Multivariate Hawkes processes (MHPs) are versatile probabilistic tools used to model various real-life phenomena: earthquakes, operations on stock markets, neuronal activity, virus propagation and many others. In this paper, we focus on MHPs with exponential decay kernels and estimate connectivity graphs, which represent the Granger causal relations between their components. We approach this inference problem by proposing an optimization criterion and model selection algorithm based on the minimum message length (MML) principle. MML compares Granger causal models using the Occam's razor principle in the following way: even when models have a comparable goodness-of-fit to the observed data, the one generating the most concise explanation of the data is preferred. While most of the state-of-art methods using lasso-type penalization tend to overfitting in scenarios with short time horizons, the proposed MML-based method achieves high F1 scores in these settings. We conduct a numerical study comparing the proposed algorithm to other related classical and state-of-art methods, where we achieve the highest F1 scores in specific sparse graph settings. We illustrate the proposed method also on G7 sovereign bond data and obtain causal connections, which are in agreement with the expert knowledge available in the literature.

Hawkes processes are a special class of temporal point processes which exhibit a natural notion of causality, as occurrence of events in the past may increase the probability of events in the future. Discovery of the underlying influence network among the dimensions of multi-dimensional temporal processes is of high importance in disciplines where a high-frequency data is to model, e.g. in financial data or in seismological data. This paper approaches the problem of learning Granger-causal network in multi-dimensional Hawkes processes. We formulate this problem as a model selection task in which we follow the minimum description length (MDL) principle. Moreover, we propose a general algorithm for MDL-based inference using a Monte-Carlo method and we use it for our causal discovery problem. We compare our algorithm with the state-of-the-art baseline methods on synthetic and real-world financial data. The synthetic experiments demonstrate superiority of our method in causal graph discovery compared to the baseline methods with respect to the size of the data. The results of experiments with the G-7 bonds price data are consistent with the experts’ knowledge.

Both clustering and outlier detection play an important role for meteorological measurements. We present the AWT algorithm, a clustering algorithm for time series data that also performs implicit outlier detection during the clustering. AWT integrates ideas of several well-known K-Means clustering algorithms. It chooses the number of clusters automatically based on a user-defined threshold parameter, and it can be used for heterogeneous meteorological input data as well as for data sets that exceed the available memory size. We apply AWT to crowd sourced 2-m temperature data with an hourly resolution from the city of Vienna to detect outliers and to investigate if the final clusters show general similarities and similarities with urban land-use characteristics. It is shown that both the outlier detection and the implicit mapping to land-use characteristic is possible with AWT which opens new possible fields of application, specifically in the rapidly evolving field of urban climate and urban weather.

Which joint interactions in the human gait cycle can be used as biometric characteristics? Most current methods on gait recognition suffer from the lack of inter-pretability. We propose an interpretable feature representation of gait sequences by the graphical Granger causal inference. Gait sequence of a person in the standardized motion capture format, constituting a set of 3D joint spatial trajectories, is envisaged as a causal system of joints interacting in time. We apply the graphical Granger model (GGM) to obtain the so-called Granger causal graph among joints as a discriminative and visually interpretable representation of a person's gait. We evaluate eleven distance functions in the GGM feature space by established classification and class-separability evaluation metrics. Our experiments indicate that, depending on the metric, the most appropriate distance functions for the GGM are the total norm distance and the Ky-Fan 1-norm distance. Experiments also show that the GGM is able to detect the most discriminative joint interactions and that it outperforms five related interpretable models in correct classification rate and in Davies-Bouldin index. The proposed GGM model can serve as a complementary tool for gait analysis in kinesiology or for gait recognition in video surveillance.

Causal inference by a graphical Granger model (GGM) among p variables is typically solved by p penalized linear regression problems in time series with a given lag. In practice however, the estimates of a penalized linear regression after a finite number of steps can be still far from the optimum. Furthermore, the selection of the regularization parameter, influencing the precision of the model is not trivial, especially when the corresponding design matrix is super-collinear. In this paper, for the first time we concept a graphical Granger model as an instance of combinatorial optimization. Computing maximum likelihood (ML) estimates of the regression coefficients and of the variance for each of p variables we propose an information-theoretic graphical Granger model (ITGGM). In the sense of information theory, the criterion to be minimized is the complexity of the class of the selected models together with the complexity of the data set. Following this idea, we propose four various information-theoretic (IT) objective functions based on stochastic complexity, on minimum message length, on Akaike and on Bayesian information criterion. To find their minima we propose a genetic algorithm operating with populations of subsets of regressor variables. The feature selection by the ITGGM with any of the functions is parameter-free in the sense that beside the ML estimates which are for each and within the model constant, no adjustable parameter is added into these objective functions. We further provide a theoretical analysis of the convergence properties of the GGM with the proposed IT functions. We test the performance of the functions in terms of F1 measure with respect to two common penalized GGMs on synthetic and real data. The experiments demonstrate the significant superiority of the IT criteria in terms of F1 measure over the two alternatives of the penalized GGM for Granger causal inference.