Fuzzy inference systems, genetic algorithms, and neural networks in data analysis

Mamdani Rule-Based System

Defuzzification

Genetic Algorithms

Genetic Encoding and Chromosomes

Fitness Calculation

Selection

Mutation

Crossover

Stopping Criteria

Artificial Neural Network

Perceptron Neurons

Sigmoid Neurons

Multi-Layers Network

Back Propagation Learning

This chapter discusses in detail the various approaches used to collect analyze and interpret data. It elaborates various important variables in relation to research methodology and approach. The various variables discussed under this section of the research include, fuzzy inference system, fuzzy set, fuzzy logic, fuzzification, Mamdani rule-based system, defuzzification, genetic algorithm, genetic encoding and chromosomes, fitness calculation, selection, mutation, crossover, stopping criteria, artificial neural network, perceptron neurons, sigmoid neurons, multi-layers network and back-propagation learning.

Fuzzy Inference System

Fuzzy inference or reasoning system is a process through which an individual maps out an output from a given input through the deployment of fuzzy logic. The process of fuzzy inference system consists of a number of components including, membership functions, if-then rules and fuzzy logic operators etcetera. (Faculty of International Burch University, 2012)

The process of fuzzy inference system has been applied to a number of fields in a successful manner. These fields include, computer vision, analysis of data, expert systems, classification of data, decision analysis and automatic control etcetera. (Faculty of International Burch University, 2012)

Because the fuzzy inference is multi-disciplinary in nature it is known by a number of names including fuzzy model, fuzzy rule-based system, fuzzy associative memory, fuzzy expert system, fuzzy logic controller, and the simple fuzzy system. (Faculty of International Burch University, 2012)

The figure below demonstrates the architecture of fuzzy inference systems. The main components of this architecture are fuzzifier, inference engine, defuzzifier and fuzzy knowledge base.

(Faculty of International Burch University, 2012)

The basic steps included in fuzzy inference system are listed below;

In the originating step, the input variables are compared with the membership rules for the determination of membership values for each of the linguistic labels. This step is also called fuzzification. (Faculty of International Burch University, 2012)

Combination of membership values on the part of premise for the evaluation of firing strength, also known as the degree of fulfilment, of various rules. (Faculty of International Burch University, 2012)

Generation of qualified consequents, which can be either fuzzy or crisp in nature, for each of the rules on the basis of their firing strength. (Faculty of International Burch University, 2012)

Aggregation of qualified consequents in order to produce an output that is crisp in nature. This step is often called defuzzification. (Faculty of International Burch University, 2012)

Components of Fuzzy Inference System

The four major components of fuzzy inference system are discussed in detail in the following section;

Fuzzy Knowledge Base

The rule base and the data base included in the fuzzy inference system are jointly known as the fuzzy knowledge base. A rule base in a fuzzy inference system consists of a number of if-then rules. A data base, on the other hand, defines and elaborates the membership functions associated with the fuzzy sets which deployed in the fuzzy rules. (Faculty of International Burch University, 2012)

Fuzzifier

The function of the fuzzifier is to convert he crisp input into a linguistic variable. This is done through the utilization of membership functions that are stored in the knowledge base of the fuzzy inference system. (Faculty of International Burch University, 2012)

Fuzzy Inference Engine

The fuzzy inference engine, with the help of a number of if-then rules, converts the fuzzy input into the fuzzy output. (Faculty of International Burch University, 2012)

Defuzzifier

The function of the defuzzifier is to convert the fuzzy input, which is generated by the inference engine, into a crisp output. This is done through the deployment of membership functions that are parallel to the membership functions used by the fuzzifier. The most commonly used defuzzifying methods include Bisector of area (BOA), Largest of maximum (LOM), Centroid of area (COA), Mean of maximum (MOM) and Smallest of maximum (SOM). (Faculty of International Burch University, 2012)

