Ejsr 86 3

European Journal of Scientific Research
ISSN 1450-216X Vol. 86 No 3 September, 2012, pp.443-459
© EuroJournals Publishing, Inc. 2012
http://www.europeanjournalofscientificresearch.com
Fuzzy and Adaptive Neuro-Fuzzy Inference
System of Washing Machine
Rana Waleed Hndoosh
Department of Software Engineering
College of Computers Sciences & Mathematics, Mosul University, Iraq
M. S. Saroa
Department of Mathematics, Maharishi Markandeshawar University
Mullana-133207, India
Sanjeev Kumar
Dr. B. R. Ambedkar University
Khandari Compus, Agra-282002, India
E-mail: sanjeevibs@yahoo.com
Abstract
Software estimation accuracy is among the greatest challenges for software
developers. Fuzzy set theory, Fuzzy system and Neural Networks techniques seem very
well suited for typical geotechnical problems. In conjunction with software computing and
conventional mathematical methods, hybrid methods can be developed that may prove to
be a step forward in modeling geotechnical problems. This study aimed at building two
different models, Fuzzy Inference Systems and Adaptive Neuro Fuzzy Inference System
and a comparison between them, through an application to real data of the relationship
between three inputs (time, temperature of water and the amount of washing powder)
during the washing process by using washing machine to get the best result for the
cleanness of clothes, where we apply this application at real data. It also provides a natural
relationship for combining both numerical information in the form of input/output pairs and
linguistic information in the form of If–Then rules in a uniform fashion. The proposed
algorithm is achieved by the intelligent system FIS and ANFIS.
Keywords: Fuzzy Inference Systems (FIS), Adaptive Neuro-Fuzzy Inference System
(ANFIS), Neuro-Fuzzy (NF), Neural Networks (NN), Learning Algorithm.
1. Introduction
Fuzzy system was first developed by Zadeh in the mid-1960s for representing uncertain and imprecise
knowledge. It provides an approximate but effective means of describing the behavior of systems that
are too complex, ill-defined, or not easily analyzed mathematically [9]. Fuzzy variables are processed
using a system called a fuzzy inference system. It involves fuzzification, fuzzy inference, and
defuzzification. ANFIS (Adaptive Neuro Fuzzy Inference System) is an architecture which is
functionally equivalent to a Sugeno type fuzzy rule base. Under certain minor constraints the ANFIS
architecture is also equivalent to a radial basis function network. Loosely speaking ANFIS is a method
for tuning an existing rule base with a learning algorithm based on a collection of training data. In

Fuzzy and Adaptive Neuro-Fuzzy Inference System of Washing Machine 444
order to process fuzzy rules by neural networks, it is necessary to modify the standard neural network
structure appropriately. Since fuzzy systems are universal approximates, it is expected that their
equivalent neural network representations will possess the same property. The reason to represent a
fuzzy system in terms of neural network is to utilize the learning capability of neural networks to
improve performance, such as adaptation of fuzzy systems [17]. Thus, the training algorithm in the
modified neural networks should be examined. In this work we will introduce the application of Neuro-
Fuzzy Inference System (ANFIS) for washing machine, using the Matlab toolbox. The relation of the
inputs with output has been discussed during application dependent on real data. This work presents an
alternative modeling approach to find the degree of cleanness of clothes dependent on the property of
inputs. The principal constituents of the modeling approach are fuzzy set, fuzzy system and neural
network. These are combined into the so-called hybrid modeling system (Neuro-fuzzy) [19]. In the
present work two different models have been designed using two different systems, Fuzzy Inference
System and Adaptive Neuro-Fuzzy System, and comparison is made between them to know which
better one is. The focus here is not only on how to construct the model but also on how to use this
modeling system to interpret the results and assess the uncertainty of the model.
2. Fuzzy Inference Systems
Fuzzy inference systems are also known as fuzzy-rule-based systems, fuzzy models, fuzzy associative
memories (FAM), or fuzzy controller when used as controllers [19,6]. In the field of learning systems
an interesting research subject concerns how to join the experimental knowledge of a system with the
knowledge of experts. The former, based on data collected from experiments, is commonly used to
train neural networks while the latter is used in expert systems and more recently in fuzzy systems [8].
The crisp input is fuzzified by the associated input membership function and submitted to fuzzy
inference block, which is a decision-making unit and generates fuzzy output through fuzzy reasoning.
Defuzzification block calculates crisp output from fuzzy output [17,3]. Knowledge base, composed of
data base and rule base, defines the associated membership function in fuzzification and
defuzzification blocks, and provides fuzzy rules to fuzzy inference block. (See Fig (1)):
Figure 1: The structure of the fuzzy. inference system.
There exist three main types of fuzzy systems that differ in the way they define the consequents
of their rules: Mamdani, Takagi-Sugeno, and Singleton fuzzy systems. I sketch below their main
characteristics:

