Fuzzy, neurális és genetikus mesterséges intelligencia módszerek a (VIVG9115) c. tárgy hallgatói részére Az előadások illusztrációi 2008 tavaszán
Gail GailCocker-Bogusz: Cocker-Bogusz:The TheFuzzy FuzzyFlag Flag
2008
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Fuzzy, neurális és genetikus mesterséges intelligencia módszerek (VIVG9115) TEMATIKA
Fuzzy rendszerek Bevezetés A fuzzy koncepció A fuzzy rendszerek matematikai alapjai Fuzzy halmazok Alapfogalmak: A fuzzy fogalom. A klasszikus halmazok. Fuzzy halmazok. Jelölési módok. Gyakori tagsági függvény típusok. A tagsági függvények. Műveletek halmazokkal: Klasszikus halmazok alapfogalmai. Műveletek klasszikus halmazokkal. Fuzzy halmazok kapcsolatai, műveletei. Fuzzy egyenlőség, fuzzy részhalmazok. A fuzzy halmazok struktúra tulajdonságai. Fuzzy halmazok további műveletei: A t- és s-normák. Parametrizált t- és s-normák. Kompenzátoros paraméter-operátorok. Átlagoló és kompenzátoros operátorok. Fuzzy relációk: Klasszikus relációk. Fuzzy relációk. Fuzzy reláció műveletek. A kiterjesztési elv Fuzzy számok és fuzzy aritmetika Nyelvi-lingvisztikai változók és HA-AKKOR szabályok Numerikus változóktól nyelvi változókig. Nyelvi kordonok: koncentráció, dilatáció. HA-AKKOR szabályok. Fuzzy logika és közelítő következtetés A klasszikus logikától a fuzzy logikáig. A fuzzy logika alapelvei: Az éles logikai következtetés. A fuzzy logikai következtetés. A közelítő következtetés "pontosabb" vizsgálata. A fuzzy szabályozás áttekintése: Fuzzy szabályozók áttekintése. Fuzzy rendszerek Fuzzy szabály-bázis A szabály-bázis struktúrája. A szabály-készlet tulajdonságai. A fuzzy inferencia gép A kompozíció alapú inferencia. Individuális szabályok alapú inferencia. Néhány inferencia gép. Fuzzifikátorok és defuzzifikátorok Fuzzifikátorok. Defuzzifikátorok. A defuzzifikátorok összehasonlítása. Fuzzy rendszerek mint nemlineáris leképzések Fuzzy rendszerek néhány osztályának képletei. Fuzzy rendszerek mint univerzális approximátorok. Fuzzy rendszerek tervezéséről
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Neurális hálózatok Bevezetés és áttekintés Miért használjuk őket. Hogyan működnek. Hátrányok. Néhány alkalmazási terület. Alapvető koncepciók, modellek Neuron modellek A McCullock-Pitts neuron modell. Az általános neuron szimbólum. A perceptron. Neurális hálózat modellek Az előrecsatolt hálózat. A visszacsatolt hálózat. Neurális processzálás Tanulás és adaptáció Tanulás mint approximáció. Felügyelt és nem felügyelt tanulás. Neurális hálózatok tanulási szabályai Az általánosított tanulási szabály. A Hebb-féle tanulási szabály. Az eredeti - Rosenblatt féle- perceptron tanulási szabály. A delta szabály folytonos perceptronra. A korrelációs tanulási szabály. A győztes - mindent - elvisz tanulási szabály. Többréteges előrecsatolt hálózatok Neurális hálózatok összefoglaló áttekintése Egy perceptronos hálózat tanulása A diszkrét perceptron. A folytonos perceptron. A többréteges hálózatok Az egyréteges hálózat. Kétréteges egyszerű hálózat. Az általános kétréteges hálózat. Többréteges előrecsatolt hálózat mint univerzális approximátor. Tanulási tényezők. Radiális bázisfüggvényes hálózatok Lokális és globális osztályozás. Az RBF hálózatok formális modellje. Az RBF hálózatok tanulási módjai. Kohonen önszervező térkép Az önszervező algoritmus általános képe. A súly adaptálás. Tanuló vektorkvantálás.
Genetikus algoritmusok Mi és milyen a genetikus algoritmus. Egyszerű genetikus algoritmus. A hasonló mintázatok (szkémák). A genetikus algoritmus alaptörvénye.
