# SAMoSSA: Multivariate Singular Spectrum Analysis with Stochastic Autoregressive Noise

arXiv: https://arxiv.org/abs/2305.16491 · HTML: https://mlsanity.com/paper/2305.16491/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 33 references that are in the map.
- First posted on arXiv 25 May 2023.

## Builds on (most cited first, 8 of 8)

- [DeepAR: Probabilistic Forecasting with Autoregressive Recurrent Networks](https://mlsanity.com/paper/1704.04110/) (2017)
- [Learning Without Mixing: Towards A Sharp Analysis of Linear System Identification](https://mlsanity.com/paper/1802.08334/) (2018)
- [Near optimal finite time identification of arbitrary linear dynamical systems](https://mlsanity.com/paper/1812.01251/) (2018)
- [GluonTS: Probabilistic Time Series Models in Python](https://mlsanity.com/paper/1906.05264/) (2019)
- [Finite-time Identification of Stable Linear Systems: Optimality of the Least-Squares Estimator](https://mlsanity.com/paper/2003.07937/) (2020)
- [Sample Complexity Lower Bounds for Linear System Identification](https://mlsanity.com/paper/1903.10343/) (2019)
- [Model Agnostic Time Series Analysis via Matrix Estimation](https://mlsanity.com/paper/1802.09064/) (2018)
- [A Finite-Sample Deviation Bound for Stable Autoregressive Processes](https://mlsanity.com/paper/1912.08103/) (2019)

## Built on by (most cited first, 1 of 1)

- [Counterfactual Forecasting for Panel Data](https://mlsanity.com/paper/2511.06189/) (2025)

## Abstract (arXiv)

The well-established practice of time series analysis involves estimating deterministic, non-stationary trend and seasonality components followed by learning the residual stochastic, stationary components. Recently, it has been shown that one can learn the deterministic non-stationary components accurately using multivariate Singular Spectrum Analysis (mSSA) in the absence of a correlated stationary component; meanwhile, in the absence of deterministic non-stationary components, the Autoregressive (AR) stationary component can also be learnt readily, e.g. via Ordinary Least Squares (OLS). However, a theoretical underpinning of multi-stage learning algorithms involving both deterministic and stationary components has been absent in the literature despite its pervasiveness. We resolve this open question by establishing desirable theoretical guarantees for a natural two-stage algorithm, where mSSA is first applied to estimate the non-stationary components despite the presence of a correlated stationary AR component, which is subsequently learned from the residual time series. We provide a finite-sample forecasting consistency bound for the proposed algorithm, SAMoSSA, which is data-driven and thus requires minimal parameter tuning. To establish theoretical guarantees, we overcome three hurdles: (i) we characterize the spectra of Page matrices of stable AR processes, thus extending the analysis of mSSA; (ii) we extend the analysis of AR process identification in the presence of arbitrary bounded perturbations; (iii) we characterize the out-of-sample or forecasting error, as opposed to solely considering model identification. Through representative empirical studies, we validate the superior performance of SAMoSSA compared to existing baselines. Notably, SAMoSSA's ability to account for AR noise structure yields improvements ranging from 5% to 37% across various benchmark datasets.