Fuzzy Set Theory - Science topic

There should be no difference regarding accuracy, regardless of whether you use "pof", "pofind", or "superSubset" from the QCA package or "QCAfit" from the SetMethods package.

In terms of utility, this obviously depends on your research aims. As a standard function, I recommend working with QCAfit to investigate the potential necessity of all conditions in a single analytical step. Often, this is what you want to know - whether there are any conditions in your analysis that are in themselves consistently necessary.

The superSubset function is useful but it can also result in misleading conclusions, as when researchers try to make sense of necessary disjunctions with two or more elements. Of course, if there are theoretical expectations about specific SUIN conditions, then these could be tested with superSubset and/or with a specified argument in pof.

Majid Baseer In fuzzy logic and control systems, two forms of fuzzy numbers are used: triangular fuzzy numbers and trapezoidal fuzzy numbers. Both sorts of fuzzy numbers are used to represent fuzzy sets, with a fuzzy number defining the membership function of a fuzzy set.

Three points define triangular fuzzy numbers: a left endpoint, a right endpoint, and a peak point. A triangular fuzzy number's membership function is specified as a triangle shape. In situations where the membership function is known to be symmetric and the peak point is known, these fuzzy numbers are employed.

Four points define trapezoidal fuzzy numbers: a left endpoint, a right endpoint, a left support point, and a right support point. A trapezoidal fuzzy number's membership function has a trapezoid form. In situations where the membership function is known to be asymmetric and the support points are known, these fuzzy numbers are employed.

The usage of triangular or trapezoidal fuzzy numbers is determined by the application and the information available about the membership function. In general, a triangular fuzzy number should be utilized if the membership function is known to be symmetric and the peak point is known. If the asymmetry of the membership function is known and the support points are known, then a trapezoidal fuzzy number should be used.

The findings may differ if we use a triangular fuzzy number instead of a trapezoidal fuzzy number, or vice versa. It will be determined by the particular application and the nature of the membership function. The results may not vary considerably in certain circumstances, while they may alter significantly in others. As a result, it is critical to carefully analyze the membership function's features and select the suitable type of fuzzy number.

Hi

I recommend Matlab.It includes Fuzzy Logic Toolbox

Dear Eman Mohammed,

I suggest you see links on the topic.

Best regards

My background is from government nuclear sites in both the US and Canada. First, I do recommend reviewing REGDOC-2.1.2, Safety Culture - Canadian Nuclear Safety Commission This document includes a scoring grid (similar to Malcom Baldrige for quality) on safety culture. I highly recommend it as a leading indicator.

Other leading indicators I have used are:

Management participation in observations

Timeliness of corrective actions for safety and quality issues

Employee Surveys (especially the question "Senior Management (above my manager) visits my place of work")

Interviews of workers / managers and/or focus groups

Voluntary Protection Program (US OSHA / DOE) participation

Safety training attendance / safety training conducted on schedule

Qualification compliance (did people do their re-quals)

Employee grievances (safety or non-safety) if unionized

Employee concerns (if there is a program)

Workplace tours by managers / experts from facilities other than the facility

The Red Pen - Blue Pen Excercise (From Dr. Bill Bellows Red Pen and Blue Pen Companies Notes on the “Starter” Exercise ... (yumpu.com)

Safety Council activities (if there are employee safety councils)

o.5 is central value of interval [0,1]. Most fuzzy systems are designed as 0.5 as intersection value of adjacent membership functions.

Dear Merve

I apologize for my past request of a questionnaire. I was confused with another project from another colleague that I am also examining

Lets’ see. As I see your problem, you can have an example, that is, a complete initial decision matrix with crisp values, and you want to apply the four fuzzy sets to this same scenario.

Once you do that, you want to compare the ranking when using, the four fussy sets, and using a MCDM method. Then, you will get four probably different rankings. This appears to be, the same problem that researchers are trying to solve from decades, and it is to determine which is the best method.

Trouble is, that we don not know Which the ‘true’ answer is, and therefore, we don’t have a yardstick to compare methods.

I would like to suggest a method that does not give the solution you are looking for, but that can give you a hint of which of the four fuzzy sets gives the best result.

Suppose your initial scenario has 15 alternatives.

Each fuzzy set most probably will give you a ranking of the 15 alternatives.

In my opinion, you can use entropy to determine the amount of information contained in each ranking, something that is easily done in a couple of minutes, the higher this value the better, and it will allow you to determine which is the fuzzy set that produces the highest amount of information, and this could be something than can be used as a comparison among fuzzy sets.

Remember that entropy indicates disorder or in case of quantities in vectors, it shows the amount of information that each one contains, which is (1 – entropy). The larger this value means that the values are dispersed, there is discrimination, and then it facilitates decisions. Decisions are very difficult to take if the components of a problem have similar values, and it is easier when they have dissimilar value.

Anyway, it is only and idea

This paper is from a federal university from Brazil, unfortunately is in portuguese, but it seems very complete and if you are able to translate it it should help you.

