# An Ensemble Quadratic Echo State Network for Nonlinear Spatio-Temporal Forecasting

arXiv: https://arxiv.org/abs/1708.05094 · HTML: https://mlsanity.com/paper/1708.05094/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Aug 2017.

## Built on by (most cited first, 12 of 12)

- [Statistical Deep Learning for Spatial and Spatio-Temporal Data](https://mlsanity.com/paper/2206.02218/) (2022)
- [Spatio-temporal DeepKriging for Interpolation and Probabilistic Forecasting](https://mlsanity.com/paper/2306.11472/) (2023)
- [On the Universal Transformation of Data-Driven Models to Control Systems](https://mlsanity.com/paper/2102.04722/) (2021)
- [Characterizing climate pathways using feature importance on echo state networks](https://mlsanity.com/paper/2310.08495/) (2023)
- [Echo State Networks for Spatio-Temporal Area-Level Data](https://mlsanity.com/paper/2410.10641/) (2024)
- [Data-driven and machine-learning based prediction of wave propagation behavior in dam-break flood](https://mlsanity.com/paper/2209.08729/) (2022)
- [SERT: A Transfomer Based Model for Spatio-Temporal Sensor Data with Missing Values for Environmental Monitoring](https://mlsanity.com/paper/2306.03042/) (2023)
- [Variability of echo state network prediction horizon for partially observed dynamical systems](https://mlsanity.com/paper/2306.10797/) (2023)
- [Pedestrian Volume Prediction Using a Diffusion Convolutional Gated Recurrent Unit Model](https://mlsanity.com/paper/2411.03360/) (2024)
- [Modeling High-Resolution Spatio-Temporal Wind with Deep Echo State Networks and Stochastic Partial Differential Equations](https://mlsanity.com/paper/2412.07265/) (2024)
- [CESAR: A Convolutional Echo State AutoencodeR for High-Resolution Wind Forecasting](https://mlsanity.com/paper/2412.10578/) (2024)
- [Hierarchical Count Echo State Network Models with Application to Graduate Student Enrollments](https://mlsanity.com/paper/2501.14698/) (2025)

## Abstract (arXiv)

Spatio-temporal data and processes are prevalent across a wide variety of scientific disciplines. These processes are often characterized by nonlinear time dynamics that include interactions across multiple scales of spatial and temporal variability. The data sets associated with many of these processes are increasing in size due to advances in automated data measurement, management, and numerical simulator output. Non- linear spatio-temporal models have only recently seen interest in statistics, but there are many classes of such models in the engineering and geophysical sciences. Tradi- tionally, these models are more heuristic than those that have been presented in the statistics literature, but are often intuitive and quite efficient computationally. We show here that with fairly simple, but important, enhancements, the echo state net- work (ESN) machine learning approach can be used to generate long-lead forecasts of nonlinear spatio-temporal processes, with reasonable uncertainty quantification, and at only a fraction of the computational expense of a traditional parametric nonlinear spatio-temporal models.