Graphical Modeling for Multivariate Hawkes Processes with Nonparametric Link Functions

Hawkes (1971) introduced a powerful multivariate point process model of mutually exciting processes to explain causal structure in data. In this paper it is shown that the Granger causality structure of such processes is fully encoded in the corresponding link functions of the model. A new nonparametric estimator of the link functions based on a time-discretized version of the point process is introduced by using an infinite order autoregression. Consistency of the new estimator is derived. The estimator is applied to simulated data and to neural spike train data from the spinal dorsal horn of a rat.Comment: 20 pages, 4 figure

A Primer on Causality in Data Science

Many questions in Data Science are fundamentally causal in that our objective is to learn the effect of some exposure, randomized or not, on an outcome interest. Even studies that are seemingly non-causal, such as those with the goal of prediction or prevalence estimation, have causal elements, including differential censoring or measurement. As a result, we, as Data Scientists, need to consider the underlying causal mechanisms that gave rise to the data, rather than simply the pattern or association observed in those data. In this work, we review the 'Causal Roadmap' of Petersen and van der Laan (2014) to provide an introduction to some key concepts in causal inference. Similar to other causal frameworks, the steps of the Roadmap include clearly stating the scientific question, defining of the causal model, translating the scientific question into a causal parameter, assessing the assumptions needed to express the causal parameter as a statistical estimand, implementation of statistical estimators including parametric and semi-parametric methods, and interpretation of our findings. We believe that using such a framework in Data Science will help to ensure that our statistical analyses are guided by the scientific question driving our research, while avoiding over-interpreting our results. We focus on the effect of an exposure occurring at a single time point and highlight the use of targeted maximum likelihood estimation (TMLE) with Super Learner.Comment: 26 pages (with references); 4 figure

Causal Discovery with Continuous Additive Noise Models

We consider the problem of learning causal directed acyclic graphs from an observational joint distribution. One can use these graphs to predict the outcome of interventional experiments, from which data are often not available. We show that if the observational distribution follows a structural equation model with an additive noise structure, the directed acyclic graph becomes identifiable from the distribution under mild conditions. This constitutes an interesting alternative to traditional methods that assume faithfulness and identify only the Markov equivalence class of the graph, thus leaving some edges undirected. We provide practical algorithms for finitely many samples, RESIT (Regression with Subsequent Independence Test) and two methods based on an independence score. We prove that RESIT is correct in the population setting and provide an empirical evaluation

Detecting language activations with functional magnetic resonance imaging

This thesis investigates a number of factors that affect sensitivity to language activations in functional Magnetic Resonance Imaging (fMRI). In the first part, I investigate the impact of experimental design parameters on the ability to detect language activations. These parameters include stimulus rate, stimulus duration, stimulus amplitude, epoch length and stimulus ordering. Crucially, they may affect sensitivity in multiple ways that include neurophysiological, efficiency-mediated and BOLD saturation effects. I illustrate and discuss these effects by presenting biophysical simulations and fMRI studies of single word and pseudoword reading. In addition, I focus on the differential effects of the above parameters in Positron Emission Tomography and fMRI studies. In the second part, I investigate the impact of the analysis used to estimate effects of interest from the data. I compare event-related and epoch analyses and show that, even in the context of blocked design fMRI, an event-related model may provide greater sensitivity than an epoch model. I then address the notion that experimentally-induced effects may be detected not only as task-dependent changes in regional responses but also as changes in connectivity amongst functionally connected regions. These two complementary approaches are motivated by two fundamental principles of brain organisation: functional specialisation and functional integration. I present two fMRI studies investigating the neural correlates of reading words and pseudowords in terms of functional specialisation and functional integration. Furthermore, in both studies I address the issue of inter-subject variability, which may be a critical determinant of sensitivity. Men