Random-fuzzy variable
[1] The first is the random uncertainty which is due to the noise in the process and the measurement.
The second contribution is due to the systematic uncertainty which may be present in the measuring instrument.
Hence, systematic uncertainty could be considered as a contribution of a fuzzy nature.
This systematic error can be approximately modeled based on our past data about the measuring instrument and the process.
Statistical methods can be used to calculate the total uncertainty from both systematic and random contributions in a measurement.
[7][8][9] Random-fuzzy variable (RFV) is a type 2 fuzzy variable,[10] defined using the mathematical possibility theory,[5][6] used to represent the entire information associated to a measurement result.
The internal distribution is the uncertainty contributions due to the systematic uncertainty and the bounds of the RFV are because of the random contributions.
The external distribution gives the uncertainty bounds from all contributions.
A Random-fuzzy Variable (RFV) is defined as a type 2 fuzzy variable which satisfies the following conditions:[11] An RFV can be seen in the figure.
Both the internal and external membership functions have a unitary value of possibility only in the rectangular part of the RFV.
If there are only systematic errors in the measurement, then the RFV simply becomes a fuzzy variable which consists of just the internal membership function.
Similarly, if there is no systematic error, then the RFV becomes a fuzzy variable with just the random contributions and therefore, is just the possibility distribution of the random contributions.
A Random-fuzzy variable can be constructed using an Internal possibility distribution(rinternal) and a random possibility distribution(rrandom).
rrandom is the possibility distribution of the random contributions to the uncertainty.
Any measurement instrument or process suffers from random error contributions due to intrinsic noise or other effects.
This is completely random in nature and is a normal probability distribution when several random contributions are combined according to the Central limit theorem.
The probability distribution can be modeled from the measurement data.
This distribution can be built based on the information that is available about the measuring instrument and the process.
This means that every value in the specified interval is equally possible.
This actually represents the state of total ignorance according to the theory of evidence[14] which means it represents a scenario in which there is maximum lack of information.
In this case, depending on the degrees of belief for the values, an appropriate possibility distribution could be constructed.
After modeling the random and internal possibility distribution, the external membership function, rexternal, of the RFV can be constructed by using the following equation:[15] where
An α-cut of a fuzzy variable F can be defined as [17][18] So, essentially an α-cut is the set of values for which the value of the membership function
So, this gives the upper and lower bounds of the fuzzy variable F for each α-cut.
are the lower and upper bounds respectively of the external membership function(rexternal) which is a fuzzy variable on its own.
are the lower and upper bounds respectively of the internal membership function(rinternal) which is a fuzzy variable on its own.
To build the RFV, let us consider the α-cuts of the two PDs i.e., rrandom and rinternal for the same value of α.
This gives the lower and upper bounds for the two α-cuts.
can be defined by [11] Using the above equations, the α-cuts are calculated for every value of α which gives us the final plot of the RFV.
A Random-Fuzzy variable is capable of giving a complete picture of the random and systematic contributions to the total uncertainty from the α-cuts for any confidence level as the confidence level is nothing but 1-α.
