As there are an infinite number of ways to characterize fuzziness,there are an infinite number of ways to graphically depict the membership functions that describe this fuzziness.The choice of which of the methods to use depends entirely on the problem size and problem type.

Instead of choosing shape of MF, setting the interval and number of MFs are also very important. For example to model a temperature control system by fuzzy logic, it is really important to know how many MFs are needed (low, med, high MF) and also choosing the intervals of MFs. These two factors also have a great impact on the outcome of a fuzzy logic system.

Thanks for useful information. The idea provided in Ross's book was particularly interesting.

to choose best MFs i use training data to find the best MFs in my case, compare the result with the real one e.g. MAPE (Mean Average Percentage Error) to find less % err to find best MFs.

sorry for my bad english.

There are many books giving directions of how to choose MF. I would like to summarise some points concerning the problem at hand. The basic problem with modelling a situation, is to break the 0-1 modelling. This can be done by using triangular MF. However if the situation is complex and deep, we might need a special type of MF. For example if the problem at hand is a Quantum Mechanics problem, then a special MF is needed. In order to make the best choice, one needs a lot of “experience” with the given situation. This experience will tune up and best fit, the subjective choice of the researcher with the given reality. There is no a objective way to do so. Thus a hight fidelity intuition based on sufficient experience, will give an acceptable answer.

I agree with Mr. Hossein Attari. By using NN you could even optimize the support of the functions. See for example the ANFIS (Artificial Neural Network-based Fuzzy Inference System) toolbox in MATLAB

There are two ways of mapping MFs i.e., the number of MFs required for each input variable as well as the baseline

1. knowledge elicited from experts in the field (manual mapping)

2. Knowledge extracted from trends in empirical data (Artificial neural fuzzy)

As you correctly mentioned, MFs play substantial role in the overall performance of fuzzy representation. Of highly applied MFs, the triangular membership functions are formed using straight lines. These straight line membership functions have the advantage of simplicity. Because of their smoothness and concise notation, Gaussian membership functions are popular methods for specifying fuzzy sets. These curves have the advantage of being smooth and nonzero at all points.

The shape of MFs is choosed by the experts on the subject and its framework

It really depends on the problem you have to solve. While MFs can take values between 0 and 1, it may be interesting to allows some of them never reaching 1 in order to represent information that is never certain. Similarly, if you have no apriori on the shape of your MFs, triangular or trapezoidal shapes are simple to implement and fast for computation. However, if you have some a priori on their shapes (e.g., from histograms on sampled data), it may be interested to build MFs with shapes derived from these a priori shapes, after some smoothing if needed.

I agree with Serge. It really depends on the application. However, I think machine learning can be utilized for post utilization tuning of the rules of a fuzzy expert system. Following is one of my papers in this regard:

https://www.researchgate.net/publication/228677299_Reinforcement_learning_for_fuzzy_control_with_linguistic_states?ev=prf_pub

Article Reinforcement learning for fuzzy control with linguistic states

Dear Mahdi! Thank you for the response. Unfortunately I dont have access to IEEE to see the paper. If possible please send a copy to me privately.

In the attached article, a constrainrd interpolations scheme is developed for fitting a membership function to a finite number of known membership values.

Chen, J. E., & Otto, K. N. (1995). Constructing membership functions using interpolation and measurement theory. Fuzzy Sets and Systems, 73(3), 313-327.

The knowledge of the experts is basically the aim to define fuzy sets. The range of MFs is the unit interval and the domain depends on the context

Dear Mahmoud Omid

I do agree with Ravindra Mudholkar that starting with triangle membership functions is the simplest approach. I used to employ the Gaussian once in the premise part, however - the reasons are well presented by the Hamid Taghavifar I strongly agree, moreover triangle membership function is nondifferentiable in three points thus that may lead to the problems if using classical optimization methods. Once the universe of discourse it scattered with MFs it is essential to fine tune them. You may find this paper useful on how to tune Gaussian MFs and triangle MFs using ES.

