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Application 447 cosine approach and PSD approach respectively in the considered noisy environment; q 0 = x0 y ; q 0 is the signal/noise ratio for the hypothesis H 0 ; 2 y is the interference variance. When the level of interference is relatively low, e.g. at q 0 ≥10dB, the criterion of Equation (22.29) coincides with that in Equation (22.19). The dependency between the Fisher criterion Equation (22.29) and the signal/noise ratio q 0 is shown in Figure 22.2. We find from Equations (22.29)–(22.32) and Figure 22.2 that the recognition effectiveness of the proposed approach, as well as the recognition effectiveness of the Hartley approach, depends only on the difference of the signal variances and the signal/noise ratio. It can be seen from Equations (22.29)–(22.32) and Figure 22.2 that the effectiveness of the proposed approach and the Hartley, cosine and PSD approaches decreases with decrements of the signal/noise ratio q 0 (e.g. increments of the noise variance) for arbitrary values of the parameter b; however, the use of the proposed approach in the considered noisy environment provides the same recognition effectiveness gain (see Equations (22.30) and (22.31) as in the case without a noisy environment. 3. Application We apply the generalized approach to the intelligent recognition of object damping and fatigue. We consider the two-class diagnostics of the object damping ratio j for hypothesis H j , using the forced oscillation diagnostic method [2,41]. The method, which consists of exciting tested objects into resonant oscillations and recognition, is based on the Fourier transform of the vibration resonant oscillations. The basis of the method is the fact that damping ratio changes will modify the parameters of the vibration resonant oscillations. The differential equation of motion for the tested object – a single degree of freedom linear oscillator under white Gaussian noise stationary excitation – is described as: ¨x +2 j n ˙x + n x = Atcost (22.33) where x is the object displacement, j = c j 2 √ km ; mc j k are the object mass, damping and stiffness respectively, n = √ k/m n is the circular natural frequency, At = A i t m At and A i t are the normalized and unnormalized random Rayleigh envelopes of the Gaussian excitation, t is the random phase, which is uniformly distributed in the range 0 2. From Equation (22.33), we obtain that the vibration resonant oscillations are stationary Gaussian signals with different variances for hypothesis H j and identical normalized autocovariance functions. Therefore, the diagnostic under consideration is the two-class intelligent recognition of the stationary Gaussian signals with different variances for hypothesis H j and identical normalized autocovariance functions. The recognition information is contained in the short-time Fourier transform of the resonant oscillations at the resonant frequency. We employ the following recognition (diagnostic) features: • the real and imaginary components of the short-time Fourier transform of the resonant oscillations at the resonant frequency, taking into account the covariance between features; • the PSD of the resonant oscillations at the resonant frequency. We undertake computer-aided simulation using Simulink and the Monte-Carlo procedure. The simulation parameters are: 0 = 0095 1 = 01 for hypotheses H 0 and H 1 ; the duration of the steady- state resonant oscillations is T = 0625 s; the circular natural frequency is n = 2 20 rad/s; the sampling frequency is 128 Hz, and the value b = 13 ·10 3 is used for the parameter of the pdf of the Rayleigh envelope. The number of randomly simulated samples is 5000 for every hypothesis. The estimate of the correlation coefficient between features in Equations (22.1) and (22.2) is nonzero: ˆr RI =012; the estimate of the parameter a is 0.29; the estimate of the effectiveness gain is ˆ G PSD =124. 