A survey of fuzzy logic applications and principles in wireless communications is presented, with the aim of highlighting successful usage of fuzzy logic techniques in applied telecommunications and signal processing. To the best of our knowledge, this is the first such study of its kind. This paper will focus firstly on discerning prevalent fuzzy logic or fuzzy-hybrid approaches in the areas of channel estimation, channel equalization and decoding, and secondly outlining what conditions and situations for which fuzzy logic techniques are most suited for these approaches. Furthermore, after insights gained from isolating fuzzy logic techniques applied to real problems, this paper proposes areas for further research targeted to practice-oriented researchers.
The complexity within a Wireless Personal Area Network (WPAN) increases. Several technologies have to share the same radio spectrum. In this paper we take a look at the 2.4 GHz Industrial Scientific and Medical (ISM)-band. This paper discusses a method of selecting the best wireless channel within Wireless Local Area Network (WLAN) when several technologies could be used in the same WPAN range of the needed access point. The issue is to keep away from already occupied channels. The method is divided into four steps: the passive probing of the power level detecting IEEE 802.15.4 (ZigBee) channels using a new and affordable hardware, the transformation of the probed data to a linguistic level using Fuzzy Set Theory (FST), the classification of the data, and finally the sorting and selection of channels based on whose power levels.
This paper presents an Artificial Intelligence approach towards classification of persons based on a verbal description of their facial features. Face features are extracted by use of an existing detection techniques, such as measurements of horizontal and vertical size of the facial elements like nose, eyes, etc. This approach allows to create fuzzy sets representing selected facial features. What is important, linguistic variables corresponding to the fuzzy sets conform the terminology applied by law enforcement to create an eyewitness verbal description. Then, fuzzy IF-THEN rules are employed for classification both the facial composites (sketches) and usual images from face databases. With regard to this concept, satisfactory results have been obtained and presented.
Traditional rule learners employ equality relations between attributes and values to express decision rules. However, inequality relationships, as supplementary relations to equation, can make up a new function for complex knowledge acquisition. We firstly discuss an extended compensatory model of decision table, and examine how it can simultaneously express both equality and inequality relationships of attributes and values. In order to cope with large-scale compensatory decision table, we propose a scalable inequality rule leaner, which initially compresses the input spaces of attribute value pairs. Example and experimental results show that the proposed learner can generate compact rule sets that maintain higher classification accuracies than equality rule learners.
Rule induction plays an important role in knowledge discovery process. Rough set based rule induction algorithms are characterized by excellent accuracy, but they lack the abilities to deal with hybrid attributes such as numeric or fuzzy attributes. In real-world applications, data usually exists with hybrid formats, and thus a unified rule induction algorithm for hybrid data learning is desirable. We firstly model different types of attributes in equivalence relationship, and define the key concepts of block, minimal complex and local covering based on fuzzy rough sets model, then propose a rule induction algorithm for hybrid data learning. Furthermore, in order to estimate performance of the proposed method, we compare it with state-of-the-art methods for hybrid data learning. Comparative studies indicate that rule sets extracted by this method can not only achieve comparable accuracy, but also get more compact rule sets. It is therefore concluded that the proposed method is effective for hybrid data learning.
This paper considers the spectrum sharing network consisting of a pair of primary users (PUs) and a pair of cognitive users (CRs) in a fading channel. The pair of PUs establishes a wireless link as the PU link. The pair of CRs establishes a wireless link as the CR link. The PU link and CR link utilize spectrum simultaneously with different priorities. The PU link has a higher priority to utilize spectrum with respect to the CR link. When the PU link utilizes spectrum, a desired quality of service (QoS) is given to be assured and the CR utilizes spectrum with an opportunistic power scale under this constraint, assuring the desired QoS on the PU link. To compute an optimal opportunistic power scale for the CR link, a fuzzy-based opportunistic power control strategy is proposed based on the Mamdani fuzzy control model using two input variables: the PU’s SNR and PU’s interference channel gain. By the proposed fuzzy-based power control strategy, the desired QoS could be assured on the PU link and the bit error rate (BER) is also reduced compared with the spectrum sharing network without power control strategy.
