Studying Dynamics of User Behavior. Heterocedastic Time Series Forecasting and Clustering of Inhomogeneous Poisson Process

Complex time series patterns are generated by the behavior of a large number of different users in so the called question and answering web platforms. This calls for flexible, accurate and descriptive techniques for studying the dynamics of such systems. In this study, we extend the Sparse Input Gaussian Process formalism, in order to incorporate functional description of the input dependent noise. Such procedure also provides a regularization method that improves the accuracy of the predictions. We compare our results with the results of the other Gaussian Process methods, and apply the methodology to time series from the questions and answer web site Stackoverflow. For finding the common behavior between the users we propose the scale invariant Dynamic Piecewise Similarity measures an d the K-PSC clustering algorithm for clustering time series in order to provide much more descriptive cluster centroids then the centroids from the K-Means clustering algorithm.