An overview and a derivation of interval type-2 fussy logic system (IT2 FLS), which can handle rule’s uncertainties on continuous domain, having good number of applications in real world. This work fo-cused on the performance of an IT2 FLS that involves the operations of a fuzzification, inference, and output processing. The output processing consists of Type-Reduction (TR) and defuzzification. This work made IT2 FLS much more accessible to FLS modellers, because it provides mathematical formulation for calculating the de-rivatives. Presenting extend to representation of T2 FSs on continuous domain and using it to derive formulas for operations, we developed and extended the derivation of the union of two IT2 FSs to the derivation of the intersection and union of N-IT2 FSs that is based on various concepts. The derivation of all the formulas that are related with an IT2 and these formulas depend on continuous domain with multiple rules. Each rule has multiple antecedents that are activated by a crisp number with T2 singleton fuzzification (SF). Then, we have shown how those results can be extended to T2 non-singleton fuzzification (NSF). We are derived the relation-ship between the consequent and the domain of uncertainty (DOU) of the T2 fired output FS. As well as, provide the derivation of the general form at continuous domain to calculate the different kinds of type-reduced. We have also applied an IT2 FLS to medical application of Heart Diseases (HDs) and an IT2 provide rather modest performance improvements over the T1 predictor. Finally, we made a comparison of HDs result between IT2 FLS using the IT2FLS in MATLAB and the IT2 FLS in Visual C# models with T1 FISs (Mamdani, and Takagi-Sugeno).
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Modelling of an IT2 FS on Continuous Domain with Medical Application
Rana Waleed Hndoosh1, M. S. Saroa2, Sanjeev Kumar3
Abstract—An overview and a derivation of interval type-2 fussy logic system (IT2 FLS), which can handle rule’s un- certainties on continuous domain, having good number of applications in real world. This work focused on the per- formance of an IT2 FLS that involves the operations of a fuzzification, inference, and output processing. The output processing consists of Type-Reduction (TR) and defuzzification. This work made IT2 FLS much more accessible to FLS modellers, because it provides mathematical formulation for calculating the derivatives. Presenting extend to represen- tation of T2 FSs on continuous domain and using it to derive formulas for operations, we developed and extended the derivation of the union of two IT2 FSs to the derivation of the intersection and union of N-IT2 FSs that is based on vari- ous concepts. The derivation of all the formulas that are related with an IT2 and these formulas depend on continuous domain with multiple rules. Each rule has multiple antecedents that are activated by a crisp number with T2 singleton fuzzification (SF). Then, we have shown how those results can be extended to T2 non-singleton fuzzification (NSF). We are derived the relationship between the consequent and the domain of uncertainty (DOU) of the T2 fired output FS. As well as, provide the derivation of the general form at continuous domain to calculate the different kinds of type- reduced. We have also applied an IT2 FLS to medical application of Heart Diseases (HDs) and an IT2 provide rather modest performance improvements over the T1 predictor. Finally, we made a comparison of HDs result between IT2 FLS using the IT2FLS in MATLAB and the IT2 FLS in Visual C# models with T1 FISs (Mamdani, and Takagi-Sugeno).
Keywords—Type-1 fuzzy logic system, Type-2 fuzzy sets, Type-2 fuzzy logic system, Type-2 membership functions, Interval type-2 fuzzy systems, Footprint of uncertainties, Type-reduction, Data base of Heart diseases.
