Let's say you have Crime rate in 100 cities and you want to divide them to 3 sets (Low, Med, Hi).
Let's consider the triangular membership function where the parameters are:
lower limit a, an upper limit b, and a value m, where a < m < b.
what are some of the optimal ways to define the parameters of each fuzzy set (Low, Med, Hi)?
Do I just need to do clustering then determine the lower and upper limits (min and max of each cluster) then adjust the value m manually?
any help is appreciated
The three fuzzy functions which are to define Low, Middle and High are not supposed to be disjoint. Like the age of human beings, we can define a range where the rate is low (say 0 to 5%) the membership function has a value of 1, between 5% and up to 40% it decreases to zero following a straight line joining 5% and 40%. The middle one may be starting at 20% goes to a high at 50% and comes down to zero at 80%. Similarly the third one starts at 60 % goes to 1 at 90% and continues so till 100%.
If we don't want any plateau then the first one can be started at 0% and the last one ending at 100%. This will define three triangular fuzzy sets to define the three concepts.
Now suppose we want to use clustering then how will we define the closeness of the elements? We have to keep thresholds. Perhaps the most vulnerable part of fuzzy set theory is the definition of membership functions. There is no fixed approach to define the membership functions. One has to define it depending upon he application where it is used.
Hi Mr Alateeq
There are different ways to do this, but I think the right way is this:
You have to leave the judgment to experts and criminologists.
To do this, design a series of binary survey forms so that experts can rate the parameters.
Then, according to the scores given by the experts, you can determine the low, medium and high limits for the desired parameters by manual or software methods.
Good luck
Dear Majed Alateeq ,
There are different methods like 1) Triangular membership function 2) Trapezoidal membership function 3) Gussian membership function etc. But it mainly depends upon the researcher in which way he/she wants to divide to the crisp observation.
Coming to the point finding the boundaries(i.e low and high), there are different methods available like
1) By using clustering techniques ( DOI: 10.1007/s13369-020-04772-4 and
DOI: 10.1007/s13369-020-04721-1
2) By considering the start and end point of the data set find Universe of discourse and divide it into equal or unequal length of intervals.
3) Use cumulative probability distribution approach (CPDA)
( DOI: 10.1007/978-3-030-30271-9_4)
4) By using Trend analysis approach.
By going through above concept, I hope you will get some idea regarding, how to divide the data set into some parts. Further if you required any in depth concept on this let me know about it.
Thanks
You may try to design your own FLC and optimize the MFs like this paper posted on https://www.researchgate.net/publication/344638760_Dimensioning_and_Power_Management_of_Hybrid_Energy_Storage_Systems_for_Electric_Vehicles_with_Multiple_Optimization_Criteria