It has been established that vague sets are intuitionistic fuzzy sets in an article:

H.Bustince and P.Burillo: Vague sets are intuitionistic fuzzy sets, Fuzzy sets and systems, 79(3), (1996), pp. 403-405.

This article provides the answer.

a simple algorithm of arithmetic operations as well as easy and intuitive interpretation

I recommend the following article:-

Dear Faisal A. Garba,

If you want to review papers and research on Fuzzy Expert System, it is difficult to suggest to what point you should study on the same, without fully knowing how much you know about the topic. But generally speaking, you must know the fuzzy set theory and its applications very well (to an intermediate level, at least).

But I will suggest you that start with the basics, and after you are done with it, start going through the papers. And if you face problems while reading papers do your research topic-wise and try to return to the basics, in that way I believe you will have a better grasp of the topics and more. That being said, I am suggesting you some reading materials:

I have also enclosed two great books on fuzzy logic and fuzzy set theory, and course material from MIT read them well,

Best of Luck!

Dear Mukesh,

Many thanks for raising this interesting question about what happens to so-called "peripheral conditions" when neither the conservative (QCA-CS) nor the intermediate solution of QCA (QCA-IS), but only the parsimonious solution (QCA-PS) should be used, as argued by Baumgartner and Thiem (2017). It is somewhat specific to the business and management QCA community, as no other area uses a distinction between "core" and "peripheral" conditions, but I think it is definitely worth considering the implications of Baumgartner and Thiem's (2017) findings in this connection nonetheless.

Before going deeper into the reply, let me point you to Thiem (2019, attached to this reply as an open access PDF), a very recent follow-up publication to Baumgartner and Thiem (2017). While Baumgartner and Thiem (2017) showed that QCA-CS and QCA-IS often generate false inferences, Thiem (2019) explains why this is the case. I use concepts from both Baumgartner and Thiem (2017) and Thiem (2019) in the remainder of my reply and refer to both works simply as Baumgartner and Thiem (2017/2019).

Let us first check Fiss' definition of "core" and "periperal" conditions: "I define core elements as those causal conditions for which the evidence indicates a strong causal relationship with the outcome of interest and peripheral elements as those for which the evidence for a causal relationship with the outcome is weaker" (Fiss, 2011:394). Alerted readers already note some inconsistencies in this definition: talk is of a strong causal relation with respect to "core", but of weak evidence for a causal relation, not of evidence for a weak causal relation, with respect to "periphery". These are clearly different things (there can be evidence for strong/weak causation, or strong/weak evidence for causation, or any combination of that). In the following, I assume that what Fiss means is strong and weak evidence for a causal relation (though I may be wrong, Fiss never clarified this).

At a later stage, Fiss then defines what, under the framework of QCA, he understands by "core" and "peripheral" conditions: "[...] core conditions are those that are part of both parsimonious and intermediate solutions, and peripheral conditions are those that are eliminated in the parsimonious solution and thus only appear in the intermediate solution (Fiss, 2011:403). Note that Fiss does not include QCA-CS anywhere (he mentions it in footnote 3, but brushes it aside as providing "rather little insight into causal configurations" [Fiss, 2011:403]).

If you do not question Fiss' view, and accept the argument of Baumgartner and Thiem (2017/2019) that researchers should use QCA-PS only, you may conclude that the implication is that researchers should use QCA only for identifying causal relationships for which there is strong evidence. No real problems arise in this case. You use QCA-PS, and will remain in line with Baumgartner and Thiem's findings and at least please all of the proponents of Fiss' view who favor results based on strong empirical evidence.

However, if you depart from Baumgartner and Thiem's (2017/2019) argument that QCA-CS and QCA-IS are generally unsuitable for empirical data analysis with a view to cause-effect relations, you will have to ask yourself whether QCA-IS is really based on weak evidence for causal relations. And here the problems start.

Based on Baumgartner and Thiem (2017/2019), you will have to conclude that Fiss (2011) is wrong insofar as the evidence for a causal relation for all QCA-IS-elements not contained in QCA-PS is not weak, but, in fact, nil. In other words, either you rely on empirical evidence and use QCA-PS to infer on that basis about causal relations, or you start adding made-up data to the empirical data and necessarily introduce a significant risk of producing false inferences. If I were a judge in court, a policy-maker, an officer in the Food and Drug Administration etc., I would clearly favor evidence-based results, and would get suspicious about analyses based (in large parts) on data which researchers have simply made up.

In summary, the are no "core" or "peripheral" conditions based on Baumgartner and Thiem (2017/2019). There are only conditions that can be shown to be causes based on empirical evidence, or conditions that cannot be shown to be causes based on empirical evidence.

I hope this helps, and answers your question. If not, or if you have further questions, please don't hesitate to ask.

All the best,

Alrik

Dear Prof Aswant Kumar Sharma

I think no relationship between the two.

Dear Benyamin,

Most likely, you are not presented with a parsimonious solution because your output function values do not vary. That is, you have only positive minterms in your truth table (rows showing a "1" on the output value). Under such a data situation, nothing can be inferred about potential causes because there are no difference-making situations (identifying a cause always requires some kind of difference-making).

