Nearly assumptionless screening for the mutually-exciting multivariate Hawkes process

Local Independence Testing for Point Processes

Constraint-based causal structure learning for point processes require empirical tests of local independence. Existing tests require strong model assumptions, e.g., that the true data generating model is a Hawkes process with no latent confounders. Even when restricting attention to Hawkes processes, latent confounders are a major technical difficulty because a marginalized process will generally not be a Hawkes process itself. We introduce an expansion similar to Volterra expansions as a tool to represent marginalized intensities. Our main theoretical result is that such expansions can approximate the true marginalized intensity arbitrarily well. Based on this, we propose a test of local independence and investigate its properties in real and simulated data.

Neuronal Network Inference and Membrane Potential Model using Multivariate Hawkes Processes

BACKGROUND

In this work, we propose to catch the complexity of the membrane potential's dynamic of a motoneuron between its spikes, taking into account the spikes from other neurons around. Our approach relies on two types of data: extracellular recordings of multiple spikes trains and intracellular recordings of the membrane potential of a central neuron.

NEW METHOD

We provide a unified framework and a complete pipeline to analyze neuronal activity from data extraction to statistical inference. To the best of our knowledge, this is the first time that a Hawkes-diffusion model is investigated on such complex data. The first step of the proposed procedure is to select a subnetwork of neurons impacting the central neuron using a multivariate Hawkes process. Then we infer a jump-diffusion dynamic in which jumps are driven from a Hawkes process, the occurrences of which correspond to the spike trains of the aforementioned subset of neurons that interact with the central neuron.

RESULTS

From the Hawkes estimation step we recover a small connectivity graph which contains the central neuron, and we show that taking into account this information improves the inference of membrane potential through the proposed jump-diffusion model. A goodness of fit test is applied to validate the relevance of the Hawkes model in such context.

COMPARISON WITH EXISTING METHODS

We compare an empirical inference method and two sparse estimation procedures based on the Hawkes assumption for the reconstruction of the connectivity graph using the spike-trains. Then, the Hawkes-diffusion model is competed with the simple diffusion in terms of best fit to describe the behavior of the membrane potential of a central neuron surrounded by a network.

CONCLUSIONS

The present method takes advantage of both spike trains and membrane potential to understand the behavior of a fixed neuron. The entire code has been developed and is freely available on GitHub.

Granger Causality: A Review and Recent Advances

Introduced more than a half-century ago, Granger causality has become a popular tool for analyzing time series data in many application domains, from economics and finance to genomics and neuroscience. Despite this popularity, the validity of this framework for inferring causal relationships among time series has remained the topic of continuous debate. Moreover, while the original definition was general, limitations in computational tools have constrained the applications of Granger causality to primarily simple bivariate vector autoregressive processes. Starting with a review of early developments and debates, this article discusses recent advances that address various shortcomings of the earlier approaches, from models for high-dimensional time series to more recent developments that account for nonlinear and non-Gaussian observations and allow for subsampled and mixed-frequency time series.

Joint Structural Break Detection and Parameter Estimation in High-Dimensional Non-Stationary VAR Models

Assuming stationarity is unrealistic in many time series applications. A more realistic alternative is to assume piecewise stationarity, where the model can change at potentially many change points. We propose a three-stage procedure for simultaneous estimation of change points and parameters of high-dimensional piecewise vector autoregressive (VAR) models. In the first step, we reformulate the change point detection problem as a high-dimensional variable selection one, and solve it using a penalized least square estimator with a total variation penalty. We show that the penalized estimation method over-estimates the number of change points, and propose a selection criterion to identify the change points. In the last step of our procedure, we estimate the VAR parameters in each of the segments. We prove that the proposed procedure consistently detects the number and location of change points, and provides consistent estimates of VAR parameters. The performance of the method is illustrated through several simulated and real data examples.