A nonlinear method of learning neuro-fuzzy models for dynamic control systems

Journal Pre-proof A nonlinear method of learning neuro-fuzzy models for dynamic control systems Maxim V. Bobyr, Sergey G. Emelyanov

PII: DOI: Reference:

S1568-4946(19)30812-9 https://doi.org/10.1016/j.asoc.2019.106030 ASOC 106030

To appear in:

Applied Soft Computing Journal

Received date : 27 January 2019 Revised date : 9 November 2019 Accepted date : 12 December 2019 Please cite this article as: M.V. Bobyr and S.G. Emelyanov, A nonlinear method of learning neuro-fuzzy models for dynamic control systems, Applied Soft Computing Journal (2019), doi: https://doi.org/10.1016/j.asoc.2019.106030. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Journal Pre-proof A nonlinear method of learning neuro-fuzzy models for dynamic control systems Maxim V. Bobyr*a, Sergey G. Emelyanov b a

Department of Computer Science, Faculty of Fundamental and Applied Informatics, Southwest

State University st. 50 let Oktyabrya, 94, Kursk, Russia 305040, Phone/fax +7 (4712) 22-26-65 b

Department of Unique Buildings and Structures, Faculty of Construction and Architecture,

Southwest State University st. 50 let Oktyabrya, 94, Kursk, Russia 305040, Phone/fax +7 (4712)

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22-24-61 Abstract.

The paper describes a new learning algorithm of adaptive neuro-fuzzy inference systems that is based on the method of areas' ratio (MAR-ANFIS). Using linear and nonlinear functions we obtain a generalized model for fuzzy inference. Considering various implication methods, different t- or s- norms and equations for fuzzy inference composition we can change the properties of the resulting output variable. As an example, we illustrate the proposed learning

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algorithm and show its distinctive characteristics. Firstly, MAR-ANFIS learning algorithm is additive. Secondly, soft operators provide symmetry for the output variable. Also, the proposed algorithm that allows improving accuracy when learning fuzzy system and speed of its learning. Using detailed numerically calculated RMSE and MAPE we evaluate the proposed algorithm.

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High accuracy of the proposed MAR-ANFIS is confirmed through the calculation of the learning time of neuro-fuzzy network RMSE and MAPE.

ANFIS

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Key words: ANFIS; RMSE; MAPE; Soft computing, Method of Areas’ Ratio, MAR-

1. Introduction

Approximate reasoning models based on fuzzy inference are an effective tool for creating systems with artificial Intelligence. The creation systems are often used in tasks of pattern recognition [1, 2], decision making under uncertainty [3, 4] and forecasting various situations [5, 6]. For example, in the article [7] ANFIS–NARX is used to predict mechanical displaced

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features of Ionic Polymer Metal Composite actuators. This model included two parts which were an adaptive neuro-fuzzy inference system (ANFIS) and a nonlinear auto-regressive with exogenous input (NARX) structure. The authors showed, that the estimation accuracy of the proposed ANFIS–NARX method is much better than dynamic NARX method based on an artificial neural network and classical NARX method based on polynomial functions. The positive properties of such models are the reduction of the computational complexity of the *

Corresponding author Address: proezd Svetliy 1-12, Kursk, Russia, 305046. Tel.: +79202643455. E-mail addresses: [email protected] and [email protected] (M. Bobyr)

Journal Pre-proof decision-making process and the possibility to develop a simulation model with its subsequent configuration. As a rule, Takagi-Sugeno algorithm is used to develop adaptive neuro-fuzzy inference systems. ANFIS is a combination of two technologies: artificial neural networks (ANN) and fuzzy logic (FL). It provides important advantages to the ANFIS model. Fuzzy inference is used to get the output variable. If the output variable does not coincide with the required value, then the artificial neural network trains weights of ANFIS. Dolatabadi M. et al. [8] for modelling

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simultaneous adsorption of dye and metal ion used ANN and ANFIS. So, RMSE (root mean square error) is used to evaluate two methods ANN and ANFIS. In all simulation experiments, the researches prove that ANFIS has the best results 2 times or more.

The learning of ANFIS is the process of determining the optimal values for the output parameters. The learning algorithms of ANFIS are divided into population-based algorithms (PBA) and derivative-based calculations (DBC). Population-based algorithms for learning ANFIS use hybrid technologies, such as, genetic algorithms (GA), particle swarm optimization

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algorithms (PSO), artificial bee colony algorithms (ABS) and simulated annealing algorithms (SA). Rezakazemi et al. [9] trained ANFIS model by using GA and PSO algorithms. The researches applied the proposed method for modelling mixed matrix membranes and found that

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ANFIS is two times better than GA-ANFIS and worse than PSO-ANFIS. Mottahedi et al. [10] trained ANFIS model by using PSO algorithm and they applied the proposed model to system overbreak prediction in underground excavations. They found the PSO algorithm is 2 times better ANFIS. Karaboga et al. [11] used adaptive and hybrid ABS algorithms to train ANFIS and

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used ABS algorithms for updating the antecedent and the conclusion parameters of ANFIS. The researchers found the ABS algorithm has done better in comparison with other models. Haznedar et al. [12] simulated annealing algorithm which was used in ANFIS to identify objects. They found SA algorithm better than Backpropagation (BP) algorithm and GA algorithm approximately 2 times. GA, PSO and ABS are population-based algorithms. A common disadvantage of these algorithms is long computational time to find the optimal solution.

