Algorithm for the detection of outliers based on the theory of rough sets

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 and 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.

Quantum theory of spin waves in finite chiral spin chains

We calculate the effect of spin waves on the properties of finite-size spin chains with a chiral spin ground state observed on biatomic Fe chains deposited on iridium(001). The system is described with a Heisenberg model supplemented with a Dzyaloshinskii-Moriya coupling and a uniaxial single ion anisotropy that presents a chiral spin ground state. Spin waves are studied using the Holstein-Primakoff boson representation of spin operators. Both the renormalized ground state and the elementary excitations are found by means of Bogoliubov transformation, as a function of the two variables that can be controlled experimentally, the applied magnetic field and the chain length. Three main results are found. First, because of the noncollinear nature of the classical ground state, there is a significant zero-point reduction of the ground-state magnetization of the spin spiral. Second, there is a critical external field from which the ground state changes from chiral spin ground state to collinear ferromagnetic order. The character of the two lowest-energy spin waves changes from edge modes to confined bulk modes over this critical field. Third, in the spin-spiral state, the spin-wave spectrum exhibits oscillatory behavior as function of the chain length with the same period of the spin helix.