Article Evolution-fuzzy rule based system with parameterized consequences

Dear Basil,

Would you explain more detailed what did you mean writing "overlapped by pairs"?

Thanks in advance.

Dear Basil,

Can I kindly ask you for the direct link? I'm quite new to RG and the RG's global search works so-so.

Thanks in advance,

P.

Dear Basil,

Got your papers - just started reading (not so much free time during this week) but I think I got the problem. By the "overlapped pairs" did you mean the situation as mentioned in chapter 3 figure 1 of the "New Optimal Approach..."?

Regards,

P.

Dear Basil,

I have faced this problem when calculating the parameters of the linear combination of the conclusion in my system (the Parameterized Consequence FIS is close to TS/TSK - the center of the conclusion MF position is calculated as position of the singleton in case of TSK). I've solved the problem by adding some minor distortion on the diagonal of the X^tX matrix (adding some small value on the diagonal to let the determinant be observably non-zero). Then X rows / columns are no longer linearly dependent , so it was possible to invert X^tX (eq. 6 of your paper). Surely it brought some inaccuracy to the final values of the P vectors but regarding the final results I observed that it was more than acceptable considering final experimental results on the popular benchmark databases.

Regards,

P.

What are advantages of using triangular/trapezoid / intuitionistic fuzzy numbers?

Triangular shapes represent fuzzy numbers, trapezoid shapes represent fuzzy intervals. These are the simplest shapes; other different shapes can be obtained from transformations of the triangle induced by linguistic modifiers, truth-functional modifiers, compositions, projections and other operations

Thanks.

How to choose different fuzzy numbers, according to what context?

Dear Kumari,

Your question is on fuzzy numbers or fuzzy sets membership functions on premises and conclusions of the fuzzy if-then rules? Or is it just on partitioning the universe of discourse?

In any of the above cases there is different answer but generally appreciate Mauro's answer - first of all those are simple shapes and they do deliver you simplicity (well, a little better than i.e. Gaussian) when differentiating or calculus is required (i.e. it appears in the Mamdani-Assilian system when de-fuzzyfing the crisp output). Because of this simplicity, many researchers used to employ triangle (usually isosceles once) with success.

Regards,

P.

Dear Kumari,

fuzzy numbers represent the subjective uncertainty of a decision-maker about the value of attributes. In particular, the area of a triangular fuzzy number is the numerical value U of the uncertainty;since the height of a normal fuzzy number is 1, then the support s = 2xU/1 is itself a representation of the uncertainty

Dear Kumari,

Most times  the problem you are modeling and the data pattern determine the type of MF you are to use. For instance if the data pattern  increased to a pick and starts fallen immediately then Triangular MF may be a choice but if the data pattern maintains a constant pick state before it starts fallen then Triapezoidal MF may  a choice. So, with other Membership functions the pattern determines the choice.

Dear Mahmoud

I think of your question in a different way; more specifically, you may build a fuzzy inference system by using a well-known membership function (say triangular) and let it be trained and tuned during its operation. What I mean is using learning fuzzy inference systems in order to achieve the best membership function that best suits the application. For more information in this regard, you may take a look at one of my papers entitled "Reinforcement learning for fuzzy control with linguistic states" listed below.

https://www.researchgate.net/publication/228677299_Reinforcement_learning_for_fuzzy_control_with_linguistic_states?ev=prf_pub

best regards,

Javid

Article Reinforcement learning for fuzzy control with linguistic states

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.......

If you have sufficient data, you can use ANFIS tool in Matlab. It will generate MFs by clustering/partitioning. You can then calibrate those initial MFs based on your needs.

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..

Choosing a particular type of MF is context specific. Linear MF like triangular, left arm, and right arm MFs would suffice in most of the situations. If the membership value variation is to be non-linear, any other MFs could be tried. All said-and-done, the suitability or reliability of a fuzzy system is bound by the acceptability of the output values. Output values must be sensible and plausible. After all, we can always train the system.