The question why you still get a conservative and an intermediate solution is easy to answer: these two solution types supplement your empirical data with artificial data, i.e. data that do not exist. They do so by creating negative minterms in the truth table (rows showing a "0" on the output value) without you noticing. By doing so, difference-making situations are artifically created, from which inferences are then drawn and presented. Specifically, conservative solutions put a "0" on all minterms that differ in exactly one position from any positive minterm you have. Intermediate solutions put a "0" on all minterms that differ in exactly one position from any positive minterm you have, and which you have designated as so-called "difficult counterfactuals" by using the respective radio buttons for directional expectations in the fs/QCA menu that pops up when conducting so-called "Standard Analyses".

This problem has first been discovered in Baumgartner and Thiem (2017), and has been explained in all necessary technical detail in Thiem (forthcoming).

In conclusion, do not use conservative or intermediate solutions, use parsimonious solutions if you're interested in cause-effect relations. But to do so, you need other/more empirical data. The data you currently have don't allow you to make any inferences about cause-effect relations.

I hope this helps.

Best wishes,

Alrik

It is important to consider that the performace also depends on the fuzzy system type: mandami or TS.

Dear Victor,

fs/QCA 2.5/3.0 often spits out "ERROR (Quine-McCluskey): The 1 Matrix is Empty" or "ERROR(Quine-McCluskey): The 1 Matrix Contains All Configurations".

The first error message is returned when all your output values are negative, i.e. when there is not a single row in the truth table which is accepted as being sufficient for the outcome. This happens when your consistency cut-off is higher than the consistency score of the best-performing row).

The seond error message is the opposite. It is returned when all your output values are positive, i.e. when there is not a single row in the truth table which is accepted as not being sufficient for the outcome. This happens when your consistency cut-off is lower than the consistency score of the worst-performing row).

Neither case is good of course. For QCA to work, you need at least one row that is negative and one row that is positive.

However, you will also notice that even though all of your rows are positive / negative, fs/QCA will still produce complex and intermediate solutions. This is only possible because, for producing these two solution types, fs/QCA adds artifical data to your data set through an algorithmic back door (if you want to know more about this process, let me know, it would lead too far here). In other words, these two solution types enlarge your data set without you noticing, and with some quite desastrous consequences (it can be shown that both solution types not only produce inferences beyond your data, but that these inferences are often leading you way off path).

I hope this helps.

Best wishes,

Alrik

Excellent answers

Excellent answers

{(a, 0.4),(b, 0.3), (c, 0.4)}

Thank you sir @Ganesan G

Dear Alelsandr,

I suggest you to see links and attached files in yopic.

-Journal of Intelligent & Fuzzy Systems - Volume 34, issue 1 - Journals ...

-Fuzzy Automata and Languages: Theory and Applications ...

https://www.amazon.com/Fuzzy...Languages-Applications.../15848...

-Myhill–Nerode type theory for fuzzy languages and automata ...

https://www.sciencedirect.com/science/article/.../S01650114090026...

-Fuzzy automata and languages : theory and applications in ...

-application of fuzzy languages to pattern recognition - Emerald Insight

-Applications of fuzzy languages to intelligent information retrieval ...

Best regards

I think there is not an ordered relation that is defined on neutrosophic sets.

See this:

I'm not able to sent code for you, but this should be quite easy to re-implementation.

Rough and Fuzzy set are almost same application wise. But theoretically they are different which makes Rough Sets superior than Fuzzy (Personal Opinion). Let me explain why theoretically Rough is better than Fuzzy. Fuzzy set starts with identifying a membership function a-priori and tries to fit the data in its theory, whereas Rough Set starts with no such assumption on membership function. Rough sets straightway starts fitting the data blindly from which membership function values are computed. This is why Rough sets make better explanation of uncertainty as it mimics what the data speak. As per my opinion Rough set is better suited in case of data science where prior information and knowledge about the process under consideration are not available and the analysts have to rely purely on the data.

Dear

The fuzzy set theory initiated by L.A. Zadeh provided mathematicians with an

appropriate tool for modelling the vagueness phenomenon and shed new light into

the control theory for engineers. Later, in 1985, T. Takagi and M. Sugeno invented a

particular fuzzy model which became very popular due to its approximation ability.

Finally, in the 1990’s, several studies aimed at approximation properties of the other

widely used fuzzy models.

Regards!

Adriana.

Dear Mr. Imran ,

I think the question is too general as membership function is a hard elementary problem in the fuzzy set framework. It is SUBJECTIVE and APPLICATION-DEPENDENT problem. Therefore, there are no general criteria. For example, for user-centric applications such as the applications of fuzzy rule base systems (FRBSs), their interpretability is required, including the interpretability of fuzzy partitions, called the low-interpretability of FRBSs, that forms the so-called frames of cognition of their respective linguistic variables. There have been many studies of the low-interpretability in which many criteria (constraints) have been proposed. You can easily to find such studies on internet using some specific key-words.