The figure below demonstrates how the defuzzifier and fuzzy inference engine work in a combination to convert a fuzzy input into a crisp output;

(Faculty of International Burch University, 2012)

3.2.1. Fuzzy Sets

According to the classical theory a set can be represented through the enumeration of all of its constituting elements by using (Saini & Singh, 2012);

A = {a1, a2, a3, a4a an}

(Saini & Singh, 2012)

According to the classical theory, the sets that have only two values 0 and 1 or only two values of truth are called crisp sets. The non-crisp sets are known as fuzzy sets. A characteristic function can be evaluated for fuzzy sets. This function can be developed through the generalization of the function of the classical set and is generally termed as the 'membership function.' The membership function of the fuzzy set, in contrast to the classical set theory, can have an arbitrary truth value in the closed and normalized interval. (Saini & Singh, 2012)

Each of the fuzzy set is particularly and wholly defined by a single and specific membership function. Apart from that, the symbols of the membership functions are used to label the fuzzy sets that are associated with these membership functions. (Saini & Singh, 2012)

In other words, each of the fuzzy sets and their accompanying membership functions are denoted by the same capital letter. (Saini & Singh, 2012) The figure below demonstrates the membership functions of a fuzzy set ? And a crisp set C.

(Saini & Singh, 2012)

Fuzzy sets were introduced in the year 1965 by Zadeh. The basic purpose behind the development and formulation of these sets was the mathematical representation of the vagueness and uncertainty that was associated with various variables. These sets also provide the researchers with formalized and sophisticated tools that can be used to deal the imprecision that is an inherent to a number of problems. (Fuzzy Sets and Fuzzy Logic, 2010)

The fuzzy sets and their membership functions can give a better representation of variables and elements that are vague in nature. (Faculty of Calvin College Engineering Department, 2011) For example, if we consider that all the people whose height is equal to or greater than six feet are tall then the ordinary set of these people can be demonstrated in the following manner;

(Faculty of Calvin College Engineering Department, 2011)

This type of a set consists of only two values of truth. It demonstrates that either a person is tall or he is not tall. A sharp edge fuzzy set, on the other hand, gives a more appropriate representation of the tallness of a person. (Faculty of Calvin College Engineering Department, 2011) The fuzzy set of tall people can be represented by a continuous inclining function and can be demonstrated as follows;

(Faculty of Calvin College Engineering Department, 2011)

The binary set, or the ordinary set, tells us that whether a person is tall or not. If a person is 6 feet 1 inch tall and the other is 7 feet 1 inch tall, this type of set characterizes both of these people as tall. The difference that exists between the heights of these people is ignored. In case of fuzzy sets, the degree of tallness of each person is appropriately represented. (Faculty of Calvin College Engineering Department, 2011)

3.2.3. Fuzzification

Fuzzification can be defined as a process in which the crisp quantities are converted into fuzzy quantities. The process of converting imprecise data and variables into fuzzy variables is useful but this process is not necessary when this fuzzy data is to be used in the fuzzy systems. (Faculty of Graduate School of Dalian Maritime University, 2011)

The above mentioned idea is demonstrated in the following figure in which the data has been considered as both a fuzzy reading as well as a crisp reading;

(Faculty of Graduate School of Dalian Maritime University, 2011)

In the above mentioned diagram, figure (a) represents fuzzy sets and crisp readings, whereas, figure (b) represents fuzzy sets along with fuzzy readings. (Faculty of Graduate School of Dalian Maritime University, 2011)

3.2.4. Mamdani Rule-Based System

The Mamdani Rule-based system consists of the following four major steps;

Step 1: Fuzzification

The first step involved with identifying the fuzzy inputs and then determining the degree to which these inputs belong to the appropriate fuzzy sets. (Kusiak & Negnevitsky, 2011)

