This generalized uncertainty model is capable of describing the whole range of uncertain information reaching from the special case of fuzziness to the special case of randomness. That is, it represents a viable model if the available information is too rich in content to be quantified as fuzziness without a loss in information but, on the other hand, cannot be quantified as randomness due to imprecision, subjectivity, and non-satisfied requirements, e.g., regarding environmental conditions. This is probably the most common case in engineering applications.

Uncertainty quantification with fuzzy randomness represents an imprecise probabilistic modeling, which incorporates imprecise data as well as uncertain or imprecise subjective assessments in terms of probability. The quantification procedure is a combination of established methods from mathematical statistics for specifying the random part and of fuzzification methods for describing the fuzzy part of the uncertainty.

For a numerical processing of fuzzy random vectors , fuzzy probability distribution functions  are formulated. To determine these fuzzy functions  from the available information, α-discretization is applied, and for each α-level the interval F_α(x) = [F_{α l}(x), F_{α r}(x)] is specified by considering all those originals X_j, which possibly fit the information. For a parametric description of the , both the distribution parameters and the distribution type are specified as fuzzy values. Subsequently, the fuzzy randomness quantification is addressed for four typical situations.

sample of small size

On the basis of a small sample it is generally not possible to determine the underlying probability distribution free of doubt. Usually, a variety of possible distribution function assumptions are not rejected by statistical tests, and interval estimations of the distribution parameters lead to wide confidence intervals. A conservative selection of a particular distribution with crisp parameters from the variety of choices, however, cannot be realized with respect to a subsequent nonlinear analysis with the specified random quantities. Thus, it is generally a wise decision to consider several if not all possible model variants, which can be realized with the aid of fuzzy probability distributions.

If, for example, the distribution type is known with a sufficient certainty, interval estimations on different confidence levels may be used to specify α-level sets for the parameters of a fuzzy probability distribution function. This provides a basis for a fuzzification of these parameters, which additionally includes subjective expert knowledge regarding the choice of the estimator (or several estimators for deriving variants) and regarding the selection of the interval type and the confidence levels. Merging this information mixture yields fuzzy sets that comprise all possible distribution parameter values assessed by a subjective degree of plausibility (membership value). For example, from a sample of size 20 and under the assumption of a normal distribution, the fuzzy parameters in Fig. 1 may be specified.

imprecise sample elements

If the elements of a sample are characterized by imprecision, e.g., due to inaccuracies of measuring devices, these can be described as fuzzy values. For this purpose, fuzzification procedures may be applied to the particular elements. The obtained fuzzy sample elements can then be evaluated with any arbitrary statistical estimator as mapping model within an α-level optimization to compute fuzzy distribution parameters. For example, a fuzzy expected value and a fuzzy standard deviation may be estimated according to

and

This leads to an interactive dependency between these fuzzy parameters, which can be neglected for many practical applications, see Fig. 2 left. As a result an associated fuzzy probability distribution function is obtained, see Fig. 2 right.

unknown, non-constant environmental conditions

If the environmental conditions are not constant while drawing a sample, and no concrete information is available to specify these changes, the observations are characterized by some dubiety. This dubiety can be quantified using the uncertainty model fuzziness. In this manner, subjective knowledge and experience regarding fluctuations in environmental conditions, e.g., from comparable experiments and situations, can be incorporated by fuzzifying the particular sample elements. Subsequently, the obtained fuzzy sample may be evaluated by applying statistical estimators in combination with α-level optimization as described above.

known, non-constant environmental conditions

If the environmental conditions are not constant but the reasons for the changes are known in detail when drawing a sample, this knowledge can be exploited to separate fuzziness and randomness in the statistical data bank. The environmental conditions are described by attributes. Observed realizations with the same attributes are lumped together in the same group. Each group of realizations is then treated as a random sample and evaluated using statistical methods. Each group yields one vector of distribution parameters, which leads to a set of distribution parameter vectors for all groups together. These sets together with subjective knowledge and experience may be taken as a basis for a fuzzification of the distribution parameters, optional with an additional estimation of interactive dependencies between them, see Fig. 3. As an alternative to the parametric specification, the fuzzy probability distributions may also be fuzzified directly by evaluating the sets of functional values of the empirical distributions associated with the groups.