For the applications which are required to interact with human user, the presence of linguistic words is necessary, the fuzzy sets to be designed must be associated with words, you should determine words fro every variable first for your application, noting that fuzzy theories are aimed to simulate human capabilities in handling words. They may form the so-called Linguistic Frames of Cognition (LFoCs). The meaning of the words of a determined LFoC for your application may suggest you to construct the fuzzy sets of their respective words. However, the effectiveness of your method to solve an application problems is crucial. So, adjusting or turning your fuzzy sets constructed for the LFoCs is necessary.

In the latter case, one way to construct fuzzy sets of the words of your determined LFoCs is to produce the desired fuzzy sets from just the semantics of the words of LFoCs. However, in this case, the word semantics must formally be defined and the word-domains of linguistic variables must be formalized to become (semantic) order-based structures, called hedge algebras, as linguistic hedges of every variable play algebraic unary operators. Note that hedges play a significant role to generate the words and their semantics of a (linguistic) variable in natural languages of human beings. The algebraic approach to the semantic structures of word-domains of variables is very specific and you can find its materials in my account of ReseachGate. It provides a formalism to connect words with their designed computational semantics, including their fuzzy set based semantics, in which the fuzziness parameters of their linguistic variables, comprising few fuzziness parameters of atomic (primary) words and of hedges of their variables, play a crucial role. That is, the given numeric values of these parameters of a variable do, in general, determine the computational semantics of the all words of the variable.

You could run your data through a paired comparisons model with the public vs. private as a predictor variable (possibly even an interaction variable, though I have never really tried that). There are also Rasch rating scale models that have similar functionality.

Dear Mohammed, thank´s for replying!. Sorry if I did not explained properly.

I have a linear relation between a dependent and an independent variable (x and y). I need to prove that x and y are equivalent. To do that I have already considered two ways: 1-verifying reflexivity, symmetry and transitivity; and 2- proving that it is a bijective function. If this is right, I have already done the first step. As a second task I have to extend this relation to fuzzy sets and I only need to prove min-max transitivity at present.

I need to build the adjacency matrix of such relation to demonstrate the rest of the properties and I think I should get a identity matrix representing variables x and y, but I don´t know if there is any theorem about this. That´s to say: If the linear ecuation is bijective, the adjacency matrix of such relationship is necessary an identity matrix?

.

(0, 0, 0) encodes a complete ignorance

for the other examples with such strong conflicting beliefs, you are in the realm of logical paradoxes

for instance, (1, 1, 1) could be the value of the proposition :  "This proposition is false"

(an example taken from Gershenson's paper, Comments on Neutrosophy, Proceedings of the First International Conference on Neutrosophy, Smarandache ed.)

.

Dear Maria, thank you for your question. You are, due to what has been claimed in most of the QCA literature, under the false impression that the complex/conservative solution (QCA-CS) is a subset of the intermediate solution (QCA-IS), which itself is a subset of the parsimonious solution (QCA-PS).

In fact, it is the other way around. QCA solutions claim causal relevancies, and QCA-CS makes by far most such claims. Take this simple example: QCA-CS is A*B -> Z, QCA-PS is A -> Z. Now, (A -> Z) -> (A*B -> Z) can be transformed into ¬A + Z -> ¬A + ¬B + Z. Clearly, this is true under all possible assignments of truth values to A, B and Z.

In contrast, (A*B -> Z) -> (A -> Z) can be transformed into ¬A + ¬B + Z -> ¬A + Z, which is false for A = 1, B = 0 and Z = 0. In other words, QCA-PS is a subset of QCA-CS, not vice versa.

That you occasionally see causal claims in QCA-CS that have no counterpart in QCA-PS is not surprising because QCA-CS introduces data artificially through the backdoor, which QCA-PS does not. This is another common error in the QCA literature, where you often read that QCA-CS makes no assumptions, in contrast to QCA-PS (e.g., Ragin, 2008, 173; Schneider and Wagemann, 2012, 162).

QCA-CS claims that ALL logical remainders are not sufficient for the outcome, which is much stronger than what QCA-PS claims, namely that some remainders are sufficient. Claiming non-sufficiency is much stronger than claiming sufficiency.

Take this example: QCA-CS claims for every remainder X that ¬(X -> Z). This claim is true only when X occurs together with ¬Z. In other words, QCA-CS assumes that every remainder is in fact observed as real data, and always in conjunction with the negation of the outcome. You can test this for yourself. Just add pseudo cases to your data for each remainder such that they show the negation of the outcome you analyze, and I can guarantee that the solution will ALWAYS be the same as the one for your original data.

Analogously, add exactly those cases to your data for which QCA-PS has made the assumption that the remainder is sufficient for the outcome, and I can guarantee that the solution is sometimes the same and sometimes not the same (depending on how your data looks) because the claim that X -> Z is true whenever X occurs with Z, or whenever X does not occur, which is exactly what a remainder essentially is, i.e. the non-occurrence of some combination of attributes.