Step 2: Rule Evaluation

The second step is of rule evaluation. In this steps the fuzzy inputs are taken and then applied to the antecedents of the fuzzy rules. If a fuzzy rule consists of a number of antecedents then either the fuzzy operator 'AND' or the fuzzy operator 'OR' is used to determine a single number that represents result of the evaluation of antecedents of fuzzy rules. (Kusiak & Negnevitsky, 2011)

The value determined is known as the truth value and it is then applied to the membership functions. In order to evaluate the disjunction of the antecedents of the fuzzy rule the fuzzy operator 'OR' is used. The fuzzy operator 'AND', on the other hand, is used to determine the conjunction of fuzzy rule antecedents. (Kusiak & Negnevitsky, 2011)

The typically used classical operation union of fuzzy rule-based system is;

A-B (x) = max [?A (x), ?B (x)]

(Kusiak & Negnevitsky, 2011)

The AND fuzzy operation intersection is;

A-B (x) = min [?A (x), ?B (x)]

(Kusiak & Negnevitsky, 2011)

The common method used to correlate the rule consequent with the truth value of the antecedent of the fuzzy rule is cutting the consequent membership function at a level that aligns with the truth of the antecedent of the fuzzy rules. This method is also known as clipping. (Kusiak & Negnevitsky, 2011)

The major drawback of clipping method is that the top of the membership function is sliced and hence it loses a considerable amount of information. Clipping, however, is preferred over other methods because it involves less complex and fast mathematical operations. In addition to that, the outputs produced by clipping are easy to defuzzify. (Kusiak & Negnevitsky, 2011)

Even though, clipping is a preferred approach but the scaling approach preserves the shape of the fuzzy states in an appropriate manner. In this method the rule consequents are multiplied by the truth of the rule antecedents. (Kusiak & Negnevitsky, 2011)

Step 3: Aggregation of the Rule Outputs

Aggregation is the process in which the outputs of the rules are unified. Under this step the membership functions of all the rule consequents, which were previously clipped or scaled, are and then combined into a single fuzzy set. (Kusiak & Negnevitsky, 2011)

The input in the step of rule evaluation is a list of clipped or scale consequents of membership functions and the output is unique fuzzy set for each of the variables of output. (Kusiak & Negnevitsky, 2011) The figure below is the pictorial representation of this step;

Step 4: Defuzzification

The last step of the Mamdani Rule-based system is defuzzification. Fuzziness enables the researchers to determine appropriate rules for the fuzzy variables but the final output of the Mamdani Rule-based system has to crisp in nature. The input in the process of defuzzification is the aggregate fuzzy sets and the output of this process is a single number. (Kusiak & Negnevitsky, 2011)

3.2.5. Defuzzification

It can be defined as a process of converting fuzzy quantities to crisp quantities. It is contradictory to fuzzification, which is the process of converting precise and crisp values into fuzzy quantities. (Faculty of Graduate School of Dalian Maritime University, 2011)

There various methods for defuzzifying fuzzy quantities some of these methods are mentioned below;

Max membership principle: It is also known as the height method. (Faculty of Graduate School of Dalian Maritime University, 2011) It can be mathematically and graphically represented as follows: (in the given formula z* is the defuzzified value)

(z*) ? For all

(Faculty of Graduate School of Dalian Maritime University, 2011)

Centroid Method: This method is also called center of area or the center of gravity method. (Faculty of Graduate School of Dalian Maritime University, 2011) The centroid method can be represented graphically and mathematically in the following manner;

Z* =

(Faculty of Graduate School of Dalian Maritime University, 2011)

Weighted Average Method: This method is generally restricted to the membership functions that produce symmetrical outputs. (Faculty of Graduate School of Dalian Maritime University, 2011) This method can be represented graphically and mathematically in the following manner;

Z* =

(Faculty of Graduate School of Dalian Maritime University, 2011)

Mean max membership: This method is also regarded as middle of maxima method. The maximum representation of a variable can be plateau rather than a single point. (Faculty of Graduate School of Dalian Maritime University, 2011) This method can be represented graphically and mathematically in the following manner;

