Minimax Estimation of Partially-Observed Vector AutoRegressions
- URL: http://arxiv.org/abs/2106.09327v1
- Date: Thu, 17 Jun 2021 08:46:53 GMT
- Title: Minimax Estimation of Partially-Observed Vector AutoRegressions
- Authors: Guillaume Dalle (CERMICS), Yohann de Castro (ICJ, ECL)
- Abstract summary: We study the properties of a partially-observed state-space model.
We describe a sparse estimator based on the Dantzig selector and upper bound its non-asymptotic error.
An application to open railway data highlights the relevance of this model for public transport traffic analysis.
- Score: 0.0
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: To understand the behavior of large dynamical systems like transportation
networks, one must often rely on measurements transmitted by a set of sensors,
for instance individual vehicles. Such measurements are likely to be incomplete
and imprecise, which makes it hard to recover the underlying signal of
interest.Hoping to quantify this phenomenon, we study the properties of a
partially-observed state-space model. In our setting, the latent state $X$
follows a high-dimensional Vector AutoRegressive process $X_t = \theta X_{t-1}
+ \varepsilon_t$. Meanwhile, the observations $Y$ are given by a
noise-corrupted random sample from the state $Y_t = \Pi_t X_t + \eta_t$.
Several random sampling mechanisms are studied, allowing us to investigate the
effect of spatial and temporal correlations in the distribution of the sampling
matrices $\Pi_t$.We first prove a lower bound on the minimax estimation error
for the transition matrix $\theta$. We then describe a sparse estimator based
on the Dantzig selector and upper bound its non-asymptotic error, showing that
it achieves the optimal convergence rate for most of our sampling mechanisms.
Numerical experiments on simulated time series validate our theoretical
findings, while an application to open railway data highlights the relevance of
this model for public transport traffic analysis.
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