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Khoo, Li-Pheng et al "RClass*: A Prototype Rough-Set and Genetic Algorithms Enhanced Multi-Concept Classification System for Manufacturing Diagnosis" Computational Intelligence in Manufacturing Handbook Edited by Jun Wang et al Boca Raton: CRC Press LLC,2001 19 RClass*: A Prototype Rough-Set and Genetic Algorithms Enhanced Multi-Concept Classification System for Manufacturing Diagnosis Li-Pheng Khoo Nanyang Technological University Lian-Yin Zhai Nanyang Technological University 19.1 19.2 19.3 19.4 19.5 19.6 Introduction Basic Notions A Prototype Multi-Concept Classification System Validation of RClass* Application of RClass* to Manufacturing Diagnosis Conclusions 19.1 Introduction Inductive learning or classification of objects from large-scale empirical data sets is an important research area in artificial intelligence (AI) In recent years, many techniques have been developed to perform inductive learning Among them, the decision tree learning technique is the most popular Using such a technique, Quinlan [1992] has successfully developed the Inductive Dichotomizer (ID3), and its later versions C4.5 and C5.0 (See 5.0) in 1986, 1992, and 1997, respectively Essentially, decision support is based on human knowledge about a specific part of a real or abstract world If the knowledge is gained by experience, decision rules can possibly be induced from the empirical training data obtained In reality, due to various reasons, empirical data often has the property of granularity and may be incomplete, imprecise, or even conflicting For example, in diagnosing a manufacturing system, the opinions of two engineers can be different, or even contradictory Some earlier inductive learning systems such as the once prevailing decision tree learning system, the ID3, are unable to deal with imprecise and inconsistent information present in empirical training data [Khoo et al., 1999] Thus, the ability to handle imprecise and inconsistent information has become one of the most important requirements for a classification system ©2001 CRC Press LLC Many theories, techniques, and algorithms have been developed to deal with the analysis of imprecise or inconsistent data in recent years The most successful ones are fuzzy set theory and Dempster–Shafer theory of evidence On the other hand, rough set theory, which was introduced by Pawlak [1982] in the early 1980s, is a new mathematical tool that can be employed to handle uncertainty and vagueness Basically, rough set handles inconsistent information using two approximations, namely the upper and lower approximations Such a technique is different from fuzzy set theory or Dempster–Shafer theory of evidence Furthermore, rough set theory focuses on the discovery of patterns in inconsistent data sets obtained from information sources [Slowinski and Stefanowski, 1989; Pawlak, 1996] and can be used as the basis to perform formal reasoning under uncertainty, machine learning, and rule discovery [Ziarko, 1994; Pawlak, 1984; Yao et al., 1997] Compared to other approaches in handling uncertainty, rough set theory has its unique advantages [Pawlak, 1996, 1997] It does not require any preliminary or additional information about the empirical training data such as probability distribution in statistics; the basic probability assignment in the Dempster–Shafer theory of evidence; or grades of membership in fuzzy set theory [Pawlak et al., 1995] Besides, rough set theory is more justified in situations where the set of empirical or experimental data is too small to employ standard statistical method [Pawlak, 1991] In less than two decades, rough set theory has rapidly established itself in many real-life applications such as medical diagnosis [Slowinski, 1992], control algorithm acquisition and process control [Mrozek, 1992], and structural engineering [Arciszewski and Ziarko, 1990] However, most literature related to inductive learning or classification using rough set theory is limited to a binary concept, such as yes or no in decision making or positive or negative in classification of objects Genetic algorithms (GAs) are stochastic and evolutionary search techniques based on the principles of biological evolution, natural selection, and genetic recombination GAs have received much attention from researchers working on optimization and machine learning [Goldberg, 1989] Basically, GA-based learning techniques take