Generic normal cloud model
Introduction
The representation of knowledge is a primary bottleneck in data ming and knowledge discovery. The best representation form should be natural language, or, to say the least, rules with linguistic values so that it can be better understood by human being [1], [6], [10], [28]. The natural language is the foundation of human thinking and the embodiment of human intelligence. Human cognition process is based on language and thinking, and concept could be considered as a basic unit of natural language and thinking. Many concepts in natural language are uncertain. Among various uncertainties involved in natural language, such as randomness, fuzziness, incompleteness, and imprecision, randomness and fuzziness are the two most important and have attracted more concerns of people [10], [11], [23], [36]. These uncertainties are difficult to be precisely defined, but they do not affect people’s communication [4], [8], [20], [23], [35]. For example, people do not need to know the absolutely accurate weather temperature in daily life. Without language, numbers make no sense. In the processes of human perception, identification, reasoning, decision-making, as well as abstracting, etc., people can receive, store and deal with various uncertainty information with equanimity [7], [30]. However, it is necessary to study the randomness and fuzziness of natural languages in order to make the computers have understanding and judgement abilities similar to those of humans, especially the formal representation of knowledge [1], [10], [13], [12], [28].
Probability theory and fuzzy set theory are the most two effective tools in the study ofuncertainty knowledge representation [22], [32], [38]. The probability theory mainly researches random phenomenon and can well deal with random uncertainty. In probability theory, normal distribution is the most important probability distribution and can be used as an approximation to a large number of random phenomena [32]. Since professor Zadeh proposed fuzzy sets in 1965 [38], fuzzy sets theories have been the main tools dealing with fuzzy uncertainty and have got many great achievements in theory and applications [22]. As the foundation of fuzzy sets, membership functions are used to measure fuzzy degree, and normal (Gaussian) membership function is regarded as one of the most suitable membership functions for many fuzzy concepts. Considering the randomness of membership degree, professor Li proposed cloud models as a new cognition model of uncertainty based on probability theory and fuzzy sets theory in 1995 [10], [14]. In cloud model theory, it is possible to measure the deviation of a random phenomenon from a normal distribution when the random phenomenon does not satisfy strictly a normal distribution, and it is more feasible to allow a stochastic disturbance of the membership degree encircling a determined central value. Meanwhile, cloud models can formally describe the inherent relation between randomness and fuzziness [10].
Cloud models use three numerical characteristics, namely Ex (expectation), En (entropy) and He (hyper entropy), to depict the intension of a concept, which just accords with human thought [10], [13], [12], [14], [28], [36], wherein, Ex is the most representative sample of a concept; En is used to figure the granularity scale of the concept; He is used to depict the uncertainty of the concept granularity. From the viewpoint of fuzzy set, the expectation Ex is the expected sample of a concept with membership degree 1; the entropy En is used to depict the uncertainty of samples in the concept, which can be used to calculate the membership degree, and the hyper entropy He is used to depict the uncertainty of the membership degree. Cloud models make it possible to get the distributing range of a qualitative concept. Meanwhile, two cloud transformations, namely forward cloud transformation (FCT) and backward cloud transformation (BCT), are used to realize the bidirectional cognitive transformation between the intension and extension of a concept [25], where, FCT is used to implement the transformation from intension to extension of a concept, and BCT realizes the transformation from extension to intension. The two transformations also provide a way to simulate the human cognition process for concepts by computer.