445 Rana Waleed Hndoosh, M. S. Saroa, and Sanjeev Kumar
Type1: Mamdani's-Type of FIS: In Mamdani model, both the input and output are represented
by linguistic terms. The antecedent and consequent parts of a rule are typically Boolean
expressions of simple clauses.
Type2: Sugeno-Type of FIS: This type is called also Takagi-Sugeno method of fuzzy inference.
Introduced in 1985 and it is similar to the Mamdani method in many respects. The first
two parts of the fuzzy inference process, fuzzifying the inputs and applying the fuzzy
operator, are exactly the same. The main difference between Mamdani and Sugeno is
that the Sugeno output membership functions are either linear or constant. A typical
rule in a Sugeno fuzzy model has the following form [17,6]:
If input1= x and input2= y then output is f= px + qy + r (1)
For a zero-order Sugeno model, the output level f is a constant (p=q=0). The output level fi of
each rule is weighted by the Wi of the rule. The final output f of the system is the weighted average of
all rule outputs, computed as (see Fig (2)):
N
i i i
1
w f
N
i i
Final ouput f
w

= = =
(2)
Figure 2: Zero-order TS fuzzy inference system with two inputs.
Type 3: Singleton-Type of Fuzzy Interference System: The rule consequents of this type of
systems are constant values. Singleton fuzzy systems can be considered as a particular
case of either Mamdani or TS fuzzy systems. In fact, a constant value is equivalent to
both a singleton fuzzy set i.e., a fuzzy set that concentrates its membership value in a
single point of the universe and a linear function in which the coefficients of the input
variables value 0, [2,8].
A types of fuzzy system model mostly used is based on “fuzzy conditional statement” also
called fuzzy if-then rules and originally applied for modeling, ill-defined industrial processes. A model
of a multi-input single-output system can be described by means of a set of rules [9,3]:
1 1 2 2 : ( ) ( ) ( ) i i i n in i R if x is A and x is A and x is A then y is B (3)
where xj , j = 1,..,n are input variables, y the output variable and Ai1 , Bi are the fuzzy sets. The model
composed of fuzzy if-then rules must be completed with an inference process. In the inference process
the degree of truth of the rule premise is evaluated. This value is carried out to the consequent and then
all the fuzzy output variables so obtained are joined and defuzzified. The four steps of the fuzzy
inference system applied to the Product-Sum method, that we have chosen to implement in our
architecture, are reported as follows:

2.1. Fuzzification
In the fuzzification, the crisp input values are transformed to fuzzy values. If the input has a crisp
value, the matching against the membership function of linguistic variable is shown in Fig. (3a). If the
input contains noise, it can be modeled by using a fuzzy input value. In this case the fuzzy output is the
intersection of fuzzy input and the linguistic variable membership functions as shown in Fig. (3b).
However, the crisp input value fuzzification is mostly used because of its simplicity [9,19,4].
2.2. Inference
The decision making unit performs the inference operations on the fuzzy rules. The fuzzy values within
a fuzzy rule are aggregated with connective operators like intersection (AND), union (OR) and
complement (NOT). The operation of the intersection is shown in Fig. (4) The final output fuzzy sets
are obtained either scaling (Max-Dot method) or cutting (Max-Min) according to the firing strength of
the fuzzy rules. If the output fuzzy sets are singletons, they are not handled by the firing strengths in
this stage [19,3].
Figure 3: Fuzzification of a crisp input and a fuzzy input.
Figure 4: The fuzzy inference using the Min-inference.
Figure 5: Defuzzification using the weighted
average strategy.
2.3. Defuzzification
In the defuzzification stage, the outputs of the fuzzy rules are combined to a crisp output value. Several
defuzzification strategies have been suggested. The most common method is the center of area (COA)
defuzzification strategy, illustrated in Fig. (5)[8]. Assuming a discrete universe of discount, the crisp
output F is produced by searching the center of gravity of consequence fuzzy sets according to:

m
i o
c i i
m
i i i
o
( )
( )
f f
F
f
μ
μ
=
=
=

(7)
where m is the number of quantization levels of the output, fi is the amount of output at the
quantization level i, and μi(fi) represents its membership value in C,[9,4].
If only singletons are used as the consequences of fuzzy rules, the natural defuzzification
method is the weighted average (WA). It can be considered as a special case of COA defuzzification
method. The WA method combines the consequences of the fuzzy rules to the output of the inference
system F according to:
n
i o
i i
m
i i
o
f
F
μ
μ
=
=
=

