Home > Articles

Frequently Asked Questions About Type-2 Fuzzy Logic and Fuzzy Sets

  • Print
  • + Share This
Fuzzy logic systems expert Jerry Mendel answers some frequently asked questions about type-2 fuzzy sets.
Like this article? We recommend Question: I thought that fuzzy logic is all about uncertainty. Now you tell us that it isn't. What's going on here?

Answer: That the original fuzzy logic (FL), type-1 FL, cannot handle (that is, model and minimize the effects of) uncertainties sounds paradoxical because the word fuzzy has the connotation of uncertainty. Type-1 FL handles uncertainties by using precise membership functions (MFs) that the user believes capture the uncertainties. When the type-1 MFs have been chosen, all uncertainty disappears because type-1 MFs are totally precise. Type-2 FL, on the other hand, handles uncertainties about the meanings of words by modeling the uncertainties using type-2 MFs.

Question: I have heard about FL but have never heard it referred to as type-1 FL. Why do we now need to refer to it as type-1 FL?

Answer: Before the work on type-2 FL, it was never necessary to refer to FL as type-1. Now, to distinguish between the two kinds of FL, we need to use "type-1" or "type-2."

Question: What exactly is type-2 FL?

Answer: First, let's recall that FL is all about IF-THEN rules (as in, IF the sky is blue and the temperature is between 60 and 75° Fahrenheit, THEN it is a lovely day). The IF and THEN parts of a rule are called its antecedent and consequent, and they are modeled as fuzzy sets. Rules are described by the MFs of these fuzzy sets. In type-1 FL, the antecedents and consequents are all described by the MFs of type-1 fuzzy sets. In type-2 FL, some or all of the antecedents and consequents are described by the MFs of type-2 fuzzy sets.

Question: Do the rules change as we go from type-1 FL to type-2 FL?

Answer: Good news, the rules do not change. Paraphrasing Gertrude Stein, "A rule is a rule is a rule…." What does change is the way in which we model a rule's antecedent and consequent fuzzy sets. In type-1 FL, they are all modeled as type-1 fuzzy sets, whereas in type-2 FL, some or all are modeled as type-2 fuzzy sets.

Question: What's the difference between a fuzzy set, a type-1 fuzzy set, and a type-2 fuzzy set?

Answer: The term "fuzzy set" is general and includes type-1 and type-2 fuzzy sets (and even higher-type fuzzy sets). All fuzzy sets are characterized by MFs. A type-1 fuzzy set is characterized by a two-dimensional MF, whereas a type-2 fuzzy set is characterized by a three-dimensional MF.

As an example, suppose that the variable of interest is eye contact, which we denote as x. Let's put eye contact on a scale of values 0–10. One of the terms that might characterize the amount of perceived eye contact (for example, during flirtation) is "some eye contact." Suppose that we surveyed 100 men and women and asked them to locate the ends of an interval for "some eye contact" on the scale of 0–10. Surely, we will not get the same results from all of them because words mean different things to different people.

One approach to using the 100 sets of two endpoints is to average the endpoint data and use the average values for the interval associated with "some eye contact." We could then construct a triangular (or other shape) MF whose base endpoints (on the x-axis) are at the two average values and whose apex is midway between the two endpoints. This type-1 triangle MF can be displayed in two dimensions and can be expressed mathematically as follows:

{(x, MF(x))| x an element of X}

Unfortunately, this MF has completely ignored the uncertainties associated with the two endpoints.

A second approach is to make use of the average values and the standard deviations for the two endpoints. By doing this, we are blurring the location of the two endpoints along the x-axis. Now locate triangles so that their base endpoints can be anywhere in the intervals along the x-axis associated with the blurred average endpoints. Doing this leads to a continuum of triangular MFs sitting on the x-axis—for example, picture a whole bunch of triangles all having the same apex point but different base points, as in Figure 1.

Figure 1 Triangular MFs when base endpoints (l and r) have uncertainty intervals associated with them.

For the purposes of this discussion, suppose that there are exactly N such triangles. Then at each value of x, there can be up to N MF values: MF1(x), MF2(x), …, MFN(x). Let's assign a weight to each of the possible MF values, say wx1, wx2, …, wxN (see Figure 1). We can think of these weights as the possibilities associated with each triangle at this value of x. The resulting type-2 MF can be expressed as follows:

{(x, {( MFi(x), wxi)| i = 1, …, N}| x an element of X}

Another way to write this is:

{(x, MF(x, w)| x an element of X and w an element of Jx}

MF(x, w) is a type-2 MF. It is three-dimensional because MF(x, w) depends on two variables, x and w.

Question: How are type-2 fuzzy sets visualized?