Some researches have already developed fuzzy decision making theoretical algorithms, but there are only a few medical fields of their applications. We thus make a trial of adopting the Yager decision making algorithm to extract the optimal medicine from a collection of drugs that can be given to a patient to cure him of an illness. The choice of the most efficacious remedy is based on clinical symptoms typical of the considered morbid unit.
Fuzzy set theory offers numerous methods that prove helpful in solving medical problems. They have already been successfully used for instance to fix the optimal level of drug action in patients revealing no clinical symptoms after treatment. In many morbid processes, however, although indices of measurable symptoms improve after the course of medication, the symptoms themselves do not retreat entirely. The authors have already proposed different fuzzy techniques involved in the solution of the problem described above. This time the suggestion of comparing three fuzzy decision making models aim at facilitation of the optimal drug choice in the case of symptoms that prevail after treatment.
The paper refers to earlier results obtained by the authors and constitutes their essential complement and extension by introducing to a diagnostic model the assumption that the decision concerning the diagnosis is based on observations of symptoms carried out repeatedly, by stages, which may have effect in a change of these symptoms in increasing time. The model concerns the observations of symptoms at an individual patient at a time interval. The changes of the symptoms give some additional information, sometimes very important in the diagnostic process when the clinical picture of a patient in a certain interval of time differs from that one which has been received from the beginning of the disease. It may occur that the change in the intensity of a symptom decides an acceptance of another diagnosis after some time when the patient does not feel better. The aim is to fix an optimal diagnosis on the basis of clinical symptoms typical of several morbid units with respect to the changes of these symptoms in time. In order to solve such a posed problem the authors apply the method of fuzzy relation equations, which are modelled by means of logical laws and the rules of inference. Moreover, in the final decision concerning the choice of a proper diagnosis, a normalized Euclidean distance is introduced as a measure between a real decision and an ”ideal” decision. A simple example presents the practical action of the method to show its relevance to a possible user.
Fuzzy set theory has used many auxiliary methods into the trials of solutions of some medical problems. One of the attempts was the evaluation of the optimal level of the drug action in the case when the clinical symptoms disappeared completely after the treatment. However, there can occur such a morbid process in which the symptoms prevail after the treatment. The medication improves too high or too low level of the quantitative symptom but it still indicates the presence of the illness. It is not so easy to choose the medicine, which acts best because it can happen that most of them influence the same symptoms, while they do not improve the others. A fuzzy decision making model tries to make easier to find such a drug which affects most of the symptoms in the highest degree. In the next attempt of solving the problem we propose the using of discrete membership functions in the model instead of the continuous ones. It should improve the thoroughness of the method and heighten the reliability of the accepted decision.
Fuzzy Set Theory has used many auxiliary methods in the trials of solutions of some medical problems. One of the attempts was the finding of the optimal level of drug action in the case when the clinical symptoms disappeared completely after the treatment (Gerstenkorn and Rakus, 1994; Rakus, 1991). However, there can occur such a morbid process in which the symptoms do not disappear after the treatment. The medication improves too high or too low level of the quantitative symptom but it still indicates the presence of the disease. It sometimes makes some problems to choose the medicine, which acts best because it can happen that most of them influence the same symptoms while they do not improve the others. A fuzzy decision making model tries to make easier to find such a drug which affects most of the symptoms in the highest degree. To solve this problem we propose an application of discrete values of the membership degrees in the model instead of the continuous ones that were tested in the paper of Rakus-Andersson and Gerstenkorn (1997). It should improve the thoroughness of the method and heighten the reliability of the accepted decision. It is also considered how to choose the best medicine in the circumstances when some decision-makers have different opinions about the priority of the tested drugs.
One of the most important features of fuzzy set theory is its potential for the modeling of natural language expressions. Most works done on this topic focus on some parts of natural language, mostly those that correspond to the so-called “evaluating linguistic expressions”. We build constraints for the mathematical substitutes of these expressions to mark characteristic limits on an ordered scale. In the current work we form families of constraints which originate from one function. By introducing a parameter in the initial membership function we can model the rest of family functions, whose shapes depend only on a number of functions and the length of a reference set. This procedure fits perfectly for being a segment of computer programs, where the loops providing us with many functions need only the initializations of two data values.
