## Abstract

This article proposes several advances to sparse nonnegative matrix factorization (SNMF) as a way to identify large-scale patterns in urban traffic data. The input to our model is traffic counts organized by time and location. Nonnegative matrix factorization additively decomposes this information, organized as a matrix, into a linear sum of temporal signatures. Penalty terms encourage this factorization to concentrate on only a few temporal signatures, with weights which are not too large. Our interest here is to quantify and compare the regularity of traffic behavior, particularly across different broad temporal windows. In addition to the rank and error, we adapt a measure introduced by Hoyer to quantify sparsity in the representation. Combining these, we construct several curves which quantify error as a function of rank (the number of possible signatures) and sparsity; as rank goes up and sparsity goes down, the approximation can be better and the error should decreases. Plots of several such curves corresponding to different time windows leads to a way to compare disorder/order at different time scalewindows. In this paper, we apply our algorithms and procedures to study a taxi traffic dataset from New York City. In this dataset, we find weekly periodicity in the signatures, which allows us an extra framework for identifying outliers as significant deviations from weekly medians. We then apply our seasonal disorder analysis to the New York City traffic data and seasonal (spring, summer, winter, fall) time windows. We do find seasonal differences in traffic order.

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## References

Asif MT, Kannan S, Dauwels J, Jaillet P (2013) Data compression techniques for urban traffic data. In: 2013 IEEE symposium on computational intelligence in vehicles and transportation systems (CIVTS), pages 44–49. IEEE

Ahmadi P, Kaviani R, Gholampour I, Tabandeh Mahmoud (2015) Modeling traffic motion patterns via non-negative matrix factorization. In 2015 IEEE international conference on signal and image processing applications (ICSIPA), pages 214–219. IEEE

Alonso-Mora J, Samaranayake S, Wallar A, Frazzoli E, Rus D (2017) On-demand high-capacity ride-sharing via dynamic trip-vehicle assignment. In: Proceedings of the National Academy of Sciences, page 201611675

Ban XJ, Hao P, Sun Z (2011) Real time queue length estimation for signalized intersections using travel times from mobile sensors. Trans Res Part C 19(6):1133–1156

Boquet G, Morell A, Serrano J, Vicario JL (2020) A variational autoencoder solution for road traffic forecasting systems: missing data imputation, dimension reduction, model selection and anomaly detection. Trans Res Part C 115:102622

Brunet J-P, Tamayo P, Golub TR, Mesirov JP (2004) Metagenes and molecular pattern discovery using matrix factorization. Proc Natl Acad Sci 101(12):4164–4169

Chagoyen M, Carmona-Saez P, Shatkay H, Carazo JM, Pascual-Montano A (2006) Discovering semantic features in the literature: a foundation for building functional associations. BMC Bioinformatics 7(1):41

Chen X, He Z, Sun L (2019) A Bayesian tensor decomposition approach for spatiotemporal traffic data imputation. Trans Res Part C 98:73–84

Cazabet R, Jensen P, Borgnat P (2018) Tracking the evolution of temporal patterns of usage in bicycle-sharing systems using nonnegative matrix factorization on multiple sliding windows. Int J Urban Sci 22(2):147–161

Carmona-Saez P, Pascual-Marqui Roberto D, Tirado Francisco, Carazo Jose M, Pascual-Montano Alberto (2006) Biclustering of gene expression data by non-smooth non-negative matrix factorization. BMC Bioinformatics 7(1):78

Carrasco DR, Tonon G, Huang Y, Zhang Y, Sinha R, Feng B, Stewart JP, Zhan F, Khatry D, Protopopova M et al (2006) High-resolution genomic profiles define distinct clinico-pathogenetic subgroups of multiple myeloma patients. Cancer cell 9(4):313–325

Deri JA, Moura JMF (2015) Taxi data in new york city: a network perspective. In: 2015 49th asilomar conference on signals, systems and computers, pages 1829–1833, Nov 2015

