# Learning Interpretable Deep State Space Model for Probabilistic Time Series Forecasting

arXiv: https://arxiv.org/abs/2102.00397 · HTML: https://mlsanity.com/paper/2102.00397/

## Key facts

- Cited by 29 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 35 references that are in the map.
- First posted on arXiv 31 Jan 2021.

## Builds on (most cited first, 13 of 13)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (2014)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [DeepAR: Probabilistic Forecasting with Autoregressive Recurrent Networks](https://mlsanity.com/paper/1704.04110/) (2017)
- [Importance Weighted Autoencoders](https://mlsanity.com/paper/1509.00519/) (2015)
- [A Recurrent Latent Variable Model for Sequential Data](https://mlsanity.com/paper/1506.02216/) (2015)
- [A Multi-Horizon Quantile Recurrent Forecaster](https://mlsanity.com/paper/1711.11053/) (2017)
- [Structured Inference Networks for Nonlinear State Space Models](https://mlsanity.com/paper/1609.09869/) (2016)
- [Deep Kalman Filters](https://mlsanity.com/paper/1511.05121/) (2015)
- [Sequential Neural Models with Stochastic Layers](https://mlsanity.com/paper/1605.07571/) (2016)
- [Variational inference for Monte Carlo objectives](https://mlsanity.com/paper/1602.06725/) (2016)
- [Identification of Gaussian Process State Space Models](https://mlsanity.com/paper/1705.10888/) (2017)
- [Variational Relevance Vector Machines](https://mlsanity.com/paper/1301.3838/) (2013)

## Built on by (most cited first, 29 of 29)

- [Diffusion-based Time Series Imputation and Forecasting with Structured State Space Models](https://mlsanity.com/paper/2208.09399/) (2022)
- [Large Pre-trained time series models for cross-domain Time series analysis tasks](https://mlsanity.com/paper/2311.11413/) (2023)
- [Multi-scale Attention Flow for Probabilistic Time Series Forecasting](https://mlsanity.com/paper/2205.07493/) (2022)
- [CAMul: Calibrated and Accurate Multi-view Time-Series Forecasting](https://mlsanity.com/paper/2109.07438/) (2021)
- [RNN with Particle Flow for Probabilistic Spatio-temporal Forecasting](https://mlsanity.com/paper/2106.06064/) (2021)
- [Traffic Prediction using Artificial Intelligence: Review of Recent Advances and Emerging Opportunities](https://mlsanity.com/paper/2305.19591/) (2023)
- [SSDNet: State Space Decomposition Neural Network for Time Series Forecasting](https://mlsanity.com/paper/2112.10251/) (2021)
- [A Systematic Review for Transformer-based Long-term Series Forecasting](https://mlsanity.com/paper/2310.20218/) (2023)
- [Hybrid Variational Autoencoder for Time Series Forecasting](https://mlsanity.com/paper/2303.07048/) (2023)
- [K2VAE: A Koopman-Kalman Enhanced Variational AutoEncoder for Probabilistic Time Series Forecasting](https://mlsanity.com/paper/2505.23017/) (2025)
- [A Gated Residual Kolmogorov-Arnold Networks for Mixtures of Experts](https://mlsanity.com/paper/2409.15161/) (2024)
- [When Rigidity Hurts: Soft Consistency Regularization for Probabilistic Hierarchical Time Series Forecasting](https://mlsanity.com/paper/2206.07940/) (2022)
- [A Temporal Linear Network for Time Series Forecasting](https://mlsanity.com/paper/2410.21448/) (2024)
- [Structured Kolmogorov-Arnold Neural ODEs for Interpretable Learning and Symbolic Discovery of Nonlinear Dynamics](https://mlsanity.com/paper/2506.18339/) (2025)
- [AA-Forecast: Anomaly-Aware Forecast for Extreme Events](https://mlsanity.com/paper/2208.09933/) (2022)
- [PEMS: Pre-trained Epidemic Time-series Models](https://mlsanity.com/paper/2311.07841/) (2023)
- [HiPPO-KAN: Efficient KAN Model for Time Series Analysis](https://mlsanity.com/paper/2410.14939/) (2024)
- [Deep Probabilistic Time Series Forecasting using Augmented Recurrent Input for Dynamic Systems](https://mlsanity.com/paper/2106.05848/) (2021)
- [Recurrence Boosts Diversity! Revisiting Recurrent Latent Variable in Transformer-Based Variational AutoEncoder for Diverse Text Generation](https://mlsanity.com/paper/2210.12409/) (2022)
- [DANLIP: Deep Autoregressive Networks for Locally Interpretable Probabilistic Forecasting](https://mlsanity.com/paper/2301.02332/) (2023)
- [Interpretable Latent Variables in Deep State Space Models](https://mlsanity.com/paper/2203.02057/) (2022)
- [Monitoring Time Series With Missing Values: a Deep Probabilistic Approach](https://mlsanity.com/paper/2203.04916/) (2022)
- [Why Did This Model Forecast This Future? Closed-Form Temporal Saliency Towards Causal Explanations of Probabilistic Forecasts](https://mlsanity.com/paper/2206.00679/) (2022)
- [Neural State-Space Modeling with Latent Causal-Effect Disentanglement](https://mlsanity.com/paper/2209.12387/) (2022)
- [When Rigidity Hurts: Soft Consistency Regularization for Probabilistic Hierarchical Time Series Forecasting](https://mlsanity.com/paper/2310.11569/) (2023)
- [STAN: Smooth Transition Autoregressive Networks](https://mlsanity.com/paper/2501.18699/) (2025)
- [Efficient Learning of Deep State Space Models via Importance Smoothing](https://mlsanity.com/paper/2605.21108/) (2026)
- [ProbRes: Volatility Learning for Probabilistic Time-Series Forecasting](https://mlsanity.com/paper/2606.02117/) (2026)
- [XMatch: Enhancing Covariate-Aware Time Series Forecasting through Tree-Structured Exogenous Matching](https://mlsanity.com/paper/2609.34939/) (2026)

## Abstract (arXiv)

Probabilistic time series forecasting involves estimating the distribution of future based on its history, which is essential for risk management in downstream decision-making. We propose a deep state space model for probabilistic time series forecasting whereby the non-linear emission model and transition model are parameterized by networks and the dependency is modeled by recurrent neural nets. We take the automatic relevance determination (ARD) view and devise a network to exploit the exogenous variables in addition to time series. In particular, our ARD network can incorporate the uncertainty of the exogenous variables and eventually helps identify useful exogenous variables and suppress those irrelevant for forecasting. The distribution of multi-step ahead forecasts are approximated by Monte Carlo simulation. We show in experiments that our model produces accurate and sharp probabilistic forecasts. The estimated uncertainty of our forecasting also realistically increases over time, in a spontaneous manner.