Knowing the best input membership function (MF) is problem dependent and for some general problems is still an area of research. How are membership values determined?  From: "Fuzzy Logic Toolbox For Use with MATLAB - User's Guide Version 2"  I understood the only condition a MF must really satisfy is that it must vary between 0 and 1, i.e., any curve that defines how each point in the input space is mapped to a membership value (or degree of membership)  between 0 and 1. The function itself can be an arbitrary curve whose shape we can define as a function that suits us from the point of view of simplicity, convenience, speed, and efficiency. There is a beautiful research paper that discusses which shape is best  if one uses fuzzy logic as a universal approximator:

Bart Kosko, and Sanya Mitaim, "What is the Best Shape for a Fuzzy Set in Function Approximation?", Proceedings of the 5th IEEE Int. Conf. on Fuzzy Systems (FUZZ-96), pp. 1237-1243, September 1996. 

A fuzzy set is completely characterized by its membership function (MF). Since most fuzzy sets in use have a universe of discourse X consisting of the real line R, it would be impractical to list all the pair defining a MF. The most common types of MF are:

1. Triangular MFs

2. Trapezoidal MFs

3. Gaussian MFs

4. Generalized bell MFs

5. π- Shaped Membership Function

6. S- Shaped Membership Function

The shapes of MFs are important for a particular problem since they effect on a fuzzy inference system.

Dear Dr.Omid,

If the input-output data patterns are known for any system, we can rightaway develop ANFIS controller. There the shape of the MF will be chosen based on the nature of the input-output data. But whe have to choose the type of the MF for each input and output.

As far as Fuzzy systems are concerned, the following research article may also help you with regard to MF selection. Thank you. 

Thank you for your comment and interesting article you attached. In choosing between various linear and nonlinear (triangular, trapezoidal, Gaussian, ...)  MFs to describe a given fuzzy system I am also interested to study the influence of various shapes, slopes, width distribution and relevant parameters of chosen MF too. These may include representation, performance, construction, optimization, adaptiveness, novelty, analytical structure, continuity, monotonicity, stability, robustness, computational cost,, control performance, etc. In most applications (in FLC) we prefer to use triangular MF by default. Why?

Dear, Mhamoud 

it is not my domain but I think  this paper will be useful for you

http://dee.poliba.it/LabControlli/file_pdf/wsc1.pdf

Dear Dr. Saeed Al Rashid ,Thank you for comment and introducing a useful paper. It says piece wise linear functions (e.g. triangular) give better performance. It points out that simulation is also a valuable tool for the designer and should always be made before an effective control of the system. Thank you.

At a level at which I was able to explore the topic, the kind of function must be selected each time to the problem. You have to look at what best fits to the data, or it may be best interpreted. Often, in fact it may be that the cost calculation in connection with a more complex function exceeded benefits. You can Refer to my study in area of portfolio optimization. 

Article Influence Of Membership Function’s Shape On Portfolio Optimi...

You can use some optimization algorithm to adjust the MF shape that is definitely problem-dependent. For this aim you have to choose some parametric description of the MFs. You can also refer to my works.

Conference Paper Fuzzy operations' parameters versus membership functions' pa...

Article Backpropagation through time training of a neuro-fuzzy controller

Article Membership Functions Shape and its Influence on the Stabilit...

Article Membership Functions Shape and its Influence on the Dynamica...

Dear Mahmud Omid,

Selection of membership function shape is problem specific. Based on extensive review on many literature, I found that the triangular MF is widely used because of its simplicity.

From my experience on using various MF for my research problem, Gausian and traingular MFs are found to be closely performing well and better than other types of MF. In specific, the triangular MF is found to be better than Gausian MF.

Zhao & Bose (2002) as given below compared the response of the system with various membership functions and conveyed that the Triangular membership function is superior to any other membership function.

Hope this information may be useful.

More suggestions are welcome.

In my opinion, suitable choice of the membership function always depends on the application. When you would like to use fuzzy or neuro-fuzzy system as the particular classification or recognition tool etc., you can find best shapes and parameters experimentally i.e. by using some exploration methods like the genetic approaches.

