# Fuzzy Set Theory

How do I choose membership functions in a fuzzy system?
Membership functions (MFs) are the building blocks of fuzzy set theory, i.e., fuzziness in a fuzzy set is determined by its MF. Accordingly, the shapes of MFs are important for a particular problem since they effect on a fuzzy inference system. They may have different shapes like triangular, trapezoidal, Gaussian, etc. The only condition a MF must really satisfy is that it must vary between 0 and 1. What are some other criterion that I need to be aware of to make a sensible choice of the MF?
Prem Kumar Singh · University of Malaya

It is totally based on the context...Suppose in the given context if you get an element which fully belongs to the set than it is triangular..If the context having two or more elements then trapezoidal...Similarly for others membership techniques.......

What are the areas in which IF set theory has proved its utility so far?

Intuitionistic Fuzzy Set Theory

Vilem Novak · University of Ostrava

I have serious doubts about usefulness of IF. Focus better on classical fuzzy systems and fuzzy sets, there are still interesting problems to solve.

• How can I choose membership function in fuzzy systems?

Im working on fuzzy FMEA(Failure mode and effects analysis)  project. i cant choose membership function for O,D,S and RPN. which membership function do you suggest for such a project?

plz help me

How can soft set theory be applied in game theory?

reference to a paper is given by Molodtsov in his paper, "Soft set theory first results". But the referred paper is in Russian.

B.K. Tripathy · VIT University

ear Prof.Dmitry Zhdanov,
You might be glad to know that i could get the paper in russian sent by you translated into English using web facilities.
1. In the mean time could you get the other paper!
Some more papers of Molodtsov are required by me, which are as follows:
D.A. Molodtsov, Stability of Optimality Principles, Nauka, Moscow (in Russian), (1987).
2. D.A. Molodtsev, An approximate integral in multidimensional case, Preprint of Computer Center of Russian Academy of Scieness, Moscow (in Russian) (1989).
3. D.A. Molodtsov, The law of large numbers for interval probability (mathematical apparatus), Preprint of Computer Center of Russian Academy of Sciences, Moscow (in Russian) (1992).
4. D.A. Molodtsov, The law of large numbers for interval probability (product space), Preprint of Computer
Center of Russian Academy of Sciences, Moscow (in Russian) (1992)
Once again thanks and regards,
B.K.Tripathy

Has anyone investigated the aleatory uncertainty of Kriging?

Generally, two different uncertainty sources, including aleatory uncertainty and epistemic uncertainty are studied. There are some different methods for exploring epistemic uncertainty, including random theory, fuzzy set theory, etc. Then how and what method would be suitable for revealing aleatory uncertainty when comparing two or more different models(e.g. spatial distribution mapping/predictive mapping of soils such as kriging and artificial neural networks)?

Antoon Bronselaer · Ghent University

Dear Amin, you  are right that, depending on the type of uncertainty, different frameworks to model that uncertainty are to be used.

What you call epistemic uncertainty is in general considered as uncertainty due to missing information. In that case, possibility theory (an uncertainty theory based upon fuzzy sets) is a valid framework.

What you call aleatory uncertainty is in general considered as uncertainty due to randomness in the outcome of an experiment. In that case, probability theory is the suited framework to work with.

To sum up and taking into account that Kriging relies on confidence intervals, probability theory is the framework you should use. As for you question on how to "reveal" this uncertainty, I am not quite understanding your question completely.

Best

What are the difference between Rough Set, Near Set, Shadow Set and Fuzzy Sets? With Example?

What are difference between Rough Set, Near Set, Shadow Set and Fuzzy Sets? With Example?

Davod Darvishi · Payame Noor University,Tehran,iran

if you find any reference in this case please send me

• What are the limitations of Grey set as uncertainty model?

Grey theory is an extension of fuzzy set theory and rough set theory where two memberships function a lower one and an upper one are used with interval. Grey system theory is a unique concept which deals with continuous systems including uncertain. Grey theory classifies sets into three groups: White sets, Black sets and Grey sets. White set contains objects that have complete knowledge behavior, while black set contains objects which have unknown behavior.
What are the limitation of Grey set as uncertainty model?

What is the difference between type1 - fuzzy logic and type 2 - fuzzy logic ?
Thanks.

In Type 1 fuzzy set , Expert should determine the degree of achieving the characteristics of the object. For example, if you have a 3 different red balls. The first is red by 75%, second is red 85%, Third is red 95%.
In Type 2 Fuzzy set, Expert can't determine exactly the degree of achieving the characteristics. For example, if you have a 3 different red balls. The first is red by 75%-80%, second is red 85%-90%, Third is red 95%-100%. So it presents an interval fuzzy set.