Káosz neurális és fuzzy rendszerkben Bevezetés Neurális hálózatok és káosz Fuzzy rendszerek és káosz A témához kapcsolódó irodalom: 1. Retter Gyula: Fuzzy, neurális genetikus, kaotikus rendszerek (Bevezetés a "lágy számítás" módszereibe) Akadémiai Kiadó, 2006. 2. Borgulya István: Neurális hálók és fuzzy rendszerek. Dialog Campus K., 1998. 3. Horváth Gábor szerk.: Neurális hálózatok és műszaki alkalmazásaik. Műegyetemi Kiadó, 1995. 4. Kóczy T. László, Tikk Domonkos: Fuzzy Rendszerek. Typotex Kft., 2000. 5. Várkonyiné Kóczy Annamária szerk.: Genetikus algoritmusok. Typotex Kft., 2002.
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Applications of Fuzzy Logic Source: Fuzzy Logic Laboratorium, Linz. http://www.flll.uni-linz.ac.at/aboutus/whatisfuzzy/applications01.html First, we shall look at the fitness of Fuzzy Control in general terms. The employment of Fuzzy Control is commendable... for very complex processes, when there is no simple mathematical model for highly nonlinear processes if the processing of (linguistically formulated) expert knowledge is to be performed The employment of Fuzzy Control is no good idea if... conventional control theory yields a satisfying result an easily solvable and adequate mathematical model already exists he problem is not solvable Now let's look at some examples where Fuzzy Control actually has been applied. Here are some examples of how Fuzzy Logic has been applied in reality: Automatic control of dam gates for hydroelectric-powerplants (Tokio Electric Pow.) Simplified control of robots (Hirota, Fuji Electric, Toshiba, Omron) Camera aiming for the telecast of sporting events (Omron) Substitution of an expert for the assessment of stock exchange activities (Yamaichi, Hitachi) Preventing unwanted temperature fluctuations in air-conditioning systems (Mitsubishi, Sharp) Efficient and stable control of car-engines (Nissan) Cruise-control for automobiles (Nissan, Subaru) Improved efficiency and optimized function of industrial control applications (Aptronix, Omron, Meiden, Sha, Micom, Mitsubishi, Nisshin-Denki, Oku-Electronics) Positioning of wafer-steppers in the production of semiconductors (Canon) Optimized planning of bus time-tables (Toshiba, Nippon-System, Keihan-Express) Archiving system for documents (Mitsubishi Elec.) Prediction system for early recognition of earthquakes (Inst. of Seismology Bureau of Metrology, Japan) Medicine technology: cancer diagnosis (Kawasaki Medical School) Combination of Fuzzy Logic and Neural Nets (Matsushita) Recognition of handwritten symbols with pocket computers (Sony) Recognition of handwriting, objects, voice (CSK, Hitachi, Hosai Univ., Ricoh) Recognition of motives in pictures with video cameras (Canon, Minolta) Automatic motor-control for vacuum cleaners with recognition of surface condition and degree of soiling (Matsushita) Back light control for camcorders (Sanyo) Compensation against vibrations in camcorders (Matsushita) Single button control for washing-machines (Matsushita, Hitatchi) Flight aid for helicopters (Sugeno) Simulation for legal proceedings (Meihi Gakuin Univ, Nagoy Univ.) Software-design for industrial processes (Aptronix, Harima, Ishikawajima-OC Engeneering) Controlling of machinery speed and temperature for steel-works (Kawasaki Steel, New-Nippon Steel, NKK) Controlling of subway systems in order to improve driving comfort, precision of halting and power economy (Hitachi) Improved fuel-consumption for automobiles (NOK, Nippon Denki Tools) Improved sensitiveness and efficiency for elevator control (Fujitec, Hitachi, Toshiba) Improved savety for nuklear reactors (Hitachi, Bernard, Nuclear Fuel div.)
