A $t$-Norm-Based Fuzzy Approach to the Estimation of Measurement Uncertainty

From a metrological point of view, a measurement process rarely consists of a direct measurement but, rather, of digital signal processing (DSP) performed by one or more instruments. The measurement algorithm makes the numerical results available as functions of acquired samples from input signals. Moreover, when repeated direct measurements are performed, one may speak about interactions in subsequent results (and it may be dependent on the type of instrument being used). With mathematical formalism, the complex relations involved can be described, although again, an indirect measurement result would be obtained. Regardless, no matter what kind of process is being examined, the distribution of the uncertainty associated with the measurement needs to be known. To express a measurement result with its associated uncertainty, the recommendations of the ISO Guide need to be met. Many published papers have proposed the use of fuzzy intervals to describe both the systematic and statistical effects of repeated measurements on the distribution of their results. In this paper, we use a random-fuzzy model, the single measure is represented as a fuzzy set, and the propagation of the possibility distribution through the DSP stage (which simply consists of an average operation) is performed using the extension principle of Zadeh based on a particular triangular norm: the so-called Dombi's.