Donovan B Mori A, Agrawal N, Meng Y, Lee J, Work D (2016) New York City hourly traffic estimates (2010-2013). https://doi.org/10.13012/B2IDB-4900670_V1

Dueck D, Morris Quaid D, Frey BJ (2005) Multi-way clustering of microarray data using probabilistic sparse matrix factorization. Bioinformatics 21(suppl-1):i144–i151

Donovan B, Work Daniel B (2015) Using coarse GPS data to quantify city-scale transportation system resilience to extreme events. arXiv preprint arXiv:1507.06011

Djenouri Y, Zimek A, Chiarandini M (2018) Outlier detection in urban traffic flow distributions. In 2018 IEEE international conference on data mining (ICDM), pages 935–940

Ermagun A, Levinson D (2018) Spatiotemporal traffic forecasting: review and proposed directions. Trans Rev 38(6):786–814

Ferreira N, Poco J, Vo HT, Freire J, Silva CT (2013) Visual exploration of big spatio-temporal urban data: a study of new york city taxi trips. IEEE Trans Vis Comput Graph 19(12):2149–2158

Gao Y, Church G (2005) Improving molecular cancer class discovery through sparse non-negative matrix factorization. Bioinformatics 21(21):3970–3975

Guan X, Chen C, Work D (2016) Tracking the evolution of infrastructure systems and mass responses using publicly available data. PloS one 11(12):e0167267

Geroliminis N, Daganzo CF (2008) Existence of urban-scale macroscopic fundamental diagrams: some experimental findings. Trans Res Part B 42(9):759–770

Guo J, Huang W, Williams BM (2015) Real time traffic flow outlier detection using short-term traffic conditional variance prediction. Trans Res Part C 50:160–172

Gong Y, Li Z, Zhang Jian, Liu W, Zheng Y, Kirsch C (2018) Network-wide crowd flow prediction of sydney trains via customized online non-negative matrix factorization. In: Proceedings of the 27th ACM international conference on information and knowledge management, pages 1243–1252. ACM

Hofleitner A, Herring R, Bayen A, Han Y, Moutarde F, De La Fortelle A (2012) Large scale estimation of arterial traffic and structural analysis of traffic patterns using probe vehicles. In Transportation Research Board 91st Annual Meeting (TRB’2012)

Han Y, Moutarde F (2011) Analysis of network-level traffic states using locality preservative non-negative matrix factorization. pages 501–506, 10

Han Y, Moutarde F (2013) Statistical traffic state analysis in large-scale transportation networks using locality-preserving non-negative matrix factorisation. IET Intell Trans Syst 7(3):283–295

Han Yufei, Moutarde Fabien (2016) Analysis of large-scale traffic dynamics in an urban transportation network using non-negative tensor factorization. Int J Intell Trans Syst Res 14(1):36–49

Hoyer PO (2002) Non-negative sparse coding. In: Neural Networks for Signal Processing, 2002. Proceedings of the 2002 12th IEEE Workshop on, pages 557–565. IEEE

Hoyer PO (2004) Non-negative matrix factorization with sparseness constraints. J Mach Learn Res 5(Nov):1457–1469

Herman R, Prigogine I (1979) A two-fluid approach to town traffic. Science 204(4389):148–151

Ronald H, Steffen R, Bernd S (2000) C*-algebras and numerical analysis. CRC Press, Boca Raton

Ito K, Ito M, Miyazaki K, Tanimoto K, Sezaki K (2017) Data analysis on train transportation data with nonnegative matrix factorization. In: 2017 IEEE international conference on big data (Big Data), pages 4080–4085. IEEE

Krichene W, Castillo MS, Bayen A (2016) On social optimal routing under selfish learning. In: IEEE transactions on control of network systems

Kim H, Park H (2007) Sparse non-negative matrix factorizations via alternating non-negativity-constrained least squares for microarray data analysis. Bioinformatics 23(12):1495–1502