448 Intelligent Recognition Using Equation (22.5), r x = for white noise and N = 125, we obtain that the theoretical correlation coefficient between features in Equations (22.1) and (22.2) is also nonzero: r RI = 0 13, where is the Dirac function, N is the number of periods related to the resonant frequency on the signal duration T. Using Equation (22.21), we find the theoretical effectiveness gain G PSD =131. One can see that the simulation results match the theoretical results. We consider the experimental fatigue crack diagnostics of objects using the forced oscillation method. The nonlinear equations of a cracked object motion under white Gaussian noise stationary excitation can be written as follows [2,41]: ¨x +2 S S ˙x + S x = Atcosx≥ 0 ¨x +2 C C ˙x + C x = Atcosx<0 where S = c 2 k S m C = c 2 k C m S = k S m C = k C m , k S and k C are the stiffnesses at tension and compression, S and C are the damping ratios at tension and compression. At compression, the crack is closed and the object behaves like a continuum; therefore, the stiffness is the same as that of the object without a crack, i.e. k C = k. At tension, the crack is opened and the object is discontinuous; therefore, the stiffness decreases with the quantity k = k C −k S k ∗ = k k , k ∗ is the stiffness ratio. A relative crack size characterizes the stiffness ratio [2,41,42]. The basis for using this method lies in the fact that the level of the object nonlinearity changes with the crack size [2,41,42]. We consider the two-class diagnostics of the object stiffness ratio: k ∗ = k ∗ j for class H j , j = 0 1. This consideration is generic because it is independent of the correlation between the stiffness ratio and relative crack size. We employ the following recognition (diagnostic) features: • the real and imaginary components of the short-time Fourier transform of the higher harmonic of the object resonant oscillations, taking into account the covariance between features; • the PSD of the higher harmonic of the object resonant oscillations. Experimental investigation was undertaken with uncracked and cracked turbine blades from an aircraft engine. The flexural resonant blade vibrations were generated using a shaker. Acoustics radiated from the blades were received using a microphone located near the blades at a distance of 1 m. The duration of the steady state blade resonant oscillations was t 1 = 23 s; the sampling frequency was 43 478 Hz, the leakage parameter was 0.4 and the higher harmonic number was 2. We used for comparison the effectiveness gain A, the ratio of the 95% upper confidence limit P PSD of the total error probability for the PSD-based feature and the proposed features, P NEW . The obtained gain estimate was ˆ A =17. Thus, the use of the proposed generalized approach provides an effectiveness gain in comparison with the PSD approach for the application under consideration. 4. Conclusions 1. Generalization of the feature representation approach [3,4] has been proposed and investigated. The generalized approach consists of simultaneously using two new recognition features – the real and imaginary components of the Fourier transform – taking into account covariance between these features. It was shown that the generalization (i.e. accounting for the covariance between these features) improves the recognition effectiveness. The relative effectiveness criterion increases as the correlation coefficient departs from zero. Conclusions 449 2. The covariance and the correlation coefficient between the proposed features, i.e. short-time Fourier components, were obtained for the first time for arbitrary stationary signals. The obtained generic expressions take into account the significant parameters: the signal normalized autocovariance function, signal variance, signal duration and Fourier transform frequency. 3. Recognition of the Gaussian signals was considered using the generalized approach. A comparison of the recognition effectiveness of the generalized approach and the Hartley, cosine and PSD approaches was carried out. 4. Comparing the generalized approach to the Hartley approach shows that: — the recognition effectiveness of the proposed approach, as well as the recognition effectiveness of the Hartley approach, depends only on the difference of the signal variances and does not depend on the correlation coefficient between the new features and the features’ variances; — the Hartley approach is not an optimal feature representation approach and does not represent even a particular case of the proposed approach; — the use of the proposed approach provides an essential constant effectiveness gain in comparison with the Hartley approach for arbitrary values of the correlation coefficient between these features and signal variances. 5. Comparing the generalized approach to the cosine approach shows that: — recognition effectiveness of the proposed approach, as well as recognition effectiveness of the cosine approach, depends only on the difference of the signal variances and does not depend on the features’ variances; — the cosine approach is not an optimal feature representation approach and does not represent even a particular case of the proposed approach; — the use of the proposed approach provides an essential constant effectiveness gain in comparison with the cosine approach for arbitrary values of the correlation coefficient between features and signal variances. 