Matrix factorization for multivariate time series analysis

Abstract

Matrix factorization is a powerful data analysis tool. It has been used in multivariate time series analysis, leading to the decomposition of the series in a small set of latent factors. However, little is known on the statistical performances of matrix factorization for time series. In this paper, we extend the results known for matrix estimation in the i.i.d setting to time series. Moreover, we prove that when the series exhibit some additional structure like periodicity or smoothness, it is possible to improve on the classical rates of convergence.Acknowledgements. The authors gratefully acknowledge Maxime Ossonce for discussions which helped to improve the paper. The first author was working at CREST, EN-SAE Paris when this paper was written; he gratefully acknowledges financial support from Labex ECODEC (ANR-11-LABEX-0047).

Tight risk bound for high dimensional time series completion

“…Under the assumption that the univariate series are φ-mixing, we prove that we can reconstruct the matrix with a similar error than in the i.i.d case in [30]. If, moreover, the time series has some additional properties, as the ones studied in [3] (periodicity or smoothness), the error can even be smaller than in the i.i.d case. This is confirmed by a short simulation study.…”

“…The noise can exhibit some temporal dependence: ε j,t will not be independent from ε j,t in general. Moreover, as discussed in [3], Θ 0 j,. can have some more structure: Θ 0 j,.…”

“…Other papers focused on a simpler model where the series is represented by a deterministic low-rank trend matrix plus some possibly correlated noise. This model was used by [51] to perform prediction, and studied in [3].…”

Tight Risk Bound for High Dimensional Time Series Completion

“…Under the assumption that the univariate series are φ-mixing, we prove that we can reconstruct the matrix with a similar error than in the i.i.d case in [28]. If, moreover, the time series has some additional properties, as the ones studied in [3] (periodicity or smoothness), the error can even be smaller than in the i.i.d case. This is confirmed by a short simulation study.…”

“…Other papers focused on a simpler model where the series is represented by a deterministic low-rank trend matrix plus some possibly correlated noise. This model was used by [45] to perform prediction, and studied in [3]. It is thus tempting to use low-rank matrix completion algorithms to recover partially observed highdimensional time series, and this was indeed done in many applications: [44,42,18] used low-rank matrix completion to reconstruct data from multiple sensors.…”