Kim H, Park H (2008) Nonnegative matrix factorization based on alternating nonnegativity constrained least squares and active set method. SIAM J Matrix Anal Appl 30(2):713–730

Kim PM, Tidor B (2003) Subsystem identification through dimensionality reduction of large-scale gene expression data. Genome Res 13(7):1706–1718

Karve V, Yager D, Abolhelm M, Work D, Sowers R NYC Traffic Patterns cSNMF Source Code. https://gitlab.engr.illinois.edu/TrafficPatterns/CSNMF.git

Liu Z, Cao J, Yang J, Wang Q (2017) Discovering dynamic patterns of urban space via semi-nonnegative matrix factorization. In: 2017 IEEE international conference on big data (Big Data), pages 3447–3453. IEEE

Lv Y, Duan Y, Kang W, Li Z, Wang FY (2015) Traffic flow prediction with big data: a deep learning approach. IEEE Trans Intell Trans Syst 16(2):865–873

Li Stan Z, Hou XW, Zhang HJ, Cheng QS (2001) Learning spatially localized, parts-based representation. In: Computer Vision and Pattern Recognition, 2001. CVPR 2001. Proceedings of the 2001 IEEE Computer Society Conference on, volume 1, pages I–I. IEEE

Li Q, Jianming H, Yi Z (2007) A flow volumes data compression approach for traffic network based on principal component analysis. In: 2007 IEEE intelligent transportation systems conference, pages 125–130

Lee T, Matsushima S, Yamanishi K (2016) Traffic risk mining using partially ordered non-negative matrix factorization. In: 2016 IEEE international conference on data science and advanced analytics (DSAA), pages 622–631. IEEE

Lee DD, Seung HS (2001) Algorithms for non-negative matrix factorization. In: Advances in neural information processing systems, pages 556–562

Li L, Xiaonan S, Zhang Y, Lin Y, Li Z (2015) Trend modeling for traffic time series analysis: an integrated study. IEEE Trans Intell Trans Syst 16(6):3430–3439

Maher EA, Brennan C, Wen PY, Durso L, Ligon KL, Richardson A, Khatry D, Feng B, Sinha R, Louis DN et al (2006) Marked genomic differences characterize primary and secondary glioblastoma subtypes and identify two distinct molecular and clinical secondary glioblastoma entities. Cancer Res 66(23):11502–11513

Ma X, Li Y, Chen P (2018) Identifying spatiotemporal traffic patterns in large-scale urban road networks using a modified nonnegative matrix factorization algorithm. Journal of Traffic and Transportation Engineering (English Edition)

Mahmassani HS, Williams JC, Herman R (1984) Investigation of network-level traffic flow relationships: some simulation results. Trans Res Record 971:121–130

Nagy AM, Simon V (2018) Survey on traffic prediction in smart cities. Pervasive Mobile Comput 50:148–163

Pavlyuk D (2019) Feature selection and extraction in spatiotemporal traffic forecasting: a systematic literature review. Euro Trans Res Rev 11(1):6

Caltrans Performance Measurement System. http://pems.dot.ca.gov/

Alberto P-M, Maria CJ, Kieko K, Dietrich L, Pascual-Marqui RD (2006) Nonsmooth nonnegative matrix factorization (nsnmf). IEEE Trans Pattern Anal Mach Intell 28(3):403–415

Paul Pauca V, Piper J, Plemmons RJ (2006) Nonnegative matrix factorization for spectral data analysis. Linear algebra and its applications 416(1):29–47

Pauca VP, Shahnaz F, Berry MW, Plemmons RJ (2004) Text mining using non-negative matrix factorizations. In :Proceedings of the 2004 SIAM international conference on data mining, pages 452–456. SIAM

