# Fuzzy Set Theory

4
Belief Level of a Fuzzy Multiset?

What is the correct way of assigning a belief level to a multiset? Lets say we have the following multiset: {(0.2,x),(0.8,y),(0.5,z)}. Is it just the minimum like it would be for the intersection of fuzzy sets, 0.2? (here x,y,z are different variable with different membership functions).

I need this to then compare belief level of different multisets. (I will also be very greatfull if you share the reference that explains the correct way to do this).

Thanks!

Dr. Tripathy,

Yes, I now see that I was wrong to consider my example as a multi-set. What I was considering was a "function" that depended of multiple fuzzy sets (each one with its own MF) to define it's value and I needed a belief degree for that value (to later get a fuzzy expected value of all the results).

After more investigation I am pretty sure that the extension principle of Zadeh in fuzzy set theory defines that that belief degree would be the minimum value of the membership values given by each fuzzy set (given by crips values) .

Please correct me if I am wrong and thanks for the answers!

2
What are the differences between these two concepts for fuzzy numbers: “L is decreasing over [0, +∞)” and “R is decreasing on [0, +∞)”?

What are the differences between these two concepts for fuzzy numbers:  “L is decreasing over [0, +∞)” and “R is decreasing on [0, +∞)”.

Thank you very much Sir

4
Regarding this abstract, is there a system with failure interaction so that a failure of component impacts on the other component failure?

Repairable system is commonly used in different industries and has been become more complex. In complex system, in addition to the random component failure, the effect of two component is random. The impact of failure is uncertain and can be considered as fuzzy. Fuzzy set theory has been the most important approach used to deal with uncertainty in problems. The repair of complex system divided to soft and hard. The hard failure causes the system stop and the soft failure does not, but it increases the system operating costs too. In this paper assumed: when the first component fails, it remains in a failed state until the next inspection time. Therefore, if the first component failed in each inspection interval, a downtime penalty cost is incurred. The cost is proportional to the elapsed time from failure time to its detection at inspection time. The short inspection interval increase the number of inspection and cause the extra cost for system. As well, long inspection interval will cause the greater cost due to long elapsed between real occurrence of the failure time and the failure detection (penalty cost). On a finite time horizon, the objective is to find the optimal inspection interval for the soft failure component so that the expected total cost will be minimized.

One of my work was about system interaction consideration with fuzzy cognitive map. Below is the link, it might help

http://www.inderscienceonline.com/doi/abs/10.1504/IJQET.2015.071652

4
Sum of memebership functions for a special X value?

As you know membership functions can overlap, which means a value of X can belong to more than one fuzzy set.

The attached image is my membership functions for a variable.

My question is that should the sum of membership values for a special X, be equal to 1?

As you see in the image sum of two memberships at X=30 isn't equal to 1.

what is wrong with this variable fuzzification?

What do you mean from complete fuzzy?

4
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?

Rough set and Fuzzy set are different types of uncertainty. Rough set is more flexible and occupy more data than fuzzy set. Also, optimal solution extracted from an optimization problem  rough data is better than fuzzy data.

4
Does anyone have a good reference on fuzzy Case based reasoning ?

Does any one have a good reference on fuzzy Case based reasoning with an illustrative example ?

Thank you Midya for the selected Papers

Best Regards

4
How can I tranform crisp data into intutionistic fuzzy sets (IFS)?

Does anybody can suggest me how to convert crisp data into intuitionistic fuzzy sets (IFS)?

For example I have quantitative data about CO2, energy consumption, noise etc. How can transform these data into μ (x) and ν (x)?

Fuzzy rules in the fuzzy set part of matlab toolbox is the very instructive document to understand fuzzy data  concepts;

I agree with Pr. Mahmoud;

4
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)?

1988 Warrick, A.W., Zhang, R., El-Haris, M.K. and Myers, D.E., Direct comparisons between kriging and other interpolators in Proceedings of the Validation of Flow and Transport Models in the Unsaturated Zone, Ruidoso, NM, 23-26 May, 1988, 505-510

If you don't find it elsewhere then go to my homepage, look at the link for "papers" and scroll down to this paper, right click and choose "save link as"

My point above is that there are various sources of uncertainty, some pertain to choosing the interpolation method, some to choices once the methodology has been chosen, some due to whether the method is based on theoretical assumptions and whether these are satisfied, there may even be uncertainties associated with the data (not only measurement/observation/analytical but spatial location uncertainties. I suggest that in most cases the real question is whether the results are useful and make sense. For more philosophical discussions I refer you to Matheron's "Estimating and Choosing", also see the last part of his 1971 notes on "The theory of regionalized variables" or perhaps "Geostatistics" by J.-P. Chiles and P. Delfiner (J. Wiley), both were early students of Matheron's

6
Anyone familiar with fuzzy membership functions?

Is there any restrictions in the number of membership functions of FLC?

Is it must to have all the output MFs  related with rule base if we have more than what we need?

Say I have 2 input variables with 4 subsets each which results in 16 rules. I have selected 40 membership functions in the output with very small values and amongst them only 12 are related to my rule base. Other 28 M.Fs lies between the range.. Can I implement a FLC like that?

Yes, there is no restriction regarding the no. of MFs. But the case that you mentioned is not a good FLC, because when we design a FLC, we consider only those MFs for output parameter which we desire for our context.  Now, if any MF of output parameter is absent in the consequent part of fuzzy if-then rules throughout the whole rule-base, then it implies that in our context, there will be no such  situation where I have to correspond our output to that particular MF. Thus,  there is no need for that particular MF to consider for our output parameter at all.

9
What are the areas in which IF set theory has proved its utility so far?

Intuitionistic Fuzzy Set Theory

It's a very intresting question, this question seemingly also exists in other fields, such as fuzzy sets, soft sets, rough sets etc. Maybe you will argue that IFS, FS, Soft sets can be used to solve many practical problems, and there are already many such papers in these fields. But we should notice that as a kind of mathematical tool,  IFS, FS, Soft sets are not exclusive in these applications; in other words, usually it's okay to substitue other methods for IFS, FS, Soft sets. This is very different from traditional maths.

35
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?

You can use related matlab toolbox but first you need data collected from real-life system to which you will apply fuzzy. You can try various fuzzy functions from among triangular or other forms and make decision about which reflects the case better. Generally speaking triangular is one of most encountered one in practice. Good luck..

8
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

3
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?

9
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.

13
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/

1
What are the basics of fuzzy Mic-Mac analysis (ISM Modelling)?

Researchers in Decision modelling and Fuzzy Logic

Dear friend

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

11
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...

6
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?

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

4
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.

2
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?

8
What is the best approach for generation of fuzzy rule ?
.

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

1
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.)

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

5
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?

8
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?

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?

3
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?

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]

6
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..

6
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.

5
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.