(8)
where n is number of fuzzy rules, μi
is the firing strength of the rule, and fi is the output value of the ith
singleton[13,9].
3. Adaptive Neuro-Fuzzy Inference Systems (ANFIS)
The ANFIS is an adaptive network of nodes and directional links with associated learning rules. The
approach learns the rules and membership functions MFs from the data [9,20,4]. Jang in 1993
introduced architecture and learning procedure for the FIS that uses a neural network learning
algorithm for constructing a set of fuzzy if-then rules with appropriate MFs from the specified input–
output pairs. This procedure of developing a FIS using the system of adaptive neural networks is called
an adaptive neuro-fuzzy inference system (ANFIS). There are two methods that ANFIS learning
employs for updating membership function parameters [10,13]:
1) Backpropagation method (BP) for all parameters (a steepest descent method).
Backpropagation is probably the most popular neural learning method. It is an application
of gradient descent algorithm originally for multilayer perceptron network. On research of
neuro-fuzzy systems, the gradient descent algorithm is used by several authors and it is
discussed widely in neural network literature. Usually, the initial fuzzy sets and rules are
first given by user. After that, the fuzzy rules are updated by a gradient descent algorithm.
The slow convergence speed near the minima is the biggest drawback of the
backpropagation [19,1].
2) Hybrid method consisting of backpropagation for the parameters associated with the input
MFs and least squares estimation for the parameters associated with the output MFs. In this
approach, both fuzzy and neural networks techniques are used independently, becoming, in
this sense, a hybrid system. Each one does its own job in serving different functions in the
system, incorporating and complementing each other in order to achieve a common goal
[4,21]. The idea of a hybrid model is the interpretation of the fuzzy rule-base in terms of a
neural network. In this way the fuzzy sets can be interpreted as weights, and the rules,
input variables, and output variables can be represented as neurons. The learning algorithm
results, like in neural networks, in a change of the architecture, i.e. in an adaption of the
weights, and/or in creating or deleting connections. These changes can be interpreted both
in terms of a neural net and in terms of a fuzzy controller [22].
The hybrid learning algorithm of ANFIS in Matlab can be explained as follows: each epoch is
composed from a forward pass and a backward pass [9,5]. In particular, the learning process consists of
a forward pass and back-propagation, where in the forward pass, functional signals go forward, and the
consequent parameters are identified by the least-square estimate. In the backward pass, the error rates
propagate backwards and the premise parameters are updated by the gradient descent shown through
the Fig.(6)[20,10].

Although adaptive networks cover a number of different approaches, for our purposes, we will
conduct a detailed investigation of the method proposed by with the architecture [19,13]. The network
can be regarded both as an adaptive fuzzy inference system with the capability of learning fuzzy rules
from data, and as a connection architecture provided with linguistic meaning shown in Figure (7).
Figure 6: Learning algorithm forward and backward
passes.
Figure 7: ANFIS architecture.
ANFIS Optimal consequence
Adjust optimally the premise paramise
parameters (a,b,c)
parameters are found (p, q, r)
Forward pass
Backward pass
The circular nodes have a fixed input-output relation; whereas the square nodes have
parameters to be learnt. Typical fuzzy rules are defined as a conditional statement in the form [5]:
If (x is A1) then (y is B1) (9)
2 2 If (x is A ) then (y is B ) (10)
where X and Y are linguistic variables; Ai and Bi are linguistic values determined by fuzzy sets on the
particular universes of discourse X and Y respectively. However, in ANFIS we use the first order
Takagi-Sugeno system which is:
1 1 1 1 1 1 If (x is A ) and (y is B ) then f = p x + q y + r ) (11)
2 2 2 2 2 2 If (x is A ) and (y is B ) then f = p x + q y + r ) (12)
where A1, A2 and B1, B2 are the MFs for inputs x and y, respectively, p1, q1, r1 and p2, q2, r2 are the
parameters of the output function. The functioning of the ANFIS is described as:
Layer 1: Every node in this layer produces membership grades of an input parameter. The node
output O1,i is explained by [21,1]:
O = μ ( x ) for i = 1,2
(13a)
1, i A i or
1, -2 ( ) 3,4 i Bi O = μ y for i = (13b)
where x (or y) is the input to the node i; Ai (or Bi–2) is a linguistic fuzzy set associated with this node.
O1,i is the MFs grade of a fuzzy set and it specifies the degree to which the given input x (or y) satisfies
the quantifier. MFs can be any functions that are Triangular, Gaussian, Bell shaped or Trapezoidal
shaped function [17].
Layer 2: Every node in this layer is a fixed node, whose output is the product of all incoming
signals:
2, ( ) ( ) , 1,2
i i i i A B O =w =μ x μ y i = (14)
Layer 3: The ith node of this layer, calculates the normalized firing strength as,
3,
, 1,2 i
= = =
1 2
i i
w
O w i
w w
+
(15)

Layer 4: Every node i in this layer is an adaptive node with a node function,
( ) O4,i = wi fi = wi pi x + qi y + ri (16)
where 1 w is the output of layer 3 and {pi, qi, ri} is the parameter set of this node.
Layer 5: The single node in this layer is a fixed node, labeled , which computes the overall
output as the summation of all incoming signals [19,22]:
w f
(17)
5, i i i
i i i i
i i
Overall output O w f
w