6. Comparing the generalized approach to the PSD approach shows that: — the PSD approach generally is not an optimal feature representation approach and represents only a particular case of the generalized approach; — the use of the PSD approach is optimal only if simultaneously: the correlation coefficient between Fourier components is equal to zero and the standard deviations of the Fourier components are equal; — the use of the generalized approach provides an effectiveness gain in comparison with the PSD approach for arbitrary values of the correlation coefficient between new features and the difference between feature variances (except for the above-mentioned case); — the effectiveness gain increases as the correlation coefficient departs from zero and as the parameter that characterizes the difference between variances of the features departs from unity. 7. Comparing the generalized approach to the Hartley, cosine and PSD approaches in a noisy environment shows that the recognition effectiveness of the proposed approach and the Hartley, cosine and PSD approaches decreases with decrements of the signal/noise ratio (e.g. increments of the noise variance) for arbitrary values of the signal difference. However, the use of the proposed approach provides the same essential effectiveness gain in comparison with the Hartley, cosine and PSD approaches as in the case without a noisy environment. 8. Application of the generalized approach was considered for vibration diagnostics of object damping and fatigue. The simulation and experimental results agree with the theoretical results. Thus, we recommend considering simultaneous usage of the Fourier components, taking into account covariance between these components, as the most generic feature representation approach. 450 Intelligent Recognition Acknowledgment The authors are very grateful to Mr Petrunin for assistance with experimental validation. References [1] Gelman, L. and Braun, S. “The optimal usage of the Fourier transform for pattern recognition,” Mechanical Systems and Signal Processing, 15(3), pp. 641–645, 2001. [2] Gelman, L. and Petrunin, I. “New generic optimal approach for vibroacoustical diagnostics and prognostics,” Proceedings of the National Symposium on Acoustics, India, 2, pp. 10–21, 2001. [3] Alam, M. and Thompson, B (Eds) Selected Papers on Optical Pattern Recognition Using Joint Transform Correlation, SPIE International Society for Optical Engineering, 1999. [4] Arsenault, H., Szoplik, T. and Macukow, B. Optical Data Processing, Academic Press, 1989. 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[40] Young, T.Y. and Fu, K.S. Handbook of Pattern Recognition and Image Processing, Academic Press, Inc., 1986. [41] Dimarogonas, A. “Vibration of cracked structures: a state of the art review,” Engineering Fracture Mechanics, 55 (5), pp. 831–857, 1996. [42] Gelman, L. and Gorpinich, S. “Non-Linear vibroacoustical free oscillation method for crack detection and evaluation,” Mechanical Systems and Signal Processing, 14(3), pp. 343–351, 2000. 23 Conceptual Data Classification: Application for Knowledge Extraction Ahmed Hasnah Ali Jaoua Jihad Jaam Department of Computer Science, University of Qatar P.O.Box 2713, Doha, Qatar Formal Concept Analysis (FCA) offers a strong background for data classification. FCA is increasingly applied in conceptual clustering, data analysis, information retrieval and knowledge discovery. In this chapter, we present an approximate algorithm for the coverage of a binary context by a minimal number of optimal concepts. We notice that optimal concepts seem to help in discovering the main data features. For that reason, they have been used several times to successfully extract knowledge from data in the context of supervised learning. The proposed algorithm has also been used for several applications, such as for discovering entities from an instance of a relational database, simplification of software architecture or automatic text summarization. Experimentation on several cases proved that optimal concepts exhibit the main invariant data structures. 