Z* =

(Faculty of Graduate School of Dalian Maritime University, 2011)

Center Of Sums: This method is faster than many of the methods of defuzzification. In addition to that, this method is not restricted to the membership functions that have symmetric outputs. Apart from that, this method involves the algebraic sums of individual fuzzy outputs rather than the union of these outputs. (Faculty of Graduate School of Dalian Maritime University, 2011) Graphically and mathematically, this method can be represented as follows;

Z* =

(Faculty of Graduate School of Dalian Maritime University, 2011)

In the above diagram, figure (a) represents the first membership function. The second membership function is represented by figure (b). Figure (c), on the other hand, represents the defuzzification step of the center of sum method. (Faculty of Graduate School of Dalian Maritime University, 2011)

3.3. Genetic Algorithms

The genetic algorithms were first introduced and deployed by John Holland. John Holland's book 'Adaptation in Natural and Artificial Systems' was the basis for the creation of a flourishing field in research and application that spreads beyond the field of genetic algorithms. (Reeves, 2008)

A biology analogy acted as the basic approach for the motivation of the approach of genetic algorithms. The fundamental and unique characteristics of genetic algorithms is their usage and utilization of population strings. (Reeves, 2008)

Genetic algorithm belongs to the set of computational models that are inspired by evolution and development. These algorithms work by encoding the solution to a particular problem. This encoding in done on a simple data structure that is chromosome like in nature. (Matthew, 2011)

This approach then applies a combination of operations on the chromosome like data structures to preserve the critical information carried by these structures. The implementation process of genetic algorithms begins with a population set of chromosomes that are typically random. (Matthew, 2011)

The next step is the evaluation of these chromosomes. The next step is the allocation of reproductive opportunities in such a manner that increases the probability of the reproduction of chromosomes that are proper solutions to the problem under consideration. (Matthew, 2011)

In addition to that, proper allocation of reproductive opportunities also decreases the chances of the reproduction of the chromosomes that are poorer solutions to the problem under consideration. The goodness or appropriateness of a solution is generally defined on the basis of the target population. (Matthew, 2011)

The genetic algorithms are generally regarded as function optimizers in various fields. It shall, however, be kept under consideration that the range of fields and areas on which the approach of genetic algorithms can be implemented is very broad in nature. The vocabulary used in genetic algorithms is quite different, therefore, it is necessary for a researcher to understand this vocabulary before deploying this approach. (Matthew, 2011) The table below elaborates the vocabulary of genetic algorithms;

Genetic Algorithms

Explanation

Chromosome (string, individual)

Solution (coding)

Genes (bits)

Part of solution

Locus

Position of gene

Alleles

Values of gene

Phenotype

Decoded solution

Genotype

Encoded solution

(Matthew, 2011)

Genetic algorithms are regarded as search algorithms that rely on the concepts of natural selection and natural genetics. The basic purpose behind the evolution of genetic algorithms was the stimulation of some of the processes that were observed in the natural evolution. (Matthew, 2011)

The approach of genetic algorithms is different from other approaches because it operates on a population. In addition to that, this approach also encodes the parameters rather than working on the original parameter values. (Matthew, 2011)

Apart from that, the traditional approaches generally use gradient approach, whereas, the approach of genetic algorithm is probabilistic in nature. Because of these features, the genetic algorithms are usually deployed as optimization algorithms for general purposes. These algorithms also enable the concerned people to search irregular space and hence these are also implemented to a wide variety of function optimization, applications in relation to machine learning and estimation of parameters. (Matthew, 2011)

The working principle of a genetic algorithm consists of the following steps;

Determination and identification of an appropriate population of solutions. (Matthew, 2011)

Evaluation of the objective function and the fitness function as well. (Matthew, 2011)

Finally, the application of genetic operators. (Matthew, 2011)

The figure below demonstrates the working principle of genetic algorithms;

