Fuzzy neural networks with genetic algorithm-based learning method

This thesis is on the reasoning of artificial neural networks based on granules for both crisp and uncertain data. However, understanding the data in this way is difficult when the data is so complex. Reducing the complexity of the problems that these networks are attempting to learn as well as decreasing the cost of the learning processes are desired for a better prediction. A suitable prediction in artificial neural networks depends on an in-depth understanding of data and fine tracking of relations between data points. Inaccuracies of the prediction are caused by complexity of data set and the complexity is caused by uncertainty and quantity of data. Uncertainties can be represented in granules, and the reasoning based on granules is known as granular computing. This thesis proposed an improvement of granular neural networks to reach an outcome from uncertain and crisp data. Two methods based on genetic algorithms (GAs) are proposed. Firstly, GA-based fuzzy granular neural networks are improved by GA-based fuzzy artificial neural networks. They consist of two parts: granulation using fuzzy c-mean clustering (FCM), and reasoning by GAbased fuzzy artificial neural networks. In order to extract granular rules, a granulation method is proposed. The method has three stages: construction of all possible granular rules, pruning the repetition, and crossing out granular rules. Secondly, the two-phase GA-based fuzzy artificial neural networks are improved by GA-based fuzzy artificial neural networks. They are designed in two phases. In this case, the improvement is based on alpha cuts of fuzzy weight in the network connections. In the first phase, the optimal values of alpha cuts zero and one are obtained to define the place of a fuzzy weight for a network connection. Then, in the second phase, the optimal values of middle alpha cuts are obtained to define the shape of a fuzzy weight. The experiments for the two improved networks are performed in terms of generated error and execution time. The results tested were based on available rule/data sets in University of California Irvine (UCI) machine learning repository. Data sets were used for GA-based fuzzy granular neural networks, and rule sets were used for GA-based fuzzy artificial neural networks. The rule sets used were customer satisfaction, uranium, and the datasets used were wine, iris, servo, concrete compressive strength, and uranium. The results for the two-phase networks revealed the improvements of these methods over the conventional onephase networks. The two-phase GA-based fuzzy artificial neural networks improved 35% and 98% for execution time, and 27% and 26% for the generated error. The results for GA-based granular neural networks were revealed in comparison with GA-based crisp artificial neural networks. The comparison with other related granular computing methods were done using the iris benchmark data set. The results for these networks showed an average performance of 82.1%. The results from the proposed methods were analyzed in terms of statistical measurements for rule strengths and classifier performance using benchmark medical datasets. Therefore, this thesis has shown GA-based fuzzy granular neural networks, and GA-based fuzzy artificial neural networks are capable of reasoning based on granules for both crisp and uncertain data in artificial neural networks

On the Suitability of the Bandler–Kohout Subproduct as an Inference Mechanism

Fuzzy relational inference (FRI) systems form an important part of approximate reasoning schemes using fuzzy sets. The compositional rule of inference (CRI), which was introduced by Zadeh, has attracted the most attention so far. In this paper, we show that the FRI scheme that is based on the Bandler-Kohout (BK) subproduct, along with a suitable realization of the fuzzy rules, possesses all the important properties that are cited in favor of using CRI, viz., equivalent and reasonable conditions for their solvability, their interpolative properties, and the preservation of the indistinguishability that may be inherent in the input fuzzy sets. Moreover, we show that under certain conditions, the equivalence of first-infer-then-aggregate (FITA) and first-aggregate-then-infer (FATI) inference strategies can be shown for the BK subproduct, much like in the case of CRI. Finally, by addressing the computational complexity that may exist in the BK subproduct, we suggest a hierarchical inferencing scheme. Thus, this paper shows that the BK-subproduct-based FRI is as effective and efficient as the CRI itself

Fuzzy Dynamic Discrimination Algorithms for Distributed Knowledge Management Systems

