Stochastic measurement procedure

A measurement is used to determine the actual value of a characteristic that is usually called measurand. A measurement is possible only if the measurand had been quantified prior to measurement by means of a suitable unit so that each value of the measurand is represented by a unique real number. For example, the characteristic "length" of a material object is quantified by the unit "meter", or the characteristic "(time) duration" of a development is quantified by the unit "second". Any measurement assumes a measurement process which is subject to randomness resulting in uncertainty about its indeterminate future outcome. Because of this uncertainty with respect to the future outcome of the measurement process, it is generally impossible to determine the true value of the measurand.

The related problems are addressed in the ISO Guide to the Expression of Uncertainty in Measurement^[1] (GUM) which was first published in 1993. However, since the GUM was published, complaints and critiques about it did not cease.^[2] The weaknesses of the GUM were one reason that stochastic measurement procedures^[3] were introduced in 2001. They are based on a rigorous introduction of the concepts of randomness and uncertainty

Measurement procedure

The measurand is given by a variable with fixed, i.e., determinate, but unknown value, and it is therefore called deterministic variable. A measurement itself is performed by a measurement device that defines a measurement process. In contrast to the measurand the measurement process is subject to randomness and its future outcome is therefore indeterminate. Consequently, the measurement process is represented by a variable X which is called a random variable.

The task is to conclude the unknown value of the measurand from the observed outcome of the measurement process. It is impossible to determine the true value of the measurand by means of the measurement process because of randomness. It is only possible to specify a set of values that includes the true value. Such a set constitutes the measurement result and it is called "correct" if it contains the true value of the measurand and wrong if not.

A measurement procedure is specified by the measurand given by the deterministic variable D, a measurement device that defines the admitted measurement range denoted $\mathfrak{D}$ and the measurement process represented by the random variable X, and finally the measurement function $C_D^{(\beta)}$ which assigns to each observation $\{x\}$ with respect to the random variable X an measurement result $C_D^{(\beta)}(\{x\})$ .^[4]

Reliability of a measurement procedure

The symbol $\beta$ is called reliability level of the stochastic measurement procedure and specifies a lower bound of the probability of obtaining a correct result when applying the measurement procedure. The larger the required reliability level $\beta$ the larger are the sets $C_D^{(\beta)}(\{x\})$ that constitutes the possible measurement results.

How to select the reliability level $\beta$ depends on the consequence of wrong measurement results. For measurement in safety related areas high values of $\beta$ up to $\beta = 1.0$ may be necessary, but if the consequences of wrong results are less serious, smaller values of $\beta$ may be justified.

Accuracy of a measurement procedure

In traditional metrology, "measurement precision" and "measurement accuracy" are distinguished, this somewhat confusing differentiation is not necessary for stochastic measurement procedures, since they distinguish between correct and wrong measurement results and meet a reliability specification given by the reliability level $\beta$ . The accuracy of a stochastic measurement procedure is defined by the average size of the measurement results, i.e., by the average size of the sets $C_D^{(\beta)}(\{x\})$ for all possible observations $\{x\}$ .

Measurement procedure and prediction procedure

Any stochastic measurement procedure is based on a stochastic model of the measurement process. This stochastic model is called Bernoulli space and enables the development of reliable and accurate stochastic prediction procedures given by the function $A_X^{(\beta)}$ with the domain being the possible values of the measurand $D$ where the reliability level $\beta$ is a lower bound for the probability that an obtained prediction will actually occur.^[5]

The measurement function $C_D^{(\beta)}$ can be reduced to the prediction function $A_X^{(\beta)}$ as given below:

C_D(\{x\}) = \{d | x \epsilon A_X^{(\beta)}(\{d\}) \}

From this relation it is seen, that the measurement result $C_D(\{x\})$ consists of those values d of the measurand D for which the observed outcome $\{x\}$ had been predicted.

The prediction procedure to be derived based on the Bernoulli space $\mathfrak{B}_{X,D}$ of the measurement process must meet the following three requirements:

The prediction procedure $A_X^{(\beta)}$ must meet the reliability requirement given by the reliability level $\beta$ .
The predictions $A_X^{(\beta)}(\{d\})$ for $d \epsilon \mathfrak{D}$ must cover all possible observations $\{x\}$ .
The predictions $A_X^{(\beta)}(\{d\})$ must be determined in a way that on average the measurement results have a minimum size.

References

↑ Guide to expression of uncertainty in measurement, .
↑ Elart von Collani, A critical note on the Guide to the Expression of Uncertainty in Measurement (GUM), Economic Quality Control, Vol. 23, 123−149, 2008.
↑ Elart von Collani and Monica Dumitrescu, Complete Neyman measurement procedures, Metrika, Vol. 54, 111−139, 2001.
↑ Elart von Collani, The Neyman theory − a scienfific measurement theory, in: V.P. Bulatov, I.G. Friedlaender (eds.): Basic Problems of Precision Theory, Nauka, St. Petersburg, pp. 74−94, 2001 (in Russian).
↑ Elart von Collani and Dmitri Stübner, Defining of reliability and precision of measurements, in: V.P. Bulatov (ed.), ``Problems of Mechanical Engineering: Precision, Friction and Depreciation, Nauka, St. Petersburg, 290−317, 2005 (in Russian).

External links

Stochastikon Ecyclopedia,
E-Learning Programme Stochastikon Magister,
Homepage of Stochastikon GmbH,
Economic Quality Control,
Journal of Uncertain Systems,

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