= = =

This is how the input vector is typically fed through the network layer by layer. We then
consider how the ANFIS learns the premise and consequent parameters for the MFs and the rules.
4. Neuro-Fuzzy Systems
Hybrid systems combining fuzzy system, neural networks, genetic algorithms, and expert systems are
proving their effectiveness in a wide variety of real-world problems. Every intelligent technique has
particular computational properties that make them suited for particular problems and not for others
[21,2]. Fuzzy systems, which can reason with imprecise information, are good at explaining their
decisions but they cannot automatically acquire the rules they use to make those decisions [17,16]. These
limitations have been a central driving force behind the creation of intelligent hybrid systems where
two or more techniques are combined in a manner that overcomes the limitations of individual
techniques. There are three types of neuro-fuzzy systems, first: neural fuzzy systems (see Fig (8a)),
second: fuzzy neural networks (see Fig (8b)) and third: fuzzy-neural hybrid systems [14,17,19]. Hybrid
systems are very important when considering the varied nature of application domains. Many complex
domains have many different problems, each of which may require different types of processing. If
there is a complex application which has two distinct sub problems, say a signal processing task and a
serial reasoning task, then a neural network and an expert system respectively can be used for solving
these separate tasks [16,14]. The use of intelligent hybrid systems is growing rapidly with successful
applications in many areas including process control, engineering design, credit evaluation, medical
diagnosis, and cognitive simulation. The main advantage of neural systems is their ability to learn from
numerical data. However, the knowledge of them is distributed into the whole network as synaptic
weights [9,11].
Figure 8: Neural fuzzy system and Fuzzy neural network.

Fuzzy system contains if-then rules, which are linguistic interpretable and easily incorporate a
prior knowledge from a human expert. Neuro-fuzzy modeling refers to the way of applying various
learning techniques developed in the neural network literature to fuzzy modeling or to FIS [6]. The
basic structure of a FIS consists of three conceptual components [14, 2]: a rulebase, which contains a
selection of fuzzy rules; a database which defines the MFs used in the fuzzy rules; and a reasoning
mechanism, which performs the inference procedure upon the rules to derive an output. To utilize both
advantages within a single system, various architectures called neuro-fuzzy systems or fuzzy neural
networks have been proposed as a hybrid of fuzzy systems and neural networks [19, 17, 7].
5. Application Example: Neuro-Fuzzy System of Washing Machines
Washing machines are a common feature today in the all household. The most important utility, a
customer can derive from a washing machine is that he saves the effort he/she had to put in brushing,
agitating and washing the cloths. Most of the people wouldn’t have noticed (but can reason out very
well) that different of all one from amount of washing time, temperature of water and amount of
washing powder, claim to the different degrees of cleanness of clothes which depends directly on the
dirt in clothes, amount of dirt, cloth quality etc.. [14]. The washing machines that are used today (the
one not using fuzzy system) serves all the purpose of washing, but which cleanness of cloths needs
what amount of agitations (washing time, temperature of water, amount of washing powder) is a
business which has not been dealt with properly[3]. In most of the cases either the user is compelled to
give all the cloths same agitation or is provided with a restricted amount of control [14]. The thing is that
the washing machines used are not as automatic as they should be and can be. This work aims at
presenting the idea of controlling the cleanness of clothes using fuzzy system, where it describes the
procedure that can be used to get a suitable cleanness of clothes for different washing time, temperature
of water and amount of washing powder [5]. The process is based entirely on the principle of taking no
precise inputs from the sensors, subjecting them to fuzzy arithmetic and obtaining a crisp value of the
cleanness. It is quite clear from this work itself that this method can be used in practice to further
automate the washing machines. Never the less, this method, though with much larger number of input
parameters and further complex situations, is being used by the giants. When one uses a washing
machine, the person generally select the length of wash time based on the amount and dirt of clothes
he/she wish to wash and degree of dirt cloths have. To automate this process, we use sensors to detect
these inputs (i.e. washing time, temperature of water, amount of washing powder). The cleanness is
then determined from this data. Unfortunately, there is no easy way to formulate a precise
mathematical relationship between time, temperature powder with the degree of cleanness required.
Consequently, this problem has remained unsolved until very recently. Conventionally, people simply
set cleanness by hand and from personal trial and error experience. The real data we used had
practically reached a group of experts in one of the giant companies to 50 cases. Washing machines
were not as automatic as they could be [3,5]. The sensor system provides external input signals into the
machine from which decisions can be made. It is the controller's responsibility to make the decisions
and to signal the outside world by some form of output. Because the input/output relationship is not
clear, the design of a washing machine controller has not in the past lent itself to traditional methods of
control design [14]. We address this design problem using fuzzy system, fuzzy system has been used
because it controlled washing machine controller gives the correct cleanness even though a precise
model of the input/output relationship is not available. The problem in this work has been simplified by
using three variables for inputs and one output variable (Cleanness of clothes) depends upon three
inputs variable(see Table(1)).