A reduction of the algorithmic complexity of the fuzzy inference engine has the following property: the inputs (the fuzzy rules and the fuzzy facts) can be divided in two parts, one being relatively constant for a long a time (the fuzzy rule or the knowledge model) when it is compared to the second part (the fuzzy facts) for every inference cycle. The occurrence of certain transformations over the constant part makes sense, in order to decrease the solution procurement time, in the case that the second part varies, but it is known at certain moments in time. The transformations attained in advance are called pre-processing or knowledge compilation. The use of variables in a Business Rule Management System knowledge representation allows factorising knowledge, like in classical knowledge based systems. The language of the first-degree predicates facilitates the formulation of complex knowledge in a rigorous way, imposing appropriate reasoning techniques. It is, thus, necessary to define the description method of fuzzy knowledge, to justify the knowledge exploiting efficiency when the compiling technique is used, to present the inference engine and highlight the functional features of the pattern matching and the state space processes. This paper presents the main results of our project PR356 for designing a compiler for fuzzy knowledge, like Rete compiler, that comprises two main components: a static fuzzy discrimination structure (Fuzzy Unification Tree) and the Fuzzy Variables Linking Network. There are also presented the features of the elementary pattern matching process that is based on the compiled structure of fuzzy knowledge. We developed fuzzy discrimination algorithms for Distributed Knowledge Management Systems (DKMSs). The implementations have been elaborated in a prototype system FRCOM (Fuzzy Rule COMpiler).Fuzzy Unification Tree, Dynamic Discrimination of Fuzzy Sets, DKMS, FRCOM

Interval-based uncertain reasoning

This thesis examines three interval based uncertain reasoning approaches: reasoning under interval constraints, reasoning using necessity and possibility functions, and reasoning with rough set theory. In all these approaches, intervals are used to characterize the uncertainty involved in a reasoning process when the available information is insufficient for single-valued truth evaluation functions. Approaches using interval constraints can be applied to both interval fuzzy sets and interval probabilities. The notion of interval triangular norms, or interval t-norms for short, is introduced and studied in both numeric and non-numeric settings. Algorithms for computing interval t-norms are proposed. Basic issues on the use of t-norms for approximate reasoning with interval fuzzy sets are studied. Inference rules for reasoning under interval constraints are investigated. In the second approach, a pair of necessity and possibility functions is used to bound the fuzzy truth values of propositions. Inference in this case is to narrow the gap between the pair of the functions. Inference rules are derived from the properties of necessity and possibility functions. The theory of rough sets is used to approximate truth values of propositions and to explore modal structures in many-valued logic. It offers an uncertain reasoning method complementary to the other two

Indicator of inclusion grade for interval-valued fuzzy sets. Application to approximate reasoning based on interval-valued fuzzy sets

AbstractWe begin the paper studying the axioms that the indicators of the grade of inclusion of a fuzzy set in another fuzzy set must satisfy. Next, we present an expression of such indicator, first for fuzzy sets and then for interval-valued fuzzy sets, analyzing in both cases their main properties. Then, we suggest an expression for the similarity measure between interval-valued fuzzy sets. Besides, we study two methods for inference in approximate reasoning based on interval-valued fuzzy sets, the inclusion grade indicator and the similarity measure. Afterwards, we expose some of the most important properties of the methods of inference presented and we compare these methods to Gorzalczany's. Lastly, we use the indicator of the grade of inclusion for interval-valued fuzzy sets as an element that selects from the different methods of inference studied, the one that will be executed in each case

Reducing the Memory Size of a Fuzzy Case-Based Reasoning System Applying Rough Set Techniques

Early work on case-based reasoning (CBR) reported in the literature shows the importance of soft computing techniques applied to different stages of the classical four-step CBR life cycle. This correspondence proposes a reduction technique based on rough sets theory capable of minimizing the case memory by analyzing the contribution of each case feature. Inspired by the application of the minimum description length principle, the method uses the granularity of the original data to compute the relevance of each attribute. The rough feature weighting and selection method is applied as a preprocessing step prior to the generation of a fuzzy rule system, which is employed in the revision phase of the proposed CBR system. Experiments using real oceanographic data show that the rough sets reduction method maintains the accuracy of the employed fuzzy rules, while reducing the computational effort needed in its generation and increasing the explanatory strength of the fuzzy rules

Rough-fuzzy rule interpolation

AbstractFuzzy rule interpolation forms an important approach for performing inference with systems comprising sparse rule bases. Even when a given observation has no overlap with the antecedent values of any existing rules, fuzzy rule interpolation may still derive a useful conclusion. Unfortunately, very little of the existing work on fuzzy rule interpolation can conjunctively handle more than one form of uncertainty in the rules or observations. In particular, the difficulty in defining the required precise-valued membership functions for the fuzzy sets that are used in conventional fuzzy rule interpolation techniques significantly restricts their application. In this paper, a novel rough-fuzzy approach is proposed in an attempt to address such difficulties. The proposed approach allows the representation, handling and utilisation of different levels of uncertainty in knowledge. This allows transformation-based fuzzy rule interpolation techniques to model and harness additional uncertain information in order to implement an effective fuzzy interpolative reasoning system. Final conclusions are derived by performing rough-fuzzy interpolation over this representation. The effectiveness of the approach is illustrated by a practical application to the prediction of diarrhoeal disease rates in remote villages. It is further evaluated against a range of other benchmark case studies. The experimental results confirm the efficacy of the proposed work