advantage of the unique search engine of GAs to perform machine learning or to glean probable decision rules from its search space This chapter describes the work that leads to the development of RClass*, a prototype multi-concept classification system for manufacturing diagnosis RClass* is based on a hybrid technique that combines the strengths of rough set, genetic algorithms, and Boolean algebra In the following sections, the basic notions of rough set theory and GAs are presented Details of RClass*, its validation, and a case study using the prototype system are also described 19.2 Basic Notions 19.2.1 Rough Set Theory Large amounts of applications of rough set theory have proven its robustness in dealing with uncertainty and vagueness, and many researchers attempted to combine it with other inductive learning techniques to achieve better results Yasdi [1995] combined rough set theory with neural network to deal with learning from imprecise training data Khoo et al [1999] developed RClass*, a prototype system based on rough sets and a decision-tree learning methodology, and the predecessor of RClass*, for inductive learning under noisy environment Approximation space and the lower and upper approximations of a set form two important notions of rough set theory The approximation space of a rough set is the classification of the domain of interest into disjoint categories [Pawlak, 1991] Such a classification refers to the ability to characterize all the classes in a domain The upper and lower approximations represent the classes of indiscernible objects that possess sharp descriptions on concepts but with no sharp boundaries The basic philosophy behind rough set theory is based on equivalence relations or indiscernibility in the classification of objects Rough set theory employs a so-called information table to describe objects The information about the objects are represented in a structure known as an information system, which can be viewed as a table with its rows and columns corresponding to objects and attributes, respectively (Table 19.1) For example, an information system (S) with 4-tuple can be expressed as follows: S = 〈 U, Q, V, ρ 〉 ©2001 CRC Press LLC TABLE 19.1 A Typical Information System Used by Rough Set Theory Objects Attributes Decisions U q1 q2 d x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 1 0 0 0 2 0 1 0 1 0 where U is the universe which contains a finite set of objects, Q is a finite set of attributes, V = U q ∈QVq Vq is a domain of the attribute q, ρ : U × Q→V is the information function such that ρ(x, q) ∈ for every q ∈ Q and x ∈ U and ∃(q, v), where q ∈ Q and v ∈ Vq is called a descriptor in S Table 19.1 shows a typical information system used for rough set analysis with xi s (i = 1, 2, 10) representing objects of the set U to be classified; qi s (i = 1, 2) denoting the condition attributes; and d representing the decision attribute As a result, qi s and d form the set of attributes, Q More specifically, { } Q = {q1 ,q ,d } ; and V = {V q1 ,V q ,V d } = {{0 ,1} ,{0 ,1 , } ,{0 ,1}} U = x , x … x 10 ; A typical information function, ρ(x1,q1), can be expressed as ( ) {} ρ x ,q1 = Any attribute-value pair such as (q1,1) is called a descriptor in S Indiscernibility is one of the most important concepts in rough set theory It is caused by imprecise information about the observed objects The indiscernibility relation (R) is an equivalence relation on the set U and can be defined in the following manner: If x, y ∈ U and P ∈ Q, then x and y are indiscernible by the set of attributes P in S Mathematically, it can be expressed as follows ( ) ( ) ˆ xPy if ρ x ,q = ρ y ,q for ∃q ∈ P For example, using the information system given in Table 19.1, objects x5 and x7 are indiscernible by ˆ the set of attributes P = {q ,q } The relation can be expressed as x Px because the information functions for the two objects are identical and are given by ©2001 CRC Press LLC ( ) { } ) ( ρ x ,q1 ,q = ρ x ,q1 ,q = ,0 Hence, it is not possible to distinguish one from another using attributes set {q1,q2} ˆ The equivalence classes of relation, P , are known as P-elementary sets in S Particularly, when P = Q, these Q-elementary sets are known as the atoms in S In an information system, concepts can be represented by the decision-elementary sets For example, using the information system depicted in Table 19.1, the {q1}-elementary sets, atoms, and concepts can be expressed as