Supervised Estimation of Granger-Based Causality between Time Series

Abstract

Brain effective connectivity aims to detect causal interactions between distinct brain units and it is typically studied through the analysis of direct measurements of the neural activity, e.g., magneto/electroencephalography (M/EEG) signals. The literature on methods for causal inference is vast. It includes model-based methods in which a generative model of the data is assumed and model-free methods that directly infer causality from the probability distribution of the underlying stochastic process. Here, we firstly focus on the model-based methods developed from the Granger criterion of causality, which assumes the autoregressive model of the data. Secondly, we introduce a new perspective, that looks at the problem in a way that is typical of the machine learning literature. Then, we formulate the problem of causality detection as a supervised learning task, by proposing a classification-based approach. A classifier is trained to identify causal interactions between time series for the chosen model and by means of a proposed feature space. In this paper, we are interested in comparing this classification-based approach with the standard Geweke measure of causality in the time domain, through simulation study. Thus, we customized our approach to the case of a MAR model and designed a feature space which contains causality measures based on the idea of precedence and predictability in time. Two variations of the supervised method are proposed and compared to a standard Granger causal analysis method. The results of the simulations show that the supervised method outperforms the standard approach, in particular it is more robust to noise. As evidence of the efficacy of the proposed method, we report the details of our submission to the causality detection competition of Biomag2014, where the proposed method reached the 2nd place. Moreover, as empirical application, we applied the supervised approach on a dataset of neural recordings of rats obtaining an important reduction in the false positive rate.

Classification-Based Prediction of Effective Connectivity Between Timeseries With a Realistic Cortical Network Model

“…The remaining part of the dataset was used to estimate the quality of predictions, both for the proposed method and for the Granger Causality Analysis (GCA, see Barnett and Seth, 2014 ) method (see section 3), for comparison. In the second part of this section, we briefly present the proposed classification-based method for predicting causality, following Benozzo et al ( 2016 , 2017 ). Before the second part, we define the multivariate autoregressive (MAR) model, that we use for generating a second dataset to further characterize the proposed method, and a traditional causality measure: the Geweke index, on which GCA is based.…”

“…Recently, the machine learning literature has started to address the problem of causality (see Schölkopf et al, 2013 ). In Benozzo et al ( 2017 ), we presented the first method to detect causality among timeseries, based on supervised learning and tested on the MAR model. Here we summarize that method, which we propose to use in conjunction with the NN model of section 2.2, instead of the MAR model.…”

“…The method presented in this work builds on our previous work (see Benozzo et al, 2016 , 2017 ), where we developed a conceptually similar paradigm applied to a simple MAR model, rather than to a complex biophysical model of neural interactions. There, we proposed a novel feature space to encode the multivariate timeseries and showed that effective connectivity could be predicted at even better rates than those of state-of-the-art solutions.…”

“…In order to set up a classification problem, here we adopt the feature space that we designed in Benozzo et al ( 2016 , 2017 ), which is based on the Granger principles of temporal precedence and predictability. With such feature space, we obtain a more convenient representation of the multivariate timeseries generated by the neural network model.…”

“…In the remaining part of the article, we first describe the proposed neural network model derived from Mazzoni et al ( 2008 ), that we extended by adding connections between microcircuits rather than studying single isolated microcircuits as in the original model (see section 2). Then, we briefly describe the proposed method for detecting the pattern of effective connectivity from observed timeseries, following Benozzo et al ( 2016 , 2017 ). In section 3, we report the experiments conducted in this work to support our claims.…”