The fuzzy if-then rules



This chapter provides some reviews on Artificial Neural Networks, the Fuzzy IF-THEN rules, fuzzy inference system (FIS) and the Neuro-Fuzzy Inference System in order to develope better understanding of this research. This chapter will focus on the potentials of Fuzzy Inference System (FIS) and the hybrid approach to handle uncertainties in the decision making.

Artificial Neural Network

Artificial neural network, or simply neural network (NNs), have been studied for more than three decades since Rosenblatt first applied single-layer perceptrons to pattern classification learning in the late 1950s (Rosenblatt, 1962). Neural nets have gone through two major development periods -the early 60 s and the mid 80's. They were a key development in the field of machine learning. Artificial Neural Networks were inspired by biological findings relating to the behavior of the brain as a network of units called neurons. The human brain is estimated to have around 10 billion neurons each connected on average to 10,000 other neurons. Each neuron receives signals through synapses that control the effects of the signal on the neuron. These synaptic connections are believed to play a key role in the behavior of the brain. The fundamental building block in an Artificial Neural Network is the mathematical model of a neuron as shown in Figure 2.1. The three basic components of the (artificial) neuron are:

  1. The synapses or connecting links that provide weights, wj , to the input values, xj for j = 1, ...m;
  2. An adder that sums the weighted input values to compute the input to the activation function , where w0 is called the bias is a numerical value associated with the neuron. It is convenient to think of the bias as the weight for an input x0 whose value is always equal to one,so that ;
  3. An activation function g that maps v to g(v) the output value of the neuron.

Fuzzy If-Then Rules

Fuzzy if-then rules or fuzzy conditional statements are expressions of the form IF A THEN B, where A and B are labels of fuzzy sets characterized by appropriate membership functions (Zadeh, 1965). Due to their concise form, fuzzy if-then rules are often employed to capture the imprecise modes of reasoning that play an essential role in the human ability to make decisions in an environment of uncertainty and imprecision (Jang, 1993). An example that describes a simple fact is "If pressure is high, then volume is small" where pressure and volume are linguistic variables, high and small are linguistic values or labels that are characterized by membership functions (Zadeh, 1973).

Another form of fuzzy if-then rule, proposed by Takagi and Sugeno in year 1983, has fuzzy sets involved only in the premise part. By using Takagi and Sugeno's fuzzy if-then rule, we can describe the resistant force on a moving object, "If velocity is high, then force = k * (velocity)2" where high is the premise part is a linguistic label characterized by an appropriate membership function. However, the consequent part is described by a nonfuzzy equation of the input variable, velocity (Jang, 1993).

We able to see that both types of fuzzy if-then rules have been used extensively in both modeling and control. Through the use of linguistic labels and membership function, the fuzzy if-then rule can easily capture the spirit of a "rule of thumb" used by humans. From other perspective, due to the qualifiers on the premise parts, each fuzzy if-then rule can be viewed as a local description of the system under consideration. Thus, the fuzzy if-then rules form a core part of the fuzzy inference system (FIS).

Fuzzy Inference Systems

The Fuzzy inference system is a popular computing framework based on concepts of fuzzy set theory, fuzzy if-then rules, and fuzzy reasoning. It has been successfully applied in fields such as automatic control, data classification, decision analysis, expert systems, and computer vision. Due to its multi-disciplinary nature, the fuzzy inference system is known by a number of names, such as fuzzy rule-based system, fuzzy expert system, fuzzy model (Takagi & Sugeno, 1985), fuzzy associative memory (Kosko, 1991), fuzzy logic controller (Lee, 1990) and simply fuzzy system. Basically, a fuzzy inference system consists of three conceptual components: a rule base, which contains a selection of fuzzy rules, a database or dictionary, which defines the membership functions used in the fuzzy rules, and a reasoning mechanism, which performs the inference procedure upon the rules and a given condition to derive a reasonable output (Jang & Sun, 1995).

A fuzzy inference system is composed of five functional blocks as shown in Figure 2.2.

Figure 2.2 Fuzzy inference systems

  • A rule base containing a number of fuzzy if-then rules;
  • A database which defines the membership functions of the fuzzy sets used in the fuzzy rules;
  • A decision-making unit which performs the inference operations on the rules;
  • A fuzzification interface which transforms the crisp inputs into degrees of match with linguistic value;
  • A defuzzification interface which transform the fuzzy results of the inference into a crisp output.
  • The rule base and the database are jointly referred to as the knowledge base. The steps of fuzzy reasoning performed by fuzzy inference systems are (Jang, 1993):

  • Compare the input variables with the membership functions on the premise part to obtain the membership values of each linguistic label. This is step called fuzzification.
  • Combine the membership values through a specific T-norm operator, usually multiplication or min on the premise part to get firing strength (weight) of each rule.
  • Generate the qualified consequent either fuzzy or crisp of each rule depending on the firing strength.
  • Aggregate the qualified consequents to produce a crisp output. This is final step called defuzzification.
  • Type 1 : Tsukamoto Fuzzy Model
  • The overall output is the weighted average of each rule's crisp output induced by the rule's firing strength (the product or minimum of the degrees of match with the premise part) and output membership functions. The output membership functions used in this scheme must be monotonic function (Tsukamoto, 1979).

  • Type 2 : Mamdani Fuzzy Model
  • The overall fuzzy output is derived by applying max operation to the qualified fuzzy outputs (each of which is equal to the minimum of firing strength and the output membership function of each rule). Various schemes have been proposed to choose the final crisp output based on the overall fuzzy output; some of them are centroid of area, bisector of area, mean of maxima, maximum criterion, etc (Lee, 1990)

  • Type 3 : Sugeno Fuzzy Model

Takagi and Sugeno's fuzzy if-then rules are used. The output of each rule is a linear combination of input variables plus a constant term, and the final output is the weighted average of each rule's output (Jang, 1993).