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The Helicopter that does what it's told Mastering a remote-controlled aircraft can take more than 70 hours training. But now the Japanese have built a mini-helicopter that anyone can fly - as long as they can talk. The little aircraft is designed to carry microphones,cameras and other equipment into places too hazardous for human pilots to visit. The pilot on the ground steers the helicopter in the air simply by speaking any of 14 commands into a microphone,such as "up", "down", "hover", and "turn". A camera aboard the aircraft relays the view from the nose to a monitor in front of the pilot. But a helmet with a screen inside and with camera controls operated by movements of the pilot's head is being developed. The voice-controlled system uses "fuzzy logic" to help identify the command. Fuzzy logic is a type of computer program in which a microprocessor makes choices and takes action on the basis of probabilities - in this case the likeliest voice command. June 1994 p30
Symbol Technologies Symbol LS 3000 Scanner Series The Symbol LS 3000 Barcode Scanner Series are designed to meet the exacting requirements of rugged outdoor and industrial work environments. The Symbol LS 3603 Barcode Scanner Series features Symbol's patented "fuzzy logic" technology with artificial intelligence for "smart" scanning of all types of difficult-to-read and damaged barcodes. No other scanner is better at reading low-contrast and poorly printed barcodes, including high-density and dot matrix symbols. Forrás: http://www.racoindustries.com/syls3000.htm
AEG LAV86741 - Factsheet Front Loading Washing Machine. Advanced Rinse Technology. Advanced Fuzzy Logic intelligence. VARIOMATIC spin cycles. Microprocessor update facility. Child safety door lock. Aqua Control flood protection and Aqua Lock. Variable spin. 3 time save options. LED showing programme length/wash time remaining. Forrás: http://www.comparestoreprices.co.uk/washing-machines/aeg-lav86741.asp
The Small camera that gets you close to the action The Samsung ECX-1 is for camera users who want the benefits of a decent zoom lens but in compact design. Designed by Porsche, the camera has a 4x zoom lens that ranges from 35mm to a full 140mm for real telephoto work. You don't have to worry about camera shake or underexposed shots: the built in microcomputer works out and adjusts the zoom,shutter speed and flash when the camera is in fuzzy logic mode. Other modes include one for professional - looking portraits,and a step zoom,which automatically shoots the same subject in up to three focal lengths.The dioptre adjuster on the viewfinder sets the camera to your individual eyesight requirement. July 1994 p62
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Three ways of controlling a train. Forrás: Reinfark, M.: Fuzzy Control Systems: Clear Advantages. Siemens Review Vol. 58, 6/91, pp. 28-32.
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Applications of neural networks Neural Networks in Practice (by Christos Stergiou and Dimitrios Siganos) Forrás: http://www.doc.ic.ac.uk/~nd/surprise_96/journal/vol4/cs11/report.html#Neural%20networks%20in%20medicine
Given this description of neural networks and how they work, what real world applications are they suited for? Neural networks have broad applicability to real world business problems. In fact, they have already been successfully applied in many industries. Since neural networks are best at identifying patterns or trends in data, they are well suited for prediction or forecasting needs including: - sales forecasting - industrial process control - customer research - data validation - risk management - target marketing But to give you some more specific examples; ANN are also used in the following specific paradigms: recognition of speakers in communications; diagnosis of hepatitis; recovery of telecommunications from faulty software; interpretation of multimeaning Chinese words; undersea mine detection; texture analysis; threedimensional object recognition; hand-written word recognition; and facial recognition. Neural networks in medicine Artificial Neural Networks (ANN) are currently a 'hot' research area in medicine and it is believed that they will receive extensive application to biomedical systems in the next few years. At the moment, the research is mostly on modelling parts of the human body and recognising diseases from various scans (e.g. cardiograms, CAT scans, ultrasonic scans, etc.). Neural networks are ideal in recognising diseases using scans since there is no need to provide a specific algorithm on how to identify the disease. Neural networks learn by example so the details of how to recognise the disease are not needed. What is needed is a set of examples that are representative of all the variations of the disease. The quantity of examples is not as important as the 'quantity'. The examples need to be selected very carefully if the system is to perform reliably and efficiently. Modelling and Diagnosing the Cardiovascular System Neural Networks are used experimentally to model the human cardiovascular system. Diagnosis can be achieved by building a model of the cardiovascular system of an individual and comparing it with the real time physiological measurements taken from the patient. If this routine is carried out regularly, potential harmful medical conditions can be detected at an early stage and thus make the process of combating the disease much easier. A model of an individual's cardiovascular system must mimic the relationship among physiological variables (i.e., heart rate, systolic and diastolic blood pressures, and breathing rate) at different physical activity levels. If a model is adapted to an individual, then it becomes a model of the physical condition of that individual. The simulator will have to be able to adapt to the features of any individual without the supervision of an expert. This calls for a neural network. Another reason that justifies the use of ANN technology, is the ability of ANNs to provide sensor fusion which is the combining of values from several different sensors. Sensor fusion enables the ANNs to learn complex relationships among the individual sensor values, which would otherwise be lost if the values were individually analysed. In medical modelling and diagnosis, this implies that even though each sensor in a set may be sensitive only to a specific physiological variable, ANNs are capable of detecting complex medical conditions by fusing the data from the individual biomedical sensors.
