Characterization and Inference of Graph Diffusion Processes from Observations of Stationary Signals

Many tools from the field of graph signal processing exploit knowledge of the underlying graph's structure (e.g., as encoded in the Laplacian matrix) to process signals on the graph. Therefore, in the case when no graph is available, graph signal processing tools cannot be used anymore. Researchers have proposed approaches to infer a graph topology from observations of signals on its nodes. Since the problem is ill-posed, these approaches make assumptions, such as smoothness of the signals on the graph, or sparsity priors. In this paper, we propose a characterization of the space of valid graphs, in the sense that they can explain stationary signals. To simplify the exposition in this paper, we focus here on the case where signals were i.i.d. at some point back in time and were observed after diffusion on a graph. We show that the set of graphs verifying this assumption has a strong connection with the eigenvectors of the covariance matrix, and forms a convex set. Along with a theoretical study in which these eigenvectors are assumed to be known, we consider the practical case when the observations are noisy, and experimentally observe how fast the set of valid graphs converges to the set obtained when the exact eigenvectors are known, as the number of observations grows. To illustrate how this characterization can be used for graph recovery, we present two methods for selecting a particular point in this set under chosen criteria, namely graph simplicity and sparsity. Additionally, we introduce a measure to evaluate how much a graph is adapted to signals under a stationarity assumption. Finally, we evaluate how state-of-the-art methods relate to this framework through experiments on a dataset of temperatures.Comment: Submitted to IEEE Transactions on Signal and Information Processing over Network

Tracking time evolving data streams for short-term traffic forecasting

YesData streams have arisen as a relevant topic during the last few years as an efficient method for extracting knowledge from big data. In the robust layered ensemble model (RLEM) proposed in this paper for short-term traffic flow forecasting, incoming traffic flow data of all connected road links are organized in chunks corresponding to an optimal time lag. The RLEM model is composed of two layers. In the first layer, we cluster the chunks by using the Graded Possibilistic c-Means method. The second layer is made up by an ensemble of forecasters, each of them trained for short-term traffic flow forecasting on the chunks belonging to a specific cluster. In the operational phase, as a new chunk of traffic flow data presented as input to the RLEM, its memberships to all clusters are evaluated, and if it is not recognized as an outlier, the outputs of all forecasters are combined in an ensemble, obtaining in this a way a forecasting of traffic flow for a short-term time horizon. The proposed RLEM model is evaluated on a synthetic data set, on a traffic flow data simulator and on two real-world traffic flow data sets. The model gives an accurate forecasting of the traffic flow rates with outlier detection and shows a good adaptation to non-stationary traffic regimes. Given its characteristics of outlier detection, accuracy, and robustness, RLEM can be fruitfully integrated in traffic flow management systems