The fuzzy technique revealed in this paper is a new attempt of solving the problem of appreciating the effectiveness level of a medicine when using it against the symptoms typical of a morbid unit. To obtain a satisfactory result, a fuzzy relation with elements equal to fuzzy numbers in L-R representation is introduced. The fuzzy numbers that appear in the relation replace the verbal expressions decided by physicians in accordance with the definition of the relation. The fuzzy relation filled with the fuzzy numbers, which is a counterpart of the average fuzzy relation with the real numbers, also has its eigen fuzzy set with membership degrees as fuzzy numbers. These, after taking appropriate places in the eigen fuzzy set, appreciate the levels of the drug influence on the clinical symptoms.
The paper refers to earlier results obtained by authors and constitutes their essential complement by introducing to a diagnostic model the assumption that a decision concerning the diagnosis is based on observations of symptoms in some stages at an individual patient. To fix the optimal diagnosis we introduce the normalized Euclidean distance between fuzzy sets representing the ideal decision and a possible real decision stated by studying clinical symptoms.
The compositional rule of inference, grounded on the modus ponens law, is one of the most effective fuzzy systems. We modify the classical version of the Zadeh rule to propose an original model, which concerns determining an operation chance for gastric cancer patients. The operation prognosis will be dependent on values of biological markers indicating the progress of the disease.
Approximate reasoning is one of the most effective fuzzy systems. The compositional rule of inference founded on the logical law modus ponens is furnished with a true conclusion, provided that the premises of the rule are true as well. Even though there exist different approaches to an implication, being the crucial part of the rule, we modify the early implication proposed by Zadeh in our practical model concerning a medical application. The approximate reasoning system presented in this work considers evaluation of a risk in the situation when physicians weigh necessity of the operation on a patient. The patient’s clinical symptom levels, pathologically heightened, indicate the presence of a disease possible to recover by surgery. We wish to evaluate the extension of the operation danger by involving particularly designed fuzzy sets in the algorithm of approximate reasoning.
It is a privilege for the author to be involved in composing a book chapter in the anthology devoted to the life and scientific occupation of Professor Zdzisław Pawlak. The author made a personal acquaintance with the outstanding scientist Professor Pawlak and still remembers him as a warm and gentle human being. Professor Pawlak’s theory of rough sets was taught to students during the courses in Computational Intelligence established at Blekinge Institute of Technology in Karlskrona, Sweden. In some Master of Science theses, the principles of rough set theory were discussed in the aspects of technical applications. In this context, we can feel that the theory is still alive and very useful. In this work, we recall again the basics of rough sets to apply them to the classification of discrete two dimensional point sets, which form the shapes resembling some letters. These possess very irregular patterns and cannot be approximated by standard curves without committing large errors. Since the approximation of letter-like point sets is required by the latter classification of their shapes then we, due to own model, wish to find a continuous curve which fits best for each distribution of points. To accomplish the thorough approximation of finite point sets, we test parametric s-truncated functions piecewise, which warrants a high accuracy of approximating. By operating on the functions, replacing samples of points obtained during experiments carried out, we are able to adopt the rough set technique to verify decisions about the primary recognitions of the curves’ appearance as letter shapes. Even if the curves are stretched and shaped differently in the plane, we will divide them in classes gathering similar objects. Our investigations have not a character of pure art — on the contrary— their results are utilized in the classifications of internet packet streams or the analysis of wave signals typical of, e.g., medical examinations.
This chapter has a theoretical character and can be studied by some medical staff researchers that seek methods of approximation of very irregular point sets. When the shape of an obtained polygon based on the point set is similar to a chain of bells, then it will be difficult to find a continuous standard curve that should approximate the polygon without making a large approximation error. The studies of some medical data give rise to the creation of polygons consisting of finite numbers of points tied together. Since the polygons are not formalized by some mathematical expressions, we suggest creating continuous functions that approximate them thoroughly in spite of their irregular shapes. To warrant a high accuracy of approximation, otherwise impossible to obtain when using standard curves, we test a continuous function composed of joined truncated π-functions or joined truncated s-functions. © 2007 Springer.