follows: {q1}-elementary sets E1 = {x1,x2,x3,x9} for ρ(x,q1) = {1} E1 = {x4,x5,x6,x7,x8,x10} for ρ(x,q1) = {0} Atoms A1 = {x1, x 9} A2 = {x2} A3 = {x3} A4 = {x4, x 10} A5 = {x5} A6 = {x6} A7 = {x7} A8 = {x8} Concepts C1 = {x1,x4,x5,x8,x9,x10} ⇒ Class = (d = 0) C2 = {x2,x3,x6,x7} ⇒ Class = (d = 1) Table 19.1 shows that objects x and x are indiscernible by condition attributes q1 and q2 Furthermore, they possess different decision attributes This implies that there exists a conflict (or inconsistency) between objects x and x Similarly, another conflict also exists between objects x and x Rough set theory offers a means to deal with inconsistency in information systems For a concept (C), the greatest definable set contained in the concept is known as the lower approximation of C (R(C)) It represents the set of objects (Y) on U that can be certainly classified as belonging to concept C by the set of attributes, R, such that ( ) { } R C = U Y ∈U / R :Y ⊆ C where U/R represents the set of all atoms in the approximation space (U, R) On the other hand, the least definable set containing concept C is called the upper approximation of C (R(C)) It represents the set of objects (Y) on U that can be possibly classified as belonging to concept C by the set of attributes R such that ( ) { R C = U Y ∈U / R :Y ∩ C ≠ ∅ } where U/R represents the set of all atoms in the approximation space (U, R) Elements belonging only to the upper approximation compose the boundary region (BNR) or the doubtful area Mathematically, a boundary region can be expressed as ( ) ( ) ( ) BN R C = R C – R C A boundary region contains a set of objects that cannot be certainly classified as belonging to or not belonging to concept C by a set of attributes, R Such a concept, C, is called a rough set In other words, rough sets are sets having non-empty boundary regions ©2001 CRC Press LLC Using the information system shown in Table 19.1 again, based on rough set theory, the upper and lower approximations, concepts C1 for d = and C2 for d = 1, can be easily obtained For example, the lower approximation of concept C1 (d = 0) is given by ( ) { } R C1 = x , x , x , x 10 ; and its upper approximation is denoted as ( ) { } R C1 = x , x , x , x , x , x , x , x 10 Thus, the boundary region of concept C1 is given by ( ) ( ) ( ) { } BN R C1 = R C1 – R C1 = x , x , x , x As for concept C2 (d = 1), the approximations can be similarly obtained as follows ( ) { } R (C2 ) = {x , x3 , x5 , x , x7 , x8 }; and BN R (C2 ) = R (C2 ) – R(C2 ) = {x5 , x , x7 , x8 } R C2 = x , x ; As already mentioned, rough set theory offers a powerful means to deal with inconsistency in an information system The upper and lower approximations make it possible to mathematically describe classes of indiscernible objects that possess sharp descriptions on concepts but with no sharp boundaries For example, universe U (Table 19.1) consists of ten objects and can be described using two concepts, namely “d = 0” and “d = 1.” As already mentioned, two conflicts, namely objects x5 and x7, and objects x6 and x8, exist in the data set These conflicts cause the objects to be indiscernible and constitute doubtful areas, which are denoted by BNR(0) or BNR(1), respectively (Figure 19.1) The lower approximation of concept “0” is given by object set {x1,x4,x9,x10}, which forms the certain training data set of concept “0.” On the other hand, the upper approximation is represented by object set {x1,x4,x5,x6,x7,x8,x9,x10}, which contains the possible training data set of concept “0.” Concept “1” can be similarly interpreted 19.2.2 Genetic Algorithms As already mentioned, GAs are stochastic and evolutionary search techniques based on the principles of biological evolution, natural selection, and genetic recombination They simulate the principle of “survival of the fittest” in a population of potential solutions known as chromosomes Each chromosome represents one possible solution to the problem or a rule in a classification The population evolves over time through a process of competition whereby the fitness of each chromosome is evaluated using a fitness function During each generation, a new population of chromosomes is formed in two steps First, the chromosomes in the current population are selected to reproduce on the basis of their relative fitness Second, the selected chromosomes are recombined using idealized genetic operators, namely crossover and mutation, to form a new set