Transition Matrix Estimation in High Dimensional Time Series.

Transcript

1. Transition Matrix Estimation in High Dimensional Time Series http://proceedings.mlr.press/v28/han13a.pdf Presenter: Elizabeth Ramirez - @eramirem
2. whoami Electrical Engineer Digital Signal Processing Applied Mathematician Computational Science and Engineering Applied Scientist Commodities. Spend a lot of my time worrying about how to process remote sensing data computationally efficient.
4. what is a transition matrix Matrix whose product with the initial state vector produces the state vector at a later time. This is not the same as a stochastic matrix, which represents the transition probabilities in a Markov chain. The transition matrix is TIME INVARIANT.
5. what is a vector autoregressive (VAR) model Model that capture linear interdependencies among mutivariable timeseries. Each variable evolves based on its own lagged values, lagged values of other variables and error terms.
6. what is spectral norm The maximum singular value of a matrix, i.e. maximum scale by which the matrix can stretch a vector. A.K.A. natural norm induced by L2-norm
7. what is a stationary process Stochastic process in whose unconditional joint probability distribution does not change when shifted in time, i.e. mean and variance don't change over time. As you might imagine, many stochastic processes are non-stationary, but as we do in CSE world: non-linear becomes linear non-stationary becomes stationary Most common cause of stationarity violation: trend
8. what is marginal / lagged covariance Marginal: the opposite to conditional. Direct representation of the covariance of the timeseries, without the effects of other variables. Relies on the marginal distribution of the variables. Lagged: Covariance of time series with time-lagged time series
9. why we wanna estimate a transition matrix. You might know the equations that govern your system. If you know the dynamics of your system you should be able to calculate your matrix in a closed form. If you have non-linearities in your system you try to linearize and calculate the matrix. But if you HAVE NO IDEA of the dynamics of your system, you need a good estimator.
10. what we want to prove 1) The error of the proposed method calculating the transition matrix of a VAR model is smaller than the one obtained using LSE and ridge/lasso penalty 2) Than this error is bounded
11. 1. Introduction Time series where Follows a stationary process given by ,
12. 1. Introduction Optimization problem using Least Squares Estimator
13. 1. Introduction Build a new estimator that satisfies:
14. 2. Background Induced Matrix Norm
15. Proposition 2.1. Suppose we have Let: Then:
16. Proposition 2.1. Proof
17. 3. Methods and Algorithms Marginal and lag one covariance matrices:
18. 3. Methods and Algorithms Using Proposition 2.1. results subject to
19. 3. Methods and Algorithms Equivalent to calculate column by column: subject to
20. 3. Methods and Algorithms Vector decomposition subject to and
21. 3. Methods and Algorithms subject to
22. 4.1. Main Result Classes of Matrices
23. Theorem 4.1. With
24. thank you!