We should admit that the case of patient P1 in Ex. 3.16 has not been very easy to solve especially when you consider the proper interpretation of PD3. By equipping us with equal values of the membership degrees it has not made it easy enough to make the proper choice of an unknown diagnosis. © 2007 Springer.
In this paper we propose a complex system, involving two control algorithms, to provide a final estimation of Resort Management System (RMS). This distinct RMS quality value depends on some individual appreciations, assigned by customers to basic services. In order to improve the qualities of control actions, we intend to add parametric membership functions of fuzzy sets to the fuzzification part. Another modification considers the newly designed technique of determining some essential estimates in the processing part of control to employ all entry data in the result of final decision.
In the first part of this study we explore continuous fuzzy numbers in the interval- and the alpha-cut forms to detect their similar nature. The conversion from one form to the other is a question of using the appropriate apparatus, which we also provide. Since the fuzzy numbers can reproduce fuzzy events we then will make a trial of extending the concept of fuzzy probability, defined by R. Yager for discrete fuzzy events, on continuous fuzzy events. In order to fulfill the task we utilize conclusions made about fuzzy numbers to propose an initial conception of approximating the Gauss curve by a particularly designed function originated from the pi-class functions. Due to the procedure of the approximation, characterized by an irrelevant cumulative error, we expand fuzzy probabilities of continuous fuzzy events in the form of continuous fuzzy sets. Furthermore, we assume that this sort of probability holds some conditions formulated for probabilities of discrete fuzzy events.
Theoretical fuzzy decision-making models mostly developed by Zadeh, Bellman, Jain and Yager can be adopted as useful tools to estimation of the total effectiveness-utility of a drug when appreciating its positive influence on a collec-tion of symptoms characteristic of a considered diagnosis. The expected effectiveness of the medicine is evaluated by a physician as a verbal expression for each distinct symptom. By converting the words at first to fuzzy sets and then numbers we can regard the effectiveness structures as entries of a utility matrix that constitutes the common basic component of all methods. We involve the matrix in a number of computations due to different decision algorithms to obtain a sequence of tested medicines in conformity with their abilities to soothe the unfavorable impact of symptoms. An adjustment of the large spectrum of applied fuzzy decision-making models to the extraction of the best medicines provides us with some deviations in obtained results but we are thus capable to select this method whose effects closest converge to the physicians’ judgments and expectations. In the current speech we apply fuzzy decision making algorithms to ranking medicines in multifocal toxoplasmosis.
In the previous chapters, we have discussed some ways of determining the most credible diagnosis in a patient who could be identified by his set of clinical symptoms. The same symptoms are usually found in several illnesses. Therefore, it is often difficult to recognize the value of each of their deterministic yet individual characteristics all at once. After improving the diagnostic model by adding complementary solutions we are at last aware of a diagnosis of the patient. The next step would be to prescribe him medication that will lead to a cure. It is seldom possible to give the patient only one remedy to remove completely all unfavourable symptoms. In order to broaden a list of medicines that complement each other, we usually want to evaluate levels of one medicine and its impact on all of the symptoms. Preferably, we want to estimate the lowest and the highest levels of effectiveness of the medicines tested, one by one, when considering their curative powers. © 2007 Springer.
The classical crisp version of Factor Analysis seldom is used in the case of qualitative factors, which often are presented by codes. It is rather difficult to divide codes in level groups without possessing appropriate criteria. To omit this obstacle, we thus propose a fuzzy application of Factor Analysis, which gives a possibility to investigate the strength of an influence of qualitative factors on a tested qualitative variable. When making a new approach to the analysis of factors, we introduce a space of verbal fuzzy numbers that first are expressed as descriptions coming from a natural language and then designed in L-R form. Since the definition of newly created verbal fuzzy numbers deviates from the general conception of fuzzy numbers, we also will check effects of other operations performed on verbal numbers, which are different from the arithmetic based on the extension principle. The verbal fuzzy numbers represent both the qualitative variable and the qualitative factors in all computations that follow the Factor Analysis algorithm. For the first time we also formulate the Yager probability of an event expanded as the verbal fuzzy number.