of chromosomes that are to be evaluated as the new solution of the problem GAs are conceptually simple but computationally powerful They are used to solve a wide variety of problems, particularly in the areas of optimization and machine learning [Grefenstette, 1994; Davis, 1991] Figure 19.2 shows the flow of a typical GA program It begins with a population of chromosomes either generated randomly or gleaned from some known domain knowledge Subsequently, it proceeds to evaluate the fitness of all the chromosomes, select good chromosomes for reproduction, and produce ©2001 CRC Press LLC Concept ‘0’ R (0) BN R (0) = BN R (1) R (1) 10 Concept ‘1‘ FIGURE 19.1 Basic notions of rough sets Start Generation of a random population of chromosomes Computation of the fitness of individual chromosome Selection of chromosomes with good fitness Reproduction of next generation of chromosomes/population No Limit on number of generation reached? Yes End FIGURE 19.2 A typical GA program flow ©2001 CRC Press LLC R (0) U R (1) Crossover Before Crossover Chromosome 1 After Crossover Crossover site 1 0 ⇒ 1 New chromosome Crossover Chromosome 0 1 ⇒ 0 1 0 New chromosome Mutation 0 0 Before Mutation 1 0 After Mutation FIGURE 19.3 Genetic operators the next generation of chromosomes More specifically, each chromosome is evaluated according to a given performance criterion or fitness function, and assigned a fitness score Using the fitness value attained by each chromosome, good chromosomes are selected to undergo reproduction Reproduction involves the creation of offspring using two operators namely crossover and mutation (Figure 19.3) By randomly selecting a common crossover site on two parent chromosomes, two new chromosomes are produced During the process of reproduction, mutation may take place For example, the binary value of bit in Figure 19.3 has been changed from to The above process of fitness evaluation, chromosome selection, and reproduction of next generation of chromosomes continues for a predetermined number of generations or until an acceptable performance level is reached 19.3 A Prototype Multi-Concept Classification System 19.3.1 Twin-Concept and Multi-Concept Classification The basic principle of rough set theory is founded on a twin-concept classification [Pawlak, 1982] For example, in the information system shown in Table 19.1, an object belongs either to “0” or “1.” However, binary-concept classification, in reality, has limited application This is because in most situations, objects can be classified into more than two classes For example, in describing the vibration experienced by a rotary machinery such as a turbine in a power plant or a pump in a chemical refinery, it is common to use more than two states such as normal, slight vibration, mild vibration, and abnormal, rather than just normal or abnormal to describe the condition As a result, the twin-concept classification of rough set theory needs to be generalized in order to handle multi-concept problems Based on rough set theory, Grzymala-Busse [1992] developed an inductive learning system called LERS to deal with inconsistency in training data Basically, LERS is able to perform multi-concept classification However, as observed by Grzymala-Busse [1992], LERS becomes impractical when it encounters a large training data set This can possibly be attributed to the complexity of its computational algorithm Furthermore, the rules induced by LERS are relatively complex and difficult to interpret 19.3.2 The Prototype System — RClass* 19.3.2.1 The Approach RClass* adopts a hybrid approach that combines the basic notions of rough set theory, the unique searching engine of GAs, and Boolean algebraic operations to carry out multi-concept classification It possesses the ability of ©2001 CRC Press LLC U B A C ¬A FIGURE 19.4 Partitioning of universe U Handling inconsistent information This is treated by rough set principles Inducing probable decision rules for each concept This is achieved by using a simple but effective GA-based search engine Simplifying the decision rules discovered by the GA-based search engine This is realized using the Boolean algebraic operators to simplify the decision rules induced Multi-concept classification can be realized using the following procedure Treat all the concepts (classes) in a training data set as component sets (sets A, B, C ) of a universe, U (Figure 19.4) Partition the universe, U, into two