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DBC uses the calculation of the gradient in each step. To train ANFIS of BP, Least Squares (LS), Gradient Descent (GD) are used. Boulkaibet et al. [14, 15] trained ANFIS model by using least squares support vector machines (LS-SVM) and Takagi–Sugeno system based on multi-kernel least squares support vector regression (TS-LSSVR) algorithms. They proposed to use these algorithms in the generalized predictive controller. It was found that TS-LSSVR algorithm has better characteristics in comparison with ANFIS and LS-SVM models about 3 times. It should be noted that the calculation of the gradient in each step is a difficult problem. Also, the gradient method has a slow convergence of parameters [13].

Journal Pre-proof ANFIS has several drawbacks including the "curse of dimension" [16, 17, 18] which leads to increase in time learning of neuro-fuzzy network. The curse of dimension increases the number of conclusions of fuzzy inference in a geometric progression with increasing the number of input variables or their membership functions (MF). Consequently, the time of the decisionmaking in ANFIS increases. This mistake can be corrected by using different fuzzy implications in the second layer of ANFIS [19, 20, 21]. For example, in the second layer of ANFIS t-norm PROD is used to multiply the degrees of the membership function [22]. If one of the degrees of

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the membership function is zero, then zero will be generated on the output of ANFIS. So, ANFIS increases the number of rules in its structure that leads to the appearance of a curse of dimension. One way to eliminate the increase in the number of fuzzy rules is to use soft arithmetic operations during fuzzy implication [23, 24, 25, 26]. The peculiarity of these formulas is the output will be different from zero, even in the case of an equality of two degrees of the membership functions to zero.

It should be noted that the accuracy of neuro-fuzzy models depends on the defuzzification

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of models used in the fifth layer of ANFIS. In article [27] detailed analysis of traditional and non-traditional defuzzification models is presented and a model of defuzzification based on the method of areas’ ratio is given. This model possesses improved characteristics in comparison

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with traditional and non-traditional defuzzification of models. The accuracy of a neuro-fuzzy system increases with using a nonlinear function in MAR and it is shown in the current research (see Section 4). Analysis of existing neuro-fuzzy models showed that their accuracy is no more than 3 times in comparison with classical ANFIS model. Therefore, the main goal of the

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scientific research is to increase the accuracy of ANFIS with the use of soft operators and a linear/non-linear defuzzifier based on the method of areas' ratio and shorten its learning time. To accomplish this goal, soft operations are used in the second layer of ANFIS. In the fifth layer of MAR-ANFIS with a nonlinear function is used. The main feature of this learning neuro-fuzzy model is the use of triangular membership functions on the output of the fuzzy inference system. 2. Modelling of adaptive neuro-fuzzy inference system (ANFIS)

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Artificial neural networks and Sugeno type fuzzy system create ANFIS structure. The combination of two methods is called a neuro-fuzzy model (NFM). The main goal of ANFIS is to optimize the parameters of fuzzy inference system using one of the learning algorithms. Minimum error value between the target output and the factual output is the parameter of optimization estimated, for instance, by using RMSE value. ANFIS consists of five layers where the degrees of MF use fuzzy rules and determine output crisp value with Takagi-Sugeno model [28, 29, 30]: If x  Ai and y  Bi, then z = pix + qiy + ri ,

Journal Pre-proof where x and y are inputs of NFM; z is the output of NFM; A and B are fuzzy sets; p, q and r are parameters of NFM. To compare the developed learning method (MAR-ANFIS) in Matlab learning ANFIS model was conducted. Fuzzy MISO-system with two input variables was formed in module «anfiseditor». Each of the input variables had three membership functions (Fig. 1a). Also, in auto mode, nine fuzzy rules were generated. b)

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a)

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Fig. 1. ANFIS: (a) Structure ANFIS. (b) Test sample For learning ANFIS 441 points were used. Learning was carried out for 2000 epochs and simulation results are presented in Figure 2a. The learning error was equal to 4.2778 (Fig. 2a).

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RMSE was not changed with an increase in the number of epochs to 5000. The surface obtained during learning ANFIS is presented in Figure 2b. b)

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a)

Fig. 2. Learning ANFIS: (a) Learning error. (b) Surface To minimize RMSE, it is necessary to increase the number of membership functions of

input variables and the number of fuzzy rules. The simulation of this process is summarized in table 1.

Journal Pre-proof Table 1. Comparing RMSE to the learning algorithm

№

Number

Number

of MF

of FR

ANFIS

ANN-

PSO-

TS-

ANFIS,

ANFIS

LSSVR

0.8114

GA-

Surface

4

5

x2=7

x1=11, x2=11

x1=15, x2=15

2.4342

4.8684

1.2171

49

1.6896

3,3792

0.8448

121

0.4016

x1=21, x2=21

0.5632

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x1=7,

25

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3

x2=5

0.8032

0.2008

0.133867

0.2432

0.0608

0.040533

0.0044

0.0011

0.000733

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2

x1=5,

225

0.1216

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1

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ANFIS

441

0.0022

As expected, the minimum RMSE value in the simulation in MatLAB was obtained when the number of fuzzy rules equals to the number of learning points. The recommended minimum number of fuzzy rules [22] is determined by formula x = lz = 212 = 441, where l is the number of

Journal Pre-proof membership function, z is the number of input variables. Considering the data in Table 1 for experiment #5, RMSE value is minimal when the number of fuzzy rules equals 441 and the number of membership functions for each input variables is equal to 21. Learning time was 10.2 minutes. It should be noted that such a large number of conclusions reduces the speed of ANFIS and leads to appearing of the curse of dimension. It will be necessary to increase the number of membership functions in input variables equal to 39 to obtain the minimum RMSE value if the number of learning points is 1500 and learning time will be about 2 hours. As noted in the

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Introduction, the accuracy of the modified ANFIS models does not exceed the accuracy of the classical ANFIS model by more than 3 times.

