International Journal of Computer Applications "Fuzzy Mathematical Models of Type-1 and Type-2 for Computing the Parameters and Its Applications"

1. International Journal of Computer Applications (0975 – 8887)
Volume 104 – No.14, October 2014
17
Fuzzy Mathematical Models of Type-1 and Type-2 for Computing the Parameters and Its Applications
Rana Waleed Hndoosh
Dept. of Software Engineering, College of Computers Sciences and Mathematics, Mosul University, Iraq.
M. S. Saroa
Dept. of Mathematics, Maharishi Markandeshawar University, Mullana, 133207, India
Sanjeev Kumar
Dept. of Mathematics, Dr. B. R. Ambedkar University, Khandari Campus, Agra-282002, India.
ABSTRACT
This work provides mathematical formulas and algorithm in order to calculate the derivatives that being necessary to perform Steepest Descent models to make T1 and T2 FLSs much more accessible to FLS modelers. It provides derivative computations that are applied on different kind of MFs, and some computations which are then clarified for specific MFs. We have learned how to model T1 FLSs when a set of training data is available and provided an application to derive the Steepest Descent models that depend on trigonometric function (SDTFM). This work, also focused on an interval type-2 non-singleton type-2 FLS (IT2 NS-T2 FLS) in order to determine how to assign all the parameters of the antecedent and consequent MFs using the set of 푛 input-output and build mathematical formulas to calculate the derivatives 휕cosh(훼)휕휃 depend on general formula of SDTFM. Additionally , we showed how to complete the calculations for input measurement and antecedent Gaussian primary MFs with uncertain standard deviations and means.
General Terms
Fuzzy modeling, fuzzy logic system, uncertainty
Keywords
Type-2 fuzzy sets, interval type-2 membership functions, type-2 fuzzy logic system, steepest descent models, interval type-2 non-singleton type-2 FLS, derivative, uncertainty.
1. INTRODUCTION
In many mathematical and engineering problems, the computation of certain solutions depends on the availability of exact values for the variables of model equations. Because the existing information usually is incomplete, inaccurate, fuzzy or linguistic, then the accurate values cannot be obtained. Therefore, it is necessary to introduce uncertain variables for modeling the available information [1], [22]. Both T1 FLS and T2 FLS include fuzzifier, rule-base, fuzzy inference engine, and output processor [2]. The output processor in T2 FLS includes type-reducer and defuzzifier while the output processor in T1 FLS includes a crisp number from the defuzzifier, [28], [4-8]. If–then rules, but its antecedent describes a T2 or consequent sets are now T2, [13], [18], [25]. T2 FLSs can be used when the cases are so uncertain to determine exact membership degrees such as when training data is corrupted by noise [3], [16], [17]. The most popular one to date, uses back propagation models (steepest descent models) for tuning all model parameters, which require the computation of first derivatives of an objective function with respect to each model parameter for making T2 FLSs much more accessible to FLS designers [23]. When all T2 FSs are modeled as interval sets, and then we obtain an IT2 FLS. We have focused on an interval type-2 non-singleton type-2 FLS (IT2 NS-T2 FLSs) [9], [26]. The T1 FLS is described using a fuzzy basis function extension that is useful for computing the output of that FLS, and used during its model for computing derivatives of an objective function with respect to MF parameters, [19]. By “model”, we specify the parameters that describe the interval T2 FLS [10], [11]. A T2 FLS model method builds how to determine all the parameters of the antecedent and consequent MFs using the training pairs [30], [4-8].
In the first part of work, we will focus on rule-based FLSs when no uncertainties are present [21], [27]. This is similar to first studying deterministic systems before studying random systems [12]. Then we will learn about extension of rule- based FLSs, ones that can directly model uncertainties [29]. The major purpose of this work is to learn how to model non- singleton type-1 fuzzy logic systems (NS-T1 FLSs) when a set of training data is available. Recount how many model parameters there can be in a specific model and describe the relation of that number to the number of possible rules in the NS-T1 FLS [20]. Explain how to calculate the derivatives that are needed for the backpropagation, such as Steepest Descent model that depend on trigonometric function (SDTFM), for updating the MF parameters. The training data is used to tune the input measurement, antecedent, and consequent MF parameters. Here mathematical formulas are built to calculate the derivatives 휕cosh 푗 휕휃 .
The structure of this work provides an introduction in this Section. In Section 2, we will learn how to model T1 FLSs when a set of training data is available [3], [2]; Section 3 provides an application to derive the SDTFM, [19], [15]. Generalized bell-shaped MF is chosen for the antecedent and the consequent [24]. Section 4, focused on an IT2 NS-T2 FLS in order to determine how to assign all the parameters of the antecedent and consequent MFs using the set of 푛 input- output and build mathematical formulas to calculate the derivatives 휕cosh(훼)휕휃 depend on general formula of SDTFM [13]. Also, provides general formulas for the left and right end-points of the type reduced set [26]. Section 5 provides computation of 휕cosh 훼 휕휃푖,푘 푙 for antecedent and consequent parameters for derivatives of 휕cosh 훼 with respect to antecedent MF parameters and provided an algorithm of the derivatives for antecedent parameters to calculate 휕cosh 훼 휕휃푖,푘 푙 [19], [13]. Section 6 provides an application in order to calculate 휕휇푂 푖 푙 푥푖,푠 푙 휕휃푖,푘 푙 , and 휕휇푂 푖 푙 푥푖,푠 푙 휕휃푖,푘 푙 for antecedent Gaussian primary MFs with uncertain means, and input measurement Gaussian primary

2. International Journal of Computer Applications (0975 – 8887)
Volume 104 – No.14, October 2014
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MFs with uncertain standard deviations [30]. Finally, Section 7, conclusions are presented, and handles the derivatives of the left end-point of the type-reduced set.
2. MODEL TYPE-1 FUZZY LOGIC SYSTEMS
The main purpose of this Section is to learn how to model T1 FLSs when a set of training data is available. By “model”, we mean specify the parameters that describe the FLS. We collect all of the equations that are needed to implement NS-T1 FLSs, [16], [17]. These equations require the modeler to make many choices. The general equations for inference system as follows [29], [19]: 휇푂푖 푙 푥푖,푠 푙 =sup 푥푖∈푋푖 휇푋푖 푥푖 ⋆휇퐴푖푙 푥푖 , (1)
where, 푥푖,푠 푙=arg sup 푥푖∈푋푖 휇푋푖 푥푖 ⋆휇퐴푖푙 푥푖 (2) 휇퐷푙 푦 =휇퐵푙 푦 ⋆푇푖=1 푛휇푂푖 푙 푥푖,푠 푙 (3)
The input–output equation, 푦 x , for the FLS is given by: 푓푛푠 x = 푦푙 min 푖=1,…,푛 휇퐴푖푙 (푥푖)푀푙 =1 min 푖=1,…,푛 휇퐴푖푙 (푥푖)푀푙 =1= 푦푙 휑푙 x 푀 푙=1 (4)
There are many ways to optimize a function [18], but we will describe a very popular way that uses the value of the function being optimized and its first derivative. Methods that use this information are called steepest descent (SDM). Here suppose that the function being minimized depends on the parameter 휃, and the function is denoted 퐽 휃 that is called an objective function [1], [22]. We have a mathematical formula for 퐽 휃 , and we do not know the shape of 퐽 휃 , but available only a set of training data 푥(푗),푦(푗) , 푗=1,…,푛 , here 퐽= 퐽 퐷,휃 . Let 퐷푇 is used by the SDM because that model is based on minimizing 퐷푇=퐽 퐷푇,휃 . The general structure of a SDM for minimizing objective function is as [19], [20]: 휃푗+1=휃푗−훾휃 derivatives of 퐽(퐷푇,휃) 휃푗 , (5)
where 훾휃 is a learning parameter, and 푗=0,1,….. In this tuning procedure, a trigonometric function is used, as: 퐽 퐷푇,휃 =cosh 퐷푇,휃푗 , (6)
where cosh 퐷푇,휃 = 푦 퐷푇,휃 −푦(푗)(퐷푇) 22 −1+ 푦(퐷푇,휃)−푦(푗)(퐷푇) 22 2 = 1 푦 퐷푇,휃 −푦(푗)(퐷푇) 2+ 푦 퐷푇,휃 −푦(푗)(퐷푇) 24 (7) 푦 퐷푇,휃 =푓 퐷푇,휃 (8)
From (8), note that 푓 퐷푇,휃 is the output of a T1 FLS. It is easy to calculate the derivatives of 퐽 퐷푇,휃 that are needed in (5), using (6)–(8), 휕 퐽 퐷푇,휃 휕휃 = 휕 푐표푠푕(퐷푇,휃) 휕휃 = 12 푦 퐷푇,휃 −푦(푗)(퐷푇) − 2 푦 퐷푇,휃 −푦(푗)(퐷푇) 3 휕 푦(퐷푇,휃) 휕휃 = 12 푦 퐷푇,휃 −푦 푗 퐷푇 − 2 푦 퐷푇,휃 −푦(푗)(퐷푇) 3 휕 푓(퐷푇,휃) 휕휃 (9)
For follow up, the specified FLS choices mentioned above must be made. Those choices will let us assign analytical formulas for 휕 푓(퐷푇,휃)휕휃 . We continue to complete these computations for a specific set of choices through an application in the next Section [24], [18].
