The random distribution(r_random)

r_random 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.^[12]

But, there can also be random contributions from other probability distributions such as a uniform distribution, gamma distribution and so on.

The probability distribution can be modeled from the measurement data. Then, the probability distribution can be used to model an equivalent possibility distribution using the maximally specific probability-possibility transformation.^[13]

Some common probability distributions and the corresponding possibility distributions can be seen in the figures.

Normal distribution in probability and possibility.

Uniform distribution in probability and possibility.

Triangular distribution in probability and possibility.

The internal distribution(r_internal)

r_internal is the internal distribution in the RFV which is the possibility distribution of the systematic contribution to the total uncertainty. This distribution can be built based on the information that is available about the measuring instrument and the process.

The largest possible distribution is the uniform or rectangular possibility distribution. 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.

This distribution is used for the systematic error when we have absolutely no idea about the systematic error except that it belongs to a particular interval of values. This is quite common in measurements.

But, in certain cases, it may be known that certain values have a higher or lower degrees of belief than certain other values. In this case, depending on the degrees of belief for the values, an appropriate possibility distribution could be constructed.

The construction of the external distribution(r_external) and the RFV

After modeling the random and internal possibility distribution, the external membership function, r_external, of the RFV can be constructed by using the following equation:^[15]

${\displaystyle r_{\textit {external}}(x)=\sup _{x^{\prime }}T_{min}[r_{\textit {random}}(x-x^{\prime }+x^{*}),r_{\textit {internal}}(x^{\prime })]}$

where ${\displaystyle x^{*}}$ is the mode of ${\displaystyle r_{\textit {random}}}$, which is the peak in the membership function of ${\displaystyle r_{random}}$ and T_min is the minimum triangular norm.^[16]

RFV can also be built from the internal and random distributions by considering the α-cuts of the two possibility distributions(PDs).

An α-cut of a fuzzy variable F can be defined as ^[17][18]

${\displaystyle F_{\alpha }=\{a\,\vert \,\mu _{\rm {F}}(a)\geq \alpha \}\qquad {\textit {where}}\qquad 0\leq \alpha \leq 1}$

So, essentially an α-cut is the set of values for which the value of the membership function ${\displaystyle \mu _{\rm {F}}(a)}$ of the fuzzy variable is greater than α. So, this gives the upper and lower bounds of the fuzzy variable F for each α-cut.

The α-cut of an RFV, however, has 4 specific bounds and is given by ${\displaystyle RFV^{\alpha }=[X_{a}^{\alpha },X_{b}^{\alpha },X_{c}^{\alpha },X_{d}^{\alpha }]}$.^[11] ${\displaystyle X_{a}^{\alpha }}$ and ${\displaystyle X_{d}^{\alpha }}$ are the lower and upper bounds respectively of the external membership function(r_external) which is a fuzzy variable on its own. ${\displaystyle X_{b}^{\alpha }}$ and ${\displaystyle X_{c}^{\alpha }}$ are the lower and upper bounds respectively of the internal membership function(r_internal) which is a fuzzy variable on its own.

To build the RFV, let us consider the α-cuts of the two PDs i.e., r_random and r_internal for the same value of α. This gives the lower and upper bounds for the two α-cuts. Let them be ${\displaystyle [X_{LR}^{\alpha },X_{UR}^{\alpha }]}$ and ${\displaystyle [X_{LI}^{\alpha },X_{UI}^{\alpha }]}$ for the random and internal distributions respectively. ${\displaystyle [X_{LR}^{\alpha },X_{UR}^{\alpha }]}$ can be again divided into two sub-intervals ${\displaystyle [X_{LR}^{\alpha },x^{*}]}$ and ${\displaystyle [x^{*},X_{UR}^{\alpha }]}$ where ${\displaystyle x^{*}}$ is the mode of the fuzzy variable. Then, the α-cut for the RFV for the same value of α, ${\displaystyle RFV^{\alpha }=[X_{a}^{\alpha },X_{b}^{\alpha },X_{c}^{\alpha },X_{d}^{\alpha }]}$ can be defined by ^[11]

${\displaystyle X_{a}^{\alpha }=X_{LI}^{\alpha }-(x^{*}-X_{LR}^{\alpha })}$

${\displaystyle X_{b}^{\alpha }=X_{LI}^{\alpha }}$

${\displaystyle X_{c}^{\alpha }=X_{UI}^{\alpha }}$

${\displaystyle X_{d}^{\alpha }=X_{UI}^{\alpha }-(X_{UR}^{\alpha }-x^{*})}$

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.^[17][18]

An example for the construction of the corresponding external membership function(r_external) and the RFV from a random PD and an internal PD can be seen in the following figure.

Construction of an external membership function and the RFV from internal and random possibility distributions.