3. The proposed method of learning the neuro-fuzzy system

MAR-ANFIS of a fuzzy MISO system is implemented as follows:

Step 1. Fuzzification of input variables. At this step input and output membership functions are formed. Let the inputs of the fuzzy system be defined as follows: Y={y1}+{y2}+{y3}+…+{yk},

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Xi={x11}+{x12}+{x13}+…+{xij},

(1) (2)

where Xi are inputs of the fuzzy system (i=1…n, is the number of input signals; j=1…m, is the

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number of terms of every input signal); Y is the output of the fuzzy system (k=1…l, is the number of terms of an output signal).

For example, if fuzzy MISO-system has two input variables with three membership functions and one output variable with five membership functions and using formula (1) we get:

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X1={x11}+{x12}+{x13}; X2={x21}+{x22}+{x23}; Y={y1}+{y2}+{y3}+{y4}+{y5}. The input and output fuzzy variables are shown in Figure 3.

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(a)

(b)

(c)

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Fig.3. MF: (a) The first input MF. (b) The second input MF. (c) The output MF. The calculation time of the area of the transformed membership functions increases (see Eq. 16) when using g-bell membership functions. Consequently, the learning time of the fuzzy system in MAR-ANFIS method also increases. So, we will use triangular membership functions. Step 2. Calculate degrees of membership functions by the following equations

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 X i ( xij )  gbellmf ( xij ; a , b , c) 

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Trapezoidal MF

0, x  a  ij ,  b  a  X i ( xij )  trapf ( xij ; a , b , c, d )  1,  d  xij ,   xij  c 0,

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Triangular MF

0, x a  ij ,   X i ( xij )  trimf ( xij ; a , b , c)   b  a c  xij  ,  c b 0,

Gaussian (gbell) MF

xij  a ;

a  xij  b ;

b  xij  c ;

(3)

c  xij . xij  a ; a  xij  b ; b  xij  c ;

,

(4)

c  xij  d ; d  xij .

1  xij  c   1    a 

2b

.

(5)

where a, b, c, d are linguistic labels of the triangular, trapezoidal or gbell membership functions [22], xij is the crisp value that belongs to the support of fuzzy set. Step 3. Create fuzzy rules (FR). It is necessary to create rules “if … then …” to form the

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rule base which defines a premise conclusion relation between input and output variables of fuzzy inference system: FRp: If x1 is X1(xij) t x2 is X2(xij) t … t xi is Xn(xnm) Then y is yk,

(6)

where t is a sign denoted one operation from t-norms (see Table 2); p is the number of fuzzy rules. The degrees of activation of the premises of fuzzy rules (6) are calculated using formula:  p   X1 ( xij ) t  X 2 ( xij ) t  t  X n ( xnm ) .

(7)

Journal Pre-proof If several fuzzy rules are referenced to the same output term, then a fuzzy rule is activated and it has a maximum degree of the conclusion of fuzzy rules with one of s-norm using formula: hk  1  s  2  s   s  p ,

(8)

where s is a sign denoted one operation from s-norms (see Table 2)

Table 2. T and s-norms

norm

Lukasiewicz norm IL Bounded difference norm IBD Soft norm ISOFT

if I  I M x1 j  x2 j ; if I  I PROD min 1,1  x1 j  x2 j  ;

I  I BD



norm



x1 j  x2 j  δ 2  x1 j  x2 j  δ 2

Product IPROD

norm

p

1

p



p

; if

I  IM

I  I PROD

Algebraic norm IAS

sum

 p   p 1   p   p 1 ;

Bounded norm IL

sum

if I  I AS min 1,  p   p 1 ;

Soft norm ISOFT

2

,

if

I  I BS





 p   p1  δ 2   p   p1  δ 2 2 δ  0.05 if I  I SOFT

2

,

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2 δ  0.05 if I  I SOFT

max  p ; if

1

if I  I L max 0, x1 j  x2 j  1 ; if

Mamdani IM

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Product IPROD

s-norm (s)

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t-norm (t) Mamdani norm IM min( x1 j ; x2 j ) ;

Operations t-norm and s-norm (IM, IPROD, IL, IBD) are used in the original Zadeh rule that leads to the error of additivity property [22]. The property of additivity occurs when the output of fuzzy system changes in proportion to the change in the input signals. Zero signal is generated on

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the output of fuzzy system if one of the degrees of membership functions is equal to 0 and using one of IM, IPROD, IL, IBD operators. For example, Im(x1; x2)=min(0; 0.7)=0. So, the appearance of zero result at the output of fuzzy system during defuzzification is possible. We can see the absence of the additivity property of hard fuzzy system in Fig. 7a. The output of fuzzy system is not changed in the range of values [0; 5][15; 20] when using one of IM, IPROD, IL, IBD operators. Non-zero signal is generated on the output of fuzzy system if one of degrees of membership

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functions is equal to 0 with using a soft operator. For example, Isoft(x1; x2)=Isoft(0; 0.7)=0.000358. The output of fuzzy system changes in the range of values [0; 5][15; 20] while using soft implication Isoft (see Fig. 7a). In this case, fuzzy systems will have the property of additivity. Based on this information we can conclude that calculations of hard operators are carried out quickly and in a simple method, result is in decreasing load on computers. But, in general, accuracy of hard fuzzy models is worse than when soft operators are used. Let MISO-system have nine fuzzy rules shown in Table 3.