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Electronic noses ANNs are used experimentally to implement electronic noses. Electronic noses have several potential applications in telemedicine. Telemedicine is the practice of medicine over long distances via a communication link. The electronic nose would identify odours in the remote surgical environment. These identified odours would then be electronically transmitted to another site where an door generation system would recreate them. Because the sense of smell can be an important sense to the surgeon, telesmell would enhance telepresent surgery. Instant Physician An application developed in the mid-1980s called the "instant physician" trained an autoassociative memory neural network to store a large number of medical records, each of which includes information on symptoms, diagnosis, and treatment for a particular case. After training, the net can be presented with input consisting of a set of symptoms; it will then find the full stored pattern that represents the "best" diagnosis and treatment. Neural Networks in business Business is a diverted field with several general areas of specialisation such as accounting or financial analysis. Almost any neural network application would fit into one business area or financial analysis. There is some potential for using neural networks for business purposes, including resource allocation and scheduling. There is also a strong potential for using neural networks for database mining, that is, searching for patterns implicit within the explicitly stored information in databases. Most of the funded work in this area is classified as proprietary. Thus, it is not possible to report on the full extent of the work going on. Most work is applying neural networks, such as the Hopfield-Tank network for optimization and scheduling. Marketing There is a marketing application which has been integrated with a neural network system. The Airline Marketing Tactician (a trademark abbreviated as AMT) is a computer system made of various intelligent technologies including expert systems. A feedforward neural network is integrated with the AMT and was trained using back-propagation to assist the marketing control of airline seat allocations. The adaptive neural approach was amenable to rule expression. Additionaly, the application's environment changed rapidly and constantly, which required a continuously adaptive solution. The system is used to monitor and recommend booking advice for each departure. Such information has a direct impact on the profitability of an airline and can provide a technological advantage for users of the system. [Hutchison & Stephens, 1987] While it is significant that neural networks have been applied to this problem, it is also important to see that this intelligent technology can be integrated with expert systems and other approaches to make a functional system. Neural networks were used to discover the influence of undefined interactions by the various variables. While these interactions were not defined, they were used by the neural system to develop useful conclusions. It is also noteworthy to see that neural networks can influence the bottom line. Credit Evaluation The HNC company, founded by Robert Hecht-Nielsen, has developed several neural network applications. One of them is the Credit Scoring system which increase the profitability of the existing model up to 27%. The HNC neural systems were also applied to mortgage screening. A neural network automated mortgage insurance underwritting system was developed by the Nestor Company. This system was trained with 5048 applications of which 2597 were certified. The data related to property and borrower qualifications. In a conservative mode the system agreed on the underwritters on 97% of the cases. In the liberal model the system agreed 84% of the cases. This is system run on an Apollo DN3000 and used 250K memory while processing a case file in approximately 1 sec.
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Applications of Genetic Algorithms (by Naranker Dulay) Forrás: http://www.doc.ic.ac.uk/~nd/surprise_96/journal/vol4/tcw2/report.html Genetic Algorithms (GAs) are adaptive heuristic search algorithm premised on the evolutionary ideas of natural selection and genetic. The basic concept of GAs is designed to simulate processes in natural system necessary for evolution, specifically those that follow the principles first laid down by Charles Darwin of survival of the fittest. As such they represent an intelligent exploitation of a random search within a defined search space to solve a problem. If the conception of a computer algorithms being based on the evolutionary of organism is surprising, the extensiveness with which this algorithms is applied in so many areas is no less than astounishing. These applications, be they commercial, educational and scientific, are increasingly dependent on this algorithms, the Genetic Algorithms. Its usefulness and gracefulness of solving problems has made it the a more favourite choice among the traditional methods, namely gradient search, random search and others. GAs are very helpful when the developer does not have precise domain expertise, because GAs possess the ability to explore and learn from their domain. GA on optimisation and planning: Travelling Salesman Problem The TSP is interesting not only from a theoretical point of view, many practical applications can be modelled as a travelling salesman problem or as variants of it, for example, pen movement of a plotter, drilling of printed circuit boards (PCB), real-world routing of school buses, airlines, delivery trucks and postal carriers. Researchers have tracked TSPs to study biomolecular pathways, to route a computer networks' parallel processing, to advance cryptography, to determine the order of thousands of exposures needed in X-ray crystallography and to determine routes searching for forest fires (which is a multiple-salesman problem partitioned into single TSPs). Therefore, there is a tremendous need for algorithms. In the last two decades an enormous progress has been made with respect to solving travelling salesman problems to optimality which, of course, is the ultimate goal of every researcher. One of landmarks in the search for optimal solutions is a 3038-city problem. This progress is only party due to the increasing hardware power of computers. Above all, it was made possible by the development of mathematical theory and of efficient algorithms. GA in Business and Their Supportive Role in Decision Making Genetic Algorithms have been used to solve many different types of business problems in functional areas such as finance, marketing, information systems, and production/ operations. Within these functional areas, GAs have performed a variety of applications such as tactical asset allocation, job scheduling, machine-part grouping, and computer network design. Finance Applications Models for tactical asset allocation and international equity strategies have been improved with the use of GAs. They report an 82% improvement in cumulative portfolio value over a passive benchmark model and a 48% improvement over a non-GA model designed to improve over the passive benchmark. Information Systems Applications Distributed computer network topologies are designed by a GA, using three different objective functions to optimise network reliability parameters, namely diameter, average distance, and computer network reliability. The GA has successfully designed networks with 100 order of nodes. GA has also been used to determine file allocation for a distributed system. The objective is to maximise the programs' abilities to reference the file s located on remote nodes. Production/Operation Applications Genetic Algorithm has been used to schedule jobs in a sequence dependent setup environment for a minimal total tardiness. All jobs are scheduled on a single machine; each job has a processing time and a due date. The setup time of each job is dependent upon the job which immediately precedes it. The GA is able to find good, but not necessarily optimal schedules, fairly quickly.