Different situation is when you want to define inference rules according to yours or some experts knowledge. Then, suitable approach is to use more intuitive shapes like triangular or trapezoidal. They are suitable for a human interpretation. Nonetheless they are not continuous functions hence the optimization of their parameters can be problematic when some derivative-based method is used.

In the not far past I have tested different membership functions in the medical diagnosis support domain. Main conclusion of the research is that each membership function shape is best for particular tested medical data set. Nonetheless, since then I use only triangular shapes as the most popular shape and their continuous analogue -- the Gaussian combination shapes.

People have already cited multiple sources with several different techniques for finding suitable MFs. The short answer is: ideally, you know your data distribution. Either you have a significant amount of data and or you have enough expertise to model your functions. People often use triangular functions because it is intuitive to divide your non-linear space into locally linear structures, which triangular functions represent very well. However, most of the fuzzy systems that are built automatically from data will use a kernel-based function, like Gaussian. In fact, if you have enough data, you do not need to use any existing function that tries to approximate your data distribution. Instead, you can create a MF that follows the distribution of your data. If you want to get to know more about that, I recommend reading about AnYa fuzzy systems.

Membership functions may include multiple subsets each representing a different degree of the linguistic term or value. Multiple overlapping membership functions which spans the entire universe of discourse are useful.

This is a long process of: (1) "choose" (triangle, trapezoidal, Gaussian, etc.) membership function, (2) "do control" to compare results (previously known results), (3) if they are not, or a little coincidence, then start over, etc. As for the boundaries of the fuzzy variables, a specialist-expert on the subject can help here.

The number of MF parameters must be minimal for proper tuning, s and z mfs are suitable choices. 

The number of MF and determination of MF parameters depend on the characteristic of collected data in the research.

Hello, 

So if I want to design a system  by which it predicts various of attacks that may occur in my network, which mf should I use?

Regards 

For predicting fuzzy systems are not so good. Tray use ANN for this aim.

If you want still to design a fuzzy system for this purpose you can take triangle mf.

Can i use polynomial regression for predicting time series?

You may try to design your own FLC and optimize the MFs like this paper posted on https://goo.gl/HrFxnY

Though there is no hard and fast rule to selecting the type of membership function, the probability distribution associated with the variable often gives a clue to it.

However, it should be noted that probability distribution and membership functions are clearly different and shouldn't be confused with each other. It's a technique which has often worked for me.

Looking at the distribution of the data is a good idea. Although, trial and error method is often used for MF shape, because there is no exact method for choosing the MFs. Shape of MFs depends on how one believes in a given linguistic variable. It is more a question of intuition then criteria. The only condition a MF must really satisfy is that it must vary between 0 and 1. The function itself can be an arbitrary curve whose shape we can define as a function that suits us from the point of view of simplicity, convenience, speed, and efficiency.

Therefore, I am to conclude here (based on previous answers) that type of MF doesn't play a crucial role in shaping how the model performs. However, the number of MF has greater influence as it determines the computational time. So, the optimum model can be determined by varying the number/type of MFs for achieving best system performance.

Any one tell me about fuzzy interval partitioning?

anyone know how to determine the correct membership function? I'm now in progress to design fuzzy logic controller for obstacle avoidance by using IR sensor.

input:left sensor and right sensor

MF variable (near and far)

output:left motor and right motor

MF variable ( stop, forward and reverse)

Depends on the application. For example, in control systems it is better to use trapezoidal or triangular functions

The main difficulty that hinders the intensive application of the theory of fuzzy sets in solving practical problems is that the membership function must be specified outside the theory itself and, therefore, its adequacy cannot be verified directly by the means of the theory. In each well-known method of constructing the membership function, their own requirements and justifications are formulated for choosing just such a construction

In the majority of algorithms of the so-called soft computing, in addition to the methods of the theory of fuzzy sets, robust, probabilistic, interval methods, etc. are used.

The main class of methods for constructing membership functions are the methods of expert estimates. When using these methods, one should take into account that there are two types of properties: properties that can be directly measured, and properties that are qualitative and require pairwise comparison of objects in order to determine their relative place in a number of objects that have this property. The nature of the measurements (primary or derivative) and the type of scale in which information is received from an expert and which determines the allowable type of operations applied to expert information is also important.