Contribución al estudio crítico de la inferencia borrosa y de sus aplicaciones

El trabajo descrito en esta memoria se enmarca en el campo general de la lógica borrosa. En nuestro caso se concreta en el de las relaciones (similitudes y diferencias) entre la estructura y propiedades de las teorías de conjuntos borrosos y las álgebras reticulares. Con este objetivo, se aborda en primer lugar (Cap. 1) una descripción de principios, propiedades y funciones de los conjuntos ordenados en general, retículos -el [0,1] en particular-. Además (Cap. 2) se exponen las bases conceptuales de los conjuntos borrosos y las teorías de conjuntos borrosos, también de cara al uso que se hace de ellos en el texto. La inclusión de los temas de ambos capítulos, pretende fijar el marco de trabajo en que se asienta el resto. A continuación (Cap. 3) se estudia la interrelación conceptual entre determinadas leyes reticulares, con als leyes y propiedades de teorías de conjuntos borrosos estándar, Pexider, funcionales y no funcionales. Este método permite descubrir nuevas propiedades en diferentes estructuras y deja abiertos campos por explorar, como la idea de booleanidad (gradual) de las teorías borrosas. En la misma línea metodológica, el siguiente capítulo estudia tipos de razonamiento basados en el condicional, tales como el Modus Ponens, Modus Tollens, Dilema constructivo, etc. en retículos y teorías de conjuntos borrosos. El estudio muestra algunas leyes que tienen un comportamiento restrictivo y fuerzan álgebras de Boole. En el caso de las teorías borrosas, donde el razonamiento condicional ha sido ampliamente usado, se introduce una nueva familia de implicaciones borrosas basada en un condicional ortomodular, la flecha de Dishkant. Por último, el capítulo 5 trata sobre otro modo de razonamiento, el disyuntivo, tema que está mucho menos tratado en la literatura. Este modo lleva de forma natural al estudio de la disyunción, que abordamos desde la perspectiva de la disyunción inclusiva y la exclusiva -diferencia simétrica-. Tanto en estructuras algebraicas como teorías de conjuntos borrosos se profundiza en el estudio de la diferencia simétrica. Sobre este operador se hace especial énfasis ya que tiene gran importancia en los modelos lingüísticos de la disyunción y es un operador muy poco estudiado. La memoria se completa con un breve recorrido por los hallazgos originales más destacables, algunas reflexiones y un repertorio de problemas abiertos. This work belongs to the general _eld of Fuzzy Logic. In our case it is focused on the relations (similarities and di_erences) between the structure and properties of Fuzzy Set Theories and Lattices. With this aim in mind, a description of principles, properties and functions of ordered sets, lattices -[0; 1] in particular- is broached in the _rst chapter. In the second chapter the conceptual basis of fuzzy sets and fuzzy sets theories are settled in order to be used later on. Next, in the third chapter the conceptual interrelation between some lattice laws and standard fuzzy sets theories laws or properties are studied, furthermore their interrelation with Pexider fuzzy sets theories and functional or nonfunctional theories are also studied. This way of working allow us to discover new properties in many di_erent structures, and opens up the study of the degree of \booleanity" of fuzzy set theories. Following up this way of working, the fourth chapter studies the classical ways of conditional reasoning, such as Modus Ponens, Modus Tollens, Constructive Dilemma... in lattices and Fuzzy Theories. This study shows that some laws have a restrictive behavior and force a speci_c boolean structures. The conditional reasoning is well known and many families of fuzzy implications are available. A new family of fuzzy implications, based on a classical orthomodular model, the Dishkant arrow, is introduced. Finally, in the _fth chapter we deal with a di_erent way of reasoning, the disjunctive mode, which has received less attention. This mode of reasoning requires the previous study of the disjunction, in its both facets, the inclusive and the exclusive disjunction (or symmetric di_erence). Then we have studied in depth the symmetric di_erence in algebraic structures and in fuzzy sets theories. This operator has a great relevance due to their use as a model of the linguistic disjunctions and has been infrequently studied. The work ends with a short review of the main contributions, some reections and some open problems