1. Introduction Is it not normal to expect that human intelligence organizes data in a uniform and universal way? The reason is that our natural and biological thinking structure is mostly invariant. The purpose of such a thinking process is to understand and learn from data how to recognize similar objects or situations, and create new objects in an incremental way. As a matter of fact, by analogy, most ‘intelligent’ information retrieval methods need to realize data classification, and minimization of its representation in memory, in an incremental way. Classification means pattern generation and recognition. Formal Concept Analysis (FCA) offers a simple, original, uniform and mathematically well-defined method for data clustering. A pattern is associated with a formal concept (i.e. a set of objects sharing the maximum number of properties). In this chapter, we minimize information representation by selecting Computer-Aided Intelligent Recognition Techniques and Applications Edited by M. Sarfraz © 2005 John Wiley & Sons, Ltd 454 Conceptual Data Classification only ‘optimal concepts’. We assume that data may be converted to a binary relation as a subset of the product of a set of objects and a set of properties. This hypothesis does not represent a strong constraint, because we can see that most numerical data may be mapped to a binary relation, with some approximation. In this chapter, we defend the idea that coverage of a binary context with a minimal number of conceptual clusters offers a base for optimal pattern generation and recognition [1]. We defend the idea that while we are thinking, we incrementally optimize the context space storage. We have generally different possible concepts for data coverage. Which one is the best? In recent years, we applied the idea of minimal conceptual coverage of a binary relation to supervised learning and it gave us defendable results with respect to other known methods in terms of error rate [2–7]. We also applied it for automatic entity extraction from a database. As a last important application, we used it for software restructuring by minimizing its complexity. Because of the huge amount of data contained in most existing documents and databases, it becomes important to find a priority order for concept selections to enable users to find pertinent information first. For that reason, we also exploited these patterns (i.e. concepts) for text summarization combined with a method for assessing word similarity [8]. In this chapter, we give all the steps of these conceptual methods and illustrate them with significant results. In the second section, we present the mathematical foundation of conceptual analysis. In the third section, we give a polynomial approximate algorithm for the NP-complete problem of binary context coverage with a minimal number of optimal concepts. In the fourth section, we explain how to apply the idea of the optimal concept (also called the optimal rectangle) for knowledge extraction. We give a synthesis discussion about the following applications: supervised learning, entity extraction from an instance of a database, minimization of the complexity of software and discovering the main groups of users communicating through one or different servers. 2. Mathematical Foundations Among the mathematical theories found recently with important applications in computer science, lattice theory has a specific place for data organization, information engineering, data mining and reasoning. It may be considered as the mathematical tool that unifies data and knowledge, also information retrieval and reasoning [9–13]. In this section, we define a binary context, a formal concept and the lattice of concepts associated with the binary context. 2.1 Definition of a Binary Context A binary context (or binary relation) is a subset of the product of two sets O (set of objects) and P (set of properties). Example 1 [10]: O =Leech, Bream, Frog, Dog, Spike-weed, Reed, Bean, Maize and P = a b c d efg hi where O is a set of some living things, and P the set of the following properties: a = needs water; b = lives in water; c = lives on land; d = needs chlorophyll to produce food; e = is two seed leaves; f = one seed leaf; g = Can move around; h = has limbs; i = suckles its offspring. A binary context R may be defined by Table 23.1. Mathematical Foundations 455 Table 23.1 An example of a binary context R. abcdefghi 1 Leech 110000100 2 Bream 110000110 3 Frog 111000110 4Dog 101000111 5 Spike-weed110101000 6 Reed 111101000 7 Bean 101110000 8 Maize 101101000 Let f be a function from the powerset of the set of objects O (i.e. 2 0 ) to the powerset of the set of properties P (i.e. 2 P ), such that: fA = m∀g ∈ A ⇒gm ∈R (23.1) fA is the set of all properties