Answer: We just indicated that type-2 fuzzy sets are three-dimensional, so they can be visualized as three-dimensional plots. Unfortunately, it is not as easy to sketch such plots as it is to sketch the two-dimensional plots of a type-1 MF. Another way to visualize type-2 fuzzy sets is to plot their so-called footprint of uncertainty (FOU). The type-2 MF, MF(x, w), sits atop a two-dimensional x-w plane. It sits only on the permissible (sometimes called "admissible") values of x and w. This means that x is defined over a range of values (its domain)—say, X. In addition, w is defined over its range of values (its domain)—say, W.

An example of an FOU is shown in Figure 2. It is an FOU for a Gaussian MF whose standard deviation is known with perfect certainty but whose mean, m, is uncertain and varies anywhere in the interval from m1 to m2, which can be expressed as m_[m1, m2]. The uniform shading over the entire FOU means that, for this example, we are assuming uniform weighting (possibilities). Because of the uniform weighting, this type-2 fuzzy set is called an interval type-2 fuzzy set.

Figure 2 FOU for Gaussian (primary) membership function with uncertain mean.

Question: Is there new terminology for type-2 fuzzy sets?

Answer: Yes, there is. The fact that we must now distinguish between type-1 and type-2 MFs is one example of the new terminology. A lot of the new terminology is due to the three-dimensional nature of a type-2 MF. Another term that we have already explained is the FOU. Some other new terms are: primary membership, primary MF, secondary grade, secondary MF, secondary set, upper and lower MFs, principal MF, embedded type-1 fuzzy set, and embedded type-2 fuzzy set. All of these terms can be defined mathematically and let us communicate effectively about type-2 fuzzy sets.

Question: What can we do with type-2 fuzzy sets?

Answer: Whatever we can do with type-1 fuzzy sets we can also do with type-2 fuzzy sets. It is how we do it that is different. For type-1 fuzzy sets, we perform set theoretic operations, such as union, intersection, and complement. We can do the same for type-2 fuzzy sets. Procedures for how to do this have been worked out and are especially simple for interval type-2 fuzzy sets.

Question: Where are type-2 fuzzy sets being used?

Answer: They are being used in type-2 fuzzy logic systems, and they let us model uncertainties totally within the framework of fuzzy logic. See the fourth and fifth articles in this collection—titled "Frequently Asked Questions About Rule-Based Type-2 Fuzzy Logic Systems" and "Applications for Rule-Based Type-2 Fuzzy Logic Systems"—for a description about these FLSs.

  • + Share This
  • 🔖 Save To Your Account

InformIT Promotional Mailings & Special Offers

I would like to receive exclusive offers and hear about products from InformIT and its family of brands. I can unsubscribe at any time.

Overview


Pearson Education, Inc., 221 River Street, Hoboken, New Jersey 07030, (Pearson) presents this site to provide information about products and services that can be purchased through this site.

This privacy notice provides an overview of our commitment to privacy and describes how we collect, protect, use and share personal information collected through this site. Please note that other Pearson websites and online products and services have their own separate privacy policies.

Collection and Use of Information


To conduct business and deliver products and services, Pearson collects and uses personal information in several ways in connection with this site, including:

Questions and Inquiries

For inquiries and questions, we collect the inquiry or question, together with name, contact details (email address, phone number and mailing address) and any other additional information voluntarily submitted to us through a Contact Us form or an email. We use this information to address the inquiry and respond to the question.

Online Store

For orders and purchases placed through our online store on this site, we collect order details, name, institution name and address (if applicable), email address, phone number, shipping and billing addresses, credit/debit card information, shipping options and any instructions. We use this information to complete transactions, fulfill orders, communicate with individuals placing orders or visiting the online store, and for related purposes.

Surveys

Pearson may offer opportunities to provide feedback or participate in surveys, including surveys evaluating Pearson products, services or sites. Participation is voluntary. Pearson collects information requested in the survey questions and uses the information to evaluate, support, maintain and improve products, services or sites, develop new products and services, conduct educational research and for other purposes specified in the survey.

Contests and Drawings

Occasionally, we may sponsor a contest or drawing. Participation is optional. Pearson collects name, contact information and other information specified on the entry form for the contest or drawing to conduct the contest or drawing. Pearson may collect additional personal information from the winners of a contest or drawing in order to award the prize and for tax reporting purposes, as required by law.

Newsletters

If you have elected to receive email newsletters or promotional mailings and special offers but want to unsubscribe, simply email information@informit.com.

Service Announcements

On rare occasions it is necessary to send out a strictly service related announcement. For instance, if our service is temporarily suspended for maintenance we might send users an email. Generally, users may not opt-out of these communications, though they can deactivate their account information. However, these communications are not promotional in nature.

Customer Service

We communicate with users on a regular basis to provide requested services and in regard to issues relating to their account we reply via email or phone in accordance with the users' wishes when a user submits their information through our Contact Us form.