The classical crisp version of Factor Analysis is seldom used in the case of qualitative factors by reason of the lack of appropriate level criteria referring to these factors. We now propose a fuzzy interpretation of the method, which gives a possibility to investigate the strength of the factor influence on a tested variable. By assuming that fuzzy numbers in L-R form represent both the variable and the factors, as the qualitative parameters, we are capable of performing all the operations that follow the Factor Analysis algorithm. Even the introduction of the conception proposing a new fuzzy space with particularly defined operations on fuzzy numbers helps to obtain satisfactory results.
This volume provides readers with selected fuzzy and rough tools used to medical tasks, especially diagnosing and medication. To build a link between theoretical, mathematical excerpts and practical medical applications, the contents is formed as a sequence of occurrences in which a patient appears to be diagnosed and cured. The fuzzy and rough elements are inserted in the book in the order required by the presentation of medical substance to maintain the logical unity of the book’s essence. In conformity with this pattern the essay presents in turn some necessary elements of fuzzy set theory, the classical fuzzy diagnostic model with extensions, the fuzzy diagnostic model with clinical examinations extended throughout time based on distance theory, methods of drug effectiveness measurements and algorithms selecting the optimal medicine. As the complement, the solution of an approximation problem is suggested to find a curve that surrounds two-dimensional clock-like point sets with the little approximation error. A lot of appealing examples are added to facilitate comprehension of theoretical principles for a reader, so that even a beginner in fuzzy set theory can follow calculation steps without implementing computer programs. It should be emphasized that all models are also applicable to other fields, especially to technical domains after necessary adaptations. This confirms the existence of the large spectrum of applicable fuzzy and rough methods not only in medicine but also in natural sciences.
The classical fuzzy decision-making model is now tested for qualitative compound states-symptoms to select the most efficacious medicine, acting on all symptoms. Instead of terminating the decision procedure in the way comparing values of total utilities of decisions-treatments, we test the aggregated utility values in utility levels. This activity lets us assign a verbally verified utility to each medicine.
The book presents different algorithms of diagnosing on the base of clinical symptoms. Some modern techniques as computing with words and the use of non-conventional operations on fuzzy membership degrees and fuzzy numbers are proposed as a new approach to the diagnostic problem. A large part of the work is devoted to differentiating the level of the drug action in the case of symptoms, which either disappear or still prevail after the treatment.
In the first part of this study we explore continuous fuzzy numbers in the interval and the α-cut forms to detect their similar nature. The conversion from one form to the other is a question of using the appropriate apparatus, which we also provide. Since the fuzzy numbers can reproduce fuzzy events we then will make a trial of extending the concept of fuzzy probability, defined by R. Yager (1979) for discrete fuzzy events, on continuous fuzzy events. In order to fulfil the task we utilise conclusions made about fuzzy numbers to propose an initial conception of approximating the Gauss curve by a particularly designed function originated from the π-class functions. Due to the procedure of the approximation, characterised by an irrelevant cumulative error, we expand fuzzy probabilities of continuous fuzzy events in the form of continuous fuzzy sets.
From the domain of Computational Intelligence we have selected immunological computation and fuzzy systems to combine them in a new hybrid model. This novel numerical method has been tested on patient data strings to make decisions about the choices of surgery types. The model input clinical data concerns patients who suffer from gastric cancer.
Minimization of regret has been adopted here as a decision making algo-rithm in order to select the best medicine from a collection of drugs considered in the treatment of an individual patient. We assume that symp-toms affected by medicines have a compound qualitative complexion. A model of evaluating the effectiveness of drugs, exerting an influence on qualitative features, is employed to found the utility matrix, which constitutes the input system in decision-making. We thus discuss the method that converts information obtained from a questionnaire to entries of the matrix mentioned. The last technique constitutes a main own contribution in evolving methods of fuzzy calculus.
Some researches have already developed fuzzy decision making theoretical algorithms, but there are only a few medical fields of their applications. We thus make a trial of adopting two decision-making algorithms based on unequal objectives and minimization of regret to extract the optimal medicine from a collection of drugs recommended to a patient. A choice of the most efficacious remedy is based on clinical symptoms typical of the considered morbid unit.