1. INTRODUCTION
This work, introduced a new class of fuzzy logic systems—interval type-2 fuzzy logic system (IT2 FLS), where the an- tecedent or/and consequent membership functions (MFs) are interval type-2 fuzzy sets (IT2 FSs), [10-14], which is an extension of the concept of a type-1 fuzzy set (T1 FS). In an IT2 FLS, the knowledge used to construct rules is uncertain, and this uncertainty drives to rules having uncertain antecedents and/or consequents, [21-23]. Now as MFs of a general T2 FSs are fuzzy, therefore T2 FSs are able to model as uncertainties, and their MFs are three-dimensional,[24]. T2 FSs third dimension provides additional degrees that make it possible to directly models uncertainties,[8]. T2 FSs are difficult to use and understand because: i) T2 FSs three-dimensional makes them very difficult to depict; ii) there is no simple terms set that let us effective communication about T2 FSs, and to then be mathematically accurate, and iii) us- ing T2 FSs is computationally more complex than using T1 FSs, [10-13],[17]. Most people only use an IT2 FSs in a T2 FLS, because of the computational complex of using a general T2 FS, the result being an IT2 FLS. The resulting IT2 FLS have the chance to provide better performance than a T1 FLS, and all of the results that are needed to perform an IT2 FLS can be obtained by T1 FS mathematics. The computations related with IT2 FSs are very flexible, which makes an IT2 FLS to a large degree practical,[16]. Section 2, defined a small set of concepts in a mathematically accurate way of general T2 FSs and IT2 FSs. We are extended the theorem1, which was given by Mendel et al. 2006 for discrete uni- verses of discourse, to continuous universes of discourse. Section 3, derived the formulas of the intersection and union of N-IT2 FSs that is based on different concepts: i) the concept of embedded IT2 FSs such as Theorem 3.1; ii) the con- cept of Extension Principle such as Theorem 3.2. Additionally, we derived the formulas of the meet and join of N-IT2 FSs such as Theorem 3.3, [4-7]. Section 4 has described an IT2 FLS, T2 singleton fuzzification (SF) and T2 non- singleton fuzzification (NSF). Present the derivation of all of the formulas that are related with an IT2 FLS at continu- ous domain, and handled multiple rules. Each rule has multiple antecedents that are activated by a crisp number (the case of SF), after which we shown how those results can be extended to (the case of NSF), [11-15]. Consequently, we are derived the relationship between the consequent and the domain of uncertainty (DOU) of the T2 fired output FS that summarized by Theorem 4.1 and 4.2 for SF and NSF, respectively,[1]. Section 5, showed that computation of the con- tinuous version of type-reduction that is used in going from fired-rule IT2 FSs to the defuzzified number at the final output of FLS,[20],[4],[6]. We have provided the derivation of the general form for continuous domain to calculate the different kinds of type-reduced, which was given by Karnik et al. 2004 but for discrete domain. Additionally, we are presented the term of defuzzification which using the average of endpoints to obtain the crisp output of IT2 FLS,[10],[12],[13],[18]. In Section 6, a medical application of IT2 FLS‘s to heart diseases (HDs) is applied, which demonstrated the basic ideas and the mathematical operations of IT2 fuzzy sets and systems. We also provide a Matlab performance of IT2 FLS. A comparison of HDs between IT2 FLS using the IT2FLS in MATLAB and the IT2FLS in Visual C# models with T1 FISs (Mamdani, and Takagi-Sugeno) are presented in this Section. Section 7, we draw conclusions. Finally, an Appendix is presents the concept of Extension Principle.
2. INTERVAL TYPE-2 FUZZY SETS
Most people only use interval type-2 fuzzy sets (IT2 FSs) in a type-2 fuzzy logic system (T2 FLS) because of the com- putational complexity of using a general T2 FS, the result being an interval type-2 fuzzy logic system (IT2 FLS). We define an IT2 FS and some important related concepts, to provide a simple collection of mathematically terms that will let us effectively communicate about such sets. Imagine fuzzing the type-1 membership function (MF) depicted through Fig. 1(a) by moving the points on the trapezoid either to the right or to the left with the different amounts, as in Fig. 1(b). Therefore, at a specific value of , say ′ for all ∈ , there no longer is a single value for the MF; instead, the MF takes on values wherever the vertical line intersects the fuzzy. The basic concepts of type-2 fuzzy sets are intro- duced at an Appendix A, [11-13],[17].
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Fig. 1. (a) Representing the Type-1 MF and (b) Fizzing T1 MF
Theorem 2.1: Let denote the embedded IT2 FSs for T2 FS, when and are continuous, is as follows:
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, and denotes the partition levels of secondary variable at each of the .
3. SET-THEORETIC OPERATIONS ON TYPE-2 FUZZY SETS
The main aim of this section is to derive formulas for the intersection and union of N IT2 FSs of an IT2 FS, because these operations are used in an IT2 FLS. In this Section, the derivation of the intersection and union of N IT2 FSs is based on tow concepts: 1) the concept of embedded IT2 FSs such as theorem 2.1; 2) the concept of Extension Principle such as theorem 3. Third part contains the derivation of the meet and join of N IT2 FSs, [4-7][10],[12],[17].
