The increased availability of multivariate time-series asks for the development of suitable methods able to holistically analyse them. To this aim, we propose a novel flexible method for data-mining, forecasting and causal patterns detection that leverages the coupling of Hidden Markov Models and Gaussian Graphical Models. Given a multivariate non-stationary time-series, the proposed method simultaneously clusters time points while understanding probabilistic relationships among variables. The clustering divides the time points into stationary sub-groups whose underlying distribution can be inferred through a graphical model. Such coupling can be further exploited to build a time-varying regression model which allows to both make predictions and obtain insights on the presence of causal patterns. We extensively validate the proposed approach on synthetic data showing that it has better performance than the state of the art on clustering, graphical models inference and prediction. Finally, to demonstrate the applicability of our approach in real-world scenarios, we exploit its characteristics to build a profitable investment portfolio. Results show that we are able to improve the state of the art, by going from a -%20 profit to a noticeable 80%.

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The World-Wide-Web is a complex system naturally represented by a directed network of documents (nodes) connected through hyperlinks (edges). In this work, we focus on one of the most relevant topological properties that characterize the network, i.e. being scale-free. A directed network is scale-free if its in-degree and out-degree distributions have an approximate and asymptotic power-law behavior. If we consider the Web as a whole, it presents empirical evidence of such property. On the other hand, when we restrict the study of the degree distributions to specific sub-categories of websites, there is no longer strong evidence for it. For this reason, many works questioned the almost universal ubiquity of the scale-free property. Moreover, existing statistical methods to test whether an empirical degree distribution follows a power law suffer when dealing with large sample size and/or noisy data. In this paper, we propose an extension of a state-of-the-art method that overcomes such problems by applying a subsampling procedure on the graphs performing Random Walks (RW). We show on synthetic experiments that even small variations of true power-law distributed data causes the state-of-the-art method to reject the hypothesis, while the proposed method is more sound and stable under such variations. Lastly, we perform a study on 3 websites showing that indeed, depending on the sub-categories of website we consider, some accept and some refuse the hypothesis of being power-law. We argue that our method could be used to further explore sub-categories of websites in order to better characterize their topological properties deriving from different generative principles: central or peripheral.