Approximate reasoning is one of the most effective fuzzy systems. The compositional rule of inference founded on the logical law Modus Ponens is furnished with a true conclusion, provided that the premises of the rule are true as well. One of the premises is formed as the implication, which is represented by different mathematical approaches, but we are especially fond of the results brought by the early implication proposed by Zadeh, which is modified in our practical model concerning a medical application. The approximate reasoning system, grounded on the extended and modified version of Modus Ponens law, will be employed here to predict a chance of survival after the operation for a patient who suffers from cancer. The patient’s symptom levels are the indicators of the disease. If the symptoms do not exceed their critical values there is still a chance to save the patient’s life by trying surgery. We wish to evaluate the verbal prognosis of the surgery by involving specifically designed fuzzy sets in the algorithm of approximate reasoning. Since the chance of successful surgery depends on the interactions of several symptoms then we will name the decision model multi-dimensional.
Approximate reasoning is one of the most effective fuzzy systems. The compositional rule of inference founded on the logical law Modus Ponens is furnished with a true conclusion, provided that the premises of the rule are true as well. There exist different approaches to an implication, being the crucial part of the rule, but we are especially fond of the results brought by the early implication proposed by Zadeh, which is modified in our practical model concerning a medical application. The approximate reasoning system, grounded on the extended version of Modus Ponens law, will be employed here to predict a chance of positive effects of the operation on a patient who suffers from stomach cancer. The patient’s CRP (C-reactive proteins) symptom level, pathologically heightened, indicates the presence of a disease. When the CRP-value does not exceed a critical border it can be realistic to try surgery to recover the patient from his/her illness due to Do-Kyong Kim. We wish to evaluate the verbal prognosis of the surgery by involving particularly designed fuzzy sets in the algorithm of approximate reasoning.
Rough sets, surrounded by two approximation sets filled with sure and possible members constitute perfect mathematical tools of the classification of some objects. In this work we adopt the rough technique to verify diagnostic decisions concerning a sample of patients whose symptoms are typical of a considered diagnosis. The objective is to extract the patients who surely Suffer from the diagnosis, to indicate the patients who are free from it, and even to make decisions in undefined diagnostic cases. By applying selected logical decision rules, we also discuss a possibility of reducing of symptom sets to their minimal collections preserving the previous results in order to minimize a number of numerical calculations.
Rough sets, surrounded by two approximation sets filled with sure and possible members constitute perfect mathematical tools of the classification of some objects. In this work we adopt the rough technique to verify diagnostic decisions concerning a sample of patients whose symptoms are typical of a considered diagnosis. The objective is to extract the patients who surely suffer from the diagnosis, to indicate the patients who are free from it, and even to make decisions in undefined diagnostic cases. By applying selected logical decision rules, we also discuss a possibility of reducing of symptom sets to their minimal collections preserving the previous results in order to minimize a number of numerical calculations.
Rough sets constitute helpful mathematical tools of the classification of objects belonging to a certain universe when dividing the universe in two collections filled with sure and possible members. In this work we adopt the rough technique to verify diagnostic decisions concerning a sample of patients whose symptoms are typical of a considered diagnosis. The objective is to extract the patients who surely suffer from the diagnosis, to indicate the patients who are free from it, and even to make decisions in undefined diagnostic cases. We also consider a decisive power of reducts being minimal collections of symptoms, which preserve the previous classification results. We use them in order to minimize a number of numerical calculations in the classification process. Finally, by testing influence of symptom intensity levels on the diagnosis indisputable appearance we select these standards, whose either presence or absence in the patients allows us to add complementary remarks making the classification effects even more readable.