Figure 9: Fuzzy Inference System of Washing Machine using Mamdani method.
We will now load the inputs and output variables used for this demo into the workspace, where
the real data ready got and these data to 50 case (see Table(1)). Three variables are loaded in the
workspace of inputs and one variable is loaded in the workspace of output, datin has 3 columns to
representing the 3 input variables and datout has 1 column representing the 1 output variable. The
number of rows in datin datout, 50, represent the number of observations or samples or datapoints
available. A row in datin constitutes a set of observed values of the 3 input variables (washing time,
temperature of water and amount of washing powder) and the corresponding row in datout represents
the observed value for the degree of cleanness of clothes generated given the observations made for the
input variables. The three inputs are:
1. Input 1(Washing Time)
Range of time from 0 to 15 and the unit is minute, but these data were treated and measured so that
becomes trapped between 0 and 1(see Fig. (10)). Washing Time represented two linguistic variables as:
Less Time (Ltime) and More Time (Mtime), (see Fig. (11)).
Figure 10: Represent real data of washing time. Figure 11: Figure (11): Represent input1
(Washing Time) using (FIS).

2. Input 2 (Temperature of Water)
Range of temperature 0 to 50 and the unit is ° C, but these data were treated and measured so that
becomes trapped between 0 and 1(see Fig. (12)). Temperature of water have five different degree of
linguistic variables as: Very Cold (Vcold), Cold (cold), Good (good), Hot (hot) and Very Hot (Vhot).
This variable defined below showing the membership function of powder (see Fig (13)).
Figure 12: Represent real data of Temperature of
water.
Figure 13: Represent input2 (Temperature of water).
3. Input 3 (Amount of washing powder)
Range of washing powder from 0 to 100 and the unit is gram , in this case also were measured data , so
that become trapped between 0 and 1(see Fig (14)). Washing powder also represented five linguistic
variables as: Very Less (Vless), less (less), middle (mid), more (more) and Very More (Vmore), (see
Fig (15)).
Figure 14: Represent real data of washing powder. Figure 15: Represent input3 (Washing powder).
5.1. (Part 1): Generating the Fuzzy Inference System (FIS)
We will model the relationship between the input variables and the output variable by Fuzzy Inference
System (FIS) in this part which can then be used to explore and understand cleanness patterns. It can
be used to take fuzzy or imprecise observations for inputs and yet arrive at crisp and precise values for
outputs. Also, the FIS is a simple and commonsensical way to build systems without using complex
analytical equations. The FIS will then act as a model that will reflect the relationship between inputs
and output. ‘genfis2’ is the function that creates and constructs the FIS. A FIS is composed of inputs,
outputs, rules and each input/output can have any number of MFs. The rules dictate the behavior of the

fuzzy system based on inputs, outputs and MF. ‘genfis2’ constructs the FIS in an attempt to capture the
position and influence of the inputs space. ‘myfis’ is the FIS that ‘genfis2’ has generated. Since the
dataset has 3 input variables and 1 output variable, ‘genfis2’ constructs a FIS with 3 inputs and 1
output. Each inputs and output has as many MFs. As seen previously, for the current dataset 2 sets of
Time and 5 sets for temperature of water amount of washing powder. After made relation between
the inputs and output for all possibility cases became the number of rules and therefore total 51 rules
are created.
We can now probe the FIS to understand how the sets got converted internally into MFs and
rules. ’fuzzy’ also is the function that launches the graphical editor for building fuzzy systems. As can
be seen, the FIS has 3 inputs and 1 output with the inputs mapped to the outputs through a rule base
(white box in the fig.(9)).
Output (Cleanness of Clothes)
Range of cleanness of clothes from 0 to 1 and the unit is percent (see Fig (16)). Cleanness of clothes
represents five linguistic variables as: Not Clean (Nclean), Less Average (Laverage), Average
(Average), More Average (Maverage) and Full Clean (Fclean) (see Fig (17)).
Figure 16: Represent real data of washing powder. Figure 17: Represent input3 (Washing powder).
In FIS programme, we have determined:
Name '(FIS) Washing machine' for system.
Type 'mamdani'
Number of Inputs 3
Number Outputs 1
Number Rules 51
And Method 'min'
Or Method 'max'
Implementation Method 'min'
Aggregation Method 'max'
Defuzzification Method 'centroid'
Now, let's explore how the fuzzy rules are constructed. ‘ruleedit’ is the graphical fuzzy rule
editor. As we can notice, there are exactly 51 rules. Each rule attempts to map a set in the inputs space
to a set in the output space, where first rule can be explained simply as follows Fig (18). The number
‘(1)’ at the end of the rules is to indicate that the rule has standard weight or an importance of 1,
where weights can take any value between 0 and 1. The output of the rules (Cleanness of clothes) is
then used to generate the output of the FIS through the output MFs. The one output of the FIS, number
of cleanness, has 5 Non linear MFs representing the 5 linguistic variables identified by subsets. We
have used the FIS for data exploration, where we could use the FIS that has been constructed to
understand the underlying dynamics of relationship being modeled. ‘surfview’ is the surface viewer
that helps view the input/output surface of the fuzzy system. In other words, this tool simulates the
response of the fuzzy system for the entire range of inputs that the system is configured to work for.