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GA is also used to schedule jobs in non-sequence dependent setup environment. The jobs are scheduled on one machine with the objective of minimising the total generally weighted penalty for earliness or tardiness from the jobs' due dates. However, this does not guarantee that it will generate optimal solutions for all schedules. GA is developed for solving the machine-component grouping problem required for cellular manufacturing systems. GA provides a collection of satisfactory solutions for a two objective environment (minimising cell load variation and minimising volume of inter cell movement), allowing the decision maker to then select the best alternative. Learning Robot behaviour using Genetic Algorithms Robot has become such an prominent tools that it has increasingly taken a more important role in many different industries. As such, it has to operate with great effieciency and accuracy. This may not sound very difficult if the environment in which the robot operates remain unchanged, since the behaviours of the robot could be pre-programmed. However, if the environment is ever-changing, it gets extremely difficult, if not impossible, for programmers to figure out every possible behaviours of the robot. Applying robot in a changing environment is not only inevitable in modern technology, but is also becoming more frequent. This has obviously led to the development of a learning robot. The approach to learning behaviours, which lead the robot to its goal, described here reflects a particular methodology for learning via simulation model. The motivation is that making mistakes on real system can be costly and dangerous. In addition, time constraints may limit the extent of learning in real world. Since learning requirs experimenting with behaviours that might occassionally produce undesriable results if applied to real world. Therefore, as shown in the diagram, the current best behaviour can be place in the real, on-line system, while learning continues in the off-line system. Genetic Algorithms for Object Localisation in a Complex Scene In order to provide machines with the ability to interact in complex, real-world environments, sensory data must be presented to the machine. One such module dealing with sensory input is the visual data processing module, also known as the computer vision module. A central task of this computer vision module is to recognise objects from images of the environment. There are two different parts to computer vision modules, namely segmentation and recognition. Segmentation is the process of finding interested objects while recognition works to see if the located object matches the predefined attributes. Since images cannot be recognised until they have been located and separated from the background, it is of paramount importance that this vision module is able to locate different objects of interest for different systems with great efficiency. Artificial Life Genetic algorithms are currently the most prominent and widely used computational models of evolution in artificial-life systems. This decentralised models provide a basis for understanding many other systems and phenomena in the world. Researches on GAs in alife give illustrative examples in which the genetic algorithm is used to study how learning and evolution interact, and to model ecosystems, immune system, cognitive systems, and social systems.
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Applications of chaos Applications Of Chaos Theory To Real-Life Situations Forrás: http://library.thinkquest.org/3493/noframes/chaos.html Much like physics, chaos theory provides a foundation for the study of all other scientific disciplines. It is acually a tool box of methods for incorporating nonlinear dynamics into the study of science. For many people, the work in chaos represents the reunification of the sciences. In mathematics, the use of strange attractors, fractals, and cellular automata, and other nonlinear, graphical models are used for studying data that was previously thought of as random. Mathematical applications of chaos theory actually began being developed 100 years ago by the French mathematician Henre Poincare. In biology, chaos is used in the identification of new evolutionary processes leading to understanding the genetic algorithim, artificial life simulations, better understanding of learning processes in systems including the brain, and studies of such previously unresearchable areas as consciousness and the mind. This strain can be traced back to the work of Charles Darwin, and is a significant new understanding of evolutionary processes. Darwin's work also appears in direct conflict with Newton's because it changes our understanding of the nature of time, demonstrating that some time is not reversible. In physics, thermodynamics in particular, chaos is applied in the study of turbulence leading to the understanding of self-organizing systems and system states (equilibrium, near equilibrium, the edge of chaos, and chaos). Prigogine explains that the concept of entropy is actually the physicists application of the concept of evolution to physical systems. The greater the entropy of a system, the more highly evolved the system is. Chaos theory is also having a major impact on quantum physics and attempts to reconcile the chaos of quantum physics with the predictability of Newton's universe. The push for such unification cam from Einstein. Chaos theory is causing most quantum physicists to accept what Einstein rejected, that God probably did play dice with the universe. Chaos theory is already affecting the critical aspects of our lives. It greatly impacts all sciences. For example, it is answering previously unsolvable problems in quantum mechanics and cosmology. The understanding of heart arrhthmias and brain function has been revolutionized by chaos research. There have been games and toys developed from chaos research, such as the SimAnt, SimLife, SimCity, etc. series of computer games. Fractal mathematics are critiical to improved information compression and encryption schemese needed for computer networking and telecommunications. Genetic algorithims are being applied to economic research and stock predictions. Engineering applications range from factory scheduling to product design, with pioneering work being done at places such as DuPont and Deere & Co.
