Effectively Modeling Time Series with Simple Discrete State Spaces

Michael Zhang, Khaled Kamal Saab, Michael Poli, Tri Dao, Karan Goel, Christopher Re

Keywords: time series, forecasting, state-space models, time series classification

Abstract: Time series modeling is a well-established problem, which often requires that methods (1) expressively represent complicated dependencies, (2) forecast long horizons, and (3) efficiently train over long sequences. State-space models (SSMs) are classical models for time series, and prior works combine SSMs with deep learning layers for efficient sequence modeling. However, we find fundamental limitations with these prior approaches, proving their SSM representations cannot express autoregressive time series processes. We thus introduce SpaceTime, a new state-space time series architecture that improves all three criteria. For expressivity, we propose a new SSM parameterization based on the companion matrix---a canonical representation for discrete-time processes---which enables SpaceTime's SSM layers to learn desirable autoregressive processes. For long horizon forecasting, we introduce a "closed-loop" variation of the companion SSM, which enables SpaceTime to predict many future time-steps by generating its own layer-wise inputs. For efficient training and inference, we introduce an algorithm that reduces the memory and compute of a forward pass with the companion matrix. With sequence length $\ell$ and state-space size $d$, we go from $\tilde{O}(d \ell)$ naïvely to $\tilde{O}(d + \ell)$. In experiments, our contributions lead to state-of-the-art results on extensive and diverse benchmarks, with best or second-best AUROC on 6 / 7 ECG and speech time series classification, and best MSE on 14 / 16 Informer forecasting tasks. Furthermore, we find SpaceTime (1) fits AR($p$) processes that prior deep SSMs fail on, (2) forecasts notably more accurately on longer horizons than prior state-of-the-art, and (3) speeds up training on real-world ETTh1 data by 73% and 80% relative wall-clock time over Transformers and LSTMs.

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Please Choose The Closest Area That Your Submission Falls Into: Deep Learning and representational learning

TL;DR: We propose SpaceTime, a deep state space time series model that achieves state-of-the-art results on forecasting and classification benchmarks, by improving expressiveness, forecasting flexibility, and training efficiency over prior approaches.

Community Implementations: [ 1 code implementation](https://www.catalyzex.com/paper/arxiv:2303.09489/code)