Adaptive Nonlinear Vector Autoregression: Robust Forecasting for Noisy Chaotic Time Series

TL;DR

This paper introduces an adaptive nonlinear vector autoregression model that combines delay-embedded linear inputs with a trainable neural network, improving forecasting accuracy and robustness for noisy chaotic time series.

Contribution

It proposes a data-adaptive NVAR model with joint training of neural features and linear readout, enhancing scalability and robustness over traditional fixed-transform methods.

Findings

Outperforms standard NVAR and RC models in noisy conditions

Demonstrates improved predictive accuracy across multiple chaotic systems

Shows robustness to high noise levels in forecasting tasks

Abstract

Nonlinear vector autoregression (NVAR) and reservoir computing (RC) have shown promise in forecasting chaotic dynamical systems, such as the Lorenz-63 model and El Nino-Southern Oscillation. However, their reliance on fixed nonlinear transformations - polynomial expansions in NVAR or random feature maps in RC - limits their adaptability to high noise or complex real-world data. Furthermore, these methods also exhibit poor scalability in high-dimensional settings due to costly matrix inversion during optimization. We propose a data-adaptive NVAR model that combines delay-embedded linear inputs with features generated by a shallow, trainable multilayer perceptron (MLP). Unlike standard NVAR and RC models, the MLP and linear readout are jointly trained using gradient-based optimization, enabling the model to learn data-driven nonlinearities, while preserving a simple readout structure and improving scalability. Initial experiments across multiple chaotic systems, tested under noise-free and synthetically noisy conditions, showed that the adaptive model outperformed in predictive accuracy the standard NVAR, a leaky echo state network (ESN) - the most common RC model - and a hybrid ESN, thereby showing robust forecasting under noisy conditions.