Peter Stavroulakis: Chaos Applications in Telecommunications Forrás: http://www.telecommunicationsnetbase.com/ejournals/books/book_summary/summary.asp?id=3000 The concept of transmitting information from one chaotic system to another derives from the observation of the synchronization of two chaotic systems. Having developed two chaotic systems that can be synchronized, scientists can modulate on one phase signal the information to be transmitted, and subtract (demodulate) the information from the corresponding phase signal of the coupled chaotic system. Chaos Applications in Telecommunications demonstrates this technique in various applications of communication systems. This book details methods of transmitting information at much higher levels of security than what is available by current techniques. Following a detailed introduction, the book demonstrates how chaotic signals are generated and transmitted. It then details the design of chaotic transmitters and receivers, and describes chaos-based modulation and demodulation techniques. The text describes how a chaosbased spreading sequence outperforms classical pseudorandom sequences in selective and nonselective channels. It also develops channel equalization techniques designed for chaotic communications systems by applying knowledge of systems dynamics, linear time-invariant representations of chaotic systems, and symbolic dynamics representations of chaotic systems. The final chapter explains a specific application for optical communications.
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Jason Putorti: Chaos and the Logistic Map Forrás:http://pear.math.pitt.edu/mathzilla/Examples/chaos/studentReports/JasonPutorti.html Applications of Chaos Theory I found several several little tidbits around the Internet that show chaos theory applied to the world around us. The most obvious I already knew about was fractals, I have explored fractal systems for quite some time and they are truly remarkable application of mathematics. Worlds within worlds, the closer you look, the more you see, constantly changing and seemingly random patterns emerging before your eyes. Mandelbrot himself found an interesting application that I found amusing: How long is the coastline of Britain? His mathematical colleagues were miffed, to say the least, at such an annoying waste of their time on such insignificant problems. Theof their time on such insignificant problems. They told him to look it up. Of course, Madelbrot had a reason for his peculiar question, quite an interesting reason. Look up the coastline of Britain yourself, in some encyclopedia. Whatever figure you get, it is wrong. Quite simply, the coastline of Britain is infinite. You protest that this is impossible. Well, consider this. Consider looking at Britain on a very large-scale map. Draw the simplest two-dimensional shape possible, a triangle, which circumscribes Britain as closely as possible. The perimeter of this shape approximates the perimeter of Britain. However, this area is of course highly inaccurate. Increasing the amount of vertices of the shape going around the coastline, and the area will become closer. The more vertices there are, the closer the circumscribing line will be able to conform to the dips and the protrusions of Britain's rugged coast. There is one problem, however. Each time the number of vertices increases, the perimeter increases. It must increase, because of the triangle inequality. Moreover, the number of vertices never reaches a maximum. There is no point at which one can say that a shape defines the coastline of Britain. After all, exactly circumscribing the coast of Britain would entail encircling every rock, every tide pool, and every pebble that happens to lie on the edge of Britain. Thus, the coastline of Britain is infiritain is infinite.” –The Chaos Experience, Thinkquest.org Chaos theory has been used to explain nearly every aspect of human life; the famous butterfly effect details how a seemingly miniscule force could affect storms on the other side of the planet. Edward Lorenz showed how the bifurcation effect that we looked at earlier is consistent with attempts to predict the weather in any amount of time into the future. Why is the weather [forecast] right sometimes and off others? We put all the variables into the system and what happens? 1/1000 decimal place in the results dramatically diverged the results. Another interesting fact I came across had to do with stock markets and their relation to the tree-like fractals: While the branches get smaller and smaller, each is similar in structure to the larger branches and the tree as a whole. Similarly, in market price action, as you look at monthly, weekly, daily, and intra day bar charts, the structure has a similar appearance. Just as with natural objects, as you move in closer and closer, you see more and more detail. Another characteristic of chaotic markets is called "sensitive dependence on initial conditions." This is what makes dynamic market systems so difficult to predict. Because we cannot accurately describe the current situation band because errors in the description are hard to f description are hard to find due to the system's overall complexity, accurate predictions become impossible. Even if we could predict tomorrow's stock market change exactly (which we can't), we would still have zero accuracy trying to predict only twenty days ahead. A number of thoughtful traders and experts have suggested that those trading with intra day data such as five-minute bar charts are trading random noise and thus wasting their time. Over time, they are doomed to failure by the costs of trading. At the same time these experts say that longer-term price action is not random. Traders can succeed trading from daily or weekly charts if they follow trends. The question naturally arises how can short-term data be random and longer-term data not be in the same market? If short-term (random) data accumulates to form long-term data, wouldn't that also have to be random? As it turns out, such a paradox can exist.” –The Chaos Experience, Thinkquest.org Note Chaos theory is undoubtedly a hot, if not the hottest topic in modern mathematics. Ever read Jurrasic Park? The possibilities for explanation of natural phenomena are endless including religion. Truly fascinating!