3. APPLICATION
This Section will derive the Steepest Descent model that depends on trigonometric function (SDTFM). Note that the following generalized bell-shaped MF is chosen for the antecedent and the consequent [30], 휇 푥 = 11+ 푥−푚 휎 2푠 , (10)
in which 푚,휎 are used to adjust to vary the center and width of the membership function, and 푠 denotes the slop at the cross points. The final implementation of input–output equation for the FLS requires choices to be made about the MFs, where generalized bell-shaped antecedent and input MFs respectively are given as the following: 휇퐴푖푙 푥푖 = 11+ 푥푖−푚퐴푖푙 휎퐴푖푙 2푠 퐴푖푙 , 푙=1,…,푀 (11) 휇푋푖 푥푖 = 11+ 푥푖−푚푋푖 휎푋푖 2푠푋푖 , 푖=1,…,푛., (12) 휇푂푖 푙 푥푖,푠 푙 = 11+ 푚푋푖−푚퐴푖푙 휎푋푖+휎퐴푖푙 2 푠푋푖 +푠 퐴푖푙 . (13)
Equations (4) and (13) perform a non-singleton T1 FLS, the parameter 휃=푦 푙,푚푋푖,푚퐴푖푙 ,휎푋푖,휎퐴푖푙 ,푠푋푖 or 푠퐴푖푙 .
The certain parts of the computations of 휕 퐽(휃)휕휃 , where 퐽 휃 =푐표푠푕 푗 = 1 푓푛푠 푥(푗) −푦(푗) 2+ 푓푛푠 푥(푗) −푦(푗) 24, are as the following: 휕 퐽 휃 휕휃 = 휕 휕휃 1 푓푛푠 푥 푗 −푦 푗 2+ 푓푛푠 푥 푗 −푦 푗 24 = 12 푓푛푠 푥 푗 −푦 푗 − 2 푓푛푠 푥(푗) −푦(푗) 3 휕 휕휃 푓푛푠 x(푗) (14)

3. International Journal of Computer Applications (0975 – 8887)
Volume 104 – No.14, October 2014
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where, 푓푛푠 푥 푗 = 푦 푙 휑푙 푥 푗 푀 푙=1= 푦 푙 푚푖푛 푖=1,…,푛 휇푂푖 푙 푥푖,푠 푙 푚푖푛 푖=1,…,푛 휇푂푖 푙 푥푖,푠 푙 푀푙 =1 푀 푙=1
= 푦 푙 푚푖푛푖=1,…,푛 1 1+ 푚푋푖 −푚 퐴푖푙 휎푋푖 +휎 퐴푖푙 2 푠푋푖 +푠 퐴푖푙 푚푖푛푖=1,…,푛 1 1+ 푚푋푖 −푚 퐴푖푙 휎푋푖 +휎 퐴푖푙 2 푠푋푖 +푠 퐴푖푙 푀푙 =1 푀푙 =1 (15)
Derivative of the output of non-singleton T1 FLS that given by (15) with respect to each one of the parameter 휃 as the follows:
1. When 휃=푦 푙
휕 휕푦 푙푓푛푠 푥(푗) =휑푙 x(푗) , (16)
so that
푦 푙 푗+1=푦 푙 푗−훾푦 푙 휕 휕푦 푙 퐽(푦 푙) ,
=푦 푙 푗−훾푦 푙 12 푓푛푠 x 푗 −푦 푗 − 2 푓푛푠 x 푗 −푦 푗 3 휑푙 x 푗 (17)
2. When 휃=푚푋푖,
푓푛푠 푥 푗 = 푦 푙푤푙푀푙 =1 푤푙푀푙 =1
= 푦 푙 푚푖푛푖=1,…,푛 1 1+ 푚푋푖 −푚 퐴푖푙 휎푋푖 +휎 퐴푖푙 2 푠푋푖 +푠 퐴푖푙 푀푙 =1 푚푖푛푖=1,…,푛 1 1+ 푚푋푖 −푚 퐴푖푙 휎푋푖 +휎 퐴푖푙 2 푠푋푖 +푠 퐴푖푙 푀푙 =1 (18)
Therefore,
휕 푓푛푠 휕푚푋푖 = 휕 푓푛푠 휕푤푙. 휕푤푙 휕푚푋푖
휕 푓푛푠 휕푤푙= 푤푙푀푙 =1 . 휕 푦 푙푤푙푀푙 =1 휕푤푙− 푦 푙푤푙푀푙 =1 . 휕 푤푙푀푙 =1 휕푤푙 푤푙푀푙 =1 2
= 푦 푙 푤푙푀푙 =1 − 푦 푙푤푙푀푙 =1 푤푙푀푙 =1 2= 푦 푙−푓푛푠 푥(푗) 푤푙푀푙 =1 (19)
And 휕푤푙 휕푚푋푖 = 휕 휕푚푋푖 푚푖푛푖=1,…,푛 1 1+ 푚푋푖 −푚 퐴푖푙 휎푋푖 +휎 퐴푖푙 2 푠푋푖 +푠 퐴푖푙 =푚푖푛푖=1,…,푛 1 1+ 푚푋푖 −푚 퐴푖푙 휎푋푖 +휎 퐴푖푙 2 푠푋푖 +푠 퐴푖푙 ∗ −2 푠푋푖 +푠 퐴푖푙 . 푚푋푖 −푚 퐴푖푙 2 푠푋푖 +푠 퐴푖푙 −1 휎푋푖 +휎 퐴푖푙 2 푠푋푖 +푠 퐴푖푙 1+ 푚푋푖 −푚 퐴푖푙 휎푋푖 +휎 퐴푖푙 2 푠푋푖 +푠 퐴푖푙
= −2 푠푋푖 +푠 퐴푖푙 . 푚푋푖 −푚 퐴푖푙 2 푠푋푖 +푠 퐴푖푙 −1 휎푋푖 +휎 퐴푖푙 2 푠푋푖 +푠 퐴푖푙 1+ 푚푋푖 −푚 퐴푖푙 휎푋푖 +휎 퐴푖푙 2 푠푋푖 +푠 퐴푖푙 .푤푙, (20)
thus,
휕푓푛푠 휕푚푋푖 = 푦 푙−푓푛푠 푥 푗 푤푙푀푙 =1 . −2 푠푋푖 +푠 퐴푖푙 . 푚푋푖 −푚 퐴푖푙 2 푠푋푖 +푠 퐴푖푙 −1 휎푋푖 +휎 퐴푖푙 2 푠푋푖 +푠 퐴푖푙 1+ 푚푋푖 −푚 퐴푖푙휎푋푖 +휎 퐴푖푙 2 푠푋푖 +푠 퐴푖푙 .푤푙 (21)
Consequently,
푚푋푖 푗+1= 푚푋푖 푗 −훾푚 12 푓푛푠 x 푗 −푦 푗 − 2 푓푛푠 x 푗 −푦 푗 3 . 휕푓푛푠 휕푚푋푖 푚푋푖 푗+1= 푚푋푖 푗 −훾푚∗
12 푓푛푠 x 푗 −푦 푗 − 2 푓푛푠 x 푗 −푦 푗 3 푦 푙 푗−푓푛푠 푥 푗 ∗ −2 푠푋푖 +푠 퐴푖푙 . 푚푋푖 −푚 퐴푖푙 2 푠푋푖 +푠 퐴푖푙 −1 휎푋푖 +휎 퐴푖푙 2 푠푋푖 +푠 퐴푖푙 1+ 푚푋푖 −푚 퐴푖푙 휎푋푖 +휎 퐴푖푙 2 푠푋푖 +푠 퐴푖푙 . 푤푙 푗 푤푙 푗 푀푙 =1 (22)
From (15) and (18), we note that, 푤푙 푗 푤푙 푗 푀푙 =1=휑푙 푥 푗 . (23)
Replacing equation (23) into the one just before it, then we reach the SDTFM for updating 푚푗 푙 as the following: 푚푋푖 푗+1= 푚푋푖 푗 −훾푚∗

4. International Journal of Computer Applications (0975 – 8887)
Volume 104 – No.14, October 2014
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12 푓푛푠 푥 푗 −푦 푗 − 2 푓푛푠 푥 푗 −푦 푗 3 푦 푙 푗−푓푛푠 푥 푗 ∗ −2 푠푋푖 +푠 퐴푖푙 . 푚푋푖 −푚 퐴푖푙 2 푠푋푖 +푠 퐴푖푙 −1 휎푋푖 +휎 퐴푖푙 2 푠푋푖 +푠 퐴푖푙 1+ 푚푋푖 −푚 퐴푖푙 휎푋푖 +휎 퐴푖푙 2 푠푋푖 +푠 퐴푖푙 .휑푙 푥 푗 (24)
Similarly, when 휃=푚퐴푖푙 , we obtain, 푚퐴푖푙 푗+1= 푚퐴푖푙 푗 −훾푚∗
12 푓푛푠 푥 푗 −푦 푗 − 2 푓푛푠 푥 푗 −푦 푗 3 푦 푙 푗−푓푛푠 푥 푗 ∗ 2 푠푋푖 +푠 퐴푖푙 . 푚푋푖 −푚 퐴푖푙 2 푠푋푖 +푠 퐴푖푙 −1 휎푋푖 +휎 퐴푖푙 2 푠푋푖 +푠 퐴푖푙 1+ 푚푋푖 −푚 퐴푖푙 휎푋푖 +휎 퐴푖푙 2 푠푋푖 +푠 퐴푖푙 .휑푙 푥 푗 (25)
3. When 휃=휎푋푖,
The derivation of 휕 푓푛푠휕 휎푋푖 is just like the derivation of 휕 푓푛푠휕 푚푋푖 , therefore, we calculate,
휕 푓푛푠 휕휎푋푖 = 휕 푓푛푠 휕푤푙. 휕 푤푙 휕휎푋푖 , (26)
where 휕 푓푛푠휕 푤푙 has been computed through (19). So, we need only the new computation of 휕 푤푙휕 휎푋푖 .
