Fuzzy granular convolutional classifiers
Introduction
In 1985, Hobbs [1] proposed the concept of granularity, believing that human observation, measurement, conceptualization and reasoning are carried out in the sense of granularity. In 1999, granular computing was first proposed by Lin [2] and successfully applied in data mining [3]. Yao explained granular computing from the perspectives of philosophy, application, and computation. He held that granular computing is a structured thinking, problem-solving, and information processing method [4]. Miao discussed the structure of granular computing from the perspective of set theory [5]. Wang analyzed the uncertainty measure in granular computing and its application in big data [6], [7]. Yao proposed neighborhood systems and neighborhood granular computing [8], [9]. Hu analyzed the problems of neighborhood reduction and classification [10], [11], [12]. Liang and Qian proposed multi-granularity data fusion and processing methods [13], [14], and applied them in the field of artificial intelligence [15]. We analyzed the granulation in a neighborhood system and its measure with information entropy, further studied the reduction and optimization of features from the perspective of swarm intelligence [16], [17], [18], [19]. Granulation, as one of the most important features of human cognition, plays a key role in the modeling of complex data. Although information granulation methods and techniques are applied in a wide range of fields, accurate information granulation and processing methods are difficult to portray things, since that the granularity of human reasoning and concept is fuzzy in many cases.
In 1979, the famous American scientist Zadeh proposed fuzzy sets and fuzzy computing [20]. Fuzzy computing is based on fuzzy set theory and simulates the inaccurate and nonlinear information processing capabilities of human brain, including fuzzy inference systems [21], fuzzy logic [22], and fuzzy systems [23] etc. It has been widely used in many fields [24], [25], [26]. Zadeh further studied the problem of fuzzy information granulation [27], [28], [29], and believed that human cognitive ability can be summarized as three main characteristics of granulation, organization and causality. Canadian academician Pedrycz pointed out that the construction of information granule is the key to granular computing. He constructed an information granule from the perspective of fuzzy sets and used it for clustering [30] and classification [31], and proposed a variety of granular classifiers [32], [33].
Convolution is a mathematical operator that generates a new function through two functions f and g, characterizing the overlapped area of functions f and g through rotation and translation. In statistics, the weighted moving average is a convolution. In acoustics, echoes are expressed by a convolution of the source sound with a function about various reflection effects. In electronic engineering and signal processing, the output of any linear system can be obtained by convolving the input signal with a system function that is an impulse response of the system. In physics, convolution exists in any linear system, which conforms to the principle of superposition. In computer science, Convolutional Neural Network (CNN) is the cornerstone of deep learning. It has an excellent effect on feature extraction, which is widely used in pattern recognition [34], [35], image processing [36], [37] and other fields in recent years. Traditional classification algorithms include decision tree [38], KNN [39], [40], support vector machine [41], random forest [42], sparse learning [43], neural network [44], etc., but their classification performances for big data and missing data are not good. At present, deep learning has achieved a great success in the field of artificial intelligence [45], [46]. It is mainly constructed by the convolutional neural network. The convolutional operation in deep learning is the calculation of numerical values (continuous or discrete data). But it is difficult to deal with the set values.
The structure of a granule is essentially a set, and the convolution of granules is necessarily a set operation. Traditional vectors are quantities of both size and direction, so they are successfully applied in the machine learning field. The fuzzy granules are essentially sets, but sets have no vector representation. Therefore, the fuzzy granules are difficult to be employed in the machine learning field. For solving this problem, we propose some fuzzy granular vectors and define two types of convolutional operations on these granular vectors. From a new perspective, based on set theory and fuzzy granulation we also propose a new classification model: the fuzzy granular convolutional classifier. Starting from the single-atom feature fuzzy granulation of a classification system, we define some concepts of fuzzy granule, fuzzy granular vector and fuzzy granular distance. Furthermore, we propose the operators and functions of fuzzy granular convolution. We also prove the difference and derivative forms of fuzzy granular convolutional functions, and apply them to the back-propagation of residual errors. Finally, fuzzy granular convolutional classifiers are designed and their performances are verified by experiments. The theoretical analysis and experimental results show that the proposed classifiers have better effects of feature extraction and classification.
The paper is structured as follows. First, in Section 2, we introduce the fuzzy granulation and present some fuzzy granular vectors. Then, we propose a classifier model based on fuzzy granular convolution in Section 3. In Section 4, we design an algorithm for convolutional classifiers of fuzzy granules. In Section 5, we present some experimental results. A conclusion and future works are covered in Section 6.
Section snippets
Fuzzy granulation and fuzzy granular vectors
A fuzzy set is used to express a vague object. Generally, it describes fuzziness by establishing an appropriate membership function and uses its operation and transformation to analyze the uncertain objects. In the following, we fuzzily granulate the samples from the perspective of classification, and construct fuzzy granular vectors to describe uncertain samples.
Definition 1 Let be a decision-making system (decision table) [47], where is a set of samples or objects;
The classifier model based on fuzzy granular convolution
By means of fuzzy granulation, a sample is granulated into some fuzzy granules according to different features. These fuzzy granules are combined into a fuzzy granular vector. At the same time, the decision label of the sample is granulated into a fuzzy decision granule. Traditional convolutions of vectors are real-value operations, which are not suitable for computations in the form of fuzzy sets. Two new convolutions are defined for the operations between fuzzy granular vectors.
The design for convolutional classifiers of fuzzy granules
The classifiers in machine learning mainly deal with real values. Fuzzy granules of a fuzzy granular vector are sets. In this paper, we propose a fuzzy granular convolutional classifier, which relates to set operations. It has three steps: fuzzy granulation, granular learning and granular classification. The principle of fuzzy granular convolutional classifier is discussed below, and the specific algorithms of granular learning and granular classification are given.
Experimental analyses
In this paper, we use Breast Cancer, Glass, Iris and Wine as data sources to test the fuzzy granular convolutional classifier from four aspects: the parameter K, learning rate, convergence and classification accuracy. The description of four datasets is illustrated in Table 2, where F is the number of features; N means the number of samples; and C marks the number of labels. Due to different ranges of data, it is necessary to normalize the data sets. We use a maximum and minimum method to
Conclusion and future works
Traditional classifiers mainly tackle numerical values, which do not involve operations of sets. Starting from the study of fuzzy granulation of samples, this paper proposes a new classification model: the fuzzy granular convolutional classifier. Firstly, the fuzzy granulation is introduced to construct fuzzy granules and fuzzy granular vectors in a classification system. The size measures and operation rules of fuzzy granular vectors are defined. Then, two kinds of convolutional operations
Declaration of Competing Interest
We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled.
Acknowledgement
This work is supported by the National Natural Science Foundation of China (Nos. 61976183, 61672442), the Natural Science Foundation of Fujian Province (Nos. 2019J01850, 2016J01325) and the Science and Technology Planning Project of Fujian Province (No. 2020H0023).
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