Fuzzy rendszerek
2008
Neurális és fuzzy rendszerek képies bemutatása
A genetikus algoritmus vázlata Idempotencia Involució Kommutativitás Asszociativitás
A∩A=A,A∪A=A
Disztributivitás Elnyelés A komplementum elnyelés A DeMorgan törvények
A=A A∩B=B∩A, A∪B=B∪A (A∪B)∪C=A∪(B∪C) (A∩B)∩C=A∩(B∩C) A∪(B∩C)=(A∪B)∩(A∪C) A∩(B∪C)=(A∩B)∪(A∩C) A∪(A∩B)=A A∩(A∪B)=A A∪( A ∩B)=A∪B A∩( A ∪B)=A∩B A∪B= A∩B A∩B= A∪B
Halmaz műveletek alaptulajdonságai.
10. oldal
Fuzzy rendszerek
2008
1 0.8
A
B
0.6 0.4 0.2
1
1
1
1
0.8
0.8
0.8
0.8
2
4
6
8
A
B
A
B
A
B
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0
0
0
0 0
11. oldal
10
0
A és B fuzzy halmaz
2
4
6
8
10
0
drasztikus összeg
2
4
6
8
10
A
B
0 0
korlátos összeg
2
4
6
8
10
0
2
algebrai összeg
4
6
8
10
maximum
Az A és B fuzzy halmaz néhány s-normája
1 0.8
A
B
1
1
1
1
0.8
0.8
0.8
0.8
A
B
A
B
A
B
0.6
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0.2
0
0
0
0
0
2
4
6
8
10
0
2
A és B fuzzy halmaz
4
6
8
10
minimum
0
2
4
6
8
10
A
B
0 0
algebrai szorzat
2
4
6
8
10
0
korlátos szorzat
2
4
6
8
drasztikus szorzat
Az A és B fuzzy halmaz néhány t-normája
1
1
0.8
0.8
A
B
A
algs
0.6 0.4
B
algs
0.6 0.4 algt
0.2
algt
0.2
0
0 0
2
4
6
8
10
0
2
4
a)
6
8
10
b)
a) A és B fuzzy halmaz, b) a két halmaz algebrai összege és szorzata 1
1
0.8
0.8 korls
A
A
B
0.6
0.6
0.4
0.4
0.2
dras
B
drat
0.2
korlt
0
0 0
2
4
6
a)
8
10
0
2
4
6
8
b)
A és B fuzzy halmaz a) korlátos, b) drasztikus összege és szorzata
10
10
Fuzzy rendszerek
2008
µ(x)
12. oldal
µ(y)
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0.5
1
0
x
0
0.5
1
y
1
1
1
1
0.5
0.5
0.5
0.5
0 1
0 1
0 1
0 1
0.5
0.5 x
0 0
1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 ↑→y 0 x
y
1
0.5
1
1
1
1
1
0.8
0.8
0.8
0.8
1
0.6
0.6
0.6
0.8
1
0.4
0.4
0.6
0.8
1
0.2
0.4
0.6
0.8
1
0.2
0.4
0.6
0.8
1
0.5
x 0 0 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 ↑→y 0 x
0.2 0.4 0.6 0.8 1
1
0.5
y
1
1
1
1
1
0.84
0.88
0.92
0.96
1
0.68
0.76
0.84
0.92
1
0.52
0.64
0.76
0.88
1
0.36
0.52
0.68
0.84
1
0.2
0.4
0.6
0.8
1
x 0 0 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 ↑→y 0 x
0.2 0.4 0.6 0.8 1
max
y
0.5
1
0.5
x
1
1
1
1
1
1
1
1
1
1
0.8
1
1
1
1
0.6
0.8
1
1
1
0.4
0.6
0.8
1
1
0.2
0.4
0.6
0.8
1
1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 ↑→y 0 x
0.2 0.4 0.6 0.8 1
alg s
0 0
y
1
0.5
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0.2
0.4
0.6
0.8
1
0.2 0.4 0.6 0.8 1
korls
dra s
s-norma felületek és vetületeik
µ(x)
µ(y)
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0.5
1
0
x
0
0.5
1
y
1
1
1
1
0.5
0.5
0.5
0.5
0 1
0 1
0 1
0 1
0.5 x 0 1 0 0.8 0 0.6 0 0.4 0 0.2 0 0 ↑→y 0 x
