Fuzzy probabilities

Fuzzy probabilities likely, highly unlikely, very likely,... 

All of these linguistic terms except fuzzy modifiers are represented in each context by appropriate fuzzy sets. Fuzzy predicates are represented by fuzzy sets defined on universal sets of elements to which the predicates apply. Fuzzy truth values and fuzzy probabilities are represented by fuzzy sets defined on the unit interval [0,1]. Fuzzy quantifiers are either absolute or relative they are represented by appropriate fuzzy numbers defined either on the set of natural numbers or on the interval [0,1]. Fuzzy modifiers are operations by which fuzzy sets representing the various other linguistic terms are appropriately modified to capture the meaning of the modified linguistic terms. 

The orbitals in each of the four subshells have characteristic shapes. The orbited in an s-subshell does not look like the orbitals in a p-subshell, and so forth. The diagrams of orbitals shown below indicate where an electron will be 95 percent of the time. They are fuzzy, probability pictures. The nucleus is at the center of each orbital. Since subshells and orbitals are closely linked, let s look at the orbitals in terms of the subshell they are in. 

Here the basic model used is one which is tested hi the laboratory and contains as much as is known about the influences of the X and Y parameters on the fatigue life. The known variabilities of the X and Y in the WOL are then used to calculate a fuzzy probability of failure which is the chance that the actual life will be less than the design life. A fuzzy logical hierarchy is then set up exactly as in the example of Section 10.5, to allow for the uncertainty associated with the application of the model in the laboratory. This new fuzzy probabUity is then again truth functionally modified to allow for the uncertainty of the matching with the WOL The procedure would then be exactly as for that example, so that a fuzzy truth restriction upon the statement, the structure is perfectly safe would result and this is the final measure of structural safety. 

These ideas, of course, still require development. Just as we have theorems of probability theory, decision theory, reliability theory, it will be possible to develop fuzzy probability theory, fuzzy decision theory and fuzzy reliability theory perhaps based on the measures presented here. 

If the experiment is highly repeatable, the repeatability of the resulting state of the system can be measured by the use of probability theory. We have seen that it is possible to calculate a probability of a fuzzy event as well as the probability of a precisely defined event. In this sense, probability is a measure of chance or frequency of occurrence in a sequence of trials. Probability itself can be used for the parameters of a system, as we have seen in the voting example given in Section 2.11. It is therefore possible to have an imprecise, vague or fuzzy probability measure. In other words an event could have, for example, a probability of highly likely. 

Consider the function Pf e F(R), defined as a measure of fuzzy probability in (f2, e). This results in an extension to the probability axioms ... 

Notice that (f2, e, P/) refers to a fuzzy probability space, denotes one of the fuzzy arithmetical operations, i.e., addition, subtraction, multiplication and division. It should also be noted that lx and Ox refer to the fuzzy subset of the real numbers 1 and 0, respectively. 

Figure 4. Fuzzy probability distribution function to the human error. 

Notice that the fuzzy probability distribution function presented in Fig. 4 represents the possibility function related to the probabilities of operator success and failure. Nevertheless, these possibilities functions are in the linguistic domain and must be translated to the real domain. To achieve this, the defuzzification process presented by Chen and KJien (1997) has to be used. 

In the present case, c = 0 and d = 1, which delimitates the domain of probabilities. Using equation (15), one cans defuzzify the fuzzy probability distributions of success and error presented in Fig. 4. The results obtained are 17.5% for error probability and 8.,5% for success probability. Based on this result, which was considered by the Petrobras managers as very high unsuccessful chances, one decided to not proceed with optical monitoring. The reservoir control will be done by traditional electrical cable and sensors. 

In this paper we presented an approach where fuzzy logic and BBN concepts are combined to estimate human error probability. This combination leads to a fuzzy Bayesian network approach based on the concept o fuzzy number and on extension principles applied to discrete fuzzy probabilities calculation. 

FPA Functional Process Analysis FPE Fuzzy Probability Estimator HEP Human Error Probability HF Human Factors... 

These are linguistic terms that modify other linguistic terms, for example, very, more or less, fairly, extremely. They may be used to modify fuzzy predicates, fuzzy truth-values, and fuzzy probabilities. The fuzzy set that defines the hedge is called a modifier. 

Kala, Z. 2007. Influence of partial safety factors on design reliability of steel structures Probability and fuzzy probability assessments. Journal of Civil Engineering and Management 13(4) 291 296. 

Earthquake Fuzzy analysis Fuzzy probability-based randomness Robustness Uncertain design Uncertain process Uncertainty... 

But the unfulfillable conditions to apply random variables in an engineering application need models to describe epistemic tmcertainty, as expert knowledge and small sample sizes therefore, the uncertainty model fuzziness is used. The combination of both models, fuzzy probability-based randomness, combines the advantages and is a sophisticated tmcertainty model. The necessary numerical algorithms are formulated in section Uncertain Structural Analysis. ... 

With generalized uncertainty models, more than one uncertainty characteristic can be considered. This contribution will present fuzziness and randomness as basic models and furthermore fuzzy probability-based randomness as a polymorphic model (see Pannier et al. 2013). 

A type of a polymorphic uncertainty model is the fuzzy probability-based randomness, taking variability and incompleteness into account For fuzzy probability-based random variables, the probability measure P of the random number, Eq. 9, is defined as an evaluated set of probability functions. This means that every event is represented by a fuzzy value and not by a real number. The fuzzy probability space is the triple (f2, H,P). Q and H are the same as in the random number definition. The fuzzy probability. P is a family of mappings... 

Robust Design Optimization for Earthquake Loads, Fig. 5 Example of the fuzzy probability-based random density function... 

Uncertain Analysis with Fuzzy Probability-Based Random Quantities... 

The application of polymorphic uncertainty models needs specific sequential numerical algorithms for COTisidering fuzzy probability-based random variables. To apply the approach, the fp-r variables need to be defined by the bunch parameters s (Eq. 22). The basic uncertainty models and the other polymorphic uncertainty models are included as special cases. 

The consideration of polymorphic uncertain variables, e.g., fp-r variables, in a numerical procedure, can be accomplished by a sequential reduction of the level of uncertainty see section Uncertain Analysis with Fuzzy Probability-Based Random Quantities. ... 

In this contribution, parameterized robust design of structures under earthquake loads is shown. The necessary capturing of uncertainty is done by using fuzzy, random, and fuzzy probability-based random variables. The numerical algorithms are shown in detail, with the focus of... 

Applying Equation (6.12) to a series system of two components A and B with fuzzy probabilities... 

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