Objective uncertainty in the form of observed/measured data is modeled as randomness, whereas subjective uncertainty, e.g., due to a lack of trustworthiness or imprecision of measurement results, of distribution parameters, of environmental conditions, or of the data sources, is described as fuzziness. The model fuzzy randomness then combines but not mixes objectivity and subjectivity; these are separately visible at any time. It may be understood as an imprecise probabilistic model, which allows for simultaneously considering all possible probability models that are relevant to describing the problem.

Fuzzy random vectors

In accordance with the traditional probability theory, the space of the random elementary events Ω and the fundamental set X = ⁿ are introduced. Instead of a real-valued realization, a fuzzy realization is assigned to each elementary event ω ∈ Ω, see Fig. 1.

A fuzzy random variable or fuzzy random vector is the fuzzy result of the uncertain mapping

On this basis, particular events can be assessed in terms of probability. In view of a numerical treatment, α-discretization is applied. This leads to random α-level sets X_α of the fuzzy random vectors. With A_i being a crisp set on X fuzzy probability is defined as

with

That is, the imprecision of the fuzzy realizations is transferred to the probability measure. For each α-level a probability interval with the bounds P_αl(A_i) and P_αr(A_i) is obtained. The fuzzy probability comprises all possible assessment results for A_i which can be obtained with respect to the fuzzy set of possible probability models accounted for with the fuzzy random vector .

Fuzzy probability distributions

The fuzzy probability distribution function of a fuzzy random vector on X = ⁿ is defined in extension of the traditional probability theory by applying the concept of fuzzy probability. For each specified x ∈ X the fuzzy functional value

is defined for all α-levels by

and

see Fig. 2.

For determining the interval bounds F_αl(x) and F_αr(x) for each α-level, all originals X_j of that are contained in X_α must be taken into account. The fuzzy probability distribution function of may thus be interpreted as being the set of the probability distribution functions F_j(x) of all originals X_j of with the membership values μ(F_j(x)). Each original X_j specifies precisely one trajectory F_j(x) ∈ . On a trajectory-by-trajectory basis associated fuzzy probability density functions are also defined. The bunch of trajectories F_j(x) or f_j(x) contained in or comprises all possible probability models which are accounted for with the fuzzy random vector as an imprecise probabilistic description of the problem.

Fuzzy random functions

In analog to the traditional probability theory the concept of fuzzy randomness can be extended to dealing with time-dependent and spatial problems.

A fuzzy random function is a function whose functional values are fuzzy random vectors. These functional values may depend on the spatial coordinates θ = (θ₁,θ₂,θ₃), the time τ, and occasionally further parameters φ=(φ₁, φ₂, ...) from the parameter space T ⊆ ^m. In general, these parameters may represent fuzzy variables. With the fuzzy parameter vector characterizes a fuzzy random function defined on the space F(T) × Ω. Thereby Ω denotes the space of the random elementary events and F(T) represents the set of all fuzzy vectors on T ⊆ ^m.

A fuzzy random function is defined as the fuzzy result of the uncertain mapping

in which F(X) characterizes the set of all fuzzy vectors on ⁿ. For each specified point t ∈ T a fuzzy random function represents a fuzzy random vector . Thus, a fuzzy random function may also be defined as a set of fuzzy random vectors on the parameter space T

.

For a numerical treatment α-discretization is applied again. The formulas can be derived straightforward from the specifications for fuzzy random vectors. Also, multi-dimensional fuzzy probability distributions can be defined for fuzzy random functions as an extension of traditional probabilistics. Special cases of fuzzy random functions are obtained for the restriction of the parameter space to exclusively time-dependence
t = τ as fuzzy random processes and to exclusively spatial coordinated t = θ = (θ₁,θ₂,θ₃) as fuzzy random fields.