# Regularized estimation in sparse high-dimensional time series models

@article{Basu2013RegularizedEI, title={Regularized estimation in sparse high-dimensional time series models}, author={Sumanta Basu and George Michailidis}, journal={arXiv: Statistics Theory}, year={2013} }

Many scientific and economic problems involve the analysis of high-dimensional time series datasets. However, theoretical studies in high-dimensional statistics to date rely primarily on the assumption of independent and identically distributed (i.i.d.) samples. In this work, we focus on stable Gaussian processes and investigate the theoretical properties of $\ell _1$-regularized estimates in two important statistical problems in the context of high-dimensional time series: (a) stochastic…

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## References

SHOWING 1-10 OF 59 REFERENCES

Transition Matrix Estimation in High Dimensional Time Series

- Mathematics, Computer ScienceICML
- 2013

The results show that the spectral norm of the transition matrix plays a pivotal role in determining the final rates of convergence in the estimation of transition matrices under a high dimensional doubly asymptotic framework.

Modeling and Estimation of High-dimensional Vector Autoregressions.

- Computer Science
- 2014

This thesis explores modeling and estimation of highdimensional VAR from short panels of time series, with applications to reconstruction of gene regulatory network from time course gene expression data and proposes a thesholded group lasso regularization framework to incorporate a priori available pathway information in the model.

Estimation of (near) low-rank matrices with noise and high-dimensional scaling

- Mathematics, Computer ScienceICML
- 2010

Simulations show excellent agreement with the high-dimensional scaling of the error predicted by the theory, and illustrate their consequences for a number of specific learning models, including low-rank multivariate or multi-task regression, system identification in vector autoregressive processes, and recovery of low- rank matrices from random projections.

Covariance and precision matrix estimation for high-dimensional time series

- Mathematics
- 2013

We consider estimation of covariance matrices and their inverses (a.k.a. precision matrices) for high-dimensional stationary and locally stationary time series. In the latter case the covariance…

A unified framework for high-dimensional analysis of $M$-estimators with decomposable regularizers

- Computer Science, MathematicsNIPS
- 2009

A unified framework for establishing consistency and convergence rates for regularized M-estimators under high-dimensional scaling is provided and one main theorem is state and shown how it can be used to re-derive several existing results, and also to obtain several new results.

Sparse Vector Autoregressive Modeling

- Mathematics
- 2012

The vector autoregressive (VAR) model has been widely used for modeling temporal dependence in a multivariate time series. For large (and even moderate) dimensions, the number of the AR coefficients…

High-dimensional regression with noisy and missing data: Provable guarantees with non-convexity

- Computer Science, MathematicsNIPS
- 2011

This work is able to both analyze the statistical error associated with any global optimum, and prove that a simple algorithm based on projected gradient descent will converge in polynomial time to a small neighborhood of the set of all global minimizers.

Asymptotic Equivalence of Regularization Methods in Thresholded Parameter Space

- Mathematics
- 2013

High-dimensional data analysis has motivated a spectrum of regularization methods for variable selection and sparse modeling, with two popular methods being convex and concave ones. A long debate has…

Regularity Properties of High-dimensional Covariate Matrices ∗

- Mathematics
- 2013

Regularity properties such as the incoherence condition, the restricted isometry property, compatibility, restricted eigenvalue and lq sensitivity of covariate matrices play a pivotal role in…

Large Vector Auto Regressions

- Mathematics, Economics
- 2011

One popular approach for nonstructural economic and financial forecasting is to include a large number of economic and financial variables, which has been shown to lead to significant improvements…