0.5 0 0
y
1
0.5
0.2
0.4
0.6
0.8
1
0
0
0
0
0.8
0
0
0
0
0.6
0
0
0
0
0.4
0
0
0
0
0.2
0
0
0
0
0
0.2 0.4 0.6 0.8 1 dra t
0.5
x 0 0 0 1 0 0.8 0 0.6 0 0.4 0 0.2 0 0 ↑→y 0 x
y
1
0.5
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
0
0
0.2
0.4
0.6
0
0
0
0.2
0.4
0
0
0
0
0.2
0
0
0
0
0
0.2 0.4 0.6 0.8 1
0.5
x 0 0 0 1 0 0.8 0 0.6 0 0.4 0 0.2 0 0 ↑→y 0 x
1
0.5
y
x 0 0
0.2
0.4
0.6
0.8
1
0.16
0.32
0.48
0.64
0.8
0.12
0.24
0.36
0.48
0.6
0.08
0.16
0.24
0.32
0.4
0.04
0.08
0.12
0.16
0.2
0
0
0
0
0
0.2 0.4 0.6 0.8 1
korlt
t-norma felületek és vetületeik
alg t
0 1 0 0.8 0 0.6 0 0.4 0 0.2 0 0 ↑→y 0 x
1
0.5
y 0.2
0.4
0.6
0.8
1
0.2
0.4
0.6
0.8
0.8
0.2
0.4
0.6
0.6
0.6
0.2
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0.2
0
0
0
0
0
0.2 0.4 0.6 0.8 1 min
Fuzzy rendszerek
2008
A fuzzy aggregációs operátorok teljes befutási tartománya.
A Hamacher operátorok tartományai.
Kompenzátoros paraméteres operátorok befutási tartományai.
13. oldal
Fuzzy rendszerek
2008
14. oldal
Az inferencia kompozíciós szabálya
1
1
0.5
0.5
0 40
0 40 30
30 20
20 10
Y
0
0
2
6
4
8
10
B 5
Y
0 0
X
a) fuzzy reláció X és Y alaphalmazon
X
b) X-en definiált "B" halmaz hengeres kiterjesztése
1
1
0.5
0.5
0 40
0 40 30
10
10
30 20
20 10
10 Y
5 0 0
X
"a)" és "b)" minimuma
10
10 Y
5 0 0
X
"c)" vetítése az Y tengelyre
Fuzzy relációk kompozíciójának „szemléletes” bemutatása
Fuzzy rendszerek
2008
15. oldal
Fuzzy rendszerek
2008
16. oldal
Fuzzy módszerek
2008
17. oldal
Fuzzy módszerek
2008
a)
18. oldal
b) a) rács, b) fa particionálás
a)
b) a) rács, b) klaszter alapú, c) "sorbaállítós" particionálás
A Mamdami és a Lukasiewicz inferencia összehasonlítása
c)
Fuzzy módszerek
2008
19. oldal
Fuzzy módszerek
2008
20. oldal
Neurális hálózatok
2008
21. oldal
Alapvető neurális hálózati topológiák
Felügyelt tanulás blokkdiagramja
Nem felügyelt tanulás blokk-vázlata Megerősítő tanulás blokk-diagrammja
Neurális hálózatok
2008
22. oldal
Neurális hálózatok
2008
23. oldal
Neurális hálózatok
2008
24. oldal
Neurális hálózatok
2008
25. oldal
Neurális hálózatok
2008
26. oldal
Neurális hálózatok
2008
A hiba-visszaterjesztéses módszer
27. oldal
Neurális hálózatok
2008
28. oldal
Neurális hálózatok
2008
29. oldal
Neurális hálózatok
2008
30. oldal
Neurális hálózatok
2008
31. oldal
Neurális hálózatok
2008
32. oldal
Kaotikus rendszerek
2008
A logisztikai egyenlet különbözõ paraméter értékeknél
A logisztikai egyenlet viselkedése a p bifurkációs paraméter függvényében
33. oldal
Kaotikus rendszerek
2008
34. oldal
Kaotikus rendszerek
2008
35. oldal
Kaotikus rendszerek
2008
36. oldal