Due to the latest research the subject of Computational Intelligence has been divided into five main regions, namely, neural networks, evolutionary algorithms, swarm intelligence, immunological systems and fuzzy systems. Our attention has been attracted by the possibilities of medical applications provided by immunological computation algorithms. Immunological computation systems are based on immune reactions of the living organisms in order to defend the bodies from pathological substances. Especially, the mechanisms of the T-cell reactions to detect strangers have been converted into artificial numerical algorithms. Immunological systems have been developed in scientific books and reports appearing during the two last decades. The basic negative selection algorithm NS was invented by Stefanie Forrest to give rise to some technical applications. We can note such applications of NS as computer virus detection, reduction of noise effect, communication of autonomous agents or identification of time varying systems. Even a trial of connection between a computer and biological systems has been proved by means of immunological computation. Hybrids made between different fields can provide researchers with richer results; therefore associations between immunological systems and neural networks have been developed as well. In the current chapter we propose another hybrid between the NS algorithm and chosen solutions coming from fuzzy systems. This hybrid constitutes the own model of adapting the NS algorithm to the operation decisions “operate” contra “do not operate” in gastric cancer surgery. The choice between two possibilities to treat patients is identified with the partition of a decision region in self and non-self, which is similar to the action of the NS algorithm. The partition is accomplished on the basis of patient data strings/vectors that contain codes of states concerning some essential biological markers. To be able to identify the strings that characterize the “operate” decision we add the own method of computing the patients’ characteristics as real values. The evaluation of the patients’ characteristics is supported by inserting importance weights assigned to powerful biological indices taking place in the operation decision process. To compute the weights of importance the Saaty algorithm is adopted.
Some collections of two-dimensional points form very irregular shapes, which cannot be approximated by standard curves without making large errors. We approximate the sets of points to introduce formal mathematical expressions giving rise for future predictions for other points, which are not placed in data sets. To accomplish the thorough approximation of finite point sets we test parametric s-truncated functions piecewise, which warrants a high accuracy of approximating. By operating with the functions, which represent samples of points obtained during experiments carried out, and by adopting the rough set technique, we attempt a classification of curves. Even if the curves are stretched and shaped differently we will divide them in classes gathering similar objects. To confirm availability and correctitude of the approximation and the classification proposed, we consider an examination of Internet packet streams, especially a bottleneck distribution based on throughput values.
The German Enigma encoding machine and the contributions of famous cryptologists who broke it, are still topics, which fascinate both scientists and general public. After the monarchy of Kaiser Wilhelm II fell, the Weimar republic came into being, and the idea of equipping the armed forces with machine ciphers already found realization in 1926. The German cipher machine, called Enigma, alarmed the general staffs of neighbouring countries, especially Poland and France. This work intends to describe the efforts of cryptanalysts who solved the mystery of Enigma during the 30-ties before the beginning of the war.
The project titled: “The Composition of the Book in Fuzzy Logic Adapted to Stationary and Distance Courses for Advanced Students” has been granted by the Swedish Royal Academy of Sciences in 2007 The following contents of the book “Basic Concepts and Applications of Fuzzy Set Theory” is primarily designed as: Contents 1. Introduction 2. Fuzzy sets 3. Fuzzy and linguistic variables 4. Operations on fuzzy sets 5. T and S Norms 6. Fuzzy measures and measures of fuzziness 7. Extension principles 8. Fuzzy numbers and their arithmetic 9. The L-R, interval and alpha-cut representations of fuzzy numbers 10. Fuzzy relations, The compositional rule of inference 11. The eigen sets of a fuzzy relation 12. Fuzzy analysis 13. Possibility theory, Probability of fuzzy events 14. Fuzzy logic and approximate reasoning 15. Fuzzy decision making, 16. Fuzzy control, 17. Imprecise optimization 18. Choquet and Sugeno integrals 19. Rough set theory 20. Discussion of other imprecise theories
Some medical and technical experiments lead to measures regarded as the coordinates of points in the plane. We can encounter vague or imprecise data as the values of the measurements. Even if the classical numerical methods cannot be applied to fuzzy data it is still desirable to find a function that interpolates the points. We thus test the Lagrange interpolation method when assuming that the entries of an algorithm will be fuzzy numbers in L-R representation, specially designed. The equation describing the fuzzy function, which goes through the points, is also used as a prognosis in the case of other points that have only one coordinate known.