휕 푤푙 휕휎푋푖 = 휕 휕휎푋푖 푚푖푛∀ 푖 1 1+ 푚푋푖 −푚 퐴푖푙 휎푋푖 +휎 퐴푖푙 2 푠푋푖 +푠 퐴푖푙 =푚푖푛푖=1,…,푛 1 1+ 푚푋푖 −푚 퐴푖푙 휎푋푖 +휎 퐴푖푙 2 푠푋푖 +푠 퐴푖푙 ∗
2 푠푋푖 +푠 퐴푖푙 . 푚푋푖 −푚 퐴푖푙 2 푠푋푖 +푠 퐴푖푙 휎푋푖 +휎 퐴푖푙 2 푠푋푖 +푠 퐴푖푙 +1 1+ 푚푋푖 −푚 퐴푖푙 휎푋푖 +휎 퐴푖푙 2 푠푋푖 +푠 퐴푖푙 = 2 푠푋푖 +푠 퐴푖푙 . 푚푋푖 −푚 퐴푖푙 2 푠푋푖 +푠 퐴푖푙 휎푋푖 +휎 퐴푖푙 2 푠푋푖 +푠 퐴푖푙 +1 1+ 푚푋푖 −푚 퐴푖푙 휎푋푖 +휎 퐴푖푙 2 푠푋푖 +푠 퐴푖푙 .푤푙 (27)
Thus, 휕 푓푛푠 휕 휎푗 푙= 푦 푙−푓푛푠 푥 푗 푤푙푀푙 =1 ∗
2 푠푋푖 +푠 퐴푖푙 . 푚푋푖 −푚 퐴푖푙 2 푠푋푖 +푠 퐴푖푙 휎푋푖 +휎 퐴푖푙 2 푠푋푖 +푠 퐴푖푙 +1 1+ 푚푋푖 −푚 퐴푖푙 휎푋푖 +휎 퐴푖푙 2 푠푋푖 +푠 퐴푖푙 .푤푙
Therefore, 휎푋푖 푗+1=휎푋푖 푗 −훾휎
12 푓푛푠 푥 푗 −푦 푗 − 2 푓푛푠 푥 푗 −푦 푗 3 푦 푙 푗−푓푛푠 푥 푗 ∗ 2 푠푋푖 +푠 퐴푖푙 . 푚푋푖 −푚 퐴푖푙 2 푠푋푖 +푠 퐴푖푙 휎푋푖 +휎 퐴푖푙 2 푠푋푖 +푠 퐴푖푙 +1 1+ 푚푋푖 −푚 퐴푖푙 휎푋푖 +휎 퐴푖푙 2 푠푋푖 +푠 퐴푖푙 .휑푙 푥 푗 . (28)
Similarly, when 휃=휎퐴푖푙 , we obtain, 휎퐴푖푙 푗+1=휎퐴푖푙 푗 −훾휎∗
12 푓푛푠 푥 푗 −푦 푗 − 2 푓푛푠 푥 푗 −푦 푗 3 푦 푙 푗−푓푛푠 푥 푗 .2 푠푋푖 +푠 퐴푖푙 . 푚푋푖 −푚 퐴푖푙 2 푠푋푖 +푠 퐴푖푙 휎푋푖 +휎 퐴푖푙 2 푠푋푖 +푠 퐴푖푙 +1 1+ 푚푋푖 −푚 퐴푖푙 휎푋푖 +휎 퐴푖푙 2 푠푋푖 +푠 퐴푖푙 .휑푙 푥 푗 . (29)
4. When 휃=푠푋푖,
The derivation of 휕 푓푛푠휕 푠푋푖 is just like the derivation of 휕 푓푛푠휕 휎푋푖 , and we then calculate
휕 푓푛푠 휕푠푋푖 = 휕 푓푛푠 휕푤푙. 휕 푤푙 휕푠푋푖 , (30)
where 휕 푓푛푠휕 푤푙 has been computed through (19). We only need the new computation of 휕 푤푙휕 푠푋푖 .
휕 푤푙 휕푠푋푖 = 휕 휕푠푋푖 푚푖푛푖=1,…,푛 1 1+ 푚푋푖 −푚 퐴푖푙 휎푋푖 +휎 퐴푖푙 2 푠푋푖 +푠 퐴푖푙 =푚푖푛푖=1,…,푛 1 1+ 푚푋푖 −푚 퐴푖푙 휎푋푖 +휎 퐴푖푙 2 푠푋푖 +푠 퐴푖푙 ∗ 2 푚푋푖 −푚 퐴푖푙 휎푋푖 +휎 퐴푖푙 2 푠푋푖 +푠 퐴푖푙 .log 푚푋푖 −푚 퐴푖푙 휎푋푖 +휎 퐴푖푙 1+ 푚푋푖 −푚 퐴푖푙 휎푋푖 +휎 퐴푖푙 2 푠푋푖 +푠 퐴푖푙
= 2 푚푋푖 −푚 퐴푖푙 휎푋푖 +휎 퐴푖푙 2 푠푋푖 +푠 퐴푖푙 .log 푚푋푖 −푚 퐴푖푙 휎푋푖 +휎 퐴푖푙 1+ 푚푋푖 −푚 퐴푖푙 휎푋푖 +휎 퐴푖푙 2 푠푋푖 +푠 퐴푖푙 .푤푙 (31)
Thus,

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휕 푓푛푠 휕푠푋푖 = 푦 푙−푓푛푠 푥(푗) 푤푙푀푙 =1 .2 푚푋푖 −푚 퐴푖푙 휎푋푖 +휎 퐴푖푙 2 푠푋푖 +푠 퐴푖푙 .푙표푔 푚푋푖 −푚 퐴푖푙 휎푋푖 +휎 퐴푖푙 1+ 푚푋푖 −푚 퐴푖푙 휎푋푖 +휎 퐴푖푙 2 푠푋푖 +푠 퐴푖푙 .푤푙,(32)
Therefore, 푠푋푖 푗+1=푠푋푖 푗 −훾휎∗
12 푓푛푠 푥 푗 −푦 푗 −2 푓푛푠 푥 푗 −푦 푗 3 푦 푙 푗−푓푛푠 푥 푗 ∗ 2 푚푋푖 −푚 퐴푖푙 휎푋푖 +휎 퐴푖푙 2 푠푋푖 +푠 퐴푖푙 .푙표푔 푚푋푖 −푚 퐴푖푙 휎푋푖 +휎 퐴푖푙 1+ 푚푋푖 −푚 퐴푖푙 휎푋푖 +휎 퐴푖푙 2 푠푋푖 +푠 퐴푖푙 .휑푙 푥 푗 (33)
Similarly, when 휃=푠퐴푖푙 , we obtain, 푠퐴푖 푙 푗+1=푠퐴푖 푙 푗 −훾휎∗
12 푓푛푠 푥 푗 −푦 푗 −2 푓푛푠 푥 푗 −푦 푗 3 푦 푙 푗−푓푛푠 푥 푗 ∗ 2 푚푋푖 −푚 퐴푖푙 휎푋푖 +휎 퐴푖푙 2 푠푋푖 +푠 퐴푖푙 .푙표푔 푚푋푖 −푚 퐴푖푙 휎푋푖 +휎 퐴푖푙 1+ 푚푋푖 −푚 퐴푖푙 휎푋푖 +휎 퐴푖푙 2 푠푋푖 +푠 퐴푖푙 .휑푙 푥 푗 (34)
4. MODEL INTERVAL TYPE-2 NON- SINGLETON TYPE-2 FUZZY LOGIC SYSTEM
This Section will model an interval type-2 non-singleton type- 2 fuzzy logic system (IT2 NS-T2 FLS). We are given a set of data training pairs, 푥(1),푦(1) , 푥(2),푦(2) ,…, 푥(푛),푦(푛) , where 푥 is an input vector and 푦 is the scalar output of an IT2 FLS. There are many types of IT2 FLS; but we focused on an IT2 NS-T2 FLS [29]. The rule antecedents and conse- quent of FLS described by IT2 FSs and the inputs that activate the FLS are IT2 FSs [10], [11]. The MFs is denoted by 휇푋 푖 푥푖 for all input 푥푖, with lower MFs 휇푋 푖 푥푖 and upper MFs 휇푋 푖 푥푖 , and the aim is to specify the type-2 FLS by using the training data. This model determines how to assign all the parameters of the antecedent and consequent MFs using the set of 푛 input-output. Suppose a general structure that tuned all MF parameters given by the formula: 휃푗+1=휃푗−훾휃 휕 cosh(훼) 휕휃 푗 , (35)
where 휃 is a model for any parameter of the FLS, 훾휃 휕 cosh(훼) 휕휃 푛 denotes that after taking the derivative with respect to a specified. We have to replace all remaining 휃 values by 휃푗, and cosh α = e− α +e α 2 = e α −1+e α 2 , in which
푒 훼 = 푓푛푠 푥 훼 −푦 훼 22 , 훼=1,…,푛,
Then, 푐표푠푕 훼 = 푓푛푠 푥 훼 −푦 훼 22 −1+ 푓푛푠 푥 훼 −푦 훼 22 2 = 1 푓푛푠 푥 훼 −푦 훼 2+ 푓푛푠 푥 훼 −푦 훼 24 (36)