In the literature there are two main methods for constructing functions of belonging — direct and indirect.

Direct method When defining a fuzzy set, the expert or experts simply set the values ​​of the membership function for each value of the universe. It does not require too precise definition of the membership function, it is enough to fix its type and characteristic values.

Indirect method If a formalized object has measurable properties, then indirect construction methods are used. Such methods, for example, include the method of pairwise comparisons on finite discrete sets. To do this, first build the matrix A = [aij], assuming that its elements of the main diagonal are equal to one, and those symmetrical with respect to the main diagonal are mutually inverse. Note that aij = μΑ (xi) / μΑ (xj). Therefore, the definition of the membership function is reduced to finding a certain vector μ from the system of linear equations Aμ = λμ, where λ is the maximum possible matrix eigenvalue of A.

Fuzzy logic control can be implemented by the following steps

1.Defining of input and its membership function(i.e open MatLab type fuzzy and click on input to open the membership function )

2. Defining members function (input)as per your input data( drag the membership valve as per the data range)

3. Defining members function (output )as per your output data( drag the membership valve as per the data range)

4 Defining of Rules (i..e click on mandani block (select input and output as per rule requirement and click on add rule )

5.click on system(defuzzification/mandani) block to open the rule editor and then open the view tab and click rules to see the defuzzified output(crisp output)

Dear Triangular is good option.

There are different ways to select membership functions. It is easy to see with a well-known system which membership function will be defined. Sensitivity increases as you move from linear membership functions to curvilinear membership functions. However, if we are less familiar with the system, it is best to optimize the parameters of the membership function with intelligent methods. Type 2 fuzzy membership functions can also be preferred to increase sensitivity.

It is important to know the computational limits of your processor, this will allow you to have an initial guess about what type of membership function is 1) suitable for your processor given you priorities on computational efficiency, 2) as mentioned in previous comments the desired sensitivity. Another important aspect to consider is what type of data is at hand, are you dealing with a fuzzy interval set (usually associated with trapezoidal membership functions) etc... And finally how many membership functions are enough and percent overlap needed for the desired output properties. Designing a fuzzy system (control, classification, system identification, decision making) can be considered an optimization problem though usually in practice the designer relies on intuition and experience. Fuzzy systems are widely used in the industry as they are quickly implemented and in some common cases offer superior performance than "traditional" methods, an example that comes to mind is field oriented control of unidentified induction motors instead of using a proportional integral controller for speed/torque control.

Some of the Criterion that you need to be aware of to make a sensible choice of the MF are:

1. Shape of MFs this is mostly driven by your intuition and the proper understanding of the problem domain . Here you need also to analyze your choice of shapes - that is increasing/decreasing or both. It must match with your concept of fuzzy term.

2. Number of Linguistic terms(values) for the fuzzy variable

3. Percentage of overlaps for partitions

4. Check whether the membership values for some random points in the universe of discourse seems logical for the particular problem domain for which you are designing fuzzy sets.

I've recently surveyed hundreds of papers for fuzzifying parameters include its membership functions, ranges, and numbers in electrical applications. The output of this research published by International Journal of Dynamics and Control, the paper was entitled "A survey: fuzzify parameters and membership function in electrical applications". I recommend surveying this paper.

Yes! I can. If the sample is big enough, let m(Aj|x)=P(yj|x)/max(P(yj/x)=[P(x|yj)/P(x)]/max(P(x|yj)/P(x)). If the sample is not big enough, we can use a generalized Kullback-Leibler formula to optimize the membership function. Please see

Article Semantic Information G Theory and Logical Bayesian Inference...

I agree with Chenguang Lu

Check the below paper:

Comparison of Triangular and Trapezoidal Fuzzy Membership Function

https://www.ijrdo.org/index.php/cse/article/view/659/609

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

D. Bravo: Thank you!

I proposed the P-T probability framework to deal with this problem

https://www.mdpi.com/2409-9287/5/4/25

@ieee