# Forecasting Sparse Movement Speed of Urban Road Networks with Nonstationary Temporal Matrix Factorization

arXiv: https://arxiv.org/abs/2203.10651 · HTML: https://mlsanity.com/paper/2203.10651/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 35 references that are in the map.
- First posted on arXiv 20 Mar 2022.

## Builds on (most cited first, 12 of 12)

- [Recurrent Neural Networks for Multivariate Time Series with Missing Values](https://mlsanity.com/paper/1606.01865/) (2016)
- [Nonconvex Optimization Meets Low-Rank Matrix Factorization: An Overview](https://mlsanity.com/paper/1809.09573/) (2018)
- [Bayesian Temporal Factorization for Multidimensional Time Series Prediction](https://mlsanity.com/paper/1910.06366/) (2019)
- [Regularized estimation in sparse high-dimensional time series models](https://mlsanity.com/paper/1311.4175/) (2013)
- [High-dimensional vector autoregressive time series modeling via tensor decomposition](https://mlsanity.com/paper/1909.06624/) (2019)
- [Low Rank and Structured Modeling of High-dimensional Vector Autoregressions](https://mlsanity.com/paper/1812.03568/) (2018)
- [Recovery of Future Data via Convolution Nuclear Norm Minimization](https://mlsanity.com/paper/1909.03889/) (2019)
- [SSMF: Shifting Seasonal Matrix Factorization](https://mlsanity.com/paper/2110.12763/) (2021)
- [Time Series Forecasting via Learning Convolutionally Low-Rank Models](https://mlsanity.com/paper/2104.11510/) (2021)
- [Online Forecasting Matrix Factorization](https://mlsanity.com/paper/1712.08734/) (2017)
- [A Direct Estimation of High Dimensional Stationary Vector Autoregressions](https://mlsanity.com/paper/1307.0293/) (2013)
- [Traffic state estimation using stochastic Lagrangian dynamics](https://mlsanity.com/paper/1806.02692/) (2018)

## Built on by (most cited first, 5 of 5)

- [Laplacian Convolutional Representation for Traffic Time Series Imputation](https://mlsanity.com/paper/2212.01529/) (2022)
- [Discovering Dynamic Patterns from Spatiotemporal Data with Time-Varying Low-Rank Autoregression](https://mlsanity.com/paper/2211.15482/) (2022)
- [DynaConF: Dynamic Forecasting of Non-Stationary Time Series](https://mlsanity.com/paper/2209.08411/) (2022)
- [High-Dimensional Fault Tolerance Testing of Highly Automated Vehicles Based on Low-Rank Models](https://mlsanity.com/paper/2407.21069/) (2024)
- [Data-Driven Discovery of Mobility Periodicity for Understanding Urban Systems](https://mlsanity.com/paper/2508.03747/) (2025)

## Abstract (arXiv)

Movement speed data from urban road networks, computed from ridesharing vehicles or taxi trajectories, is often high-dimensional, sparse, and nonstationary (e.g., exhibiting seasonality). To address these challenges, we propose a Nonstationary Temporal Matrix Factorization (NoTMF) model that leverages matrix factorization to project high-dimensional and sparse movement speed data into low-dimensional latent spaces. This results in a concise formula with the multiplication between spatial and temporal factor matrices. To characterize the temporal correlations, NoTMF takes a latent equation on the seasonal differenced temporal factors using higher-order vector autoregression (VAR). This approach not only preserves the low-rank structure of sparse movement speed data but also maintains consistent temporal dynamics, including seasonality information. The learning process for NoTMF involves optimizing the spatial and temporal factor matrices along with a collection of VAR coefficient matrices. To solve this efficiently, we introduce an alternating minimization framework, which tackles a challenging procedure of estimating the temporal factor matrix using conjugate gradient method, as the subproblem involves both partially observed matrix factorization and seasonal differenced VAR. To evaluate the forecasting performance of NoTMF, we conduct extensive experiments on Uber movement speed datasets, which are estimated from ridesharing vehicle trajectories. These datasets contain a large proportion of missing values due to insufficient ridesharing vehicles on the urban road network. Despite the presence of missing data, NoTMF demonstrates superior forecasting accuracy and effectiveness compared to baseline models. Moreover, as the seasonality of movement speed data is of great concern, the experiment results highlight the significance of addressing the nonstationarity of movement speed data.