### 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 *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.

An application of this method to acquire low-rank spectrotemporal decompositions was published at the IEEE Workshop on Statistical Signal Processing in 2016 and can be found here.