Which is the best book to study and get solutions over maths related to Fuzzy Logic for beginners?
I am a beginner and am studying fuzzy logic from the book "Fuzzy sets and Fuzzy logic" by M. Ganesh, now there's a problem with the maths section, actually I find maths too difficult to understand from that book so would like to know some other book and prepare for that.

http://logic.harvard.edu/EFI_CH.pdf

http://www.ams.org/notices/200106/fea-woodin.pdf

http://www.ams.org/notices/200107/fea-woodin.pdf

http://www.math.unicaen.fr/~dehornoy/Surveys/DgtUS.pdf

http://logic.amu.edu.pl/images/a/a5/Foremanch.pdf

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC221287/

How true is it that fuzzy sets and rough sets are soft sets?

In his first paper on soft sets Molodtsov (1999) has argued that all fuzzy sets are soft sets and in 2010 Herawan has written a paper to show that rough sets are also soft sets.

B.K. Tripathy · VIT University

For deriving any answer one has to refer the following two papers:

D. Molodtsov, Soft set theory-First results, Computers and Mathematics with Applications, 37, (1999), pp.19-31

T.Herawan and M.deris, A direct proof of every rough set is a soft set, 3rd Asia international conference on modelling and simulation, (2009), pp.119 - 124.

• What are the basics of fuzzy Mic-Mac analysis (ISM Modelling)?

Researchers in Decision modelling and Fuzzy Logic

Sarfaraz Hashemkhani Zolfani · Amirkabir University of Technology

Dear friend

As far as I may know, Mic Mac using for Cross Impact analysis and also related to futures studies.

Fuzzy set - what is meant by level set?
I am a new student in Fuzzy sets & Fuzzy relations. I can found that the Level of a fuzzy set as
ΛA = {α/μA(x) = α for some x belongs to X} , Please provide an example of this with a set.

1. http://www.mv.helsinki.fi/home/niskanen/zimmermann_review.pdf

2. http://www2.cs.uregina.ca/~yyao/PAPERS/combination.pdf

4. debian.fmi.uni-sofia.bg/~cathy/SoftCpu/fuzzy_geometry.pdf

Hope that these references will be useful to you...

What is the best way for choosing membership function in fuzzy logic?

In order to obtain batter results in ANFIS, different membership functions are used. Is there any inductive way for obtaining best membership function based on the type of data used?

Ganesan G · Adikavi Nannaya University

It is always better using testing of hypothesis through samples to fix membership values.

Please, help me with type-reduction of interval type-2 fuzzy sets?

I have a problem with Karnik-Mendel algorithm - computing yl, yr and switch points. Please, give me example how to compute yl, yr and switch points.

Dragan Pamučar · University of Belgrade

Is the definition of Dubois to be corrected regarding Rough-Fuzzy Sets?

In Dubois Rough-fuzzy Model, Min and Max operators are used for Lower and Upper Approximations. Can we swap Min and Max operators?

What is the best approach for generation of fuzzy rule ?
.
Morteza Seidi · University of Maine

In case you have the model of the system, by using model-based fuzzy control systems such as Takagi-Sugeno, you can obtain fuzzy rules and membership functions analytically and guarantee the closed loop stability. The attached is a chapter on Takagi-Sugeno fuzzy model.

Good luck

How can I define an numerical example (specially real world problem) for temporal intuitionistic fuzzy sets.?

There are several studies about Atanassov Temporal Intuitionistic Fuzzy Sets. But I can not find an numerical example for it. (Specially real world problem.)

Nasruddin Hassan · National University of Malaysia

Look up articles written by Shawkat Alkhazaleh, Khaleed Alhazaymeh and Abdul Razak Salleh amongst others

What is the difference between Fuzzy Gray and Grey Fuzzy?

Can anybody tell me that more precisely, what is a "Fuzzy grey Set"? what is a " grey fuzzy Set"?  And what is the difference between “Fuzzy ”, “grey” and “Rough Set?” please ?

What is the difference between Fuzzy Gray and GreyFuzzy?

What is the method of constructing the membership function of a fuzzy vector?
I think in the mathematics of fuzziness the membership function of a fuzzy vector is heuristically assumed. Is there any logical method to construct the membership function of a fuzzy vector?
Lawrence Ibeh · Ludwig-Maximilian-University of Munich

I used  the  matlab fuzzy logic tool box,  after plotting a membership function how  do I derive  my membership values automatically to be used for  a further analysis? Can any one help?

What are the advantages and disadvantages of using fuzzy with rough set?
A rough membership function may be interpreted as a special kind of fuzzy membership function. Under this interpretation, is it possible to re-express the standard rough set approximations, and to establish their connection to the core and support of a fuzzy set
but what is the weakness and strengths of this model?
Ganesan G · Adikavi Nannaya University

I don't find any disadvantage. The advantage is expanding rough approximations into fuzzy environment which help to obtain solutions for various real time problems [since a good number of real time problems are fuzzy in nature]

Airborne Lidar and Hyperspectral data fusion: which software to use?