Theorem 3.1: Derivation the intersection of -T2 FSs depending on the concept of the embedded IT2 FSs The intersection and union of IT2 FSs, ( = 1, … , ) are given by (3) and (4), respectively:
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Theorem 3.2: Derivation the intersection of -T2 FSs depending on the concept of the Extension Principle
Let T2 FSs in a continuous universe . Suppose
illustration not visible in this excerpt
bership degrees of , where ∈ . Then membership degrees for intersection and union of type-2 FS have been defined as follows:
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Theorem 3.3: Derivation the meet operations of -T2 FSs depending on the concept of the secondary MF
Suppose that we have convex, normal, type-1 real fuzzy sets described by membership functions respectively. Let be real numbers such that 1 ≤ 2 ≤ ⋯ ≤ and 1 1 ≤ ⋯ ≤ , then,
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4. INTERVAL TYPE-2 FUZZY LOGIC SYSTEM
We assume that all the antecedent and consequent fuzzy sets in the rules are T2. A FLS is T2 as long as any one of its antecedent or consequent FSs is T2. The rules structure remains the same in the case of T2, but some or all of the FSs involved are T2,[16],[24]. The T2 FLS has inputs 1 ∈ 1 , … , ∈ , and output ∈ , and, is describe by rules, where the rule has the form
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If all of the antecedent and consequent T2 FSs are IT2 FSs, then we call the resulting T2 FLS an IT2 FLS. A rule-base contains four components: rules, fuzzifier, inference system, and output processing that consist of defuzzifier and typereducer. The outputs of the T2 FLS are the type-reduced set and the crisp defuzzified value,[11],[13],[17].
4.1. Type-2 Singleton Fuzzification Model
From the rule (8), let , , … , be IT2 FSs in continuous universe of discourses 1 , 2 , … , , respectively,
and be an IT2 FS in continuous universe of discourse . Decompose each into its → ∞ bedded IT2 FSs , as the following:
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We also decompose into → ∞ embedded IT2 FSs , whose domains are the embedded T1 FSs ; we see
that can be expressed as:
National Conference on Advances in Engineering, Technology & Management (AETM’15) 47 | Page
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Cartesian product of antecedents 1 × … × has
→ ∞ collections of the embedded T1 FSs . The
relationship between antecedents and consequent can be represented by:
illustration not visible in this excerpt
where it has been supposed that Mamdani implications are used, multiple antecedents are connected by or (i.e. by S- norms), is short for an S-norm and ∔ represents the max S-norms,[24].
In general, there are rules that describe an IT2 FLS and repeatedly more than one rule fires when input is applied to that system. Consequently, we have 1 × … × × collections of embedded T1 antecedent and consequent FSs, which generate all fired output sets for all collections of antecedent and consequent FSs, as the following[10],[12], and [18]:
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The relationship between the consequent in (14) and the DOU of the T2 fired output FS is made a summary by theorem 5a, [21-23].
Theorem 4.1: The output in (14) that calculated by using T1 FS is the same as the DOU of the T2 fired output FS, which is calculated by using T2 FS.
4.2. Type-2 Non-singleton Fuzzification Model
Let the -dimensional input is given by the IT2 FS, and we suppose denote the IT2 FSs describing each of the in- puts. More specifically 1, 2, … , are IT2 FSs in continuous universes of discourse 1 , 2 , … , . There are rules that described an IT2 FLS, and repeatedly more than one rule fires when input is applied to that system. Decom-
pose into their → ∞
illustration not visible in this excerpt
The domain of each is the embedded T1 FS . The Cartesian product be 1 × 2 × … × , has =1
∞ collections of the embedded T1 FSs , then the MF of a fuzzy Cartesian product is given by: 1 1 [...] 1 ==1
Since, each rule determines a fuzzy set in such that when we use Zadeh‘s sup-star composition, note that:
illustration not visible in this excerpt
Then, we have derived the formula of NSF as the following:
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1 (Dept. of Software Engineering, College of Computers Sciences & Mathematics, Musol. University, Iraq
2 (Dept. of Mathematics, Maharishi Markandeshawar University, Mullana-133207, India
3 (Dept. of Mathematics, IBS College, Dr. B. R. Ambedkar University, Khandari Campus, Agra-282002, India
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- R.W. Hndoosh (Autor:in), M.S. Saroa (Autor:in), S. Kumar (Autor:in), 2015, Modelling of an Interval Type-2 Fussy Logic System (IT2 FLS) on Continuous Domain with Medical Application, München, GRIN Verlag, https://www.grin.com/document/296058
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