Last, but not least, I would advise you not to use the fs/QCA software because it does not reveal all models that fit your data due to the use of an algorithm that is unsuitable for causal analysis. This essentially means that you are at high risk of massively over-interpreting the evidence that is contained in your data.

If you want to know more on any of these above points or need references, just let me know and I refer you to appropriate methodological publications.

References

Ragin, Charles C. 2008. Redesigning Social Inquiry: Fuzzy Sets and Beyond. Chicago: University of Chicago Press.

Schneider, Carsten Q., and Claudius Wagemann. 2012. Set-Theoretic Methods for the Social Sciences: A Guide to Qualitative Comparative Analysis (QCA). Cambridge: Cambridge University Press.

Q #1:

Answer: it depends on the number of criteria that you will use to evaluate the object under evaluation. of course if you have many criteria in this case fuzzy set based rule cannot be used. Why?? 

because the traditional fuzzy set cannot be used to measure the target for two reasons, which are: (1) its weakness in the distribution importance weights of multiple criteria and (2) its weakness in the assessment of the target with regard to each criterion. Therefore, Bellman and Zadeh, (1970) presented a methodology called fuzzy multi-criteria decision-making (FMCDM) in order to resolve these weaknesses.

Q#2:Answer: if you have 5 criteria rule construction can be done using IF-THEN statements.

Q#3: Yes, in MCDM problem the assigning weight (The degree of importance) and the Rate (The degree of Performance) depends on Human thoughts and experience.  Then, you have to use scale (numeric scale (e.g., 5 likert scale), Fuzzy linguistic variables) to assign the weight and rate. however, if the criteria that you used are subjective in nature you have to use fuzzy linguistic variables. but if they are not subjective you can use 5 likert scale of satty scale (1-9).  

 

Q#4: this is another research problem. to develop fuzzy scale for your criteria or your research case this is different and complex process and task. you need to collect huge data set to train your method based on the criteria that you have  and then come out with fuzzy scale. But in fuzzy MCDM, you need to Identify the suitable linguistic variables and relevant membership functions. you can adopt two linguistic variable sets (Weight set, Rate set) with five, seven, or nine linguistic terms corresponding to seven fuzzy numbers check the attached file (Chen, 2000).

Q#5: for your case study, you need to conduct a real case study. in the case study, the experts will assign the weight and rate to your criteria. but, first you have to collect real data regard each criterion that will help experts to assign the rate (real performance of each criterion) . the data that you will collect will be the input to your method FTOPSIS. you contribution will be more on the criteria that you will identify.

 All the best

Yes, you can. The way of implementation is same for inconsistent(boundary not equal zero) and consistent (boundary equal zero). Only difference is the total dependency for consistent data set is equal 1 and for inconsistent data set less than 1. 

A favourite - in its 3rd edition - is Nonlinear Programming: Theory and Algorithms by Bazaraa, Sherali, and Shetty, published by Wiley. It's very clear, and while not being up-to-date on the newest and most efficient methods it provides very good basic material for developing methods, because it is quite thorough in explaining what can go wrong if you do not comply with the "rules", that is, what goes wrong if theory does not support your method. A - to you - natural-looking method may get stuck at the initial point, simply you have not understood the basics of what it means to be optimal, or rather, what non-optimality means. I repeat - it is very good.

Hope these two references will help you.

Thank you Prof. Shakhtreh, I have downloaded the papers from your link.

consultez vous  ce document vous "Introduction à la logique floue en Utilisant MATLAB"

This file gives you basic answer and an example about this topic, I think...

Regards,

I've seen Monte Carlo used for contingency risk estimation, typically using an 80% confidence level.  Bear in mind this is my experience in recent years so is not a fully reliable statistic.  

The weaknesses with this approach include subjective estimation of risk quantifications and probabilities, and subjective selection of the confidence level, all of which can greatly influence the contingency amount.  

You probably already know enough about Monte Carlo but I attach the following in case it helps.  If you decide to cite it in your work please include the citation here on researchgate.  

Hope that helps.  

In MATLAB, use GLCM. For that graycomatrix and  graycoprops in order to calculate homogeniety of an image.

And let me know whether it works for you or not.

A multiset is a collection in which the multiple occurrence of elements is counted unlike sets. The example which you have provided does not fit into this. I think, you are considering fuzzy multisets, because you are taking elements with graded memberships. Even then where is the repetition?

So, is it simply a fuzzy set, which you are considering?

Thank you very much Sir

Dear Hamed

just when fuzzy membership functions is complete, the sum of membership degrees must equal to 1 and in other situations it is not a necessity .

You can study these papers:

1.Lushu, L. and Lai, K. K.: A Fuzzy Approach to the Multi-Objective Transportation

Problem," Computers and Operations Research, 27, 43-57.

2.H.J. Zimmermann, Fuzzy programming and linear programming with several objective functions, Fuzzy Sets and Syst. 1 (1978) 45-55

3.Liu, B: Theory and Practice of Uncertain Programming. Physica-Verlag, Heidelberg(

2002) .