sets using one of the concepts such as A and ‘not A’ (¬A) This implies that the rough set’s twin-concept classification can be used to treat concept A and its complement, ¬A Apply the twin-concept classification to determine the upper and lower approximations of concept A in accordance to rough set theory Use Steps and repeatedly to classify other concepts on universe U 19.3.2.2 Framework of RClass* The framework of RClass* is shown in Figure 19.5 It comprises four main modules, namely a preprocessor, a rough-set analyzer, a GA-based searching engine, and a rule pruner The raw knowledge or data gleaned from a process or experts is stored and subsequently forwarded to RClass* for classification and rule induction The preprocessor module performs the following tasks: Access input data Identify attributes and their value Perform redundancy check and reorganize the new data set with no superfluous observations for subsequent use Initialize all the necessary parameters for the GA-based search engine, such as the length of chromosome, population size, number of generation, and the probabilities of crossover and mutation The rough set analyzer carries out three subtasks, namely, consistency check, concept forming, and approximation It scans the training data set obtained from the preprocessor module and checks its consistency Once an inconsistency is spotted, it will activate the concept partitioner and the approximation operator to carry out analysis using rough set theory The concept partitioner performs set operations for each concept (class) according to the approach outlined previously The approximation operator employs the lower and upper approximators to calculate the lower and upper approximations, during which the training data set is split into certain training data set and possible training data set Subsequently, these training sets are forwarded to the GA-based search engine for rule extraction ©2001 CRC Press LLC Attributes Identifier Redundancy Analysis Pre-processor System Initializer Module 3: GA-based search engine Consistency Analysis Input Data C L A S S I F I E R Rough-set Analyzer Concept Partitioner Knowledge Extracted Fi Approximator GA Configuration GA-based Search Engine GA Operator Expert System Raw Information Pruning/Simplifying Rule Pruner Rule Evaluation FIGURE 19.5 Framework of RClass* The GA-based search engine, once invoked, performs the bespoke genetic operations such as crossover, mutation, and reproduction to gather certain rules and possible rules from the certain training data set and possible training data set, respectively The rule pruner performs two tasks: pruning (or simplifying) and rule evaluation It examines all the rules, both certain and possible rules, extracted by the GA-based search engine and employs Boolean algebraic operators such as union and intersection, to prune and simplify the rules During the pruning operation, redundant rules are removed, whereas related rules are clustered and generalized during simplification As possible rules are not definitely certain, the quality and reliability of these possible rules must therefore be assessed For every possible rule, RClass* also estimates its reliability using the following index: Reliability index = Observation_Possible_Rule Observation_Possible_Original_Data where Observation_Possible_Rule is the number of observations that are correctly classified by a possible rule, and Observation_Possible_Original_Data is the number of observations with condition attributes covered by the same rule in the original data set This index can be viewed as the probability of classifying an inconsistent training data set correctly For each certain rule extracted from the certain training data set, RClass* uses a so-called completeness index to indicate the number of observations in the original training data set that are related to the certain rule Such an index is defined as follows: ©2001 CRC Press LLC Completeness index = Observation_Certain_Rule Observation_Certain_Original_Data where Observation_Certain_Rule is the number of observations that are correctly classified by a certain rule, and Observation_Certain_Original_Data is the number of observations with condition attributes covered by the same rule in the original training data In other words, the completeness index represents the usefulness or the effectiveness of a certain rule The reliability and completeness indices are included as part