(Matthew, 2011)

3.3.1. Genetic Encoding and Chromosomes

In order to solve a specific problem through genetic algorithms the variables under consideration are first coded into some string structures. It shall, however, be taken under consideration that it is not necessary that the variables shall be coded into some string structure. (Matthew, 2011)

The binary coded strings that have 1's and 0's are generally used in genetic algorithms. In the genetic algorithms mapping rules are used in genetic algorithms to identify any point in the search space. (Matthew, 2011) The most commonly used mapping rule in genetic algorithm is;

(Matthew, 2011)

The choice of chromosome and their representation in a genetic algorithm depends on the type of the optimization problem that is to me solved by the algorithm. A large number of problems consists of a number of variables. These variables have real values and are continuous in nature. (Faculty of Electrical Engineering and Computing Department Of Electronics, Microelectronics, Computer and Intelligent Systems, 2004)

Most of the genetic algorithms generally used the binary encoding techniques that have m bits per variable. (Faculty of Electrical Engineering and Computing Department Of Electronics, Microelectronics, Computer and Intelligent Systems, 2004) In other words, if the chromosome is nm in length, then each chromosome having length m will be encoded as;

Bi =

(Faculty of Electrical Engineering and Computing Department Of Electronics, Microelectronics, Computer and Intelligent Systems, 2004)

In order to decode the variables the following expression is used;

(Faculty of Electrical Engineering and Computing Department Of Electronics, Microelectronics, Computer and Intelligent Systems, 2004)

3.3.2. Fitness Calculation

The approach of genetic algorithms mimics the principle of the survival of the fittest. Therefore, as indicated by a number of studies, the process of genetic algorithms is most suitable for solving the maximization problems. These maximization problems are generally converted into a single maximization problem through an appropriate transformation process. (Mitchell, 2009)

Typically a fitness function, F (i), is determined with the help of an objective function. This fitness function, once determined, is then used in the successive steps of the genetic algorithm approach. In biological sense, fitness can be defined as a quality value that represents the reproductive effectiveness and efficiency of the chromosomes. (Mitchell, 2009)

For the purpose of the genetic algorithms are used to allocate the reproductive traits to the individuals of the population and hence they ultimately increase the fitness of a solution. In other word, the candidates with higher fitness value have a higher probability of getting selected for further tests and examination. In some cases, the genetic operators make it mandatory for the fitness functions to be negative, whereas, most of the genetic operators do not have any such requirement. (Mitchell, 2009)

In case of the maximization problems, the fitness function can be considered as the same as the objective functions. However, in case of minimization problems, it is necessary to map the natural objective function that underlies the fitness function in order to generate negative fitness function and to represent the degree of fitness of all the individuals that are associated with the string. (Mitchell, 2009)

One of the commonly used fitness mapping for minimization problem is;

(Mitchell, 2009)

This transformation function directly converts the minimization problem to a maximization problem of an equal level without altering the location of the minimum of the given problem. (Mitchell, 2009)

The other commonly used fitness mapping for minimization function is:

(Mitchell, 2009)

This transformation again does not alter the location of minimum in a solution instead in converts a minimization problem into a maximization problem of an equal level and frequency. In case of a string, the fitness function value associated with a string is termed as string fitness. (Mitchell, 2009)

3.3.3. Selection

The selection procedures can be divided into two broad categories. These categories are discussed below;

Fitness Proportionate Selection

The two major methods included in this category are roulette-wheel selection and the stochastic universal selection. In the roulette-wheel selection each of the variable is given a proportion or slot according to its fitness. In other words, under this selection method the fittest variables have the biggest slot in the solution. (Sastry & Goldberg et al., 2009) The roulette-wheel selection can be implemented through the following steps;

In the beginning the fitness, f (i), of each of the individuals in the population is evaluated. (Sastry & Goldberg et al., 2009)

The probability or slot size of selection of each member of the population is then calculated. (Sastry & Goldberg et al., 2009) This is calculated through the following expression

