Outliers are objects that show abnormal behavior with respect to their context or that have unexpected values in some of their parameters. In decision-making processes, information quality is of the utmost importance. In specific applications, an outlying data element may represent an important deviation in a production process or a damaged sensor. Therefore, the ability to detect these elements could make the difference between making a correct or an incorrect decision. This task is complicated by the large sizes of typical databases. Due to their importance in search processes in large volumes of data, researchers pay special attention to the development of efficient outlier detection techniques. This article presents a computationally efficient algorithm for the detection of outliers in large volumes of information. This proposal is based on an extension of the mathematical framework upon which the basic theory of detection of outliers, founded on Rough Set Theory, has been constructed. From this starting point, current problems are analyzed; a detection method is proposed, along with a computational algorithm that allows the performance of outlier detection tasks with an almost-linear complexity. To illustrate its viability, the results of the application of the outlier-detection algorithm to the concrete example of a large database are presented. 1.-Introduction Decision support systems are computer-based programs that assist decision makers in effective and efficient decision-making. The proper functioning of these systems requires large amounts of precise, high-quality data. However, if the data contain abnormal, unrealistic or simply erroneous elements, it may misguide the decision-making process, thereby leading to incorrect Highlights • We propose a formal expansion to the theory of rough sets. • We propose an efficient algorithm for the detection of outliers. • We have implemented the algorithm and verified the theoretical results.

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In a data mining process, outlier detection aims to use the high marginality of these elements to identify them by measuring their degree of deviation from representative patterns, thereby yielding relevant knowledge. Whereas rough sets (RS) theory has been applied to the field of knowledge discovery in databases (KDD) since its formulation in the 1980s; in recent years, outlier detection has been increasingly regarded as a KDD process with its own usefulness. The application of RS theory as a basis to characterise and detect outliers is a novel approach with great theoretical relevance and practical applicability. However, algorithms whose spatial and temporal complexity allows their application to realistic scenarios involving vast amounts of data and requiring very fast responses are difficult to develop. This study presents a theoretical framework based on a generalisation of RS theory, termed the variable precision rough sets model (VPRS), which allows the establishment of a stochastic approach to solving the problem of assessing whether a given element is an outlier within a specific universe of data. An algorithm derived from quasi-linearisation is developed based on this theoretical framework, thus enabling its application to large volumes of data. The experiments conducted demonstrate the feasibility of the proposed algorithm, whose usefulness is contextualised by comparison to different algorithms analysed in the literature.

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