shared by all objects of A (subset of O) with respect to the context R. Let g be a function from 2 P to 2 O , such that: gB =g∀m ∈ B ⇒ g m ∈ R (23.2) gB is the set of objects sharing all the properties B (subset of P) with respect to the binary context R. We also define closureA =gfA =A , and closureB = fgB = B . The meaning of A is that a set of objects A shares the same set of properties fA with other objects A − A, relative to the context R. A is the maximal set of objects sharing the same properties as objects A. In Example 1, if A = Leech, Bream, Frog, Spike-weed then A = Leech, Bream, Frog, Spike-weed, Reed. This means that the shared properties a and b of living things in A, are also shared by a reed, the only element in A −A. The meaning of B is that if an object x of the context R verifies properties B, then x also verifies some number of additional properties B −B.B is the maximal set of properties shared by all objects verifying properties B. In Example 1, if B = a h, then B = a h g. This means that any animal that needs water (a) and has limbs (h), can move around (g). For each subset B, we may create an association rule B →B −B. The number of these rules depends on the binary context R. In [10], we find different algorithms for extracting the minimal set of such association rules. 2.2 Definition of a Formal Concept A formal concept of a binary context is the pair (A,B), such that fA = B and gB = A. We call A the extent and B the intent of the concept (A,B). If A 1 B 1 and A 2 B 2 are two concepts, A 1 B 1 is called a subconcept of A 2 B 2 , provided that A 1 ⊆ A 2 B 2 ⊆ B 1 . In this case, A 2 B 2 is a superconcept of A 1 B 1 and it is written A 1 B 1 <A 2 B 2 . The relation ‘<’ is called the hierarchical order relation of the concepts. The set of all concepts of (G, M, I) ordered in this way is called the concept lattice of the Context (G, M, I). Formal Concept Analysis (FCA) is used for deriving conceptual structures from data. These structures can be graphically represented as conceptual hierarchies, allowing the analysis of complex structures and the discovery of dependencies within the data. Formal concept analysis is based on the philosophical understanding that a concept is constituted by two parts: its extent, which consists of all objects belonging to the concept, and its intent, which 456 Conceptual Data Classification Table 23.2 Formal context K. ABCD G11100 G21100 G30110 G40111 G50011 comprises all attributes shared by those objects. One of the main objectives of this method is to visualize the data in the form of concept lattices. Let K be the formal context presented in Table 23.2. Then Figure 23.1 represents the structured set of concepts. A formal concept has been defined and introduced by different scientific communities in the world, starting in 1948 with Riguet [14] under the name of maximal rectangle. In graph theory, a maximal bipartite graph was exploited in the thesis of Le Than in 1986 [15] in a database to introduce a new kind of dependencies called ‘Iso-dependencies’, independently, Jaoua et al. introduced difunctional dependencies as the most suitable name for iso-dependencies in a database [16]. The most recent mathematical studies about formal concept analysis have been done by Ganter and Wille. More details may be found in [9–11]. What is remarkable is that concepts are increasingly used in several areas in real-life applications: text analysis, machine learning, databases, data mining, software decomposition, reasoning and pattern recognition. A complete conjunctive query and its associated answer in a database is no more nor less than a concept (i.e. an element in a lattice of concepts). Its generality and simplicity is very attractive. We almost may find a bridge between any computer science application and concepts. Combined with other methods for mapping any kind of data into a binary context, it gives an elegant base for data mining. 2.3 Galois Connection A Galois connection is a conceptual learning structure used to extract new knowledge from an existing context (database). The context is represented by a binary relation. We can decompose the context into {} {G1,G2,G3,G4,G5} {B} {G1,G2,G3,G4} {C} {G3,G4,G5} {A,B} {G1,G2} {B,C} {G3,G4} {C,D} {G4,G5} {B,C,D} {G4} {A,B,C,D} {} Figure 23.1 Concept lattice of the context K. [...]