The aim of this part is to build mathematical formulas for calculate the derivatives 휕cosh(훼)휕휃 . The kinds of primary MFs can be described mathematically such as Gaussians, triangles, trapezoids, etc [30]. We have chosen height type- reduction because there is a specific appearance of antecedent and consequent MF parameters for it. Since the centroid of an IT2 FS is an IT1 FS [9], [12], and as sets are completely described by their left-end point 푦푙 and right-end point 푦푙; then, calculating the centroid of an IT2 FS just requires calculating those two end-points [2]. All the different types of type-reduction, 푌푇−푅 푥 , can be expressed as, [13], [15], [17]: 푦푙 ,푦푟 = = … … 1 푦푖 푎푖 푀 푖=1 푎푖푀 푖=1 . 푎푀∈ 푎푀,푎 푀 . 푎1∈ 푎1,푎 1 . 푦푀∈ 푦푙 푀,푦푟 푀 . 푦1∈ 푦푙 1,푦푟 1 (36)
Here focused on the hight type-reduction; hence, 푦푙 푖= 푦푙 푖= 푦푖 be a single point in the consequent domain of the 푖푡푕 rule and treated as a consequent parameter,푎푖and 푎푖 are lower and upper firing degrees of the 푖푡푕 rule that contains antecedent MF parameters, and 푀 is a rules number. We always compute 푦푙 and 푦푟 using the Karnik–Mendel iterative procedures, [13], [14]. Therefore, reorder the 푎푖 accordingly and call them 푏푖 and we can be represented 푦푙 푎푛푑 푦푟 as [28]: 푦푙= 푦푙 푖 푏푙 푖푀푖 =1 푏푙 푖푀푖=1= 푦푙 푖 푏 푖 퐿푖 =1+ 푦푙 푖 푏푖푀푖 =퐿+1 푏 푖 퐿푖 =1+ 푏푖푀푖 =퐿+1 , (37)
And 푦푟= 푦푟 푖 푏푟푖 푀푖 =1 푏푟푖 푀푖 =1= 푦푟 푖 푏 푖 푅푖 =1+ 푦푟 푖 푏푖푀푖 =푅+1 푏 푖 푅푖 =1+ 푏푖푀푖 =푅+1 . (38)
Since 푏푙 푖,푦푙 푖, 푏푟푖 and 푦푟 푖 have been unordered during step-1 of the two iterative procedures of type-reduction, therefore, these formulas cannot be used as it is. We need to know exactly where specified antecedent and consequent MF parameters are located for computing the derivatives of 푦푙 푎푛푑 푦푟 with respect to parameters of MF. It is very difficult to satisfy when 푦푙 푎푛푑 푦푟 are not in rule-ordered format, therefore, we must first re-express (37) and (38) in rule-ordered format [11].
Now, let 퐴푙(x′) are denoted rule-ordered firing intervals, and we have described the rule-unordered firing intervals by 퐵푙 x′ , as the following [20]: 퐴푙 푥′ = 푎푙(푥′),푎푙(푥′) = 푎푙,푎푙 ,
퐵푙 푥′ = 푏푙,푏 푙 .

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Let 푦푙 푎푛푑 푦푟 denote rule-unordered values as the following [3]:
푦푙= 푦푙 1,…,푦푙 푀 푇 and 푦푟= 푦푟 1,…,푦푟 푀 푇,
푦푙= 푦푙 푖 푏 푖퐿푖 =1+ 푦푙 푖 푏푖푀푖 =퐿+1 푏 푖퐿푖 =1+ 푏푖푀푖 =퐿+1 , 푦푟= 푦푟푖 푏 푖푅푖 =1+ 푦푟푖 푏푖푀푖 =푅+1 푏 푖푅푖 =1+ 푏푖푀푖 =푅+1 , (39)
and 푧푙 푎푛푑 푧푟 denote rule-ordered counterparts as: 푧푙= 푧푙 1,…,푧푙 푀 푇 and 푧푟= 푧푟 1,…,푧푟 푀 푇
푧푙= 푧푙 푖 푎 푖퐿푖 =1+ 푧푙 푖 푎푖푀푖 =퐿+1 푎 푖퐿푖 =1+ 푎푖푀푖 =퐿+1, 푧푟= 푧푟푖 푎 푖푅푖 =1+ 푧푟푖 푎푖푀푖 =푅+1 푎 푖푅푖 =1+ 푎푖푀푖 =푅+1, (40)
in which 푦푙=푄푙 푧푙 and 푦푟=푄푟 푧푟, where 푄푙 and 푄푟 are an 푀×푀 permutation matrix.
Now, we should be going from the rule-unordered versions to the rule-ordered versions [13], [14]. For re-express 푦푟 in terms of rule-ordered quantities, we have to re-express the four sums of 푦푟 in (39) as the follows:
Let 퐛= 푏1,푏2,…,푏푀 푇 ≡푄푟.퐚 and 퐛= 푏 1,푏 2,…,푏 푀 푇 ≡푄푟.퐚 ,
퐄ퟏퟏ= 푒1 푒2 …|푒푅|0|…|0 푅×푀 푇, where 푒푖=푅×1 is the 푖푡푕 elementary vector, and
퐄ퟐퟐ= 0|…|0|휀1 휀2 …|휀푀−푅 푀−푅 ×1 푇, where
휀푖= 푀−푅 ×1 is the 푖푡푕 elementary vector, therefore 푏푖 푦푟 푖 푅 푖=1= 푒1 푒2 …|푒푅|0|…|0 푇 푏1,푏2,…,푏푀 푇 푒1 푒2 …|푒푅|0|…|0 푦푟 1,…,푦푟 푀 = 퐄ퟏퟏ퐛 푇 퐄ퟏퟏ푦푟 = 퐄ퟏퟏ푄푟.퐚 푇 퐄ퟏퟏ푄푟.푧푟
=퐚푇푄푟 푇퐄ퟏퟏ 푇퐄ퟏퟏ푄푟 푧푟=퐚푇퐻푟1푧푟 =퐚푇 휎푟. (41)
푏 푖 푦푟 푖 푀푖 =푅+1= 0|…|0|휀1 휀2 …|휀푀−푅 푇 푏 1,푏 2,…,푏 푀 푇 * 0|…|0|휀1 휀2 …|휀푀−푅 푦푟 1,…,푦푟 푀 = 퐄ퟐퟐ퐛 푇 퐄ퟐퟐ푦푙 = 퐄ퟐퟐ푄푟.퐚 푇 퐄ퟐퟐ푄푟.푧푟
=퐚푇푄푟 푇퐄ퟐퟐ 푇퐄ퟐퟐ푄푟 푧푟=퐚푇퐻푟2푧푟 =퐚푇휌푟 (42)
in which 푄푟 푇퐄ퟏퟏ 푇퐄ퟏퟏ푄푟 푀×푀 =퐻푟1, 퐻푟1푧푟 푀×1=휎푟, 푄푟 푇퐄ퟐퟐ 푇퐄ퟐퟐ푄푟 푀×푀 =퐻푟2, and 퐻푟2푧푟 푀×1=휌푟
suppose 1,1,…,1 푅 ,0,…,0 푀×1=푘1푟, 0,…,0,1,1,…,1 푀−푅 푀×1=푘2푟 푏푖 푅 푖=1= 1,1,…,1,0,…,0 푇 푏1,푏2,…,푏푀 =푘1푟 푇푄푟 .퐚 =휏푟 푇 퐚 (43) 푏 푖 푀 푖=퐿+1= 0,…,0,1,1,…,1 푇 푏 1,푏 2,…,푏 푀 =푘2푟 푇푄푟 .퐚=휗푟 푇 퐚 (44)
in which 푘1푟 푇푄푟 1×푀 =휏푟 and 푘2푟 푇푄푟 1×푀 =휗푟.