Thereafter, the output or the response of the FIS to the inputs is plotted against the inputs as a surface.
This visualization is very helpful to understand how the system is going to behave for the entire range
of values in the inputs space.
Figure 18: Represent rules editor. Figure 19: Rule view to inputs with output.
In the plots below the first surface viewer shows the output surface for two inputs Time
Powder (see Fig. (20a)), second surface viewer shows the output surface for two inputs Time
Temperature (see Fig. (20b)) and third surface viewer shows the output surface for two inputs
Temperature Powder (see Fig. (20b)). As we can see the degree of output increases with increase in
Time, Temperature and Amount of powder.
Figure (19a): Surface viewer. Figure (19b): Surface viewer. Figure (19c): Surface viewer.
Rule view is the graphical simulator for simulating the FIS response for specific values of the
input variables (see Fig. (19)). This system gives a snapshot of the entire fuzzy inference process, right
from how the MFs are being satisfied in every rule to how the final output is being generated through
defuzzification. In this generated to FIS we got the result for all data by use rule view, where all result
finding in Table (1), from this table we have seen the error = |Rv –FISv| (where Rv (Real value) and
FISv (FIS value)) is very less, where (min error=0.0045) (max error=0.07).
5.2. (Part 2): Generating the Adaptive Neuro-Fuzzy Inference System (ANFIS)
The basic structure of the type of FIS that we've seen thus far is a model that maps input characteristics
to input MFs, input MF to rules, rules to a set of output characteristics, output characteristics to output
MFs, and the output MF to a single valued output or a decision associated with the output, where we
have applied this system in part1. In this part we discuss the use of the ANFIS; these tools apply Neuro-fuzzy
inference techniques to data modeling. Neuro-adaptive learning techniques provide a method for
the fuzzy modeling procedure to learn information about a data set. Then, Fuzzy Logic Toolbox
computes the MF parameters that best allow the associated FIS to track the given input/output data.

The Fuzzy Logic Toolbox function that accomplishes this MF parameter adjustment is called anfis.
The anfis function can be accessed either from the command line or through the ANFIS Editor. Using a
given input/output data set, the toolbox function anfis constructs ANFIS whose MF parameters are
adjusted using either a backpropagation algorithm alone or in combination with a least squares type of
method. This adjustment allows our fuzzy systems to learn from the data they are modeling. In general,
this type of modeling works well if the training data presented to anfis for training (estimating) MF
parameters is fully representative of the features of the data that the trained FIS is intended to model.
Now, let's load training and checking data sets, where the checking data set is corrupted by
noise, into the ANFIS Editor from the workspace. In MATLAB command line to load these data sets
from the directory fuzzydemos into the MATLAB workspace. We could open the ANFIS Editor by
typing ‘anfisedit’ in the MATLAB command line. The training data, ‘trnData’, is a required argument
to anfis, as well as to the ANFIS. Each row of ‘trnData’ is a desired input/output pair of the target
system, where we want to model each row starts with input vectors and is followed by an output value.
Therefore, the number of rows of ‘trnData’ is equal to the number of training data pairs, and, because
there is only one output, the number of columns of ‘trnData’ is equal to the number of inputs plus one.
The training data appears in the plot as a set of circles blow ‘o’, but the checking data appears as pluses
superimposed ‘+’ on the training data (see Fig (21)). The checking data, ‘chkData’, is used for testing
the generalization capability of the ANFIS at each epoch. The checking data has the same format as
that of the training data, and its elements are generally distinct from those of the training data. The
checking data is important for learning tasks for which the inputs number is large, and/or operation the
data itself is noisy. ANFIS needs to track a given input/output data set well. Because the model
structure used for anfis is fixed, there is a tendency for the model to over fit the data on which is it
trained, especially for a large number of training epochs. If over fitting does occur, the ANFIS may not
respond well to other independent data sets, especially if they are corrupted by noise. This data set is
used to train a fuzzy system by adjusting the MF parameters that best model this data. The next step is
to specify an initial FIS for anfis to train. After that it would generate FIS. We used Gaussian MF
‘gaussmf’ to represent inputs FIS and we choice Sugeno-type systems of output MF because anfis only
operates on these type systems (see fig. (22)). we can implement FIS generation from the command
line using the command ‘genfis1’ (for grid partitioning). The output MFs must either be all constant or
all linear (as this work). To load an existing FIS for ANFIS initialization, in the Generate FIS portion,
select load from workspace or load from file. When we build the program FIS we should determined:
Figure 21: Rule view to inputs with output. Figure 22: The structure to Generate FIS of W.
M. using ST model.
Name'(ANFIS)Washing Machine'
Type 'sugeno' Number Inputs 3
Number Outputs 1 Number Rules 50
And Method 'prod'
Or Method 'probor'
Implication Method 'prod'
Aggregation Method 'sum'
Defuzzification Method 'wtaver'

