### Time-Varying Causal Inference

We consider the questions:

1. Do two observed time series have a causal influence on one another?
2. How do these influences change over time?

Using a Granger causality viewpoint, at every time $t$, the causal influence from $X$ to $Y$ is meant to represent how much better we can predict $Y$ given its past and the past of $X$ than if we were given the past of $Y$ alone. To actually acquire these measures, we leverage tools in sequential prediction and directed information.

### Non-Linear / Non-Markov Latent Time-Series Estimation

We consider the problem of estimating a latent multi-dimensional time-series given noisy measurements and knowledge of the dynamics of the signal. In the case of a Markov signal with linear dynamics and Gaussian measurements, the problem can be solved using the Kalman filter.

We consider the class of problems where the underlying signal is non-Markov and/or the measurements obey and arbitrary log-concave likelihood model. We propose a framework that uses the Alternating Direction Method of Multipliers to decompose problems of this nature into smaller, easy to solve subproblems.

Relevant Publications:

• “A Modularized Efficient Framework for Non-Markov Time Series Estimation”, IEEE Transactions on Signal Processing, In Press. [arXiv] [Code Ocean]
• “Efficient Low-Rank Spectrotemporal Decomposition using ADMM”, IEEE Statistical Signal Processing Workshop, June 2016. [IEEE Xplore]