Consequently, from (41)-(44), we obtained: 푦푟= 퐚푇 퐻푟1 푧푟+퐚푇퐻푟2 푧푟 푘1푟 푇푄푟 퐚+푘2푟 푇푄푟 퐚 (45푎) = 퐚푇 훔푟+퐚푇훒푟 훕푟 푇 퐚+훝푟 푇 퐚 = 퐚푇 훔푟+퐚푇훒푟 퐚푇 훕푟+퐚푇훝푟 (45푏) = 휎푟,푖 푏푖푀푖 =1+ 휌푟,푖 푏 푖 푀푖 =1 휏푟,푖 푏푖푀푖 =1+ 휗푟,푖 푏 푖 푀푖 =1 (45푐)
Note that Equation (45a) includes the entire 퐚, 퐚 and 푧푟 vectors. The matrices 퐻푟1,퐻푟2 and the vectors 푘1푟,푘2푟 will automatically get out of the unnecessary elements of 퐚 and 퐚 those depend on 푅. Similarly, 푦푙 can be express in terms of rule-ordered quantities, [15], [17].
5. COMPUTATION of 흏퐜퐨퐬퐡 휶 흏휽풊,풌 풍 for ANTECEDENT and CONSEQU- ENT PARAMETERS
In this Section, we can built mathematical formulas to calculate the derivatives 휕cosh(훼)휕휃 first for antecedent parameters, and second for consequent parameters as follows [13]:
5.1 Computation of 훛퐜퐨퐬퐡(훂)훛훉퐢,퐤 퐥 for Parameters of Antecedent
Parameters of an antecedent are the parameters that describe antecedent MFs [12]. Let us denote any one of the antecedent parameters that will be tuned as 휃푖,푘 푙 (푖=1,…,푛 and 푙= 1,…,푀), and 푚 denotes the number of parameters when there can be more than one parameter related with the MF of each antecedent 푖 and rule 푙. From Equation (35), where 푓푛푠 푥 is given as [Mitchell 2006]: 푓푛푠 푥 = 푦푙(푥)+푦푟(푥) 2 , (46)
Therefore, 휕cosh 훼 휕휃푖,푘 푙= 12 휕 푒 훼 −1 휕휃푖,푘 푙+ 휕 푒 훼 휕휃푖,푘 푙 = 12 휕 푒 훼 −1 휕푓푛푠 . 휕푓푛푠 휕휃푖,푘 푙+ 휕 푒 훼 휕푓푛푠 . 휕푓푛푠 휕휃푖,푘 푙 = 12 휕 푒 훼 −1 휕푓푛푠 . 휕푓푛푠 휕푦푙 . 휕푦푙 휕휃푖,푘 푙+ 휕푓푛푠 휕푦푟 . 휕푦푟 휕휃푖,푘 푙 + 휕 푒 훼 휕푓푛푠 . 휕푓푛푠 휕푦푙 . 휕푦푙 휕휃푖,푘 푙+ 휕푓푛푠 휕푦푟 . 휕푦푟 휕휃푖,푘 푙 = 12 휕 푒 훼 −1 휕푓푛푠 + 휕 푒 훼 휕푓푛푠 ∗ 휕푓푛푠 휕푦푙 . 휕푦푙 휕휃푖,푘 푙+ 휕푓푛푠 휕푦푟 . 휕푦푟 휕휃푖,푘 푙 (47)
Since, 휕푓푛푠(푥)휕푦푙(푥) =휕푓푛푠(푥)휕푦푟(푥) =12 and 휕 푒 훼 −1 휕푓푛푠 + 휕 푒 훼 휕푓푛푠 = −4 푓푛푠 푥 훼 −푦 훼 3+ 푓푛푠 푥 훼 −푦 훼 2 (48)
Then, from (47) and (48) we obtain,

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휕cosh 훼 휕휃푖,푘 푙= 푓푛푠 푥 훼 −푦 훼 4− 1 푓푛푠 푥 훼 −푦 훼 3 . 휕푦푙 휕휃푖,푘 푙+ 휕푦푟 휕휃푖,푘 푙 (49)
Now handle 푦푙 and 푦푟 in (49) such as functions of 퐚 and 퐚; therefore, 휕푦푟 휕휃푖,푘 푙= 휕푦푟 휕푎푗. 휕푎푗 휕휃푖,푘 푙+ 휕푦푟 휕푎푗. 휕푎푗 휕휃푖,푘 푙 푀 푗=1 , (50) 휕푦푙 휕휃푖,푘 푙= 휕푦푙 휕푎푗. 휕푎푗 휕휃푖,푘 푙+ 휕푦푙 휕푎푗. 휕푎푗 휕휃푖,푘 푙 . (51) 푀 푗=1
Consequently, we must estimate all of the derivatives in (50) and (51). From the formula of 푦푟 in (45), we obtain 휕푦푟 휕푎푖= 휎푟,푖 퐚푇 훕푟+퐚푇훝푟 −휏푟,푖 퐚푇 훔푟+퐚푇훒푟 (퐚푇 훕푟+퐚푇훝푟)2 ,
so that 휕푦푟 휕푎푖= 휎푟,푖 퐚푇 훕푟+퐚푇훝푟 − 푦푟 휏푟,푖 퐚푇 훕푟+퐚푇훝푟 = 휎푟,푖−푦푟 휏푟,푖 (퐚푇 훕푟+퐚푇훝푟) (52)
From (45) and (A-5) with the same manner, we have obtained 휕푦푟 휕푎푖= 휌푟,푖−푦푟 휗푟,푖 (퐚푇 훕푟+퐚푇훝푟) (53) 휕푦푙 휕푎푖= 휌푙,푖−푦푙 휗푙,푖 (퐚푇 훕푙+퐚푇훝푙) (54) 휕푦푙 휕푎푖= 휎푙,푖−푦푙 휏푙,푖 (퐚푇 훕푙+퐚푇훝푙) (55)
In order to obtain 휕푦푟휕휃푖,푘 푙 , calculate 휕푎푙휕휃푖,푘 푙 and 휕푎푙휕휃푖,푘 푙 . Now, we just need to calculate [푎푙,푎푙], then try to choose an operator t-norm and create the functions 휇푂 푖 푙 푥푖 and 휇푂 푖 푙 푥푖 , in which, 휇푂 푖 푙 푥푖 = 휇푋 푖 푥푖 ∗휇퐴 푖푙 푥푖 푥푖 . 푥푖∈푋푖 (56) 휇푂 푖 푙 푥푖 = 휇푋 푖 푥푖 ∗휇퐴 푖푙 푥푖 푥푖 . 푥푖∈푋푖 (57)
Further, calculate the values of 푥푖 that are related with sup푥푖휇푂 푖 푙 푥푖 and sup푥푖휇푂 푖 푙 푥푖 as follows:
푥푖,푠 푙=sup푥푖휇푂 푖 푙 푥푖 And 푥푖,푠 푙=sup푥푖휇푂 푖 푙 푥푖 , (58)
Where 푥푖,푠 푙 and 푥푖,푠 푙 are the maximum values of 푥푖, then we have estimated 푎푖 푙(푥푖′) and 푎푖 푙(푥푖′), in which 푎푖 푙 푥푖′ =휇푂 푖 푙 푥푖,푠 푙 , (59푎)
And 푎푖 푙 푥푖 ′ =휇푂 푖 푙 푥푖,푠 푙 . (59푏)
Observe that 푥푖,푠 푙 and 푥푖,푠 푙 are depended on measurement 푥푖 ′, and computed 푎푙 푥′ =푇푖=1 푛 푎푖 푙 푥푖 ′ =푇푖=1 푛 휇푂 푖 푙 푥푖,푠 푙 =휇푂 1푙 푥1,푠 푙 ∗…∗휇푂 푛푙 푥푛,푠 푙 (60)
and 푎푙 푥′ =푇푖=1 푛 푎푖 푙 푥푖 ′ =푇푖=1 푛 휇푂 푖 푙 푥푖,푠 푙 =휇푂 1푙 푥1,푠 푙 ∗…∗휇푂 푛푙 푥푛,푠 푙 . (61)
The major difference between calculating the firing interval for an IT2 non-singleton T2 FLS and an interval singleton T2 FLS is to calculate 푥푖,푠 푙 and 푥푖,푠 푙 [23], [26]. Because no parameters are shared across rules or MFs, then for specified values of 푖 and 푙. Parameters of antecedent 휃푖,푘 푙 can appear in 휇푂 푖 푙 푥푖,푠 푙 and 휇푂 푖 푙 푥푖,푠 푙 and cannot appear in 휇푂 푗 푙 푥푗,푠 푙 and 휇푂 푗 푙 푥푗,푠 푙 for j≠i, that means, 휕푎푗 휕휃푖,푘 푙= 휕푎푙 휕휃푖,푘 푙 ∀ 푗≠푙 0 ∀ 푗≠푙 푎푛푑 휕푎푗 휕휃푖,푘 푙= 휕푎푙 휕휃푖,푘 푙 ∀ 푗≠푙 0 ∀ 푗≠푙 (62)