I have several airborne hyperspectral 65-105 band datasets with simultaneously acquired lidar data and RGB orthophotos too. Each source has usually a different GSD. I don't have any experience with data fusion so far. My task is to evaluate currently available tools and methods e.g. algebraic procedures, Dempster-Shafer Theory, neural networks, Bayesian networks, fuzzy set theory, principle component analysis or any combined methodology. Suggestions for particular method and tool are welcome.

I have access to commercial (ENVI5, ArcGIS10.2, eCognition8) and free software (QGIS, GRASS, FUSION). Maybe you know of something better for this task..

Thanks !

I have now access to ERDAS Imagine 2014 too.. maybe helps.. It's a shame I don't have latest ENVI 5.2 with LIDAR module..

What is the most significant success of the fuzzy sets theory?
What is the greatest achievement of fuzzy theory in the all applications area and scientific work?

Words have also ability of bifurcations as it is in some cases at equations ( linear or nonlinear) . Then people must use quasi-regularization for next treatment.

What are some good function approximation methods using fuzzy sets and logic such as fuzzy expert systems, fuzzy SVR, etc.?
I have a project on function approximation by fuzzy decision trees and I want to compare my results with some other methods improved by fuzzy logic.

It will be good use the notice that the maximum of entropy product is equivalent to maximum entropy principle because of the properties of ln functions.

I'm working with fsQCA and I'm not clear on how to introduce Enhanced Standard Analysis in fsQCA software. Anyone has access to online resources?

In Schneider and Wagemann book Set- Theoretic Methods for the Social
Sciences is presented an approach to deal with logical remainder Enhanced Standard Analysis. The online resources are not available and I'm not clear how to use fsQCA for this part of the analysis. Is there anyone that can explain/share the online resources?

Annalisa Staffa · University of Milan

Thank you very much Stefan!

That is exactly what I was looking for!

What would be superiority of Sugeno's approach when establishing fuzzy set?

what would be superiority of Sugeno's approach when establishing fuzzy set?

Mahmoud Omid · University of Tehran

Both Mamdani and Sugeno FIS are universal approximators, i.e.,  they approximate any continuous functions to any degree of accuracy Mamdani type FIS gives an output that is a fuzzy set, whereas Sugeno-type inference gives an output that is either constant or a linear (weighted) mathematical expression. According to Ying et al. (1998) the minimal system configurations of the Sugeno and Mamdani FIS's  are  comparable. However, In terms of performance and adaptability to other user defined environment making Sugeno-type FIS highly flexible and optimization of FIS could be done by a well defined set of algorithms such as ANNs or GA.

Hao Ying, Yongsheng Ding, Shaokuan Li' and Shihuang Shao (1998) 'Typical Takagi-Sugeno and Mamdani Fuzzy Systems as Universal Approximators: Necessary Conditions and Comparison, IEEE  Fuzzy Systems Proceedings, PP: 824-828.

How can we use SERVQUAL with fuzzy set theory to measure teaching quality (performance) within academic institutes?

Various techniques such as SERVPERF, SERVQUAL / weighted SERVQUAL are being used for measuring service quality in banking, airlines, restaurants, etc. Is there a review on evaluation of different models of service quality using fuzzy set theory ?

Mahmoud Omid · University of Tehran

"Fuzzy" and statistics?
I am collecting material for a survey paper on the interface of statistics and fuzzy set theory since around 2001 (the date comes from the publication, after the heated debates of the 90's, of the books "Fundamentals of fuzzy sets" in 2000 and "Fuzzy logic and probability applications: Bridging the gap" in 2002).

There are two ways in which you can help:

1. I keep finding papers in some areas I have no expertise on, so suggestions of good papers as a starting point in a specific area will be very valuable.

2. What objections to fuzzy sets, raised within the statistical community or elsewhere, do you think remain valid?

The survey will focus on topics already familiar to statisticians, avoiding some popular topics in the fuzzy community like e.g. statistics with fuzzy data and Tanaka-style fuzzy regression.
Angela Stanton · Self employed

Hemanta, why would there not be fuzzy sets applicable to rain fall on any given day the same way as with temperature? Rain fall "density" is just as fuzzy as temperature. If a single rain drop fell, it rained and thus there would be nothing fuzzy about it but we cannot be sure how many rain drops, how big, do they reach the ground or dry just before hitting the ground (typical California situation), etc. I think that rain as "density of water falling" can be fuzzy as well as temperature variations within the ranges set. Another reason why rain is suspicious becuase as clouds move, it may be raining over my head and not yours yet we walk side by side... there is something special about rain that is perhaps even harder to define than temperature. I am not very familiar with using fuzzy sets for any of my medical experiments since they all have been quite simple yes/no type experiments where even regression is meaningless, so if I am off the subject in some way, I apologize.