# Detecting IoT Anomalies Using Fuzzy Subspace Clustering Algorithms
## 1 Introduction
Also, the traditional \(k\)-means algorithm has a wide range of applications, but it is not free from difficulties such as difficulties in determining the number of clusters, sensitivity to initial cluster-centers, low accuracy rate, etc. Some of the aforesaid issues were addressed nicely in [14; 36; 37; 38; 39; 40] up to some extent. But there is still room for improvement. Anomaly detection models based on the fuzzy \(c\)-means clustering algorithm [32; 41; 42; 43; 44; 45; 46; 47] can be a better solution for the aforesaid issues for three primary reasons. Firstly, fuzzy clustering allows for the overlapping of clusters useful in dealing with the complex structure or ambiguity or overlapping class boundaries available in the datasets. Secondly, they are more robust to the anomalies and noise, as the transition from one cluster to another is gradual. Thirdly, because it enables a more thorough depiction of the relationship between data points and clusters, fuzzy clustering offers a more nuanced view of the data's structure. In [48], the authors proposed a new algorithm, MSRFCM (Mahalanobis Shadowed Rough Fuzzy Clustering Method), which uses Mahalanobis distance to improve the accuracy of intrusion detection. Using the principal component analysis for selecting most discriminative features, a fuzzy \(c\)-means clustering approach was presented in [41] for intrusion detection in network data. In any IoT application, the data are high-dimensional. Also, the computation of high-dimensional correlation matrices for Mahalanobis distance is almost impossible, so it does not work well for high-dimensional data. In this article, most of the shortcomings of all of the aforesaid methods are addressed in an efficient manner, and some hybrid approaches are proposed which use nano topology and a couple of fuzzy clustering algorithms for the efficient detection of anomalies in high dimensional IoT data. This paper's objective is described as follows:  
* A nano topology [9; 49] along with its basis is constructed to identify a subspace using the Nano Topology-based Subspace Generation Algorithm. * Secondly, a couple of well-known fuzzy clustering approaches are proposed to generate soft clusters. * A comparative analysis is conducted among all of the proposed approaches along with the traditional fuzzy clustering approaches. The methods initially find a smaller dimensional space by deleting unnecessary features using a rough set theoretical approach. Then, fuzzy clustering-based approaches, namely the fuzzy \(c\)-means algorithm (FCM) [41; 47], Gustafson-Kessel Algorithm (GK) [42; 43; 44; 45; 46], Gath-Geva Algorithm (GG) [43; 47], Mahalanobis Distance-based Fuzzy \(c\)-Means algorithm (M-FCM) [47], and Common Mahalanobis Distance-based Fuzzy \(c\)-Means algorithm (CM-FCM) [47], are used on it to identify the fuzzy clusters. The approaches time complexities are also calculated. The proposed approaches are then tested with the help of MATLAB and the datasets KDDCup'99 [50] and Kitsune Network Attack Dataset [51] ([[https://github.com/ymirsky/Kitsune-py](https://github.com/ymirsky/Kitsune-py)]([https://github.com/ymirsky/Kitsune-py](https://github.com/ymirsky/Kitsune-py)) (accessed on 12 December 2021)), and comparisons are also made. The results convincingly show that the nano topology-based approaches outperform the traditional fuzzy clustering approaches in many parameters, and the nano topology-based CM-FCM (NT-CM-FCM) is the most efficient one. The paper is prescribed in the following manner. The problem statement is presented in Section 2. The proposed methods are discussed in Section 3. The complexity analysis of the methods is presented in Section 4. The experimental results and discussions are presented in Section 5, and the paper's conclusions, limitations, and recommendations for further research are presented in Section 6.