Journal Pre-proof Table 3. The base of fuzzy rules FR FR1 FR2 FR3

If x11 x11 x11

Then y5 y4 y3

x21 x22 x23

FR FR4 FR5 FR6

If x12 x12 x12

x21 x22 x23

Then y4 y3 y2

FR FR7 FR8 FR9

If x13 x13 x13

x21 x22 x23

Then y3 y2 y1

Step 4. Defuzzification with the method of areas' ratio [27]. The method of areas’ ratio (MAR) is based on calculating the areas of triangular and / or

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trapezoidal membership functions with a universal formula for calculation of their areas (Fig.4) as follows S

h d1  4d 2  d 3 , 6

(9)

where h is the height of the geometric shape; d1, d2, d3 are the lengths of the lower, middle and

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upper basis of the geometric shape.

Fig. 4. Calculation of areas of the geometric shapes. Let the height of the geometric shape h be equal to the height of the degrees of fuzzy rules conclusion hk determined by the formula (8), h=hk.

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Formula (9) should be converted in order to be used in MAR: – for trapeze MF:

S trapeze 

hd1  d 3  d  d3 , if h =1 then S trapeze  1 . 2 2

(10)

– for triangle MF:

S triangle 

d hd1 , if h =1 then S triangle  1 . 2 2

(11)

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Step 4.1. Use formula (9) to determine the total area of the shape of output value:

S1  k  w 

d1 , 2

(12)

where k is the number of terms of output MF (k = 5); w is weight coefficient. Step 4.2. Calculate the area of the transformed membership functions which depend on

the height values of the premise fuzzy rules:

Journal Pre-proof  S1i  0 , if h  0 ; k  d S1i  S1i  1 , if h  1; 2 i 1  h S1i  d1  d 3 , if h  (0, 1) .  2

(13)

Step 4.3. Determine the total area of the shape with transformed membership functions as follows: k

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S 2   S1i . i 1

(14)

Step 4.4. Determine the ratio of the total area of the shape of the output variable (see Eq. 12) and the total area of the shape with transformed membership functions (see Eq. 9 and 14) as follows: D

S2 . S1

(15)

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The variable D defined by Eq. (15) is always in the range of values from 0 to 1. To get a crisp value at the output of fuzzy system, we must reflect the value of the variable D to the support of the output fuzzy variable.

Step 4.5.1 Calculate crisp value at the output of fuzzy system with linear function as follows:

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

ydefuz_ line  D  y fin  уst   уst ,

(16)

where yst, yfin are the start and the final values in the range of crisp values of the output fuzzy set (see Fig.3c).

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Step 4.5.2 Calculate crisp value at the output of fuzzy system with nonlinear function. Let the nonlinear function be given as follows: b ( y

y

 c)

e defuz _ nonline fin D , b ( y  y  c) 1  e defuz _ nonline fin

(17)

where b’, c’ are parameters of a nonlinear function. It is necessary to determine the value of a crisp value at the output of fuzzy system from

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formula (17). Later, we transform formula (17) into the equation:



D  1 e

b ( y defuz _ nonline  y fin  c  )

 e

b ( y defuz _ nonline  y fin  c  )

.

(18)

Denote l as

l  b( ydefuz_ nonline  y fin  c) .

(19)

Substituting the value of formula (19) into (18), we obtain:

D  D  el  el . From formula (20) we define D:

(20)

Journal Pre-proof D  el  D  el  el 1  D .

(21)

From formula (21) we define el and get:

el 

D . 1  D 

(22)

By the property of exponent ln(ex) = x we obtain the formula:

 D  l  ln  .  1  D 

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(23)

Substitute equation (23) into equation (19) and get  D  ln    b( ydefuz_ nonline  y fin  c) .  1  D 

From formula (24) we define ydefuz_nonline  D  ln  1  D  ydefuz_ nonline  y fin  c   . b

(24)

(25)

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For nonlinear defuzzification the formula for calculating crisp value will be recorded as follows:

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 D  ln  1  D  ydefuz_ nonline    y fin  c. b

(26)

Step 5. Learning the neuro-fuzzy system. On this step, the neuro-fuzzy system is learned using the method of backpropagation error by formula

where

output

until ydefuz  yexp  T ,

(27)

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woutput  woutput1  ( ydefuz  yexp ) ,

is weight coefficient;  is a learning step to the neuro-fuzzy system (by default is

equal 0.07); T is a threshold (by default is equal 0.01). MAR-ANFIS structure is shown in Figure 5. Layer 1. Fuzzification of input variables xintput = f (X1, X2) using formula (1) where input values can be obtained from sensors of dynamic control systems.

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Layer 1. Vector youtput = f (Y) output fuzzy variable is formed. Layer 2. Calculation of the degrees of membership functions by formulas (3)(5). Layer 2. Fuzzification of output variable using formula (2). Layer 3. Calculation of the degrees of activation of the premise of fuzzy rules by formula

(7). Layer 4. Calculation degrees of the conclusion of fuzzy rules by formula (8). Layer 5. Calculation of areas of the transformed membership functions by formula (13).

Journal Pre-proof Layer 5. Calculation of weight coefficient by formula (27). On the first step of learning, we recommend accepting the value of weight coefficient equal to woutput-1=k/2 from formula (12). Layer 6. Calculation of the total area of the shape with transformed membership functions by formula (14). Layer 6. Calculation of the total area of the shape of output value by formula (12). Layer 7. Determination of the areas’ ratio by formula (15).

