# Time Series Forecasting Using Manifold Learning

arXiv: https://arxiv.org/abs/2110.03625 · HTML: https://mlsanity.com/paper/2110.03625/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 102 references that are in the map.
- First posted on arXiv 7 Oct 2021.

## Builds on (most cited first, 10 of 10)

- [LSTM: A Search Space Odyssey](https://mlsanity.com/paper/1503.04069/) (2015)
- [Data-Driven Forecasting of High-Dimensional Chaotic Systems with Long Short-Term Memory Networks](https://mlsanity.com/paper/1802.07486/) (2018)
- [Backpropagation Algorithms and Reservoir Computing in Recurrent Neural Networks for the Forecasting of Complex Spatiotemporal Dynamics](https://mlsanity.com/paper/1910.05266/) (2019)
- [SciPy 1.0--Fundamental Algorithms for Scientific Computing in Python](https://mlsanity.com/paper/1907.10121/) (2019)
- [Coarse-scale PDEs from fine-scale observations via machine learning](https://mlsanity.com/paper/1909.05707/) (2019)
- [Molecular enhanced sampling with autoencoders: On-the-fly collective variable discovery and accelerated free energy landscape exploration](https://mlsanity.com/paper/1801.00203/) (2017)
- [Time series forecasting with Gaussian Processes needs priors](https://mlsanity.com/paper/2009.08102/) (2020)
- [A Nonlinear Dimensionality Reduction Framework Using Smooth Geodesics](https://mlsanity.com/paper/1707.06757/) (2017)
- [Inverting Nonlinear Dimensionality Reduction with Scale-Free Radial Basis Function Interpolation](https://mlsanity.com/paper/1305.0258/) (2013)
- [Can social microblogging be used to forecast intraday exchange rates?](https://mlsanity.com/paper/1310.5306/) (2013)

## Built on by (most cited first, 7 of 7)

- [Double Diffusion Maps and their Latent Harmonics for Scientific Computations in Latent Space](https://mlsanity.com/paper/2204.12536/) (2022)
- [RandONet: Shallow-Networks with Random Projections for learning linear and nonlinear operators](https://mlsanity.com/paper/2406.05470/) (2024)
- [Data-driven Control of Agent-based Models: an Equation/Variable-free Machine Learning Approach](https://mlsanity.com/paper/2207.05779/) (2022)
- [Data-Driven Stochastic Closure Modeling via Conditional Diffusion Model and Neural Operator](https://mlsanity.com/paper/2408.02965/) (2024)
- [Slow Invariant Manifolds of Singularly Perturbed Systems via Physics-Informed Machine Learning](https://mlsanity.com/paper/2309.07946/) (2023)
- [Data-driven modelling of brain activity using neural networks, Diffusion Maps, and the Koopman operator](https://mlsanity.com/paper/2304.11925/) (2023)
- [Next Generation Equation-Free Multiscale Modelling of Crowd Dynamics via Machine Learning](https://mlsanity.com/paper/2508.03926/) (2025)

## Abstract (arXiv)

We address a three-tier numerical framework based on manifold learning for the forecasting of high-dimensional time series. At the first step, we embed the time series into a reduced low-dimensional space using a nonlinear manifold learning algorithm such as Locally Linear Embedding and Diffusion Maps. At the second step, we construct reduced-order regression models on the manifold, in particular Multivariate Autoregressive (MVAR) and Gaussian Process Regression (GPR) models, to forecast the embedded dynamics. At the final step, we lift the embedded time series back to the original high-dimensional space using Radial Basis Functions interpolation and Geometric Harmonics. For our illustrations, we test the forecasting performance of the proposed numerical scheme with four sets of time series: three synthetic stochastic ones resembling EEG signals produced from linear and nonlinear stochastic models with different model orders, and one real-world data set containing daily time series of 10 key foreign exchange rates (FOREX) spanning the time period 03/09/2001-29/10/2020. The forecasting performance of the proposed numerical scheme is assessed using the combinations of manifold learning, modelling and lifting approaches. We also provide a comparison with the Principal Component Analysis algorithm as well as with the naive random walk model and the MVAR and GPR models trained and implemented directly in the high-dimensional space.