After we generated the FIS, we can view the model structure, as follows Fig. (23). the branches
in this graph are color coded to indicate whether and, not, or are used in the rules. If clicking on the
nodes indicates information about the structure. To ANFIS training the two anfis parameter
optimization method options available for ANFIS training are ‘hybrid’ (the default, mixed least squares
and backpropagation) and ‘backpropa’ (backpropagation). Error Tolerance is used to create a training
stopping criterion, which is related to the error size. The training will stop after the training data error
remains within this tolerance.
Figure 23: The ANFIS model structure. Figure 24: The ANFIS editor to Train FIS.
To start the training: Leave the optimization method at hybrid or backpropa, set the number of
training Epochs to (33) and select Train Now. The following window appears on our screen (see fig.
(24)), after we performed this step, the program would find:
Number of nodes: 130
Number of linear parameters: 200
Number of nonlinear parameters: 36
Total number of parameters: 236
Number of training data pairs: 50
Number of checking data pairs: 50
Number of fuzzy rules: 50
Note:
1. At second row there are find 50 rules every rule has output and every output (f50) has 4 parameter (a1, a2, a3, b).
2. Third row there are find 3 inputs first input has 2 Mfs, second and third has 5 Mfs, every Mf has 3 parameters.
3. Fourth row there are find 50 cases.
4. Fifth row there are find 50 cases same training data but with noise.
The checking error decreases up to a certain point in the training. This application shows why
the checking data option of anfis is useful. After we have performed the program steps is got very less
error (0.0026402). The last step in this system Testing Data against the trained ANFIS, to test ANFIS
against the checking data. When we test the checking data against the ANFIS, it looks satisfactory,
where we got less Average testing error for cheking data (0.0090721) and we got less Average testing
error for Training data (2.0654e-006= 0.0053) (see Fig. (25)).
In this part we can getting the result for all data by use rule view special to this system ANFIS,
where all result finding in Table (1), from this table we have seen the Error = |Rv – ANFISv| (where Rv
(Real value) and ANFISv (ANFIS value)) is very less where (min error=0) (max error=0.0005), and
this result is better with compare to error in FIS.

Figure 25: The ANFIS editor to Test FIS.
6. Conclusion
This application has attempted to convey how fuzzy system can be employed as effective techniques
for data modeling and analysis. The ANFIS apply either a backpropagation or a combination of
backpropagation and the least-squares method to estimate membership function parameters. The
training error is the difference between the training data output value and the output of the fuzzy
inference system corresponding to the same training data input value. The ANFIS editor plots the
training error versus epochs curve as the system is trained. The checking error is the difference
between the checking data output value, and the output of the fuzzy inference system corresponding to
the same checking data input value, which is the one associated with that checking data output value.
Table 1: Cleanness Training data
No. Time Temperature powder
Real
Value
Result
(FIS)
Error=
Result
(ANFIS)
Error=
1 0.1493 0.108 0.9273 0.5911 0.582 0.0091 0.591 0.0001
2 0.5995 0.8846 0.5154 0.753 0.725 0.028 0.753 0 Min
3 0.0752 0.7195 0.9848 0.8212 0.84 0.0188 0.821 0.0002
4 0.8426 0.7561 0.2094 0.5995 0.588 0.0115 0.599 0.0005 Max
5 0.7407 0.6704 0.1124 0.4893 0.517 0.0277 0.489 0.0003
6 0.9487 0.532 0.3006 0.5976 0.557 0.0406 0.598 0.0004
7 0.5048 0.3364 0.4008 0.4706 0.442 0.0286 0.471 0.0004
8 0.671 0.682 0.4254 0.6493 0.689 0.0397 0.649 0.0003
9 0.9121 0.1212 0.3149 0.4504 0.444 0.0064 0.45 0.0004
10 0.5518 0.1523 0.265 0.3419 0.377 0.0351 0.342 0.0001
11 0.2892 0.0204 0.0929 0.1325 0.128 0.0045min 0.133 0.0005
12 1 0.7937 0.465 0.7948 0.688 0.0454 0.795 0.0002
13 0.0373 1 0.2984 0.5301 0.588 0.0579 0.53 0.0001
14 0.3343 0.1276 0.3098 0.3025 0.288 0.0145 0.302 0.0005
15 0.997 0.4817 0.1067 0.4845 0.425 0.0595 0.485 0.0005
16 0.374 0.6767 0.0603 0.3691 0.326 0.0431 0.369 0.0001
17 0.3168 0.0299 0.2858 0.25 0.310 0.06 0.25 0
18 0.1239 0.7776 0.2569 0.4502 0.412 0.0382 0.45 0.0002
19 0.6983 0.5752 0.3407 0.5714 0.616 0.0446 0.571 0.0004
20 0.1788 0.4409 0.2867 0.3612 0.328 0.0332 0.361 0.0002
21 0.3403 0.2754 0.5568 0.4936 0.458 0.0356 0.494 0.0004
22 0.9196 0.7767 0.8804 0.9988 0.972 0.0268 0.999 0.0002
23 0.512 0.9258 0.5427 0.7604 0.746 0.0144 0.76 0.0004
24 0.6304 0.7507 0.9146 0.9349 0.975 0.0401 0.935 0.0001
25 0.6075 0.4193 0.4324 0.5437 0.535 0.0087 0.544 0.0003
26 0.7155 0.967 0.8428 0.9935 0.974 0.0195 0.993 0.0005
27 0.0301 0.2632 0.8092 0.5502 0.53 0.0202 0.55 0.0002