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Layer 8. Calculation of crisp value on output fuzzy system. In the case of usage of linear function crisp value on output fuzzy system is calculated by the formula (16). In the case of usage of nonlinear function crisp value on output neuro-fuzzy system is calculated by the formula (26). Layer 1

Layer 2

Layer 3

Layer 4

Layer 5 Layer 6

1 2 x11

h1

S11

h2

S12

X1

x12 4 x13 5

h3

x21 6 X2

h4

x22 7 8 9 y1

Y

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y2

h5

S13

S2

S14

y3

Layer 8

yexp

D

ydefuz

youtput

= (ydefuz – yexp)

S15

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x23

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3

Layer 7

woutput-1

Learning

woutput  woutput 1    ( y defuz  yexp ) S1

y4

y5

Layer 1 Layer 2

Layer 5 Layer 6

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Fig. 5. Structure of the method of learning MAR-ANFIS. Layer 9. Learning the neuro-fuzzy system using formula (27), i.e., selecting weight

coefficient for each learning point until the difference between given value yexp and the fuzzy output ydefuz become equal to the threshold value. 4. Numerical simulation 4.1 Analysis of the properties of MAR-ANFIS MAR-ANFIS software model with linear and nonlinear functions (Fig. 6) to analyze its properties was created in Simulink. In the block “Input” the values of the input signals are

Journal Pre-proof generated: for variable x1 in the range from 110 to 190, for variable x2 from 200 to 280. In the blocks "Degrees X1" and "Degrees X2" the degrees of membership functions are calculated by the formula (3). In the block “Implication” using the base of fuzzy rules from Table 2 the degrees of activation of the premises of fuzzy rules are calculated: 1  t ( x13 , x23 );  2  t ( x13 , x22 );  3  t ( x13 , x21);  4  t ( x23 , x23 );  5  t ( x23 , x22 );  6  t ( x23 , x21);  7  t ( x13 , x23 ); 8  t ( x13 , x22 );  9  t ( x13 , x21).

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If fuzzy rules refer to the same output term, then in the block “Aggregation” with using the base of fuzzy rules (see Table 3) and one of s-norm (see Table 2), maximum degree of activation is by formulas: h1   9 , h2   s ( 6 ,  8 ) , h3   s ( 3 ,  5 ,  7 ) , h4   s ( 2 ,  4 ) ,

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h5  1 .

Fig. 6. Structural scheme of a fuzzy model

In the block "MAR linear" the output value of the fuzzy system is calculated by the formula (16). In the block "MAR nonlinear" the output value of the fuzzy system is calculated by the formula (26). In the block "CoG singltone MF" the output value is calculated using the center

Journal Pre-proof of gravity method. Moreover, the output variable has singleton membership functions which are defined by the following vectors Y = {Y1 = 415, Y2 = 430, Y3 = 445, Y4 = 460, Y5 = 475}. In the blocks "Global aggregation" and "CoG triangular MF" the output value is calculated using the centre of gravity method. The output variable has triangular membership functions (see Fig. 3c). The blocks “simout1”, “simout2”, “simout3” and “simout4” are used to plot the resulting fuzzy variable. The fuzzy system using Mamdani implication (IM) is shown in Fig 7a and fuzzy

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system using Soft implication (ISOFT) is shown in Fig 7b.

(b)

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(a)

Fig.7. Output fuzzy system: (a) With implication Mamdani. (b) With soft implication. Analysis of the graphs presented in Figure 7a showed that the classical models of fuzzy

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inference with using the centre of gravity method are not additive in the range from 0 to 5 and from 15 to 20. That is, 50% of the values in the fuzzy output graph will not be changed when one of the input variables x1 or x2 changes. The disadvantage of MAR model is breaking symmetry in the time interval from 3 to 4.5 seconds. MAR model is symmetrical while using soft implication

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(ISOFT). Thus, the large learning time and low accuracy of the classical ANFIS model is explained by the presence of dead zones in the time interval from 0 to 5 or from 15 to 20 seconds.

The positive feature of the MAR model is the ability to change the output of the fuzzy system which depends on the change of the structure of fuzzy rules. For example, during aggregation we will use the following formulas:

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h1   9 ,

h2  max( 5 ,  6 ,  8 ) , h3  max( 3 ,  5 ,  7 ) ,

h4  max( 5 ,  2 ,  4 ) , h5  1 .

In this case output of the fuzzy system, the graphics of MAR models will vary in the time interval from 5 to 15, and the graphs of the classical model will not change significantly (see Fig. 8).

Journal Pre-proof (b)

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(a)

Fig.8. Fuzzy: (a) With Mamdani implication. (b) With Soft implication. For example, when using ANFIS model with t-norm IPROD and s-norm IM for 2000 epochs of learning RMSE is equal 4.2778. When using MAR-ANFIS model with similar indicators its accuracy is equal 0.37. Therefore, the accuracy of the proposed learning method MAR-ANFIS is

4.2 Modeling of MAR-ANFIS

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higher than ANFIS by 4.2778 / 0.37 = 11.56 times.

Consider the simulation procedure. Let the neuro-fuzzy system have triangular membership functions (see Fig. 3). Initially, the research of neuro-fuzzy system was conducted

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without learning function (27). In each experiment different implication models (t-norm) were used (IM, IPROD, IL, IBD, ISOFT) (see Table 2). To obtain the resultant variable with the help of MAR both linear (p. 4.5.1) and nonlinear (p. 4.5.2) functions were used. The resulting surfaces of the resultant variable are presented in Figure 9.