Use ANFIS in matlab. Via anfisedit  command you can upload your crisp data and make automatic fuzzy rules and save them as, data.fis.

Dear Sheik Mohammed.

There are no restrictions on the number of membership functions you can use in your FLC, however, think about how the FLC works and what it is used for.

When you think about how it works: When you have 28 (output) membership functions not represented in the rule base, it means your FLC will not produce any of them in the fuzzy output. However, after defuzzification, you can still get a crisp value which has a high membership degree in one (or more) of the 28 membership functions than in the ones used in the rule-base (subject to the defuzzification you choose). As an example, imagine your rule-base contains the outputs "high" and "low", but your output membership functions are "high", "medium" and "low". It's easy to imagine that defuzzifying an output of equal "high" and "low" may give a crisp value which is better fit for the set "medium".

When you think about what it is used for: Granular computing, where you can use the number of membership functions to increase or decrease the "precision" of your FLC. I.e. you control how how well the FLC can approximate the underlying function. More membership functions makes the system less "human" friendly, but more "precise".

I would consider if all 40 output membership functions are needed, or, if you can reconstruct them, so you only use the ones in the rule-base. More information about the problem you're trying to solve with a FLC could give more insight into whether it's a good suggestion or not.

Best regards.

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..

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.

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

As models of uncertainty, grey sets are very close (one could even argue formally identical) to interval-value fuzzy sets, and therefore share most of their limitations. I guess some good starting references to grasp thir advantages and limitations are

and

and references within them or references citing them.

This is a good question, covering a lot of ground.

Here are brief descriptions of each type of set.

1.   Rough set.   A set A is rough, provided the difference between the lower approximation of A (written $A_*$, also written $A^{''}$) and upper approximation of A (written $A^) is nonempty.    The set $A^ - A_*$ is called the boundary region of the approximation of the set A.   The boundary region is nonempty whenever the set A is roughly approximated.   This very useful view of any nonempty set was introduced by Zdzislaw Pawlak during the early 1980s.   See the attached figure for an example from

2.  Near sets.   Unlike the notions of rough set, shadow set and fuzzy set, the notion of nearness is applied to a pair of sets.   With near sets, we always think in terms of a pair of sets that are in some sense close to each other.   For example, if there is an overlap between the  upper approximation $A^*$ of a set A, the set $A^*$ is considered near the rough set $A$.   In general, a pair of sets A and B are near sets, provided A and B have elements in common, i.e., the intersection of A and B is nonempty.    See the attached figure for an example of 3D near sets, i.e., all points in the spherical neighbourhood C of the point p are in the interior of the spherical neighbourhood U of the point p.   This is an example of what are known as strongly near sets.    A paper on strongly near sets will be posted on RG in the near future.   For more about near sets,  see

J.F. Peters, Applications of near sets, Notices of the Amer. Math. Soc. 59, 2012, no. 4, 536-542:

and Near Sets web page:

3.  Shadow set.   For each fuzzy set (FS), a shadow set (SS) , any membership values between a lower bound $\lambda$ and upper bound $1-\lambda$ belong a completely uncertain, shadow set SS region of the fuzzy set FS.    Shadow sets were introduced by Witold Pedrycz in 1998.   See the attached figure for an example from

4.  Fuzzy set.   Let A be a subset in a space X.   The set A is fuzzy, provided every element of A has a degree of membership in the interval [0,1].    The degree of membership of an element x in A is characterized by the membership function

\[

\mu_A: X \righarrow [0,1]. \ \mbox{(degree of membership function)}

\]

The attached image for a shadow set also illustrates a fuzzy set.    Fuzzy sets were introduced by Lotfi Zadeh in his seminal 1965.    For more about this, see

This is purely a design question. I doubt that any expert could provide you with advice with so few details.

Dear friend

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

My 15-year experience with ANFIS tells that experimentation is the best way to come up with best system performance. It is a design process. However, type of MF doesn't usually play a vital role in shaping how the system performs. Number of MF absolutely does. I always tend to try Gaussian MF first as they are represented by the least number of parameters (Only two parameters: mean and variance). This makes the number of modifiable parameters minimum if you keep everything else unchanged. The number of modifiable parameters, which is affected by type and number of MF as well as system order, determines the computational time and burden required for conversion. I hope this helps.

Good luck,

Mohamed

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

I'm not sure that I understand your question. If you swap min and max, then the lower approximation will be bigger than the upper approximation. With the Dubois-Prade definition, the lower approximation is always contained in the fuzzy set, which is contained in the upper approximation. 

IMHO combining fuzzy and rough sets generates useless generalizations.  Just like "intuitionistic" fuzzy sets do not really broaden our understanding of vagueness.

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.

You can read my papers about infinite fuzzy logic controllers (Fuzzy Sets and Systems, 1995, InterStat 2003) if you want work with the fuzzy approximations of Lebesque functions or fuzzy probability)

Hi

Much material available in the pages of Small-N COMPASSS: http://www.compasss.org/

Pentti

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.

