### Time-Varying Causal Inference

We consider the questions:

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

Using a Granger causality viewpoint, at every time , the causal influence from to is meant to represent how much better we can predict given its past *and* the past of than if we were given the past of 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]