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(a)

(b)

(d)

(e)

(f)

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(c)

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(h)

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(g)

(j)

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(i)

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Fig. 9. Method of areas’ ratio without learning. (а) Implication IM with a linear function. (b) Implication IM with nonlinear function. (c) Implication IPROD with a linear function. (d) Implication IPROD with nonlinear function. (e) Implication IL with a linear function.

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(f) Implication IL with nonlinear function. (g) Implication IBD with a linear function. (h) Implication IBD with nonlinear function. (i) Implication ISOFT with a linear function. (j) Implication ISOFT with nonlinear function.

In articles [30, 31] it was recommended to use t-norms in the second layer of ANFIS. In Figure 9 we investigated the use of t-norms on the second ANFIS layer and presented the results (Fig. 9) to indicate that on graphs a, b, c, d, e, f, g and h at the borders of the support of the input

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variables sup(x1) and sup(x2), the degrees of membership functions are equal 0. Therefore, there is the possibility of an error called the curse of dimensionality during learning of the fuzzy system. It leads to the increasing of RMSE and MAPE coefficients. When using soft operators (Fig. 9, i and j) the degrees of membership functions are not equal to 0. Therefore, the appearing of the curse of dimensionality is excluded. For further research of nonlinear learning of neuro-fuzzy system, an experimental selection (see Fig.10), consisting of 441 points, was constructed by using formula

Journal Pre-proof yexp 

1  x  c  1  2   a 

2 b

 x1d   yst ,

(28)

where x1 and x2 are support of fuzzy input variables; с is the centre of support of input variable x2 (с=250); а is the parameter of function from range [5; 50]; b is the parameter of function

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(b = 3); d is the scaling factor from range [0.1; 0.7].

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Fig. 10. Experimental selection Accuracy of the implication models and the usage of linear (16) and nonlinear functions (26) in MAR-ANFIS without learning was estimated with RMSE and MAPE. To improve the reliability of the obtained results the performance of MAR-ANFIS model based on two indicators RMSE and MAPE was evaluated. The best-recognized model is with the lowest value

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of RMSE and MAPE coefficients.

RMSE and MAPE value are determined by formulas 1 N

1 N

n

y i 1

n



defuz

 yexp

ydefuz  yexp

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RMSE 

MAPE 

i 1

yexp

2

 min ,

(29)

 min ,

(30)

where ydefuz is a defuzzification output (Eq. 16 or 26); yexp is a multiplicity of learning points (Eq. 28); N is several points in a test sample (N=441). RMSE and MAPE calculation were carried out in the range of values for x1 [100; 200] and x2 [200; 300] (Table 4).

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Table 4. Calculation of RMSE and MAPE without learning neuro-fuzzy systems

Function MAR

in

IM

IPROD

linear nonlinear

17.8 25.29

19.47 22.78

linear nonlinear

2.78 4.21

3.13 3.77

Implication IL RMSE 66.21 60.44 MAPE 14.77 13.4

IBD

ISOFT

43.07 20.25

21.45 28.06

1.41 5,12

3.73 5.71

Journal Pre-proof Next, RMSE and MAPE coefficients were calculated during learning the neuro-fuzzy system without limiting the range of input variables. During learning of the neuro-fuzzy system in formula (26) the following parameter values were used: yfin = 496, b = 0.05, c = 1. Besides, during the experiment, the learning time of the neuro-fuzzy system was estimated. The calculation is shown in Table 5. Table 5. Calculation of RMSE and MAPE with learning the neuro-fuzzy system (exp. #1)

linear nonlinear

linear nonlinear

IM

IPROD

12.3 (1.06 min) 0.03 (0.19 min)

12.3 (1.09 min) 0.001 (0.14 min)

1.08 0.0006

1.07 0.0001

Implication IL RMSE / tlearning, min 1.79 (1.40 min) 4.81 (1.25 min) MAPE 0.26 0.83

IBD

ISOFT

26.09 (4.13 min) 0.44 (1.10 min)

0.44 (0.36 min) 0.02 (0.33 min)

2.69 0.002

0.055 0.0006

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Function in MAR

The analysis of table 6 showed that RMSE with using nonlinear function in MAR-ANFIS

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is smaller compared with the linear function used in MAR-ANFIS. Time learning of a neurofuzzy system is shorter when a non-linear function in MAR-ANFIS is used. The learning process of the neuro-fuzzy system stopped when fuzzy implications IM, IPROD, and IBD were used in MAR-ANFIS. It is because in formula (13), in the case when the values of the degrees of

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membership are equal to 0, S2 value also equals 0. Consequently, in formula (14) the value of D will become equal to 0 and call an error at the substitution of this value in formula (15). To eliminate this drawback by the method of mathematical programming value S2=0.25 is replaced, if its value is equal to 0. After that, the learning process was continued.

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The graphical interpretation of the results obtained by using a linear function in learning NFS is shown in Figure 11. The graphical interpretation of the results obtained by using a nonlinear function in learning NFS is shown in Figure 12. (b)

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(c)

(d)

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(e)

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Fig. 11. Learning ANFIS with a linear function. (а) Implication IM. (b) Implication IPROD. (c) Implication IL. (d) Implication IBD. (e) Implication ISOFT.

(b)

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(e)

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Fig. 12. Learning ANFIS with nonlinear function. (а) Implication IM. (b) Implication IPROD. (c) Implication IL. (d) Implication IBD. (e) Implication ISOFT.