At present, normal cloud models based on normal distribution function and Gaussian membership function are the most important cloud models. Especially, the -order normal cloud model with the three numerical characteristics has been widely studied and applied into intelligent control [5], [9], [39], data mining [18], [31], system evaluation [15], qualitative evaluation [19], image segmentation [21], [33], multi-criterion group decision making [28], [36], and so on. The -order forward normal cloud transformation algorithm automatically generates sample data (called cloud drops) by two normal random number generators according to the three characteristics (). The distribution of cloud drops has many excellent mathematical properties [10], [17]. In this paper, a generic normal cloud model is proposed, and the main contents and contributions are as follows. First, a -order generic normal cloud model, which establishes a relationship between normal cloud and normal distribution, is proposed, and the -order generic forward normal cloud transformation algorithm (-GFCT) is presented, which reveals that the -order normal cloud and the normal distribution are both special cases of the -order generic normal cloud. Second, following the mutually inverse features of FCT and BCT, an ideal backward cloud transformation algorithm of the -order generic normal cloud model (-GIBCT) is developed, in which the prior knowledge of the distribution of all the cloud drops generated by the -GFCT is used. After that, a -order generic backward cloud transformation algorithm (-GBCT), which does not require the prior knowledge of the distribution of cloud drops, is proposed to solve real life problems since it is impossible to know the distribution of all the cloud drops in advance in real life applications. The relationships between the -GIBCT and the -GBCT are studied, which help reach the finding that the two backward cloud transformation algorithms presented by Wang and Xu [26], [34] are two special cases of the -GBCT. Third, according to the recursive definition of -order normal cloud proposed by Wang [27], the -order generic normal cloud model is further generalized into a -order generic normal cloud model, and the -order generic forward normal cloud transformation algorithm (-GFCT) and backward cloud transformation algorithm (-GBCT) are presented. Finally, the comparative illustration for the -GIBCT and the -GBCT is made through experiments, and the results show that that the -GBCT can be used to approximately calculate the unknown parameters of the -order generic normal cloud in view of the fact that the -GIBCT does not exist in in real life applications. The effectiveness of the -GBCT is verified by the results of image segmentation.
The remainder of this paper is organized as follows. In Section 2, some basic concepts regarding the -order normal cloud and its algorithm are briefly reviewed. Section 3 proposes -GFCT, -GIBCT and -GBCT, and the relationship between the -GIBCT and the -GBCT is also analyzed in more detail. In Section 4, the -order generic normal cloud, including -GFCT and -GBCT, is proposed. Section 5 provides two experiments to show the effectiveness of the proposed method. Final conclusions are presented in Section 6.
Section snippets
-Order normal cloud
A concept is composed of its intension and extension, where the intension refers to the sum of the essential attributes of a concept reflecting the nature of things, that is, the content of the concept, while the extension refers to the set of all instances of the concept. In cloud model, the intension of a concept is depicted and expressed by some numerical characteristics, and its extension is the set of all the cloud drops with different certainty degrees. Therefore, we can use the numerical
-Order generic forward normal cloud transformation algorithm
Algorithm 1 shows that each cloud drop is generated by two normal random numbers, and is the input to generate . Therefore, n cloud drops are generated by n normal distributions , wherein each is generated by the normal distribution . In Algorithm 1, we find that Step 1 only generates one random number being related with En and He each time, and for each , there is only one random number (cloud drop) be obtained in Step 2. Based on this
-Order generic normal cloud
The -order cloud model uses three numerical characteristics to represent the intension of a concept. If the intension of a concept is represented by multiple numerical characteristics, it will be got the higher-order cloud model. This section will propose the -order generic normal cloud model which is expressed by numerical characteristics.
Ref. [27] proposed the recursive definition of -order normal cloud model and discussed its statistical properties. The recursive
Experiment analysis
The differences between the -GIBCT and the -GBCT will be illustrated and compared by the following experiments, and the -GBCT will be applied to image segmentation.
Conclusions
The cloud model realizes the bidirectional cognitive transformation between the intension and extension of a concept through cloud transformation. It provides a way of using the computer to research the human cognition process for concept. As one of the most important models, the -order normal cloud model has been used in many applications and research. In this paper, the -order generic normal cloud model is proposed. Meanwhile, the -GFCT, which establishes the relationship between the
Acknowledgements
The authors would like to thank the anonymous reviewers and the editor for their constructive and valuable comments. This work is supported by the Natural Science Foundation of China under Grant (No. 61272060), the Hundred Talents Program of Chinese Academy of Sciences, the Key Natural Science Foundation of Chongqing of China under Grant (No. CSTC2013jjB40003), and Chongqing Key Laboratory of Computational Intelligence (No. CQ-LCI-2013-08).
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