Therefore, from (62), simplify (50) and (51) as the following: 휕푦푟 휕휃푖,푘 푙= 휕푦푟 휕푎푙. 휕푎푙 휕휃푖,푘 푙+ 휕푦푟 휕푎푙. 휕푎푙 휕휃푖,푘 푙 (63) 휕푦푙 휕휃푖,푘 푙= 휕푦푙 휕푎푙. 휕푎푙 휕휃푖,푘 푙+ 휕푦푙 휕푎푙. 휕푎푙 휕휃푖,푘 푙 (64)
Thus, the last part in (49) become: 휕푦푙 휕휃푖,푘 푙+ 휕푦푟 휕휃푖,푘 푙= = 휕푦푙 휕푎푙. 휕푎푙 휕휃푖,푘 푙+ 휕푦푙 휕푎푙. 휕푎푙 휕휃푖,푘 푙 + 휕푦푟 휕푎푙. 휕푎푙 휕휃푖,푘 푙+ 휕푦푟 휕푎푙. 휕푎푙 휕휃푖,푘 푙 = 휕푦푙 휕푎푙+ 휕푦푟 휕푎푙 휕푎푙 휕휃푖,푘 푙+ 휕푦푙 휕푎푙+ 휕푦푟 휕푎푙 휕푎푙 휕휃푖,푘 푙 (65)
Consequently, from (49) and (64), we obtained, 휕푐표푠푕 훼 휕휃푖,푘 푙= 푓푛푠 푥 훼 −푦 훼 4− 1 푓푛푠 푥 훼 −푦 훼 3 ∗ 휕푦푙 휕푎푙+ 휕푦푟 휕푎푙 휕푎푙 휕휃푖,푘 푙 + 휕푦푙 휕푎푙+ 휕푦푟 휕푎푙 휕푎푙 휕휃푖,푘 푙 (66)
Now, since the parameters of antecedent 휃푖,푘 푙 can appear in 휇푂 푖 푙 푥푖,푠 푙 and 휇푂 푖 푙 푥푖,푠 푙 and cannot appear in 휇푂 푗 푙 푥푗,푠 푙 and 휇푂 푗 푙 푥푗,푠 푙 for 푗≠푖 with Equations (60) and (61) [27], and we obtain for t-norm, 휕푎푙 푥푖 훼 휕휃푖,푘 푙= 푇푗=1 푗≠푖 푛 휇푂 푗 푙 푥푗,푠 푙 × 휕휇푂 푖 푙 푥푖,푠 푙 휕휃푖,푘 푙 (67) 휕푎푙 푥푖 훼 휕휃푖,푘 푙= 푇푗=1 푗≠푖 푛 휇푂 푗 푙 푥푗,푠 푙 × 휕휇푂 푖 푙 푥푖,푠 푙 휕휃푖,푘 푙 (68)

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Next, the remaining computation of 휕휇푂 푖 푙 푥푖,푠 푙 휕휃푖,푘 푙 and 휕휇푂 푖 푙 푥푖,푠 푙 휕휃푖,푘 푙 need designation of antecedent MFs and their related FOUs, therefore, we can be summarized the steps for calculation 휕푐표푠푕 훼 휕휃푖,푘 푙 as follows:
5.2 Algorithm (The Derivatives for Antecedent Parameters to Calculate 훛퐜퐨퐬퐡 훂 훛훉퐢,퐤 퐥 )
Step 1: For input vector x α , specify active states for all i=1,…,n.
Step 2: For all i=1,…,n and for all 푙=1,…,푀, using (58)- (59) calculate xi,sl,xi,sl,μO il xi,sl and μO il xi,sl .
Step 3: For all 푙=1,…,푀,
a) Using (61) to calculate 푎푙 푥 훼 , thus depend on 휃푖,푘 푙 푎푛푑 휕휇푂 푖 푙 푥푖,푠 푙 휕휃푖,푘 푙 with (67), calculate 휕푎푙 푥 훼 휕휃푖,푘 푙 .
b) Using (62) to calculate 푎푙 푥 훼 , thus depend on 휃푖,푘 푙 푎푛푑 휕휇푂 푖 푙 푥푖,푠 푙 휕휃푖,푘 푙 with (68), calculate 휕푎푙 푥 훼 휕휃푖,푘 푙 .
Step 4: Using the KM iterative procedure
a) Depending on 푧푙 푖, calculate 푦푙,퐿 and 푄푙.
b) Depending on 푧푟푖 , calculate 푦푟,푅 and 푄푟.
From (a) and (b) with (46), we can calculate 푓푛푠 푥 훼 , and thus, calculate the first term in (65),
푓푛푠 푥 훼 −푦 훼 4− 1 푓푛푠 푥 훼 −푦 훼 3 .
Step 5: For 풂= 푎1,푎2,…,푎푀 푇 and 풂= 푎1,푎2,…,푎푀 푇
a) Using definitions of 푬ퟏ and 푬ퟐ in order to calculate 퐻푙1,퐻푙2,푘1푙,푘2푙,흈푙,흆푙,흉푙 푇 푎푛푑 흑푙 푇.
b) Using definitions of 퐄ퟏퟏ and 퐄ퟐ2 in order to calculate 퐻푟1,퐻푟2,푘1푟,푘2푟,흈푟,흆푟,흉푟 푇 푎푛푑 흑푟 푇.
Step 6:
a) Using (54) and (55) to calculate ∂yl∂ai and ∂yl∂ai , respectively.
b) Using (52) and (53) to calculate ∂yr∂ai and ∂yr∂ai , respectively.
Step 7: From the results of step 3 and step 4 with using (66), we can be calculated 휕푐표푠푕 훼 휕휃푖,푘 푙 .
5.3 Computation of 훛퐜퐨퐬퐡(훂)훛훉퐣 for Parameters of Consequent
Parameters of consequent are the parameters that describe consequent MFs. When, we use height type-reduction, then those parameters can be replaced by the two end-points of the T2 consequent sets [28], [25], and this can reduce the number of model parameters [3], [18]. Observe that the parameters of consequent do not need the "푖 " and "푘" subscript symbols in 휃푖,푘 푗 .
Since, 휃푗=푧푙 푗 표푟 휃푗=푧푟 푗 and from (36) and (46), we obtain, 휕푐표푠푕 훼 휕푧푙 푗= 휕푐표푠푕 훼 휕푓푛푠 푥 훼 . 휕푓푛푠 푥 훼 휕푦푙 . 휕푦푙 휕푧푙 푗 = 푓푛푠 푥 훼 −푦 훼 4− 1 푓푛푠 푥 훼 −푦 훼 3 . 휕푦푙 휕푧푙 푗 , (69)
and 휕푐표푠푕 훼 휕푧푟 푗= 푓푛푠 푥 훼 −푦 훼 4− 1 푓푛푠 푥 훼 −푦 훼 3 . 휕푦푟 휕푧푟 푗 . (70)
From (45a) and (A-5) with the fact 휕 휕푧 (훽푇 푧)=훽, we obtain, 휕 푦푙 휕 푧푙 = 풂푇 퐻푙1 +풂푇퐻푙2 푘1푙 푇푄푙 풂+푘2푙 푇푄푙 풂 푇 = 퐻푙1 푇푇 풂+퐻푙2 푇 풂 푘1푙 푇 푄푙 풂+푘2푙 푇 푄푙 풂 (71) 휕 푦푟 휕 푧푟 = 풂푇 퐻푟1 +풂푇퐻푟2 푘1푟 푇푄푟 풂+푘2푟 푇푄푟 풂 푇 = 퐻푟1 푇푇 풂+퐻푟2 푇 풂 푘1푟 푇 푄푟 풂+푘2푟 푇 푄푟 풂 (72)
Since, 휕 푦푙 휕 푧푙 = 휕 푦푙 휕 푧푙 1, 휕 푦푙 휕 푧푙 1,…, 휕 푦푙 휕 푧푙 1 푇 , so that 푒푗 푇. 휕 푦푙 휕 푧푙 = 휕 푦푙 휕 푧푙 푗. Consequently, re-write the Equations (71) and (72) as the following: 휕 푦푙 휕 푧푙 푗=푒푗 푇 퐻푙1 푇푇 풂+퐻푙2 푇 풂 푘1푙 푇 푄푙 풂+푘2푙 푇 푄푙 풂 (73) 휕 푦푟 휕 푧푟 푗=푒푗 푇 퐻푟1 푇푇 풂+퐻푟2 푇 풂 푘1푟 푇 푄푟 풂+푘2푟 푇 푄푟 풂 (74)
This completes the derivations of derivative models for calculate ∂cosh α ∂θi,kl .