Here, at RG data base there are many papers available on the application of SEVQUAL in educational services! Not bad metrics at all! Here is a paper!

@Sujit; No. Both are correct.

I. Beg and T. Rashid: A democratic preference aggregation model, J. Uncertainty Anal. and Appl., 1:5 (2013) 11 pages

You do have few choices:

1. Ask experts to let them deliver information on the nature of the problem and possible classification or at least the amount of the variables to partition your universe of discourse (most useful but usually impossible ;-) )

2. Take a look into the training / sample set for the problem and try to find out the local / global maximums / minimums then try i.e. fuzzy-C-means clustering on the training dataset (requires at least a little knowledge on the problem and its nature)

3. Evolve the correct set using GA/ES/EA to let the heuristics find the best suitable set and number of the linguistic variables itself (requires the function to estimate the quality of the classification) - this automated model limits the required knowledge but you do need to be aware that searching full R may be problematics, the limits may be found with ease if you have the training data set (it is advised to extend its min/max values at least a little to let the border curves have the possibility to evolve correctly.

A nice elementary book; A first course in Fuzzy Logic by H.T. Nguyen and E.A. Walker

According to my knowledge the categorization of 'Nomimal, Ordinal, Interval and Ratio' was popularized by this publication:

Stevens, S.S. (1946), "On the Theory of Scales of Measurement." Science, New Series, Vol. 103, No. 2684 (Jun. 7, 1946), pp. 677-680.

Stevens discusses the characteristics of each category. Ordinal-category allows *only* median and percentile. Means are explicitly considered *illegal* because successive intervals are not equal in size. On the other hand, it is mentioned that there might be pragmatic reasons to break rule for "fruitfull results", as long as interpretations are done carefully.

I happen to share this opinion and Ordinal means are quite often questionable non-sense. Consider, for example, the attitude against "death penalty". What is the the distance between Completely accept and Completely disagree. What is the distance between 'Completely accept' vs 'Accept'?

In what conditions do these persons accept death penalty? For example, someone might be very strongly in favor of 'Death Penalty' but only for very limited cases like 'genocide'. Someone might be very much against 'Death Penalty' in civil court but actually accept the existence of military service and military court during war time. The actual opinion distance between 1 and 5 might be minimal and much less than the distance between 5 and 3. The one who does not care about death penalties at all. No matter how easily death penalties are given for many subjective or cultural reasons.

The distance between 'Completely Accept' and 'Accept', on the other hand, is related to many conditions that are attached to the interpretation of the question. Therefore, the distance between the intervals is related to the interpretations of each answer rather than a characteristic of question or scale. One would need to have a massive amount of additional information about each answer to be able to develop metrics for answers. Even then it could be, that distances between intervals are not linear but a non-ordered network of independent distance values. Such as in the dilemma of Death Penalty -> the distance between 5 and 1 is less than 5 and 3.

In my opinion, Likert-scales about qualitative issues are always very vulnerable to a non-sense interpretations. Therefore one should avoid weights and means. Interpretations about means and linear interval orders usually just do not pass critical analysis about the phenomena.

Thanks to both of you.

Piegat had a website with a lot of useful links some years ago (unless I'm mistaking him for someone else now) but it seems to exist only in the past. I'll have a look at his book.

Andrew, unfortunately a full account of the interaction of fuzzy and statistical ideas over half a century is a project exceeding my capability, and it couldn't be squeezed into paper length (but it looks like a beautiful book project). The 2000 book I mentioned contains a 96-page chapter titled "Possibility Theory, Probability and Fuzzy Sets: Misunderstandings, Bridges and Gaps". It is reasonable to take the story at the point those books left it. Moreover, I am specially interested in the appearence of fuzzy-like objects, of course under other names, in mainstream statistical journals, which is mainly a recent phenomenon.

In the 90's there was a full-fledged attack on fuzzy sets by well-known Bayesian statisticians. It actually originated in the 80's within the artificial intelligence community, but the arrival of "fuzzy" electronic devices to the market, the hype surrounding it, and a number of overblown or wrong claims by some fuzzy proponents triggered a very strong response from respected members of the statistical community.

Since then, both communities have more or less ignored each other, to the point that Andrew Gelman recently referred to fuzzy sets as an "obscure statistical method", even if Zadeh's seminal paper is among the ten most cited ever O_o

Due to this rift, it is difficult to me to realize how the issue is viewed today. It could be hoped that fuzzy sets are more familiar now and some arguments would not find support.

Although the aim of the survey is not to debunk old arguments, I expect a quite hard time trying to publish the paper in a statistics journal. Understanding what reasons would be presented today against fuzzy sets should help me structure the paper.

Statisticians' objections twenty years ago were typically not concerned with questions such as "Are fuzzy connectives and linguistic hedges a faithful model of how language / cognition works?" or "Does fuzzy logic advance to what logicians consider is their aim?" which were raised in other communities.

