# Detecting IoT Anomalies Using Fuzzy Subspace Clustering Algorithms
## 3 Proposed Methods
### Gustafson-Kessel (GK) Algorithm [42; 47]
Finally, any NT-based method is a combination of two algorithms where Algorithm 1 returns subspaces of the datasets. So, the run-time complexity of Algorithm 1 depends on the data size and dimensions. It is quadratic to the dataset size and linear to the dimension. Since the size of any dataset is bigger than its dimension size, and the dimension size of the subspace is quite small, the time complexity of all of the NT-based algorithms depends on the time complexity of Algorithm 1 and the dataset size. It has been found that the NT-M-FCM and NT-CM-FCM run in cubic time, but the others run in biquadratic time. ### Limitations and Lines for Future Works

Though the NT-based approaches are performing better than the traditional fuzzy clustering approaches, they are not free from limitations. Firstly, through using Algorithm 1, the computational cost of any NT-based algorithm can be reduced up to some extent, but they are still more expensive than non-fuzzy clustering, as they require optimization over multiple membership grades. Secondly, choosing the number of clusters and membership functions is the most challenging task which requires either trial/error approach or a domain expert. The future lines of work can be focused on the following. * In the future, the time attribute can be addressed separately to find fuzzy clusters along with lifetimes which may provide detailed insights of the IoT system. * In the future, detecting anomalies from high-dimensional data may be accomplished with an effective supervised approach. Conceptualization, M.S., F.A.M. and A.S.W. ; methodology, M.S., F.A.M. and A.S.W. ; software, M.S., F.A.M. and A.S.W. ; validation, M.S., F.A.M. and A.S.W. ; formal analysis, M.S. and F.A.M. ; investigation, M.S. and F.A.M. ; resources, M.S. and F.A.M. ; data curation, F.A.M. ; writing--original draft preparation, M.S., F.A.M. and A.S.W. ; writing--review and editing, M.S., F.A.M. and A.S.W. ; visualization, F.A.M., M.S. and A.S.W. ; supervision, F.A.M. ; project administration, F.A.M. ; funding acquisition, M.S. All authors have read and agreed to the published version of the manuscript. This research received no external funding. ## References

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