(Sastry & Goldberg et al., 2009)

The cumulative probability for each individual is then calculated. (Sastry & Goldberg et al., 2009) This is through the following expression

(Sastry & Goldberg et al., 2009)

A uniform random number is then generated. (Sastry & Goldberg et al., 2009)

If the random number r < qi then the first individual is selected. But if this is not the case then the individual qi is selected, such that qi -- 1 < r ? qi. (Sastry & Goldberg et al., 2009)

Step number 4 and 5 are repeated to create n number of candidates in the mating pool. (Sastry & Goldberg et al., 2009)

The figure below illustrates the above mentioned selection method for a population of five members that is n = 5. The total fitness of the population is 95. (Sastry & Goldberg et al., 2009) This fitness is calculated through the following expression;

(Sastry & Goldberg et al., 2009)

The probability of selection of each of the individual along with the corresponding cumulative probabilities are demonstrated in the following table;

Chromosome #

1

2

3

4

5

Fitness, f

28

18

14

9

26

Probability, pi

28/95 = 0.295

0.189

0.147

0.095

0.274

Cumulative Probability, qi

0.295

0.484

0.631

0.726

1.000

(Sastry & Goldberg et al., 2009)

Ordinal Selection

Another method for selection is Ordinal selection. This category consists of various methods such as tournament selection and truncation selection. In tournament selection s number of chromosomes are selected at random. This random selection can either be with replacement or without replacement. (Sastry & Goldberg et al., 2009)

These selected chromosomes are then entered into tournament against each other. The individual who is fittest in the group of k chromosomes wins the tournament and is then selected as the parent. The most commonly and generally used value for s is 2. (Sastry & Goldberg et al., 2009)

3.3.4. Mutation

Mutation can be as a genetic operator that converts or alters the value of one or more genes into chromosomes from its originating stage. This alteration can result in the addition of new genes values in the already existing genes pool. Mutation is an important step in genetic research as it prevents the population from stagnating at typical or local outcome. (NeuroDimension, 2012)

Mutation generally occurs in evolution according to the probability defined by the user. This probability shall be fairly low, 0.01 is regarded as a good point for starting the search. If this probability is set too high than the genetic algorithm search is converted into random search. (NeuroDimension, 2012) There are various types of mutation processes, some of these are discussed below;

Flip Bit: in this method the researcher inverts the value of the gene under consideration. In other words, zero is converted to one and one is converted to zero. (NeuroDimension, 2012)

Boundary: it can be defined as a mutation value that replaces the current value of the genes with its upper or lower bound. This upper or lower bound is selected randomly. (NeuroDimension, 2012)

Non-Uniform: In can be defined as the mutation operator that enhances the probability that the amount of mutation will be closer to zero as there will be an increase in the generation number. (NeuroDimension, 2012)

Uniform: it can be defined as a mutation operator that replaces the gene value with the upper and lower bounds that specified by the user in relation to that genes. These bounds are selected randomly. (NeuroDimension, 2012)

3.3.5. Crossover

Crossover can be defined as a genetic operator that exchanges the information between two parent strings and develop two off springs for the next generation of the population. (Obtiko, 2012) There are various kinds of crossover techniques, some of these are listed below:

Single Point Crossover: in this technique a single cross over point is selected. A binary string that extends from the beginning of the chromosome to the cross over point is selected from one parent, whereas, the other strings are copied from the other parent. (Obtiko, 2012) The pictorial representation of this category is as follows:

(Obtiko, 2012)

Two Point Crossover: in this category two cross over points are selected. A binary string that extends from the beginning of the chromosome to the cross over point is selected from one parent. The point from first crossover point to the second crossover point is selected from the second parent, whereas, the other strings are copied from the first parent. (Obtiko, 2012) The pictorial representation of this category is as follows:

(Obtiko, 2012)

