Future Generation Computer Systems 83 (2018) 564–581 Contents lists available at ScienceDirect Future Generation Computer Systems journal homepage: www.elsevier.com/locate/fgcs Rule acquisition and optimal scale selection in multi-scale formal decision contexts and their applications to smart city Junping Xie a,d,∗ , Minhua Yang a,d , Jinhai Li b , Zhong Zheng c a School of Geosciences and Info-Physics, Central South University, Changsha, Hunan 410083, PR China b Faculty of Science, Kunming University of Science and Technology, Kunming, Yunnan 650500, PR China c College of Resources and Environment, Chengdu University of Information Technology, Chengdu, Sichuan 610103, PR China d Key Laboratory of Metallogenic Prediction of Nonferrous Metals and Geological Environment Monitoring (Central South University), Ministry of Education, PR China highlights • Three new types of rules and their extraction methods are explored. • Three new types of consistencies and their relationships are investigated. • Three new types of optimal scales and their relationships are discussed. • A method of optical scale selection is put forward. • The applications of the proposed methods to real world problems are given. article info Article history: Received 27 August 2016 Received in revised form 19 December 2016 Accepted 8 March 2017 Available online 23 March 2017 Keywords: Multi-scale formal decision context Concept lattice Rule acquisition Optimal scale selection abstract In order to enrich the existing rule acquisition theory in formal decision contexts, this study puts forward three new types of rules: decision association rules, non-redundant decision association rules and simplest decision association rules. Then, we analyze the relationship among these three types of rules, and develop methods to acquire them from single-scale formal decision contexts. Some numerical experiments are also conducted to compare the performance of the method of acquiring the simplest decision association rules with that of the existing one of acquiring the non-redundant decision rules. Moreover, the new three types of rules are employed to introduce three types of consistencies in multi- scale formal decision contexts. In addition, the notion of an optimal scale is defined by each type of consistency, and how to select an optimal scale is investigated as well. Finally, two applications in smart city for the proposed rule acquisition and optimal scale selection methods are applied to smart city. © 2017 Elsevier B.V. All rights reserved. 1. Introduction Formal concept analysis (FCA), proposed by Wille [1], is an ef- fective mathematical tool for knowledge processing and concep- tual data analysis. Formal contexts, formal concepts and concept lattices [2] are three basic notions in this theory. The concept lattice of a formal context reflects the relationship of generalization and specialization among the formal concepts. To handle uncertainty, FCA has been extended with fuzzy sets [3,4], interval-valued fuzzy ∗ Corresponding author at: School of Geosciences and Info-Physics, Central South University, Changsha, Hunan 410083, PR China. Fax: +86 0871 63965760. E-mail address: hnxiejunping@163.com (J. Xie). sets [5], bipolar fuzzy sets [6], linked fuzzy sets [7], possibility the- ory [8] and rough sets [9–12]. To handle incomplete data, large data and multi-source data, FCA has been augmented with incomplete contexts [13–16], granular computing [17,15,18–21], multi-scale contexts [22,23] and triadic contexts [24]. In recent years, knowl- edge reduction in FCA [25,26], three-way FCA [27–29], and concept learning [30,31] have received much attention. So far, its applica- tions cover many domains such as data mining [32], machine learn- ing [33] and knowledge discovery [34–36]. In FCA, a useful way of characterizing dependencies between the attributes of a formal context is via implications [37,38] or association rules [39]. Note that directly acquiring these types of rules from a formal context takes a lot of calculations and the number of rules is usually large. So, how to efficiently mine implications or association rules from a formal context and eliminate superfluous rules has been discussed http://dx.doi.org/10.1016/j.future.2017.03.011 0167-739X/© 2017 Elsevier B.V. All rights reserved.