# Explainable nonlinear modelling of multiple time series with invertible neural networks

arXiv: https://arxiv.org/abs/2107.00391 · HTML: https://mlsanity.com/paper/2107.00391/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 21 references that are in the map.
- First posted on arXiv 1 Jul 2021.

## Builds on (most cited first, 7 of 7)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Independent Innovation Analysis for Nonlinear Vector Autoregressive Process](https://mlsanity.com/paper/2006.10944/) (2020)
- [Online Topology Identification from Vector Autoregressive Time Series](https://mlsanity.com/paper/1904.01864/) (2019)
- [Online Non-linear Topology Identification from Graph-connected Time Series](https://mlsanity.com/paper/2104.00030/) (2021)
- [Semi-Blind Inference of Topologies and Dynamical Processes over Graphs](https://mlsanity.com/paper/1805.06095/) (2018)
- [An Interpretable and Sparse Neural Network Model for Nonlinear Granger Causality Discovery](https://mlsanity.com/paper/1711.08160/) (2017)

## Built on by (most cited first, 1 of 1)

- [Efficient Interpretable Nonlinear Modeling for Multiple Time Series](https://mlsanity.com/paper/2309.17154/) (2023)

## Abstract (arXiv)

A method for nonlinear topology identification is proposed, based on the assumption that a collection of time series are generated in two steps: i) a vector autoregressive process in a latent space, and ii) a nonlinear, component-wise, monotonically increasing observation mapping. The latter mappings are assumed invertible, and are modelled as shallow neural networks, so that their inverse can be numerically evaluated, and their parameters can be learned using a technique inspired in deep learning. Due to the function inversion, the back-propagation step is not straightforward, and this paper explains the steps needed to calculate the gradients applying implicit differentiation. Whereas the model explainability is the same as that for linear VAR processes, preliminary numerical tests show that the prediction error becomes smaller.