Table 1: Cleanness Training data - Continued
28 0.5409 0.5494 0.9279 0.848 0.871 0.023 0.848 0
29 0.359 0.239 0.4138 0.406 0.36 0.046 0.406 0
30 0.8475 0.2759 0.2303 0.4419 0.382 0.0599 0.442 0.0001
31 0.3484 0.4377 0.2294 0.3715 0.329 0.0425 0.371 0.0005
32 0.8676 0.956 0.3755 0.769 0.699 0.07 max 0.769 0
33 0.2521 0.0707 0.5815 0.4121 0.381 0.0311 0.412 0.0001
34 0.5958 0.3085 0.4128 0.4906 0.518 0.0274 0.491 0.0004
35 0.9608 0.6096 0.2168 0.5817 0.587 0.0053 0.582 0.0003
36 0.059 0.2095 0.5256 0.3811 0.439 0.0579 0.381 0.0001
37 0.0853 0.6553 1 0.8094 0.84 0.0306 0.809 0.0004
38 0.0211 0.8227 0.4953 0.5723 0.615 0.0427 0.572 0.0003
39 0.6983 0.517 0.0702 0.4006 0.414 0.0134 0.401 0.0004
40 0.6134 0.6707 0.4159 0.6253 0.686 0.0607 0.625 0.0003
41 0.1168 0.3203 0.0352 0.163 0.129 0.034 0.163 0
42 0.8159 0.2405 0.296 0.4577 0.442 0.0157 0.458 0.0003
43 0.6331 0.6192 0.8102 0.8309 0.85 0.0191 0.831 0.0001
44 0.0719 0.6359 0.4303 0.4829 0.531 0.0481 0.483 0.0001
45 0.1091 0.9451 0.0284 0.3791 0.412 0.0329 0.379 0.0001
46 0.1394 0.8634 0.5167 0.6288 0.64 0.0112 0.629 0.0002
47 0.8083 0.9489 0.3708 0.7488 0.689 0.0598 0.749 0.0002
48 0.0947 0.5134 0.7476 0.6213 0.588 0.0333 0.621 0.0003
49 0.2437 0.2861 0.5305 0.4581 0.438 0.0201 0.458 0.0001
50 0.2496 0.6724 0.8133 0.7537 0.706 0.0477 0.754 0.0003
The ANFIS plots the checking error versus epochs curve as the system is trained. When the
checking data option is used with ANFIS, either via the command line, or using the ANFIS editor, the
checking data is applied to the model at each training epoch. The FIS membership function parameters
computed using the ANFIS editor when both training and checking data are loaded are associated with
the training epoch that has a minimum checking error. The checking data is similar enough to the
training data that the checking data error decreases as the training begins and increases at some point in
the training after the data over fitting occurs.
In fact, the main purpose is to have a comparison between FIS and ANFIS with an application
to washing machine. The output of the rules is used to generate the output of the FIS through the output
MFs. The one output of the FIS, number of cleanness, has 5 non linear MFs representing the 5
linguistic variables identified by subsets, and we could use the FIS that has been constructed to
understand the underlying dynamics of relationship being modeled. The surface viewer is very helpful
to understand how the system is going to behave for the entire range of values in the inputs space.
After generated to FIS we got the result for all data by use rule view and the result is less, where (min
error=0.0045) (max error=0.07), but the result from ANFIS very less where (min error=0) (max
error=0.0005). This result is best with compare to error in FIS. After we have performed the program
ANFIS got very less checking error (0.0026402) then the checking data option of ANFIS is useful, and
when we test data we got less Average testing error for checking data (0.0090721) and we got less
Average testing error for training data (2.0654e-006= 0.0053). This meaning the result of ANFIS is
best compare with FIS.
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