page_title: Random-fuzzy variable
section_title: The construction of the external distribution(rexternal) and the RFV
hierarchical_section_title: Random-fuzzy variable / Construction / The construction of the external distribution(rexternal) and the RFV
caption_attribution_description: English: Construction of an external membership function and the RFV from internal and random possibility distributions
context_page_description: In measurements, the measurement obtained can suffer from two types of uncertainties. 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. Systematic errors, if detected, can be easily compensated as they are usually constant throughout the measurement process as long as the measuring instrument and the measurement process are not changed. But it can not be accurately known while using the instrument if there is a systematic error and if there is, how much? 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. But, the computational complexity is very high and hence, are not desirable.
  L.A.Zadeh introduced the concepts of fuzzy variables and fuzzy sets. Fuzzy variables are based on the theory of possibility and hence are possibility distributions.
context_section_description: After modeling the random and internal possibility distribution, the external membership function, rₑₓₜₑᵣₙₐₗ, of the RFV can be constructed by using the following equation:
  where  is the mode of , which is the peak in the membership function of  and Tₘᵢₙ is the minimum triangular norm.
  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 
  So, essentially an α-cut is the set of values for which the value of the membership function  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 .  and  are the lower and upper bounds respectively of the external membership function(rₑₓₜₑᵣₙₐₗ) which is a fuzzy variable on its own.  and  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. Let them be  and  for the random and internal distributions respectively.  can be again divided into two sub-intervals  and  where  is the mode of the fuzzy variable. Then, the α-cut for the RFV for the same value of α,  can be defined by 
  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.
  An example for the construction of the corresponding external membership function(rₑₓₜₑᵣₙₐₗ) and the RFV from a random PD and an internal PD can be seen in the following figure.
caption: Construction of an external membership function and the RFV from internal and random possibility distributions.
exif: {}