All the solutions to the problems sketched below will be created on the basis of Fuzzy Set Theory. Fuzzy Set Theory is applied instead of the classical set theory when data involved in the problem to solve is imprecise, verbally described or cannot be measured exactly. The models, which are the contents of the project should contain some proposed solutions to such problems as: 1) the approximation of the mean value and the standard deviation for some imprecise data, 2) a probability distribution filled with fuzzy probability sets (Yager’s probabilities) that replace the probability values from the normal distribution, 3) the application of the last distribution to statistical tests with imprecise data, 4) the development of Factor Analysis for qualitative variables and factors provided that the data is collected by means of a questionnaire, 5) the interpolation of a set of points with imprecise coordinates by a fuzzy function.
The eigen fuzzy set of a given fuzzy relation often corresponds to an occurrence of invariability in natural sciences. By determining the fuzzy relations as connections between pairs of symptoms we utilize the greatest and the least eigen fuzzy sets in order to find the estimates of the medicine effectiveness levels.
Fuzzy numbers constitute a helpful tool for some users who deal with imprecise data. To facilitate a procedure of performing operations on fuzzy numbers for these users who do not possess deep mathematical qualifications, we introduce a special space of fuzzy numbers verbally defined. This consists of families of fuzzy numbers in the L-R form provided that every family has a common mean value. We not only discuss some properties of the space, but also propose new conceptions of the arithmetical operations on the verbally expressed fuzzy numbers. The classification of the IP-traffic can act as a practical aspect, which clarifies the theoretical assumptions formulated in the paper.
Strict analytic formulas are the tools usually derived for determining the formal relationships between a sample of independent variables and a variable which they affect. If we cannot formalize the function tying the independent and dependent variables then we will utilize some expert-system control actions. We often adopt their fuzzy variants developed by Mamdani, Sugeno and Takagi. Fuzzy expert-system algorithms are furnished with softer mechanisms, when comparing them to crisp versions. An efficient action of these softer mechanisms depends on the proper fuzzification of variables. At the stage of fuzzifying the variable levels we will prove some parametric expressions, which rearrange one function to several forms needed by the expert-system algorithm. The general parametric equation of membership functions allows creating arbitrary lists without any intuitive assumptions. The fuzzy expert-system algorithms are particularly adaptable to support medical tasks to solve. These tasks often cope with uncertain premises and conclusions. From the medical point of view it would be desirable to prognosticate the survival length for patients suffering from gastric cancer. We thus formulate the objective of the current chapter as the utilization of the Mamdani fuzzy control actions as a methodology adapted for the purpose of making the survival prognoses.
It has passed over twenty five years since I encountered fuzzy mathematics for the first time but I can still feel this excitement, which I have experienced during that meeting with an unknown mathematical domain. This was different from other classical mathematical fields but I immediately felt that the concept of imprecision would be expected to have a fine future. Since that day in 1987, when I held in my hand the first available paper on the medical applications of fuzzy relations, it was obvious for me that I could devote my time to investigate the topic much deeper. I made an acquaintance with the creator of fuzzy mathematics, Professor Lotfi Zadeh, in Budapest in 1999. I expected to be confronted with a self-confident and outstanding scientist who was not going to talk to unknown people. Instead I saw a very nice and modest man speaking to everybody in a very friendly manner, especially to new members of the fuzzy society. His behavior awoke in me sympathy and gratitude for his kindness and I wanted to meet him again. The idea of editing a book, based on our memories and experiences concerning fuzzy mathematics, is really great. Let me thus tell you about my own scientific carrier in which fuzzy mathematics has played a dominant role.
The current research is devoted to developing methods of a novel mathematical interpretation of term-sets of linguistic variables. To the term-sets of the linguistic variables fuzzy sets are assigned. We intend to adopt the π-functions and the π-functions to derive formulas of membership functions of these sets. The fuzzy sets are divided in three families in the case of an odd number of the term-sets. To each family, we assign only one parametric formula, which depends on two parameters: the width of a non-fuzzy set, which contains all supports of the fuzzy sets being representatives of the term-sets, and a number of the term-sets. Provided that the supports of fuzzy sets will be unequal, the membership function of the set, belonging to one of the families, is computed by means of a functional modifier, inserted in the common equation typical of this family. Medical examples explain how to use cumulated membership functions practically. The procedure can be easily computerized.