Rather, the impossibility that fuzzy sets and fuzzy logic could make any sense or have any usefulness *at all* was posited. In 1996, a top statistics journal did not find it necessary to edit a line of an interview in which the interviewer, a widely known British probabilist, called working on fuzzy sets "a fate worse than death". That viciousness is really far from Susan Haack's solidly argued "We don't need fuzzy logic". But I am clearly digressing now--

Rough sets and fuzzy sets are two distinct (but complementary) approaches to data modelling as a result of uncertainty in knowledge. The advantages are that one potentially has the ability to model two different types of uncertainty; indiscernibility for rough sets and vagueness for fuzzy sets.

Many of the characteristics of the hybridisation (fuzzy and rough) depends on which interpretation you use. It is important to note of course that rough-fuzzy sets and fuzzy-rough sets are two different things. A fuzzy-rough set can be defined in a number of ways, but should always collapse to a rough set when all of the objects under consideration are crisp or discrete. Have a read of Dubois and Prades seminal work in this area, it will provide much more detail.

Dubois, Didier, and Henri Prade. "Putting rough sets and fuzzy sets together." Intelligent Decision Support. Springer Netherlands, 1992. 203-232.

TSK is a little more complicated (understand and develop), basically it was formulated for solving problems with multiple inputs and outputs (MISO, MIMO) by a 'consequent' in linear equation form. Mamdani is simple and intuitive, commonly is used in easy problems with two inputs and one output. Its 'consequent' normally is a linguistic term like 'high', 'low', 'zero', 'right', etc.. Greetings.

Can you help me by providing code of fuzzy c-means?

Fuzzy rules may have , as I would see it, at least two goals

a) Making a model of a process (physical, chemical, weather forecast, climate, human society etc. ) with an declarative goal

b) Making a control strategy having a process model already in mind.

Both of these rule-based approaches can be again described in two ways.

1. Fuzzy rules based on expert knowledge

2. Fuzzy rules based on data

Fuzzy rules based on expert knowledge are obtained on the basis of observations by human operators. There is always to consider a trade-off between the complexity of the model/control (number of variables, number of rules, etc.) and its clarity. This is because both from a declarative model or from a set of control rules one would like to get some more or less simple advise to act. So, to get a reasonable number of variables and rules depends again on expert knowledge regarding the process to be modeled or controlled.

Fuzzy rules based on data is also indirectly based on expert knowledge because the type of the variables selected and the number of data points are (hopefully) done by an expert. After that in most cases follows a clustering of data which corresponds to a constrution of local models (mostly local linear) or controllers. Also here the number of local models/controllers can be chosen in advance or automatically obtained.

I personally used a Gustafson/Kessel approach for clustering and modeling of various processes with success (see attachment).

Problem like Safety, Purchasing etc... this can be used

a-level sets of a fuzzy set shows objects which have the membership function more than a. This set is a crisp set.

If A={(x,0.1) , (y,0.5) , (z,0.88)}, then 0.4-lelel set A={y,z}

For a comprehensive study of various clustering methods, including fuzzy k-means, set the attached paper.

Hello, I agree with opinion of Pushpinder Singh and I would like to add something:

Advantages of polygonal membership functions:

1. a small amount of data is needed to define the membership function,

2. easy of modification of parameters (modal values) of membership functions on the basis of measured values of the input --> output of a system,

3. the possibility of obtaining input --> output mapping of a model which is a hypersurface consisting of linear segments,

4. polygonal membership functions mean the condition of a partition of unity (it means that the sum of membership grades for each value x amounts to 1) is easily satisfied.

Disadvantages of polygonal membership functions:

1. Polygonal membership functions are not continuously differentiable.

The main disadvantage of other membership functions is fact, that majority of other membership functions are usually symmetric and it means that the sum of membership grades for each value x doesn't everytime amount to 1. Additionally, these functions are much harder to identification.

The p-fuzzy dynamical systems obtained by means of a Mamdani controller,

evidently produce approximate solutions of the studied phenomenon and such solutions are more trustworthy if we get more information on the phenomenon. Thus, as much information we have, more refined will be the families of successive fuzzy subsets used in the base of fuzzy rules.

The Mamdani method can induce non linearity in the response (output), but this fact does not harm the induced dynamic process in the p-fuzzy system because it simply uses the values of

response (output) after the defuzzification method. Moreover, the Mamdani method favors the attainment of a field of directions (responses) that can be used to establish parameters of deterministic systems originated by variational equations.

The number of rules (crisp or fuzzy) depends on the problem to be solved so that the question about the optimal number of rules cannot be answered in general.

Since a fuzzy rule base, e.g. for a control problem, has the property of interpolation between regions in the state space one is tempted to pick a high number of rules. The disadvantage is that for a high number of variables the fuzzy rule base "explodes". So one should make a trade-off between the number of rules and the required interpolation performance. For example: for a simple diagonal rule base a rule matrix of 5x5 or 7x7 may be enough. If one is not quite sure about this trade-off, one should check the performance of the rule base (e.g. the fuzzy controller) starting with an initial number and increase/decrease the number of rules by comparing the performances of the results (error, overshoot, stability of the system, interpolation quality etc).