With the successful application of granular computing in anomaly detection, a variety of tools including fuzzy information entropy can achieve superior detection results. However, fuzzy information entropy calculates fuzzy similarity through a global strategy, ignoring the local information in the data. To address this deficiency, this paper constructs a fuzzy$k$NN entropy theory and applies it to identify anomalies. Firstly, fuzzy$k$-similarity and fuzzy$k$NN are defined, and$k$NN entropy theory and the related information-theoretic metrics are proposed. Then, the relevant definitions and propositions of fuzzy$k$NN entropy, fuzzy$k$-joint entropy, fuzzy$k$-conditional information entropy, as well as fuzzy$k$-mutual information are elaborated. Based on the proposed theory, an anomaly detection model is constructed. At first, the fuzzy$k$-similarity relation matrix is constructed based on the fuzzy$k$-similarity in the proposed theory, and the relative fuzzy$k$NN entropy is calculated. Based on the relative fuzzy$k$NN entropy, the fuzzy$k$-relation anomaly degree is defined to characterize the anomaly intensity of fuzzy$k$NN information granules. Then, the anomaly factor based on fuzzy$k$NN entropy is built to represent the anomaly degree of data objects. Finally, the corresponding Fuzzy$k$NN Entropy-based Anomaly Detection algorithm (F$k$EAD) is designed. Comparative experiments are conducted with 11 state-of-the-art anomaly detection methods on thirty public datasets. The results reveal that the proposed method achieves better performance.

In information systems lacking decision-making information, effectively leveraging fuzzy rough sets for outlier detection in complex data is challenging, especially in capturing inherent uncertainty and multi-granularity characteristics to construct discriminative outlier scores. However, existing fuzzy rough sets-based outlier detection methods often suffer from three key limitations: (1) Local data distributions are often ignored when calculating fuzzy relation matrices, resulting in inaccurate fuzzy similarity representations; (2) Use of all objects in fuzzy upper and lower approximations can weaken noise resistance and increase computational complexity; (3) Single-granularity data processing reduces efficiency and may fail to capture the multi-granularity nature of data, thereby limiting the adaptability of these methods in complex data environments. To address these issues, we propose to fusesNatural neighbor fuzzy approximations withGranular-ball representation forOutlierDetection (NGOD), which integrates the multi-granularity granular-ball representation and fuzzy rough sets to improve the effectiveness and robustness of unsupervised outlier detection. Specifically, we first define a local distribution-aware fuzzy relation, enabling more discriminative similarity calculations between samples. To improve the effectiveness and robustness of fuzzy upper and lower approximations, we propose a multi-granularity natural neighbor fuzzy approximation model, which effectively utilizes the inherent uncertainty and local abnormal information of data in approximations. Moreover, by introducing natural neighbors, NGOD can adaptively capture local abnormal information in the data without setting neighborhoods manually. Finally, the outlier factors of each sample are calculated in NGOD to measure their outlier degrees. Extensive experiments on diverse datasets demonstrate that NGOD outperforms state-of-the-art methods, validating its superior performance and adaptability. The NGOD code and associated datasets are publicly available athttps://github.com/Mxeron/NGOD.

Outlier Detection (OD) has attracted extensive research due to its application in many fields. The idea of neighborhood computing is one of the widely used methods in outlier analysis. Nevertheless, these methods mainly use certainty strategies to model outlier detection, so they cannot effectively handle the fuzzy information in the dataset. Moreover, they mainly focus on dealing with outlier detection in numerical data and cannot effectively find outliers in mixed-attribute data. Fuzzy information granulation theory is an effective granular computing model that allows objects to belong to a set to a certain extent (i.e., membership degree), which makes it possible to better handle uncertainty problems such as fuzziness. In this work, we propose an outlier detection model based on fuzzy neighborhoods. First, a hybrid fuzzy similarity is constructed to granulate the set of objects to form fuzzy information granules. Second, the fuzzy$k$-nearest neighbor is defined to describe the fuzzy local information. Then, the fuzzy neighborhood density is defined to indicate the degree of aggregation of each object. The smaller the fuzzy neighborhood density of an object, the more likely it is to be an outlier. Based on this idea, the fuzzy neighborhood deviation degree is defined to quantify the degree of outliers of objects. Finally, the fuzzy deviation degree on the set of conditional attributes is constructed to indicate the outlier scores of objects. Experimental comparisons with state-of-the-art methods show that the proposed method has a significant improvement on the AUC index and applies to three types of data.

Outliers carry significant information to reflect an anomaly mechanism, so outlier detection facilitates relevant data mining. In terms of outlier detection, the classical approaches from distances apply to numerical data rather than nominal data, while the recent methods on basic rough sets deal with nominal data rather than numerical data. Aiming at wide outlier detection on numerical, nominal, and hybrid data, this paper investigates three-way neighborhood characteristic regions and corresponding fusion measurement to advance outlier detection. First, neighborhood rough sets are deepened via three-way decision, so they derive three-way neighborhood structures on model boundaries, inner regions, and characteristic regions. Second, the three-way neighborhood characteristic regions motivate the information fusion and weight measurement regarding all features, and thus, a multiple neighborhood outlier factor emerges to establish a new method of outlier detection; furthermore, a relevant outlier detection algorithm (called 3WNCROD) is designed to comprehensively process numerical, nominal, and mixed data. Finally, the 3WNCROD algorithm is experimentally validated, and it generally outperforms 13 contrast algorithms to perform better for outlier detection.