of RClass *’s output and are displayed in the parentheses following the rules induced 19.4 Validation of RClass* The training example on the classification of hypothermic post-anesthesia patients used by GryzmalaBusse [1992] for the verification of LERS is adopted here to validate the prototype system RClass* Briefly, the attributes (symptoms) used to describe the condition of patients are body temperature, hemoglobin, blood pressure, and oxygen saturation Attributes body temperature and blood pressure can be represented by three discrete conditions — namely, low, normal, and high Attributes hemoglobin and oxygen saturation can be expressed using linguistic terms such as poor, fair, or good The level of comfort experienced by the patients may be clustered into three different classes or concepts — namely, very low, low, and medium In this example, nine observations are recorded and summarized in Table 19.2 The linguistic description of the condition of patients (symptoms) and the level of comfort experienced by them (decision) need to be transformed into real numbers The transformation is achieved by using the following conversion scheme For attributes/symptoms Low/Poor ⇒ 1; Normal/Fair ⇒ 2; High/Good ⇒ ⇒ 1; Low ⇒ 2; Medium ⇒ For decision/concept Very low The results of the conversion are depicted in Table 19.3 It is clear that the comfort levels experienced by patients and contradicts one another As an inconsistency is detected in this information system, the rough set analyzer proceeds to perform concept forming and carry out approximation Three concepts, namely C1(Comfort = Very low), C2(Comfort = Low), and C3(Comfort = Medium) can be formed The lower and upper approximations of these concepts are then calculated At the same time, the certain and possible training data sets are identified Upon completion, the GA-based search engine is invoked to look for classification rules from the certain and possible training data sets obtained from the rough set analyzer It randomly generates 50 chromosomes to form an initial population of possible solutions (chromosomes) These chromosomes are coded using the scheme shown in Table 19.4 For chromosome representation and genetic operations, RClass* adopts the traditional binary string representation, and its corresponding crossover and mutation operators Using this scheme, each chromosome is expressed as a binary string comprising “0” and “1” genes As a result, a classification rule can be represented by an 8-bit chromosome For instance, the rule “If (Body Temperature = low) and (Hemoglobin = fair) Then (Comfort = low)” can be coded as 01100000 Such a representation is rather effective in performing crossover and mutation operations Other than choosing a good scheme for chromosome representation, it is important to define a reasonable fitness function that rewards the right kind of chromosomes The objective of using GAs here is ©2001 CRC Press LLC TABLE 19.2 Training Data Set for the Validation of RClass* Attributes/Symptoms Patient Body Temperature Hemoglobin Low Low Normal Normal Low Low Normal Normal High Fair Fair Good Good Good Good Fair Poor Good Decision/Concept Blood Pressure Oxygen Saturation Comfort Low Normal Low Low Normal Normal Normal High High Fair Poor Good Good Good Fair Good Good Fair Low Low Low Medium Medium Medium Medium Very low Very low TABLE 19.3 Results of Conversion Attributes/Symptoms Patient Body Temperature Hemoglobin 1 2 1 2 2 3 3 Decision/Concept Blood Pressure Oxygen Saturation Comfort 1 2 3 3 3 2 2 3 3 1 TABLE 19.4 Chromosome Coding Scheme Bit Number Bit 1–2: Attribute value of Body Temperature Bit 3–4: Attribute value of Hemoglobin Bit 5–6: Attribute value of Blood Pressure Bit 7–8: Attribute value of Oxygen Saturation Interpretation 00 = Ignore this attribute 01 = Low 10 = Normal 11 = High 00 = Ignore this attribute 01 = Poor 10 = Fair 11 = Good 00 = Ignore this attribute 01 = Low 10 = Normal 11 = High 00 = Ignore this attribute 01 = Poor 10 = Fair 11 = Good to extract rules that can maximize the probability of classifying objects correctly Thus, the fitness of a chromosome is calculated by testing the rules using existing training data set Mathematically, it is given by  number of examples classified correctly by the rule fitness of chromosome =   number of examples related to the rule   ©2001 CRC Press LLC The above fitness function favors rules that classify examples correctly It satisfies both