... Rome Athens London Athens London Athens Rome Athens 200 700 400 200 200 500 600 400 800 100 200 500 300 300 100 0 100 200 1200 500 400 100 800 200 500 Smith Smith Durand Durand Durand Durand Durand Durand Durand Durand Dupont Dupont Clark Clark Kurt Kurt Kurt Kurt Kurt Kurt Kurt Kurt Kurt Kurt 20 20 10 10 10 10 10 10 10 10 30 30 20 20 30 30 30 30 30 30 30 30 30 30 London London Paris Paris Paris Paris Paris... many kilometers, and for retrieval of the information at the end point On the other hand, the users of this new technology demand effective protection of their information While companies of all sizes increasingly are utilizing the Internet and Web technologies to lower costs and increase efficiencies and revenues, the open architecture of the Internet and Web opens organizations and users up to an... based on biometric technology, such as fingerprint recognition, face recognition, iris recognition, hand shape recognition, signature recognition, voice recognition, etc (see, for example, [2] to get a quick insight about some recent advances on this topic) The core of this new paradigm for security is to use human body characteristics to identify and authenticate users Unlike ID cards, passwords or... Design Techniques and Analysis, Word Scientific, 1999 [9] Davey, B A and Priestley, H A Introduction to Lattices and Order, Cambridge Mathematical Textbooks, 1990 [10] Ganter, B and Wille, R Formal Concept Analysis, Springer Verlag, 1999 [11] Schmidt, G and Ströhlein, S Relations and Graphs, Springer Verlag, 1989 [12] Jaoua, A., Bsaies, K and Consmtini, W “May Reasoning be Reduced to an Information Retrieval... cryptographic schemes, are also discussed Computer-Aided Intelligent Recognition Techniques and Applications © 2005 John Wiley & Sons, Ltd Edited by M Sarfraz 470 Communications With Chaotic Semiconductor Lasers 1 Introduction In the last few years we have witnessed an extraordinary worldwide growth of digital communications Extraordinary advances in both laser and semiconductor technologies have favored... Knowledge and Data Engineering, 7(5), pp 824–829, 1995 [4] Ben Yahia, S., Arour, K., Slimani, A and Jaoua, A “Discovery of Compact Rules in Relational Databases,” Information Journal, 3(4), pp 497–511, 2000 [5] Ben Yahia, S and Jaoua, A “Discovering Knowledge from Fuzzy Concept Lattice,” in Kandel, A Last, M and Bunke, H (Eds), Data Mining and Computational Intelligence, Studies in Fuzziness and Soft... Al-Ghanim, N and Al-Misaifri, S Knowledge Extraction and Reduction System (K.E.R.S.), Senior project, Computer Science Department, University of Qatar, June 2001 [7] Maddouri, M., Elloumi, S and Jaoua, A “An Incremental Learning System for Imprecise and Uncertain Knowledge Discovery,” Information Science Journal, 109 , pp 149–164, 1998 [8] Alsuwaiyel, M H Algorithms, Design Techniques and Analysis,... number Symbol n Lext I q e g s N0 Value 5 2 ns 0 0238 ps−1 200 ps 1 cm 1 5 × 10 9 ps−1 44 mA 1 602 × 10 19 C 2 ns 1 5 × 10 8 5 × 10 7 1 5 × 10 8 Semiconductor Lasers with Optical Feedback 475 dynamical behavior when varying an accessible parameter (for example, the distance between the laser and the mirror, Lext Other laser and mirror parameters cannot be chosen but only determined after construction... 106 Intensity 600 000 2.5 × 106 500 000 2 × 106 1.5 × 106 400 000 1 × 106 300 000 500 000 200 000 100 0 2000 3000 4000 5000 10 (a) 20 30 40 50 60 Frequency (in GHz) time (in ps) (b) Figure 24.3 Chaotic behavior of the diode laser with PCF for orbit during 5 ns; (b) Fourier spectrum = 0 0238 (a) Intensity of the chaotic 2.4 Step 4: Synchronization of the Chaotic Transmitter and Receiver Systems The synchronization... Multilingual Documents and Optimal Rectangle, http://arxiv.org/PS_cache/ cs /pdf/ 9902/9902022 .pdf [25] Natural Language Techniques and Text Mining Applications, http://citeseer.nj.nec.com/rajman97text.html [26] Tutorial: Text Analyst, http://www.megaputer.com/products/ta/tutorial/textanalyst_tutorial_1.html [27] Mosaid, T., Hassan, F., Saleh, H and Abdullah, F Conceptual Text Mining: Application for Text Summarization, . binary context R. abcdefghi 1 Leech 1100 0 0100 2 Bream 1100 00 110 3 Frog 1 1100 0 110 4Dog 101 000111 5 Spike-weed 1101 0100 0 6 Reed 11 1101 000 7 Bean 101 1100 00 8 Maize 101 1 0100 0 Let f be a function from the. 463 Table 23.3 Relation R. P1 P2 P3 P4 P5 P6 Class O 1100 100C1 O20 1101 1C2 O 3100 100C1 O40 1101 1C2 O 5100 100C1 From this relation R between objects O1O5 and properties P1P6, by the algorithm of the. Durand 10 Paris p3 Screw Blue 17 Rome J1 Sorter Paris 400 S2 Durand 10 Paris p3 Screw Blue 17 Rome J2 Punch Rome 200 S2 Durand 10 Paris p3 Screw Blue 17 Rome J3 Reader Athens 200 S2 Durand 10
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