Other Collection and Use of Information


Application and System Logs

Pearson automatically collects log data to help ensure the delivery, availability and security of this site. Log data may include technical information about how a user or visitor connected to this site, such as browser type, type of computer/device, operating system, internet service provider and IP address. We use this information for support purposes and to monitor the health of the site, identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents and appropriately scale computing resources.

Web Analytics

Pearson may use third party web trend analytical services, including Google Analytics, to collect visitor information, such as IP addresses, browser types, referring pages, pages visited and time spent on a particular site. While these analytical services collect and report information on an anonymous basis, they may use cookies to gather web trend information. The information gathered may enable Pearson (but not the third party web trend services) to link information with application and system log data. Pearson uses this information for system administration and to identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents, appropriately scale computing resources and otherwise support and deliver this site and its services.

Cookies and Related Technologies

This site uses cookies and similar technologies to personalize content, measure traffic patterns, control security, track use and access of information on this site, and provide interest-based messages and advertising. Users can manage and block the use of cookies through their browser. Disabling or blocking certain cookies may limit the functionality of this site.

Do Not Track

This site currently does not respond to Do Not Track signals.

Security


Pearson uses appropriate physical, administrative and technical security measures to protect personal information from unauthorized access, use and disclosure.

Children


This site is not directed to children under the age of 13.

Marketing


Pearson may send or direct marketing communications to users, provided that

  • Pearson will not use personal information collected or processed as a K-12 school service provider for the purpose of directed or targeted advertising.
  • Such marketing is consistent with applicable law and Pearson's legal obligations.
  • Pearson will not knowingly direct or send marketing communications to an individual who has expressed a preference not to receive marketing.
  • Where required by applicable law, express or implied consent to marketing exists and has not been withdrawn.

Pearson may provide personal information to a third party service provider on a restricted basis to provide marketing solely on behalf of Pearson or an affiliate or customer for whom Pearson is a service provider. Marketing preferences may be changed at any time.

Correcting/Updating Personal Information


If a user's personally identifiable information changes (such as your postal address or email address), we provide a way to correct or update that user's personal data provided to us. This can be done on the Account page. If a user no longer desires our service and desires to delete his or her account, please contact us at customer-service@informit.com and we will process the deletion of a user's account.

Choice/Opt-out


Users can always make an informed choice as to whether they should proceed with certain services offered by InformIT. If you choose to remove yourself from our mailing list(s) simply visit the following page and uncheck any communication you no longer want to receive: www.informit.com/u.aspx.

Sale of Personal Information


Pearson does not rent or sell personal information in exchange for any payment of money.

While Pearson does not sell personal information, as defined in Nevada law, Nevada residents may email a request for no sale of their personal information to NevadaDesignatedRequest@pearson.com.

Supplemental Privacy Statement for California Residents


California residents should read our Supplemental privacy statement for California residents in conjunction with this Privacy Notice. The Supplemental privacy statement for California residents explains Pearson's commitment to comply with California law and applies to personal information of California residents collected in connection with this site and the Services.

Sharing and Disclosure


Pearson may disclose personal information, as follows:

  • As required by law.
  • With the consent of the individual (or their parent, if the individual is a minor)
  • In response to a subpoena, court order or legal process, to the extent permitted or required by law
  • To protect the security and safety of individuals, data, assets and systems, consistent with applicable law
  • In connection the sale, joint venture or other transfer of some or all of its company or assets, subject to the provisions of this Privacy Notice
  • To investigate or address actual or suspected fraud or other illegal activities
  • To exercise its legal rights, including enforcement of the Terms of Use for this site or another contract
  • To affiliated Pearson companies and other companies and organizations who perform work for Pearson and are obligated to protect the privacy of personal information consistent with this Privacy Notice
  • To a school, organization, company or government agency, where Pearson collects or processes the personal information in a school setting or on behalf of such organization, company or government agency.

Links


This web site contains links to other sites. Please be aware that we are not responsible for the privacy practices of such other sites. We encourage our users to be aware when they leave our site and to read the privacy statements of each and every web site that collects Personal Information. This privacy statement applies solely to information collected by this web site.

Requests and Contact


Please contact us about this Privacy Notice or if you have any requests or questions relating to the privacy of your personal information.

Changes to this Privacy Notice


We may revise this Privacy Notice through an updated posting. We will identify the effective date of the revision in the posting. Often, updates are made to provide greater clarity or to comply with changes in regulatory requirements. If the updates involve material changes to the collection, protection, use or disclosure of Personal Information, Pearson will provide notice of the change through a conspicuous notice on this site or other appropriate way. Continued use of the site after the effective date of a posted revision evidences acceptance. Please contact us if you have questions or concerns about the Privacy Notice or any objection to any revisions.

Last Update: November 17, 2020