Consider an example of learning fuzzy system using MAR-ANFIS at the point x1 = 120, x2 = 275. At this point, the value of the experimental sample is equal to yexp = 448.01. Modeling the learning process using fuzzy system (see Fig. 6) was established that the output value of fuzzy system (MAR-ANFIS linear) calculated by the formula (16) is equal to ydefuz = 422.7, and

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the output value of fuzzy system (MAR-ANFIS nonlinear) calculated by formula (26) is equal to ydefuz = 431.6. The learning process is presented in table 6. Table 6. Learning process

1

2

MAR linear (ydefuz=422.7)

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№ epoch

w1=5 – 0.07(422.7 – 448.01) = 3.232,

w1=5 – 0.07(431.6 – 448.01) = 3.854,

|ydefuz – yexp| th, |422.7 – 448.01| is false.

|ydefuz – yexp| th, |431.6 – 448.01| is false.

w2=3.232 – 0.07(435.2 – 448.01) = 2.335, w2=3.854 – 0.07(438.4 – 448.01) = 3.18,

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|435.2 – 448.01| is false. 3

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|ydefuz – yexp| th, |447.1 – 448.01| is false.

w6=2.372 – 0.07(448.0 – 448.01) = 2.368, w6=2.613 – 0.07(447.6 – 448.01) = 2.587, |448.0 – 448.01| is true.

7

|ydefuz – yexp| th, |445.8 – 448.01| is false.

w5=2.363 – 0.07(448.1 – 448.01) = 2.372, w5=2.679 – 0.07(447.1 – 448.01) = 2.613, |448.1 – 448.01| is false.

6

|ydefuz – yexp| th, |443.1 – 448.01| is false.

w4=2.384 – 0.07(447.7 – 448.01) = 2.363, w4=2.835 – 0.07(445.8 – 448.01) = 2.679, |447.7 – 448.01| is false.

5

|ydefuz – yexp| th, |440.1 – 448.01| is false.

w3=2.335 – 0.07(448.7 – 448.01) = 2.384, w3=3.18 – 0.07(443.1 – 448.01) = 2.835, |448.7 – 448.01| is false.

4

MAR nonlinear (ydefuz=431.6)

|ydefuz – yexp| th, |447.6 – 448.01| is false. w7=2.587 – 0.07(447.9 – 448.01) = 2.577, |ydefuz – yexp| th, |447.9 – 448.01| is false.

8

w8=2.577 – 0.07(448.0 – 448.01) = 2.576, |ydefuz – yexp| th, |448.0 – 448.01| is true.

Journal Pre-proof It was also found that the accuracy of learning MAR-ANFIS depends on the parameter yfin used in formula (26). For example, if to decrease this parameter accuracy of NFS increases and its learning time decreases. Table 7. Calculation of RMSE and MAPE with using learning neuro-fuzzy system and yfin=490 IM

IPROD

nonlinear

0.001 (0.13 min)

0.001 (0.10 min)

nonlinear

0.0001

0.0001

Implication IL RMSE / tlearning, min 3.33 (1.15 min) MAPE 0.56

IBD

ISOFT

0.21 (0.4 min)

0.001 (0.23 min)

0.002

0.0001

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Function in MAR

For more detailed research of the proposed MAR-ANFIS method and an analysis of the usage of both linear and non-linear functions in them, the experimental selection defined by formula (28) was changed.

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In the course of the experiment, the value of RMSE and the learning time of the neurofuzzy system were estimated. Additional experimental selections used for analysis are shown in Figure 13.

(b)

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Fig. 13. Additional experimental selections. (а) Second experimental selection. (b) Third experimental selection. Also, during experiments #2 and #3 the learning time of the neuro-fuzzy system was estimated. The calculation is shown in Table 8 and Table 9.

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Table 8. Calculation of RMSE and MAPE with using learning NFS (experiment #2) Function in MAR

IBD

ISOFT

5.64 (1.35 min)

Implication IL RMSE / tlearning, min 1.26 (2.03 min)

8.38 (1.58 min)

0.22 (0.51 min)

0.001 (0.24 min)

0.001 (0.19 min)

3.18 (1.32 min)

0.29 (0.04 min)

0.001 (0.35 min)

IM

IPROD

5.64 (1.38 min)

linear

nonlinear

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linear nonlinear

0.696 0.0001

MAPE 0.175 0.140

0.695 0.0001

1.141 0.005

0.016 0.0001

Table 9. Calculation of RMSE and MAPE with using learning NFS (experiment #3) IBD

ISOFT

23.65 (1.33 min)

Implication IL RMSE / tlearning, min 0.08 (0.29 min)

29.03 (1.57 min)

0.001 (0.08 min)

0.001 (0.07 min)

0.001 (0.05 min)

0.85 (1.32 min)

0.001 (0.04 min)

0.001 (0.14 min)

3.393 0.0001

3.393 0.0001

4.728 0.0001

0.0001 0.0001

IM

IPROD

23.65 (1.35 min) linear

5. Conclusion

MAPE 0.003 0.083

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linear nonlinear

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nonlinear

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Function in MAR

We have described the architecture of the adaptive neuro-fuzzy inference system based on the linear or nonlinear method of areas' ratio (MAR-ANFIS). A distinctive feature of the

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proposed model is the use on the output of the fuzzy system triangular membership functions. Another important issue in learning MAR-ANFIS is how to use bell-shaped membership functions. Although we did not pursue this direction in this paper, it can be achieved by changing the calculation of the area of transformed membership functions in formula (13). The analysis of the modelling process learning MAR-ANFIS based on RMSE and MAPE calculation showed that:

1. the use of a nonlinear function in MAR-ANFIS improves its accuracy by an average of

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89.4 times compared with the usage of a linear function. 2. the learning time of MAR-ANFIS with using a nonlinear function is 3.05 times less

than with using a linear function. 3. the shortest learning time of MAR-ANFIS was obtained by using soft implication

(ISOFT): relative to Mamdani implication (IM) by 2.42 times, relative to the implication of algebraic product (IPROD) by 2.34 times, relative to the implication of Lukasevich (IL) by 4.29 times, relative to the implication limited difference (IBD) by 4.78 times.