Adaptive Neuro-Fuzzy Inference System (ANFIS)

ANFIS which stands for Adaptive Network-based Fuzzy Inference System, or semantically equivalently, Adaptive Neuro-Fuzzy Inference System. The ANFIS architecture and its learning algorithm for the Sugeno fuzzy model will be described in this section.

ANFIS Architecture

Related Researches of ANFIS

ANFIS is a fuzzy inference tool implemented in the framework of adaptive network. It combines the comprehensibility of fuzzy rules and the adaptability and self-learning algorithms of neural networks. Moreover, ANFIS with zero-order Sugeno model has been proved to have universal approximation ability under certain circumstance. Because of its good characteristic, ANFIS has been widely applied in many fields, such as system identification, information fusion, fuzzy control, and data processing etc.

Case Study 1: Road Safety Evaluation from Traffic Information Based on ANFIS

In year 2008, Ma, Pan and Wang presented the road safety evaluation from traffic information based on ANFIS will be analyzed. The factors that influence the road traffic safety are complicated and relate to a great deal of aspects such as the basic traffic parameters, various peccancy behaviors, the road state and the weather conditions. The basic traffic parameters include the mean velocity, the mean density and the mean occupation coefficient. The peccancy behaviors means over speed, overweight, acceleration, changing lane and retrogradation etc. The establishment self-contained degree is one of the road state indicators. The weather conditions usually include the plane visibility, rainfall intensity and snowfall intensity on the road etc.

From this case study, ANFIS is used to realize neuro-fuzzy modeling from numerical data of the traffic information. As long as various state parameters of the traffic information are acquired, the road safety evaluating algorithm that consists of sets of fuzzy "if - then" rules will be formed. Level of safety (LOS) is defined to indicate the road safety extent of traffic state. It is a continuous number and should much fit human perception on safety. LOS is related to various traffic parameters that affecting the road safety. The ANFIS toolbox in MATLAB is used as the simulation platform.

As a conclusion, Ma, Pan and Wang had defined a new index named LOS (level of safety) to evaluate traffic safety. It is a continuous variable to express the situation from 0 (absolute safety) to 1 (deadly safety). LOS based on fuzzy logic can be given from a fuzzy inference system by inputting mean density, mean velocity and plane visibility variables of a section of a road. We analyzed the system and showed that the system is of rationality due to its coherence with human perception.

Case Study 2: ANFIS Modelling of a Twin Rotor System

In this case study, an adaptive neuro-fuzzy inference system (ANFIS) network design is deployed and used for modelling a twin rotor multi-input multi-output system (TRMS). This paper had been proposed by Toha, Tokhi and Hussain in year 2008 from University of Sheffield. The system is perceived as a challenging engineering problem due to its high nonlinearity, cross coupling between horizontal and vertical axes and inaccessibility of some of its states and outputs for measurements. Accurate modelling of the system is thus required so as to achieve satisfactory control objectives. It is demonstrated experimentally that ANFIS can be effectively used for modelling the system with highly accurate results. The accuracy of the modelling results is demonstrated through validation tests including training and test validation and correlation tests (Toha et al., 2008).

The ANFIS structure is considered for dynamic modelling of TRMS in vertical plane motion (one degree of freedom). The objective of the identification experiments is to estimate a model of the TRMS in hovering mode without any prior system knowledge pertaining to the exact mathematical model structure. The rig configuration is such that it permits open-loop system identification, unlike a helicopter which is open-loop unstable in hovering mode. The measured signals are position of the beam, namely two position angles, and the angular velocities of the rotors.

In summary, ANFIS architecture has demonstrated a good performance in modelling the TRMS in vertical plane motion. The best feature of ANFIS is that it pre-processes all the data into several membership functions before mapping the data into an adaptive neuro structure. This pre-processing feature allows ANFIS to converge faster and better. The results have been obtained in both time and frequency domains and the ANFIS modelling method has also been validated using input-output mapping and correlation tests. The resulting model will be used for further development of analysis and control strategies for twin rotor MIMO systems.

Case Study 3: A Neuro Fuzzy Inference System for Student Modeling In Web-Based Intelligent Tutoring Systems

Adaptive Neuro Fuzzy Inference System (ANFIS) has established itself as one of the popular modeling techniques in the field of control systems, expert system and complex systems modeling. Ali and Ghatol (2004) presented a fuzzy neural inference system for modeling a student in the context of wed-based intelligent tutoring system. The proposed model has been implemented and tested for the simulated data.

Basically, fuzzy logic techniques are used to provide human-like approximate diagnosis of student knowledge and cognitive abilities. Neural networks are trained to imitate human teachers' decisions regarding students' characteristics and fixed weight networks are used to evaluate and aggregate membership functions. ANFIS for student modeling is to provide input to the subsequent pedagogical module in a web-based ITS. Each student is presented with the course material in concept - example - explanation format. For the test of understanding of each concept the student is continuously evaluated through multiple choice type questions, true/false type questions and fill in the blanks type questions. The expert module creates these tests in such a manner that they can be used to test the memorizing skills, concept understandings as well as the misconceptions.

Numerical values from these individual tests are given as inputs to the neuro fuzzy inference system and used to decide the performance for that concept as poor, fair, good and excellent. Each of the inputs is fuzzified to three levels as low, medium and high. 40 input-output data pairs of simulated students performance were used, where 30 pairs for implementation and 10 pairs for testing. Finally, the testing of the proposed model is done by presenting the test data and observing the response. The responses were then evaluated by the human teachers and are found to match a human teacher decision (Ali & Ghatol, 2004).

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