ORIGINAL ARTICLE An information fusion technology for triadic decision contexts Yaqiang Tang 1 • Min Fan 1 • Jinhai Li 1 Received: 8 April 2015 / Accepted: 6 August 2015 / Published online: 22 August 2015 Ó Springer-Verlag Berlin Heidelberg 2015 Abstract In this paper, the notion of a projected context is proposed to explore a novel algorithm of computing triadic concepts of a triadic context, and a triadic decision context is defined by combining triadic contexts. Then a rule acquisition method is presented for triadic decision contexts. It can be considered as an information fusion technology for decision-making analysis of multi-source data if the data under each condition is viewed as a single- source data. Moreover, a knowledge reduction framework is established to simplify knowledge discovery. Finally, discernibility matrix and Boolean function are constructed to compute all reducts, which is beneficial to the acquisi- tion of compact rules from a triadic decision context. Keywords Triadic concept analysis  Triadic context  Triadic decision context  Rule acquisition  Information fusion 1 Introduction Formal concept analysis [1] mainly discusses how to obtain binary concepts and their hierarchy from a given binary relation between objects and attributes. Now, this theory has been applied to a variety of fields such as knowledge discovery, data mining and software engineering [2–8]. However, in the real world, the relation between objects and attribute is often established under certain conditions. In order to widen the application scope, Lehmann and Wille [9] extended classical formal concept analysis into triadic concept analysis. More studies on triadic concept analysis can be found in [10–15]. Formal decision context was proposed by Zhang and Qiu [16] in 2005 for decision-making analysis of the data, and in recent years it has been studied by many scholars. For example, Shao [17] and Qu et al. [18] discussed rule acquisition in formal decision contexts. Wei et al. [19] put forward two types of attribute reduction methods for for- mal decision contexts and Hong et al. [27] gave another one. Wang and Zhang [20] divided formal decision con- texts into two categories: consistent and inconsistent ones, and developed attribute reduction approaches for consistent decision contexts. Based on granular computing, Wu et al. [21] defined a new kind of consistent decision contexts. Shao et al [22] also investigated the issue of attribute reduction in consistent decision contexts from the view- point of rule acquisition. Considering that an inconsistent decision context appears more often than a consistent one, Li et al. [23] presented an attribute reduction technology for inconsistent decision contexts. Since computing a minimal reduct of a formal decision context is computa- tionally expensive, Li et al. [24] designed a heuristic attribute reduction method for formal decision contexts. For making better decision analysis, Li et al. [25] gave a rule acquisition oriented attribute reduction approach for general formal decision contexts. Also, it deserves to be mentioned that the issue of object reduction was discussed in [26] for formal decision contexts. Except the classical formal decision contexts, rule acquisition and attribute & Jinhai Li jhlixjtu@163.com Yaqiang Tang yaqiangtang1991@163.com Min Fan fanmin9412@sina.com 1 Faculty of Science, Kunming University of Science and Technology, Kunming 650500, Yunnan, People’s Republic of China 123 Int. J. Mach. Learn. & Cyber. (2016) 7:13–24 DOI 10.1007/s13042-015-0411-0