A Summary of F-Transform Techniques in Data Analysis

Fuzzy transform is a technique applied to approximate a function of one or more variables applied by researchers in various image and data analysis. In this work we present a summary of a fuzzy transform method proposed in recent years in different data mining disciplines, such as the detection of relationships between features and the extraction of association rules, time series analysis, data classification. After having given the definition of the concept of Fuzzy Transform in one or more dimensions in which the constraint of sufficient data density with respect to fuzzy partitions is also explored, the data analysis approaches recently proposed in the literature based on the use of the Fuzzy Transform are analyzed. In particular, the strategies adopted in these approaches for managing the constraint of sufficient data density and the performance results obtained, compared with those measured by adopting other methods in the literature, are explored. The last section is dedicated to final considerations and future scenarios for using the Fuzzy Transform for the analysis of massive and high-dimensional data.

A classification algorithm based on multi-dimensional fuzzy transforms

We present a new classification algorithm for machine learning numerical data based on direct and inverse fuzzy transforms. In our previous work fuzzy transforms were used for numerical attribute dependency in data analysis: the multi-dimensional inverse fuzzy transform was used to approximate the regression function. Also here the classification method presented is based on this operator. Strictly speaking, we apply the K-fold cross-validation algorithm for controlling the presence of over-fitting and for estimating the accuracy of the classification model: for each training (resp., testing) subset an iteration process evaluates the best fuzzy partitions of the inputs. Finally, a weighted mean of the multi-dimensional inverse fuzzy transforms calculated for each training subset (resp., testing) is used for data classification. We compare this algorithm on well-known datasets with other five classification methods.

Fuzzy Study on the Winning Rate of Football Game Betting

This study aims to find variables that affect the winning rate of the football team before a match. Qualitative variables such as venue, match importance, performance, and atmosphere of both teams are suggested to predict the outcome. Regression analysis is used to select proper variables. In this study, the performance of the football team is based on the opinions of experts, and the team atmosphere can be calculated with the results of the previous five games. ELO rating represents the state of the opponent. Also, the selected qualitative variables are expressed in fuzzy numbers using fuzzy partitions. A fuzzy regression model for the winning rate of the football team can be estimated by using the least squares method and the least absolute method. It is concluded that the stadium environment, ELO rating, team performance, and importance of the match have effects on the winning rate of Korean National Football (KNF) team from the data on 118 matches.

Dynamic programming based fuzzy partition in fuzzy decision tree induction

Fuzzy decision trees are one of the most popular extensions of decision trees for symbolic knowledge acquisition by fuzzy representation. Among the majority of fuzzy decision trees learning methods, the number of fuzzy partitions is given in advance, that is, there are the same amount of fuzzy items utilized in each condition attribute. In this study, a dynamic programming-based partition criterion for fuzzy items is designed in the framework of fuzzy decision tree induction. The proposed criterion applies an improved dynamic programming algorithm used in scheduling problems to establish an optimal number of fuzzy items for each condition attribute. Then, based on these fuzzy partitions, a fuzzy decision tree is constructed in a top-down recursive way. A comparative analysis using several traditional decision trees verify the feasibility of the proposed dynamic programming based fuzzy partition criterion. Furthermore, under the same framework of fuzzy decision trees, the proposed fuzzy partition solution can obtain a higher classification accuracy than some cases with the same amount of fuzzy items.

Triangular Fuzzy-Rough Set Based Fuzzification of Fuzzy Rule-Based Systems

In real-world approximation problems, precise input data are economically expensive. Therefore, fuzzy methods devoted to uncertain data are in the focus of current research. Consequently, a method based on fuzzy-rough sets for fuzzification of inputs in a rule-based fuzzy system is discussed in this paper. A triangular membership function is applied to describe the nature of imprecision in data. Firstly, triangular fuzzy partitions are introduced to approximate common antecedent fuzzy rule sets. As a consequence of the proposed method, we obtain a structure of a general (non-interval) type-2 fuzzy logic system in which secondary membership functions are cropped triangular. Then, the possibility of applying so-called regular triangular norms is discussed. Finally, an experimental system constructed on precise data, which is then transformed and verified for uncertain data, is provided to demonstrate its basic properties.

Functors among Relational Variants of Categories Related to L-Fuzzy Partitions, L-Fuzzy Pretopological Spaces and L-Fuzzy Closure Spaces

Various types of topological and closure operators are significantly used in fuzzy theory and applications. Although they are different operators, in some cases it is possible to transform an operator of one type into another. This in turn makes it possible to transform results relating to an operator of one type into results relating to another operator. In the paper relationships among 15 categories of modifications of topological L-valued operators, including Čech closure or interior L-valued operators, L-fuzzy pretopological and L-fuzzy co-pretopological operators, L-valued fuzzy relations, upper and lower F-transforms and spaces with fuzzy partitions are investigated. The common feature of these categories is that their morphisms are various L-fuzzy relations and not only maps. We prove the existence of 23 functors among these categories, which represent transformation processes of one operator into another operator, and we show how these transformation processes can be mutually combined.