Time series analysis aims to extract meaningful information from data that has been generated in sequence by a dynamic process. The modelling of the non-linear dynamics of a signal is often performed using a linear space with a similarity metric which is either linear or attempts to model the non-linearity of the data in the linear space. In this research, a different approach is taken where the non-linear dynamics of the time series are represented using a phase space. Training data is used to construct the phase space in which the data lies on or close to a lower-dimensional manifold. The basis of the non-linear manifold is derived using the kernel principal components derived using kernel principal component analysis where fewer components are retained in order to identify the lower-dimensional manifold. Data instances are projected onto the manifold, and those with a large distance between the original point and the projection are considered to be derived from a different underlying process. The proposed algorithm is able to perform time series classification on univariate and multivariate data . Evaluations on a large number of real-world data sets demonstrate the accuracy of the new algorithm and how it exceeds state-of-the-art performance.

Principal component analysis and the residual error is an effective anomaly detection technique. In an environment where anomalies are present in the training set, the derived principal components can be skewed by the anomalies. A further aspect of anomaly detection is that data might be distributed across different nodes in a network and their communication to a centralized processing unit is prohibited due to communication cost. Current solutions to distributed anomaly detection rely on a hierarchical network infrastructure to aggregate data or models; however, in this environment, links close to the root of the tree become critical and congested. In this paper, an algorithm is proposed that is more robust in its derivation of the principal components of a training set containing anomalies. A distributed form of the algorithm is then derived where each node in a network can iterate towards the centralized solution by exchanging small matrices with neighboring nodes. Experimental evaluations on both synthetic and real-world data sets demonstrate the superior performance of the proposed approach in comparison to principal component analysis and alternative anomaly detection techniques. In addition, it is shown that in a variety of network infrastructures, the distributed form of the anomaly detection model is able to derive a close approximation of the centralized model.

Kernel principal component analysis and the reconstruction error is an effective anomaly detection technique for non-linear data sets. In an environment where a phenomenon is generating data that is non-stationary, anomaly detection requires a recomputation of the kernel eigenspace in order to represent the current data distribution. Recomputation is a computationally complex operation and reducing computational complexity is therefore a key challenge. In this paper, we propose an algorithm that is able to accurately remove data from a kernel eigenspace without performing a batch recomputation. Coupled with a kernel eigenspace update, we demonstrate that our technique is able to remove and add data to a kernel eigenspace more accurately than existing techniques. An adaptive version determines an appropriately sized sliding window of data and when a model update is necessary. Experimental evaluations on both synthetic and real-world data sets demonstrate the superior performance of the proposed approach in comparison to alternative incremental KPCA approaches and alternative anomaly detection techniques.

Anomaly detection in a Wireless Sensor Network is an important aspect of data analysis in order to facilitate intrusion and event detection. A key challenge is creating optimal classifiers constructed from training sets in which the anomaly rates are varying due to the existence of non-stationary distributions in the data. In this paper we propose an adaptive algorithm that can dynamically adjust the anomaly rate parameter, which can be represented by a model parameter of a one-class quarter-sphere support vector machine. This algorithm operates in an online, iterative manner producing an optimal model for a training set, which is presented sequentially. Our evaluations demonstrate that our algorithm is capable of constructing optimal models for a training set that minimizes the error rate on the classification set compared to a static model, where the anomaly rate is kept stationary.