Mahalanobis distance is a useful multivariate statistic for determining how far apart two points are from one another. It is a very helpful statistic with excellent uses in multivariate anomaly detection, one-class classification, and classification on severely unbalanced datasets.

Radial basis function (RBF) networks are often viewed as instable when used in multi-layered architectures and therefore are mostly used in a single-layered manner. Universal approximation theorems for single-layered RBF networks further render deeper architectures useless. However, deep neural networks have proven their effectiveness on many different tasks. We show that deeper RBF architectures with multiple radial basis function layers are achievable and able to learn. We introduce an initialization scheme for deep RBF networks based on k-means clustering and covariance estimation. We further show how to make use of convolutions to speed up the calculation of a Mahalanobis distance in a partially-connected fashion, similar to convolutional neural networks (CNNs). Finally, we evaluate our approach on image classification as well as speech emotion recognition tasks. Our results show that deep RBF networks perform similar to simple CNNs.

There are many applications of anomaly detection in IoT domain. IoT technology consists of large number of interconnecting digital devices not only generating huge data continuously but also making real-time computations. Since IoT devices are highly exposed due to Internet, they frequently meet with the challenges of illegitimate accesses in the form of intrusions, anomaly, fraud, etc. Identifying these illegitimate accesses in IoT domain can be an exciting research problem. In numerous applications fuzzy clustering and rough set theory have been successfully employed. As the data generated in IoT domains are high-dimensional, the clustering methods used for lower dimensional data cannot be applied efficiently. In this article, mixed approaches consisting of nano topology and fuzzy clustering techniques are proposed for anomaly detection. First of all, the nano topology is generated to find lower dimensional apace and then a couple of well-known fuzzy clustering techniques are employed on it for the efficient anomaly detection. The effectiveness of the proposed approaches is evaluated using time-complexity analysis, experimental studies with a synthetic dataset and a real-life dataset along with comparative studies with traditional fuzzy clustering approaches namely fuzzy c-means clustering (FCM) algorithm, Gustafson-Kessel (GK) Algorithm, Gath-Geva (GG) Algorithm, Mahalanobis Distance based Fuzzy C-Means algorithm (M-FCM), and Common Mahalanobis Distance based Fuzzy C-Means algorithm (CM-FCM). Experimentally, it has been found that the proposed approaches outperform the aforesaid algorithms in terms of detection rates, accuracy rates, false alarm rates and computational times.