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.
Highlights
Matrix factorization is a very powerful tool in statistics and data analysis
There, matrix factorization is mainly a tool to estimate a coefficient matrix under a low-rank constraint
Nonnegative matrix factorization (NMF) was introduced by [35] as a tool to represent a huge number of objects as linear combinations of elements of “parts” of objects
Summary
Matrix factorization is a very powerful tool in statistics and data analysis. It was used as early as in the 70’s in econometrics in reduced-rank regression [27, 22, 29]. The factorization provides a decomposition of each series in a dictionary which member that can be interpreted as latent factors used for example in state-space models, see e.g. Chapter 3 in [33] For this reasons, matrix factorization was used in multivariate time series analysis beyond econometrics: electricity consumptions. Our estimator can be tuned to take into account a possible periodicity or smoothness of the series This is done by rewriting W = V Λ where Λ is a τ × T matrix encoding the temporal structure, and τ T.
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