation of every case of uncertainty, for example, by using many different makes and kinds of instruments, different methods of measurement, different applications of the methods, and different approximations in its theoretical models of the measurement. In this case all of the uncertainty components could be evaluated by the statistical analysis of series of observations. However, as it is not possible or at least economically feasible in all cases, the Guide recognises the evaluation of uncertainty components by “whatever other means is practical”.

One should also take into account that new generation intelligent instruments whose operation is based on such emerging technologies as fuzzy logic, neural networks, genetic algorithms are introduced into a practice including a measurement practice. The measurement result produced by these instruments has to be characterised with some uncertainty as well as in a case of a conventional technique. However, the uncertainty evaluation may be influenced by the absence of conventional formulae, which is commonly applied in statistical procedures. It may create another reason for alternative models application.

The uncertainty or its components estimates can be obtained from observations (single and repeated) by statistical calculation (type A) or judgements based on the experience (type B). In the second case one often deals with expert’s opinion expressed often in a form of linguistic statements or linguistic rules. Type B evaluation of the standard uncertainty supposes a very wide application of professional experience and even general knowledge. Generally this evaluation can be expressed as an expert’s opinion.

Despite some degree of flexibility allowed, the Guide developers definitely based their conclusions and recommendations on probabilistic models applied in measurement and statistical procedures. It has left some important cases beyond the Guide scope. New measurement systems including not measurements elements only but processing and decision-making units require the development of new methods for the evaluation of uncertainty of the results produced by these systems.

The ontological view of a true value to be determined by measurement is gradually superseded by a more pragmatic interpretation, according to which "measurement is the assignment of numerals to objects or events according to rule, any rule" [5]. In a number of the international and national standards, the term “measurement error” has been replaced by the term “measurement uncertainty”, which is philosophically more attuned to fuzzy systems terminology. This shift of perspective overcomes the basic criticism against the classical view of the concept of measurement, viz. the fact that it requires the use of the concept of true value, which can no longer be maintained in an operational sense [4]. The probabilistic approach of seeking to determine “a” value, rather than “the” true value, and the fuzzy sets approach of seeking to determine the degree of membership of the value of a measurand to a specified set are both in harmony with this more sophisticated view of measurement variability.

Alternative models applied for uncertainty evaluation should satisfy to the following conditions:

- be suitable to describe the uncertainty components for which statistical models and methods do not fit well (fist of all type B components based on expert’s opinion or judgements and components resulting from intelligent techniques application),

- be supported by a well developed theory providing the methods for model developing and processing,

- allow for easy implementation in modern measuring systems,

- allow for joint application and processing alongside with the statistical models,

- in practice, the difference in a point of view and in the formalisation methods should not lead to the difference in the numerical value of the measurement result or of the uncertainty assigned to that result.

Based on the given requirements the fuzzy sets models are proposed to model some components of uncertainty. Since its introduction in 1965, fuzzy logic has been applied for different purposes. However, the basic idea is its use for modelling “human like” logic and information. So it is definitely the best way to formalise and process an expert’s information especially expressed with a natural language. Recent developments of fuzzy logic theory such as fuzzy constraints theory and calculus, possibility theory and especially the new theory of fuzzy information granulation [2] have prepared a necessary framework for joint application of fuzzy and probabilistic models in measurement science. Fuzzy models calculus has been developed as well (see, for example, [3]).

Looking at the typical block diagram [4] of a measuring system (see Fig. 1), one can see that it contains processing and a decision-making sections as its basic elements. So inclusion some new calculus and decision rules based on expert system application will not change the structure of a measuring system. Fast development of microprocessors and information technology will allow to avoid implementation problems.

In practice there are many possible sources of uncertainty in measurement. The sources given in the Guide[1] are included into the table alongside with the feasible model. The comments provided tend to give some explanation of an application of the particular model type.