Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Xinyu Chen, Chengyuan Zhang, Xi-Le Zhao, Nicolas Saunier, Lijun Sun

TL;DR

The paper tackles forecasting sparse, high‑dimensional movement speeds in urban road networks by introducing NoTMF, a nonstationary temporal matrix factorization framework. NoTMF couples low‑rank matrix factorization of the data matrix $\boldsymbol{Y}$ with seasonal differencing applied to the latent temporal factors and a multivariate VAR to capture temporal dynamics, solved efficiently through alternating minimization and conjugate gradient methods. The method demonstrates superior forecasting accuracy on NYC and Seattle Uber Movement data across multiple horizons, particularly when using seasonal differencing (e.g., $m=168$) to address nonstationarity. This approach enables robust imputation and prediction under heavy missingness and meaningful seasonal patterns, with practical impact for city traffic state estimation and planning.

Abstract

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.

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

TL;DR

The paper tackles forecasting sparse, high‑dimensional movement speeds in urban road networks by introducing NoTMF, a nonstationary temporal matrix factorization framework. NoTMF couples low‑rank matrix factorization of the data matrix with seasonal differencing applied to the latent temporal factors and a multivariate VAR to capture temporal dynamics, solved efficiently through alternating minimization and conjugate gradient methods. The method demonstrates superior forecasting accuracy on NYC and Seattle Uber Movement data across multiple horizons, particularly when using seasonal differencing (e.g., ) to address nonstationarity. This approach enables robust imputation and prediction under heavy missingness and meaningful seasonal patterns, with practical impact for city traffic state estimation and planning.

Abstract

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.
Paper Structure (19 sections, 2 theorems, 35 equations, 11 figures, 3 tables, 3 algorithms)

This paper contains 19 sections, 2 theorems, 35 equations, 11 figures, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Opportunities and Challenges of Using Ridesharing Vehicle Data in Traffic State Estimation
  4. Machine Learning Methods
  5. Research Gaps
  6. Nonstationary Temporal Matrix Factorization
  7. Preliminaries: Matrix Factorization
  8. Optimization Problem of NoTMF
  9. Matrix Representation: Temporal Operator Matrices
  10. Spatial Factor Matrix Estimation
  11. Temporal Factor Matrix Estimation: Factorization Plus Autoregression
  12. Coefficient Matrix Estimation
  13. Solution Algorithm of NoTMF
  14. Rolling Forecasting with Dictionary Learning
  15. Experiments on Uber Movement Data
  16. ...and 4 more sections

Key Result

Lemma 3

Suppose $\boldsymbol{A}\in\mathbb{R}^{m\times n}$, $\boldsymbol{X}\in\mathbb{R}^{n\times p}$, and $\boldsymbol{B}\in\mathbb{R}^{p\times q}$ be three matrices commensurate from multiplication in that order, then it always hold that where $\operatorname{vec}(\cdot)$ denotes the vectorization operator, and $\otimes$ denotes the Kronecker product (see Definition kron_def). The aforementioned formula

Figures (11)

  • Figure 1: Illustration of TMF on sparse movement speed data where the symbols "?" represent the unobserved values in the data. TMF characterizes spatiotemporal patterns of the data $\boldsymbol{Y}\in\mathbb{R}^{N\times T}$ as a spatial factor matrix $\boldsymbol{W}\in\mathbb{R}^{R\times N}$ and a temporal factor matrix $\boldsymbol{X}\in\mathbb{R}^{R\times N}$ in which the temporal factor matrix is indeed a multivariate time series.
  • Figure 2: Sparse movement speed forecasting with NoTMF in which $\operatorname{VAR}(d,m)$ is the VAR with the order $d$ and the season $m$.
  • Figure 3: The missing rates of Uber movement speed data aggregated per week over the whole year of 2019. The red curve shows the average missing rates over all 52 weeks. The red area shows the standard deviation of missing rates in each hour over 52 weeks. The 168 time steps refer to 168 hours of Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday, and Monday. (a) The dataset has 98,210 road segments, and the overall missing rate is 64.43%. (b) The dataset has 63,490 road segments, and the overall missing rate is 84.95%.
  • Figure 4: Histogram of observation rate of road segment in the NYC Uber movement speed dataset. Only a small fraction of road segments have an observation rate greater than 50%, i.e., $30723/98210\approx31\%$. For the observation rates greater than 20% and 80%, there are about 49% and 17% of road segments, respectively.
  • Figure 5: Movement speed of 6 road segments of January 1, 2019 (24 hours) in NYC. Blue points indicate the observed speed from the movement dataset, while black points indicate missing values (set to 0).
  • ...and 6 more figures

Theorems & Definitions (6)

  • Definition 1: Temporal Operator Matrices
  • Remark 2
  • Lemma 3
  • Definition 4: Kronecker Product golubl2013matrix
  • Lemma 5
  • Remark 6