6. APPLICATION
Consider the case of Gaussian primary MF having an uncertain mean and standard deviation that take on values in 푚∈[푚1,푚2], and 휎∈ 휎1,휎2 respectively, as follows [19], [27]: 휇퐴 푥 =푒푥푝 − 푥−푚 22휎2 ,푚∈ 푚1,푚2 ,휎∈ 휎1,휎2 (75)
For calculate 휕푎푙휕휃푖,푘 푙 and 휕푎푙휕휃푖,푘 푙 using (67) and (68), we need to calculate 휕휇푂 푖 푙 푥푖,푠 푙 휕휃푖,푘 푙 and 휕휇푂 푖 푙 푥푖,푠 푙 휕휃푖,푘 푙 . Further, calculate 휕휇푂 푖 푙 푥푖,푠 푙 휕휃푖,푘 푙 and 휕휇푂 푖 푙 푥푖,푠 푙 휕휃푖,푘 푙 for antecedent Gaussian primary MFs with uncertain means and input measurement Gaussian primary MFs with uncertain standard deviations. Formulas for antecedent MFs and their lower and upper MFs are given as [13]: 휇푖 푙 푥 =푒푥푝 − 푥푖−푚푖 푙 22휎푖 푙2 , 푚푖 푙∈ 푚푖,1 푙,푚푖,2 푙 , 푖=1,…,푛, 푙=1,…,푀. (76)
휇푖 푙 푥푖 = 푒푥푝 − 푥푖−푚푖,1 푙 22휎푖 푙2 푥푖<푚푖,1 푙 1 푥푖∈[푚푖,1 푙,푚푖,2 푙] 푒푥푝 − 푥푖−푚푖,2 푙 22휎푖 푙2 푚푖,2 푙<푥푖 (77a)

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휇푖 푙 푥푖 = 푒푥푝 − 푥푖−푚푖,2 푙 22휎푖 푙2 푥푖≤ 푚푖,1 푙+푚푖,2 푙 2 푒푥푝 − 푥푖−푚푖,1 푙 22휎푖 푙2 푚푖,1 푙+푚푖,2 푙 2<푥푖 (77b)
Formulas for input measurement MFs and their lower and upper MFs are: 휇푖 푙 푥푖 =푒푥푝 − 푥푖−푥푖 (푗) 22휎푖 푙2 , 휎푖 푙∈ 휎푖1 푙,휎푖2 푙 , (78)
휇푖 푙 푥푖 =푒푥푝 − 푥푖−푥푖 (푗) 22휎푖,2 푙2 , 휇푖 푙 푥푖 =푒푥푝 − 푥푖−푥푖 (푗) 22휎푖,1 푙2 (79)
We have summarized the state of 푥푖 ′ and the results for 휕휇푂 푖 푙 푥푖,푠 푙 휕휃푖,푘 푙 and 휕휇푂 푖 푙 푥푖,푠 푙 휕휃푖,푘 푙 that depend on, as a function of 푥푖, in Table II.
Suppose휃푖,1 푙≡푚푖1 푙, 휃푖,2 푙≡푚푖2 푙, 휃푖,3 푙≡휎푖 푙, 휃푖,4 푙≡휎푖1, 휃푖,5 푙≡ 휎푖2. Depending on the formulas in Table I, we have calculate the exact derivatives of 휕휇푂 푖 푙 푥푖,푠 푙 휕휃푖푘 푙 and 휕휇푂 푖 푙 푥푖,푠 푙 휕휃푖,푘 푙 . Tables II and III provide nonzero or zero derivatives of 휇푂 푖 푙 푥푖,푠 푙 and 휇푂 푖 푙 푥푖,푠 푙 with respect to all 휃푖,푘 푙 where 푘=1,…,5 . The results in Table II have been provided 휇푂 푖 푙 푥푖,푠 푙 which is needed to calculate 휕푎푙(푥푖 (푗))휕휃푖,푘 푙 , and the results in Table III have been provided 휇푂 푖 푙 푥푖,푠 푙 which is needed to calculate 휕푎푙(푥푖 (푗))휕휃푖,푘 푙 . From the result of this application, note that 휇푂 푖 푙 푥푖,푠 푙 depends on 휃푖,4 푙=휎푖1 in all states, and 휇푂 푖 푙 푥푖,푠 푙 depends on 휃푖,5 푙=휎푖2 only in first and last states and no state depends on both 휃푖,4 푙 and 휃푖,5 푙.
7. CONCLUSION
This work provided methods for tuning the parameters of T1 FLS and an IT2 FLS, which made T1 and T2 FLSs much more accessible to FLS modelers by using mathematical formulas. We have modeled T1 FLSs when a collection of training data is available. Further, presented an application to derive the SDTFM depending on generalized bell-shaped MF for the antecedent and the consequent. Depending on general formula of SDTFM, we have proposed an IT2 NS-T2 FLS in order to determine all the parameters of the antecedent and consequent MFs using the set of 푛 input-output and we have built mathematical formulas to calculate the derivatives 휕cosh(훼)휕휃 . Present general formulas for the left and right end-points of the type-reduced set. Proposed mathematical formulas for derivatives of 휕cosh 훼 with respect to antecedent MF parameters and consequent MF parameters. As well as, provided an algorithm of the derivatives for antecedent parameters to calculate휕cosh 훼 휕휃푖,푘 푙 . An application is provided and showed how to complete the calculations for input measurement and antecedent Gaussian primary MFs with uncertain standard deviations and means. From the result of the application, note that 휇푂 푖 푙 푥푖,푠 푙 depends on 휃푖,4 푙 in all states, and 휇푂 푖 푙 푥푖,푠 푙 depends on 휃푖,5 푙 only in first and last states and no state depends on both 휃푖,4 푙 and 휃푖,5 푙. Future studies will try to handle, the large number of parameters or reducing the number of model parameters. If some parameters are shared across rules or MFs and if mathematical formulas for derivatives cannot be obtained then we must be modify and control some or all of the results of this work.
8. APPENDIX A
Re-expressing 풚풍 in Rule-Ordered Format
In order to re-express 푦푙 in terms of rule-ordered quantities, we have to re-express the four sums of 푦푙 in (6) as the follows [14]:
Let 퐚= 푎1,푎2,…,푎푀 푇 ,퐚= 푎1,푎2,…,푎푀 푇 ,
퐛= 푏1,푏2,…,푏푀 푇 ≡푄푙.퐚 and 퐛= 푏 1,푏 2,…,푏 푀 푇 ≡푄푙.퐚 ,
퐄ퟏ= 푒1 푒2 …|푒퐿|0|…|0 퐿×푀 푇, where 푒푖=퐿×1 is the 푖푡푕 elementary vector, and
퐄ퟐ= 0|…|0|휀1 휀2 …|휀푀−퐿 푀−퐿 ×1 푇 where 휀푖= 푀−퐿 ×1 elementary vector, therefore 푏 푖 푦푙 푖 퐿 푖=1= 푒1 푒2 …|푒퐿|0|…|0 푇 푏 1,푏 2,…,푏 푀 푇 ∗ 푒1 푒2 …|푒퐿|0|…|0 푦푙 1,…,푦푙 푀 = 퐄ퟏ퐛 푇 퐄ퟏ푦푙 = 퐄ퟏ푄푙.퐚 푇 퐄ퟏ푄푙.푧푙
=퐚푇푄푙 푇퐄ퟏ 푇퐄ퟏ푄푙 푧푙=퐚푇퐻푙1푧푙 =퐚푇 휎푙. (A-1) 푦푙 푖 푏푖 푀 푖=퐿+1= 0|…|0|휀1 휀2 …|휀푀−퐿 푇 푏1,푏2,…,푏푀 푇 ∗ 0|…|0|휀1 휀2 …|휀푀−퐿 푦푙 1,…,푦푙 푀 = 퐄ퟐ퐛 푇 퐄ퟐ푦푙 = 퐄ퟐ푄푙.퐚 푇 퐄ퟐ푄푙.푧푙
=퐚푇푄푙 푇퐄ퟐ 푇퐄ퟐ푄푙 푧푙=퐚푇퐻푙2푧푙 =퐚푇휌푙. (A-2)
in which 푄푙 푇퐄ퟏ 푇퐄ퟏ푄푙 푀×푀 =퐻푙1, 퐻푙1푧푙 푀×1=휎푙,
푄푙 푇퐄ퟐ 푇퐄ퟐ푄푙 푀×푀 =퐻푙2, and 퐻푙2푧푙 푀×1=휌푙
Suppose 1,1,…,1 퐿 ,0,…,0 푀×1=푘1푙and 0,…,0,1,1,…,1 푀−퐿 푀×1= 푘2푙 푏 푖 퐿푖 =1= 1,1,…,1,0,…,0 푇 푏 1,푏 2,…,푏 푀 =푘1푙 푇푄푙 .퐚=휏푙 푇 퐚 (A−3) 푏푖 푀 푖=퐿+1= 0,…,0,1,1,…,1 푇 푏1,푏2,…,푏푀 =푘2푙 푇푄푙 .퐚=휗푙 푇 퐚 (퐴−4)
in which 푘1푙 푇푄푙 1×푀 =휏푙 and 푘2푙 푇푄푙 1×푀 =휗푙.