Journal Pre-proof 4. The accuracy of the proposed MAR-ANFIS method with similar indicators ANFIS is higher than by 11.56 times. 6. Summary In work [2] fuzzy logic to construct a stereo vision system was used. This system was worked out to build a route for movement of a mobile robot. When the obstacles were detected, the mobile robot needed to relearn a fuzzy model and find a new route. In Matlab we modelled an adaptive neuro-fuzzy inference system consisting of two input variables and one output

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variable. After relearning ANFIS the test results showed that the accuracy is equal to 2.43 (see Table 1). We increased the number of membership functions for the input variables to 21 and the accuracy was 0.0022. However, the learning time of ANFIS increased from 30 seconds to 12.1 minutes. Therefore, we faced the problem of increasing ANFIS accuracy and reducing the time of its learning.

We found reasons for reducing the accuracy of learning of neuro-fuzzy network. Firstly, to use hard logical formulas t- and s-norms during fuzzy implication. As a result, the output of

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the fuzzy model has value equal to 0 (see Fig. 9) on the borders of support of input fuzzy variables. It is explained by the following. If one of the degrees of membership functions is equal to zero, then the output of the fuzzy model also is equal zero during hard implication. In ANFIS

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model this error is reduced by increasing the number of membership functions of input variables. Increasing the number of membership functions leads to the appearing of the curse of dimensionality and, as a consequence, increasing in learning time of a fuzzy model. We recommend eliminating the curse of dimensionality by using soft operators to find minimum and

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maximum. Secondly, some of the rules with the lowest degree of activation are excluded from the calculation during fuzzy inference. Therefore, such cases are possible when the input parameters of the fuzzy model are being changed, but the output of the fuzzy model is not being changed and the fuzzy system becomes nonadditive during defuzzification. So, in time range t(0, 5)(15, 20) output of fuzzy model using the centre of gravity method is not being changed (see Fig. 7, a and 7,b). As a rule, existing defuzzifiers based on the centre of gravity method have

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exactly the same error. In work [26] we developed a defuzzification model based on the method of areas’ ratio to compensate this error. As each rule refers to the membership function of the output variable nonadditivity is excluded. Figures 7,b and 8,b show the output of the fuzzy model which is additive in the entire time range. Considering two features, we created the unified MAR-ANFIS model (see Figs. 5 and 6), the distinguishing feature of which is the use of soft operators during fuzzy implication and linear or nonlinear model of areas' ratio, and used error backpropagation algorithm for learning neuro-fuzzy network (see table. 5). After that, we tested MAR-ANFIS model. Three test samples

Journal Pre-proof consisting of 441 points were constructed using function (28) (see Figs. 10 and 13). We conducted simulation and performed an analysis of the results based on RMSE and MAPE coefficients. The analysis showed the speed of learning MAR-ANFIS model is 10 or more times faster than the speed of ANFIS model. Based on the analysis of RMSE calculation data presented in tables 1, 5 and 7, we concluded that MAR-ANFIS method is approximately 22 times better than ANFIS algorithm, 44 times better than ANN-ANFIS and GA-ANFIS algorithms, 11 times better than PSO-ANFIS algorithm and 7.33 times better than TS-LSSVR.

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It was also found that the accuracy of learning of neuro-fuzzy model depends on the parameters of the nonlinear function. The accuracy of the neuro-fuzzy network can be increased by 2 or more times with selecting parameter yfin of function (26) according to the data presented in table 7. References

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Journal Pre-proof Graphical abstracts Layer 1

Layer 2

Layer 3

Layer 4

Layer 5 Layer 6

Layer 7

Layer 8

1 2 x11

h1

S11

h2

S12

3 X1

x12 4

yexp

x13 h3

S13

h4

S14

h5

S15

pro of

5 x21 6 X2

x22 7 x23 8 9

S2

D

youtput

= (ydefuz – yexp)

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youtput = youtput −1 +

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y1

y2 y3

youtput-1

y4 y5

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Layer 1 Layer 2

lP

Y

ydefuz

S1

Layer 5 Layer 6

+  ( ydefuz − yexp )

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Highlights: – The method presents the implementation of neuro-fuzzy system on the areas' ratio model. – The method increases the accuracy of the neuro-fuzzy system.

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– The method ensures the additivity of the neuro-fuzzy system.

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Bobyr M.V. conceived of the presented idea. Bobyr M.V. developed the theory and performed the computations. Emelyanov S.G. verified the analytical methods. Bobyr M.V. supervised the findings of this work. All authors discussed the results and contributed to the final manuscript.

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Conflict of Interest and Authorship Conformation Form Please check the following as appropriate: All authors have participated in (a) conception and design, or analysis and interpretation of the data; (b) drafting the article or revising it critically for important intellectual content; and (c) approval of the final version.

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This manuscript has not been submitted to, nor is under review at, another journal or other publishing venue.

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The authors have no affiliation with any organization with a direct or indirect financial interest in the subject matter discussed in the manuscript

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The following authors have affiliations with organizations with direct or indirect financial interest in the subject matter discussed in the manuscript:

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Author’s name Bobyr Maxim (a) (b) (c) Emelyanov Sergey (a) (b) (c)

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Affiliation Southwest State University Southwest State University