completeness and consistency criteria A rule is said to be consistent if it covers no negative samples and is complete if it covers all the positive samples [De Jong et al., 1993] Chromosomes with above-average fitness values are selected for reproduction In this case, the probability of crossover and mutation are fixed at 0.85 and 0.01, respectively The rule set induced by the GA-based search engine may contain rules with identical fitness values Some of these rules can be combined to form a more general or concise rule using Boolean operations As previously mentioned, the rule pruner is assigned to detect and solve the redundancy problem For example, two of the rules extracted are found to have the same fitness values Rule 1: If (Temperature = low) and (Hemoglobin = fair) Then (Comfort = low) Rule 2: If (Temperature = low) Then (Comfort = low) It is obvious that Rule ⊂ Rule The rule pruner proceeds to combine the rules and produce the following rule: If (Temperature = low) Then (Comfort = low) The fitness value attained by the rule remains the same The induction rules generated using the information system are depicted in Figure 19.6 As already mentioned, two kinds of rules namely certain and possible rules, are available The value in the parenthesis following each of the rules represents the bespoke completeness or reliability indices All the indices are represented in fraction form, with the numerator and denominator corresponding to the number of correctly classified observations and the number of observations whose condition attributes are covered by the rule, respectively The results show that RClass* is able to support multi-concept classification of objects It has successfully integrated the basic notions of rough set theory with a GA-based search engine and Boolean algebraic operations to yield a new approach for inductive learning under uncertainty RClass* has enhanced the performance of its predecessor, RClass [Khoo et al., 1999], and expands rough set’s twin-concept to multi-concept classification Through this integration, RClass* has combined the strengths of rough set theory and the GA-based search mechanism With the help of Boolean algebraic operations, the rules produced are simple and concise compared to those derived by LERS The ability to induce simple and concise rules has the following advantages: • Easy to understand • Easy to interpret and analyse • Easy to validate and cross-check 19.5 Application of RClass* to Manufacturing Diagnosis The diagnosis of critical equipment in a manufacturing system is an important issue Frequently, it is also a difficult task as vast amount of experience or knowledge about the equipment is needed A computerized system that can assist domain experts in extracting diagnostic knowledge from historical operation records of equipment becomes necessary Figure 19.7 shows a rotary machinery that comprises a motor and a pump Three main types of mechanical faults — namely, machine unbalance, misalignment, and mechanical loosening — are considered here All these mechanical faults will result in abnormal vibration Among them, machine unbalance is the most common fault, contributing nearly 30% of abnormal vibration Misalignment and mechanical loosening are mainly caused by improper installation of the machine Figure 19.8 shows a typical vibration signature (presented in frequency domain) of the equipment The frequency of the vibration signature can be broadly divided into five bands based on the number of ©2001 CRC Press LLC ************************************************************************************* * Rough Set - GA Enhanced Rule Induction under Uncertainty * ************************************************************************************* Last compiled on Dec 11 1998, 15:47:44 Length of chromosome: (bits) Decoding sites: Rules extracted from data file : Rule Certain rules for concept ë1í: 1: IF(Blood Pressure=3) THEN Comfort=1 2: IF(Hemoglobin=1) THEN Comfort=1 3: IF(Temperature=3) THEN Comfort=1 Certain rules for concept ë2í: 1: IF(Temperature=1)&(Hemoglobin=2) THEN Comfort=2 2: IF(Blood Pressure=1)&(Oxygen_Saturation=2) THEN Comfort=2 3: IF(Temperature=1)&(Blood Pressure=1) THEN Comfort=2 4: IF(Hemoglobin=2)&(Blood Pressure=1) THEN Comfort=2 5: IF(Hemoglobin=2)&(Oxygen_Saturation=2) THEN Comfort=2 6: IF(Oxygen_Saturation=1) THEN Comfort=2 Possible rules for concept ë2í: 7: IF(Blood Pressure=1) THEN Comfort=2 8: IF(Hemoglobin=2) THEN Comfort=2 Confidence level (2/2=100%) (1/1=100%) (1/1=100%) (2/2=100%) (1/1=100%) (1/1=100%) (1/1=100%) (1/1=100%) (1/1=100%) (2/3=67%) (2/3=67%) Certain rules for concept ë3í: 1: IF(Blood Pressure=2)&(Oxygen_Saturation=3) THEN Comfort=3 2: IF(Hemoglobin=3)&(Blood Pressure=2) THEN Comfort=3 3: IF(Temperature=1)&(Hemoglobin=3) THEN Comfort=3 4: IF(Temperature=1)&(Oxygen_Saturation=3) THEN Comfort=3 5: IF(Blood Pressure=2)&(Oxygen_Saturation=2) THEN Comfort=3 6: IF(Temperature=2)&(Hemoglobin=2) THEN Comfort=3 7: IF(Hemoglobin=2)&(Oxygen_Saturation=3) THEN Comfort=3 8: IF(Temperature=2)&(Blood Pressure=2) THEN Comfort=3 (2/2=100%) (2/2=100%) (2/2=100%) (1/1=100%) (1/1=100%) (1/1=100%) (1/1=100%) (1/1=100%) Possible rules for concept ë3í: 9: IF(Blood Pressure=2) THEN Comfort=3 10: IF(Hemoglobin=3)&(Oxygen_Saturation=3) THEN Comfort=3 11: IF(Temperature=1)&(Blood Pressure=2) THEN Comfort=3 12: IF(Oxygen_Saturation=3) THEN Comfort=3 13: IF(Hemoglobin=3) THEN Comfort=3 (3/4=75%) (2/3=67%) (2/3=67%) (3/5=60%) (3/5=60%) END - FIGURE 19.6 Rules extracted by RClass* revolution, X, of the equipment, namely, less than 1X (0 ~ 0.9X), about 1X (0.9 ~ 1.1X), 2X (1.9 ~ 2.1X), 3X (2.9 ~ 3.1X), and more than 4X Within each of the bands, the largest amplitude is indicative of a fault symptom at a particular frequency Seven attributes are used to describe the condition of the targeted equipment These attributes are defined as follows A0: the ratio of peak amplitudes in bands less than ‘1X’ and ‘1X’ A1: the ratio of peak amplitudes in band ‘1X’ and its initial record (in good condition) A2: the ratio of peak amplitudes in bands ‘2X’ and ‘1X’ A3: the ratio of peak amplitudes in bands ‘3X’ and ‘1X’ A4: the ratio of peak amplitudes in bands more than ‘4X’ and ‘1X’ A5: mode of vibration denoted by horizontal (H), vertical (V), or axial (A) A6: the overall vibration level ©2001 CRC Press LLC A Joint Vertical Rotary Pump Motor Horizontal ω Axial A Cross Section A - A Sensor Plane FIGURE 19.7 A motor and pump assembly Amplitude =4X Frequency FIGURE 19.8 A typical vibration signature Attributes A0 – A4 are continuous variables On the other hand, attributes A5 and A6 are discrete variables Attribute A6 registers the overall vibration level using three states, namely normal vibration (N), moderately high vibration (M), and extremely high vibration (E) A sample set of data with 72 observations is depicted in Table 19.5 The continuous attributes, A0 ~ A4, are first discretized into different intervals (Table 19.6) using classification codes 1, 2, 3, and Details of the discretization will not be discussed here As for attributes A5 and A6, and the decision attribute, they are transformed into integers using the following conversion scheme Discrete Attributes Attribute A5: A (Axial) ⇒ 1; H (Horizontal) ⇒ 2; V (Vertical) ⇒ Attribute A6: N (Normal) ⇒ 1; M (Moderate) ⇒ 2; E (Extreme) ⇒ Decision Attributes NC (Normal Condition) ⇒ 1; MA (Misalignment) ⇒ 2; ML (Mechanical Loosening) ⇒ 3; UB (Unbalance) ⇒ Using the classification codes and conversion scheme given above, the original data set is transformed into an information system with elements denoted by integers which is the format required by RClass* Fifty-one certain rules and forty-three possible rules are extracted from the sample data set (see Appendix for the rules generated) ©2001 CRC Press LLC TABLE 19.5 Sample Data Observation A0 A1 A2 Attributes A3 A4 A5 A6 Decision 10 11 12 0.02 0.01 0.30 0.05 0.03 0.08 0.02 0.06 0.02 0.04 0.04 0.25 1.60 2.80 1.90 1.20 2.34 1.32 2.50 1.45 2.75 1.35 1.58 1.85 1.45 0.10 0.42 0.10 0.09 1.21 0.15 0.16 0.18 0.20 1.36 0.35 0.17 0.06 0.28 0.09 0.07 0.07 0.04 0.03 0.05 0.05 0.08 0.18 0.10 0.02 0.15 0.08 0.05 0.08 0.09 0.05 0.07 0.03 0.11 0.16 A V V H A H H V V A V H E E E N M N E M E N M E MA UB ML NC UB MA UB NC UB NC MA ML M 64 65 66 67 68 69 70 71 72 0.28 0.02 0.34 0.03 0.05 0.16 0.03 0.05 0.08 1.95 1.55 1.76 2.60 1.22 2.20 1.50 1.30 1.15 0.85 1.35 0.52 0.12 0.14 0.34 1.00 0.15 0.12 0.22 0.10 0.27 0.06 0.06 0.31 0.16 0.04 0.07 0.20 0.11 0.15 0.04 0.02 0.23 0.06 0.04 0.03 V A H H H A H A V E M E E N M M N N ML MA ML UB NC ML MA NC NC Notes: MA denotes misalignment; UB stands for machine unbalanced; NC denotes normal condition; ML stands for mechanical loosening TABLE 19.6 Discretization of Continuous Attributes Classification Code Attribute A0 A1 A2 A3 A4 0

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