Consequently, from (A-1)-(A-4) we are obtained 푦푙= 퐚푇 훔푙+퐚푇훒푙 훕푙 푇 퐚+훝푙 푇 퐚 = 퐚푇 훔푙+퐚푇훒푙 퐚푇 훕푙+퐚푇훝푙 = 휎푙,푖 푏 푖 푀푖 =1+ 휌푙,푖 푏푖푀푖 =1 휏푙,푖 푏 푖 푀푖 =1+ 휗푙,푖 푏푖푀푖 =1 (A−5)

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Note that Equation (A-5) includes the entire 퐚, 퐚 and 푧푙 vectors. The matrices 퐻푙1,퐻푙2 and the vectors 푘1푙,푘2푙 will automatically get out of the unnecessary elements of 퐚 and 퐚 those depend on 퐿.
9. ACKNOWLEDGMENTS
The author wishes to thank the reviewers for their excellent suggestions that have been incorporated into this paper. I would like to acknowledge my profound gratitude to my research supervisors DR. Sanjeev Kumar, and DR. M. S. Saroa for their vigilant supervision as well as for pouring many invaluable pearls from the deep ocean of knowledge.
Table I. Definitions of the states and the 흁푶 풊풍 풙풊,풔 풍 and 흁푶 풊풍 풙풊,풔 풍
State of 풙풊′
흁푶 풊풍 풙풊,풔 풍
흁푶 풊풍 풙풊,풔 풍
푥푖 (푗)< 푚푖1 푙+푚푖2 푙 2− 휎푖1 푚푖2 푙−푚푖1 푙 2휎푖 푙 ≤푚푖1 푙
푒푥푝 − 푚푖2 푙−푥푖 (푗) 22 휎푖1+휎푖 푙
푒푥푝 − 푚푖1 푙−푥푖 (푗) 22 휎푖2+휎푖 푙
푚푖1 푙≤푥푖 (푗)< 푚푖1 푙+푚푖2 푙 2− 휎푖1 푚푖2 푙−푚푖1 푙 2휎푖 푙 ≤푚푖2 푙
푒푥푝 − 푚푖2 푙−푥푖 (푗) 22 휎푖1+휎푖 푙
1
푚푖1 푙≤ 푚푖1 푙+푚푖2 푙 2− 휎푖1 푚푖2 푙−푚푖1 푙 2휎푖 푙 ≤푥푖 (푗)≤ 푚푖1 푙+푚푖2 푙 2+ 휎푖1 푚푖2 푙−푚푖1 푙 2휎푖 푙 ≤푚푖2 푙
푒푥푝 − 푚푖2 푙+푚푖1 푙−2푥푖 (푗) 28휎푖1− 푚푖2 푙−푚푖1 푙 28휎푖 푙
1
푚푖1 푙≤ 푚푖1 푙+푚푖2 푙 2+ 휎푖1 푚푖2 푙−푚푖1 푙 2휎푖 푙 <푥푖 (푗)≤푚푖2 푙
푒푥푝 − 푚푖1 푙−푥푖 (푗) 22 휎푖1+휎푖 푙
1
푚푖1 푙+푚푖2 푙 2+ 휎푖1 푚푖2 푙−푚푖1 푙 2휎푖 푙 <푚푖2 푙≤푥푖 (푗)
푒푥푝 − 푚푖1 푙−푥푖 (푗) 22 휎푖1+휎푖 푙
푒푥푝 − 푚푖2 푙−푥푖 (푗) 22 휎푖2+휎푖 푙

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Table II. The derivatives of 흁푶 풊풍 풙풊,풔 풍 with respect to all 휽풊,풋 풍
흁푶 풊풍 풙풊,풔 풍 풎풊ퟏ 풍
흁푶 풊풍 풙풊,풔 풍 풎풊ퟐ 풍
흁푶 풊풍 풙풊,풔 풍 흈풊풍
흁푶 풊풍 풙풊,풔 풍 흈풊ퟏ
흁푶 풊풍 풙풊,풔 풍 흈풊ퟐ
0
− 푚푖2 푙−푥푖 (푗) 휎푖1+휎푖 푙 푒푥푝 − 푚푖2 푙−푥푖 (푗) 22 휎푖1+휎푖 푙
푚푖2 푙−푥푖 (푗) 22 휎푖1+휎푖 푙 2푒푥푝 − 푚푖2 푙−푥푖 (푗) 22 휎푖1+휎푖 푙
푚푖2 푙−푥푖 (푗) 22 휎푖1+휎푖 푙 2푒푥푝 − 푚푖2 푙−푥푖 (푗) 22 휎푖1+휎푖 푙
0
0
− 푚푖2 푙−푥푖 (푗) 휎푖1+휎푖 푙 푒푥푝 − 푚푖2 푙−푥푖 (푗) 22 휎푖1+휎푖 푙
푚푖2 푙−푥푖 (푗) 22 휎푖1+휎푖 푙 2푒푥푝 − 푚푖2 푙−푥푖 (푗) 22 휎푖1+휎푖 푙
푚푖2 푙−푥푖 (푗) 22 휎푖1+휎푖 푙 2푒푥푝 − 푚푖2 푙−푥푖 (푗) 22 휎푖1+휎푖 푙
0
푚푖2 푙−푚푖1 푙 4휎푖 푙 − 푚푖2 푙+푚푖1 푙−2푥푖 (푗) 4휎푖1 푒푥푝 − 푚푖2 푙+푚푖1 푙−2푥푖 푗 28휎푖1− 푚푖2 푙−푚푖1 푙 28휎푖 푙
− 푚푖2 푙+푚푖1 푙−2푥푖 푗 4휎푖1− 푚푖2 푙−푚푖1 푙 4휎푖 푙 ∗ 푒푥푝 − 푚푖2 푙+푚푖1 푙−2푥푖 푗 28휎푖1− 푚푖2 푙−푚푖1 푙 28휎푖 푙
푚푖2 푙−푚푖1 푙 28휎푖 푙2 ∗ 푒푥푝 − 푚푖2 푙+푚푖1 푙−2푥푖 푗 28휎푖1− 푚푖2 푙−푚푖1 푙 28휎푖 푙
푚푖2 푙+푚푖1 푙−2푥푖 푗 8 휎푖12 푒푥푝 − 푚푖2 푙+푚푖1 푙−2푥푖 푗 28휎푖1− 푚푖2 푙−푚푖1 푙 28휎푖 푙
0
− 푚푖1 푙−푥푖 (푗) 휎푖1+휎푖 푙 푒푥푝 − 푚푖1 푙−푥푖 (푗) 22 휎푖1+휎푖 푙
0
푚푖1 푙−푥푖 (푗) 22 휎푖1+휎푖 푙 2푒푥푝 − 푚푖1 푙−푥푖 (푗) 22 휎푖1+휎푖 푙
푚푖1 푙−푥푖 (푗) 22 휎푖1+휎푖 푙 2푒푥푝 − 푚푖1 푙−푥푖 (푗) 22 휎푖1+휎푖 푙
0
− 푚푖1 푙−푥푖 (푗) 휎푖1+휎푖 푙 푒푥푝 − 푚푖1 푙−푥푖 (푗) 22 휎푖1+휎푖 푙
0
푚푖1 푙−푥푖 (푗) 22 휎푖1+휎푖 푙 2푒푥푝 − 푚푖1 푙−푥푖 (푗) 22 휎푖1+휎푖 푙
푚푖1 푙−푥푖 (푗) 22 휎푖1+휎푖 푙 2푒푥푝 − 푚푖1 푙−푥푖 (푗) 22 휎푖1+휎푖 푙
0
Table III. The derivatives of 흁 푶 풊 풍 풙풊,풔 풍 with respect to all 휽풊,풋 풍
흁푶 풊풍 풙풊,풔 풍 풎풊ퟏ 풍
흁푶 풊풍 풙풊,풔 풍 풎풊ퟐ 풍
흁푶 풊풍 풙풊,풔 풍 흈풊풍
흁푶 풊풍 풙풊,풔 풍 흈풊ퟏ
흁푶 풊풍 풙풊,풔 풍 흈풊ퟐ
− 푚푖1 푙−푥푖 (푗) 휎푖2+휎푖 푙 푒푥푝 − 푚푖1 푙−푥푖 (푗) 22 휎푖2+휎푖 푙
0
푚푖1 푙−푥푖 (푗) 2 휎푖2+휎푖 푙 2푒푥푝 − 푚푖1 푙−푥푖 (푗) 22 휎푖2+휎푖 푙
0
푚푖1 푙−푥푖 (푗) 2 휎푖2+휎푖 푙 2푒푥푝 − 푚푖1 푙−푥푖 (푗) 22 휎푖2+휎푖 푙
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
− 푚푖2 푙−푥푖 (푗) 휎푖2+휎푖 푙 푒푥푝 − 푚푖2 푙−푥푖 (푗) 22 휎푖2+휎푖 푙
푚푖2 푙−푥푖 (푗) 22 휎푖2+휎푖 푙 2푒푥푝 − 푚푖2 푙−푥푖 (푗) 22 휎푖2+휎푖 푙
0
푚푖2 푙−푥푖 (푗) 22 휎푖2+휎푖 푙 2푒푥푝 − 푚푖2 푙−푥푖 (푗) 22 휎푖2+휎푖 푙

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