Uniform Crossover: Under this category, the bits are selected from both the parents at a random level. (Obtiko, 2012) The pictorial representation of this category is as follows:

(Obtiko, 2012)

3.3.6. Stopping Criteria

After the assurance of the convergence of a genetic algorithm, the next step is to identify the stopping criteria or stopping time of the algorithm. (Bhandari & Murthy et al., 2012) The time-based stopping criteria is of the following two types;

The first criteria identifies the number of iterations that are to be executed in relation to a genetic algorithm. (Bhandari & Murthy et al., 2012)

The second criteria is based on the time or the length of the genetic algorithm. (Bhandari & Murthy et al., 2012)

The stopping criteria based on the objective function values uses the fitness function values that underlie the objective function values to evaluate the auxiliary values. These auxiliary values are used as a measure of the state of convergences that is associated with genetic algorithm. (Bhandari & Murthy et al., 2012)

3.4. Artificial Neural Network

The artificial neural network is a computational model that is based on the structure, processing methods and the learning ability of a human brain. It consists of a large number of processing elements that are similar to neurons. In addition to that, it also consists of a large number of weighted connections that exist between these neurons. (Faculty of Computer Science and Engineering Department University of Nevada, 2010)

Apart from that in an artificial neural network knowledge is distributed equally over a large number of connections. The knowledge that is distributed over the connections is collected through a learning process. (Faculty of Computer Science and Engineering Department University of Nevada, 2010) The figure below demonstrates an artificial neural network;

(Faculty of Computer Science and Engineering Department University of Nevada, 2010)

3.4.1. Perceptron Neurons

The perceptron is a single unit that resembles neuron. It is used to compute a non-linear function y in relation to its input x. (Storkey, 2012) This can be done through the deployment of the following expression:

(Storkey, 2012)

Here the weights w are used to encode the mappings that are being used by the perceptron neurons. (Storkey, 2012) In case of multiple inputs the following expression is used;

(Storkey, 2012)

3.4.2. Sigmoid Neurons

As a perceptron uses a discontinuous step function hence it is not possible for it to address the needs of the gradient descent. In this case the sigmoid neurons are used, which are also regarded as easy to use derivatives. This computing power of the sigmoid neurons comes from non-linearity of the function used by sigmoid neurons. (Bhattacharyya, 2009) The figure below represents the operations of the sigmoid neurons;

(Bhattacharyya, 2009)

3.4.3. Multi-Layers Network

A multi-layer network can be defined as a network that consists of one layer of inputs, one layer of outputs and one or more hidden layers. A neural network that does not have any hidden layers is known as perceptron. The perceptron network, however, consists of linear functions only. A multi-layered network, on the other hand, can address a wide range of non-linear functions, therefore, it is widely used in practice. (Parker, 2006)

3.4.4. Back Propagation Learning

Back propagation learning can be defined as an algorithm that is used for the purpose of supervised learning under the multi-layered networks. The basic idea behind the back propagation learning is the continuous application of the chain rules in order to evaluate the influence of each of the weights on a defined arbitrary error function. (Chakraborty, 2011) It can expressed mathematically as:

(Chakraborty, 2011)

After the determination of error function the aim of back propagation learning is to reduce error. This aim is achieved through the performance of a simple gradient descent: (Chakraborty, 2011)

(Chakraborty, 2011)

Summary

This research aims at collecting, analyzing, interpreting and representing data through the fuzzy inference systems, genetic algorithms and artificial neurons. All of these methods are based on evolution and are quite different from the conventional search methods.

The major components of fuzzy inference systems are Fuzzy Inference System, fuzzy set, fuzzy logic, fuzzification, Mamdani rule-based system, and defuzzification. The major components of genetic algorithm are genetic encoding and chromosomes, fitness calculation, selection, mutation, crossover, stopping criteria. Whereas, those included in the artificial neural network are perceptron neurons, sigmoid neurons, multi-layers network and back-propagation learning.