Pehkonen P, Wong G, Törönen P (2005) Theme discovery from gene lists for identification and viewing of multiple functional groups. BMC bioinformatics 6(1):162

Lijun S, Kay WA. Understanding urban mobility patterns with a probabilistic tensor factorization framework. Transportation Research Part B: Methodological, 91:511–524

Hongzhi W, Mohamed JB, Mohamed H. Progress in outlier detection techniques: A survey. IEEE Access, 7:107964–108000

Wisconsin worries: Labor rallies in NY. https://nypost.com/2011/02/26/wisconsin-worries-labor-rallies-in-ny/

NY subway system shuts down due to Hurricane Irene (updated). http://www.wnyc.org

Xu L, Wang Y, Yu H, Li H (2015) Feature extraction of urban traffic network data based on locally sensitive discriminant analysis algorithm

Yangyang X, Yin W, Wen Z, Zhang Y (2012) An alternating direction algorithm for matrix completion with nonnegative factors. Front Math China 7(2):365–384

Yang S, Qian S (2019) Understanding and predicting travel time with spatio-temporal features of network traffic flow, weather and incidents. IEEE Intell Trans Syst Mag 11(3):12–28

Zhang Z, He Q, Tong H, Gou J, Li X (2016) Spatial-temporal traffic flow pattern identification and anomaly detection with dictionary-based compression theory in a large-scale urban network. Trans Res Part C 71:284–302

Zheng J Liu HX (2017) Estimating traffic volumes for signalized intersections using connected vehicle data. Trans Res Part C 79:347–362

Yuan Z, Kaan O, Kun X, Hong Y (2016) Using big data to study resilience of taxi and subway trips for hurricanes sandy and irene. Trans Res Record 2599:70–80

Zhan X, Ukkusuri SV, Zhu F (2014) Inferring urban land use using large-scale social media check-in data. Netw Spatial Econ 14(3):647–667. https://doi.org/10.1007/s11067-014-9264-4

Zhang S, Wang W, Ford J, Makedon Fillia Learning from Incomplete Ratings Using Non-negative Matrix Factorization, pages 549–553

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The authors acknowledge the Program for Interdisciplinary and Industrial Internships at Illinois (PI4) and the Illinois Geometry Laboratory (IGL). The many IGL students who have made invaluable contributions to this work are: Raghav Bakshi, James Kerns, Xinyi Li, Xinyu Liu, Yicheng Pu, Gabriel Shindnes, Haozhe Wang, Jing Wang, Ziying Wang, Yu Wu, Zeyu Wu, Bin Xu, and Dajun Xu. The authors would also like to thank the Siebel Energy Institute for its support of this work. This material is based upon work supported by the National Science Foundation under Grant Numbers CMMI 1727785 and DMS 1345032. This work was also supported by a grant from the Siebel Energy Institute. The code for this work is at https://gitlab.engr.illinois.edu/TrafficPatterns/CSNMF.git

Sandia National Laboratories is a multimission laboratory operated by National Technology and Engineering Solutions of Sandia LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy’s National Nuclear Security Administration. Sandia Labs has major research and development responsibilities in nuclear deterrence, global security, defense, energy technologies and economic competitiveness, with main facilities in Albuquerque, New Mexico, and Livermore, California.

This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government.

## Appendix

### Appendix

For completeness, let’s write down the calculations leading to the algorithm of Sect. 2.5.

Writing out \({\mathcal{E}}_{\beta ,\eta }\) of (5), we get

We seek to minimize this by alternating between minimization problems in *W* and *H*. Namely, if we start with a fixed \((W,H)\in \mathbb {R}^{T\times N}_+\times \mathbb {R}^{N\times L}_+\), we can construct a descent step for the function \({\mathcal{E}}_{\beta ,\eta }(W,\cdot )\) and then, letting \(H'\) be the result, we can construct a descent step for \({\mathcal{E}}_{\beta ,\eta }(\cdot ,H')\). This should decrease the value of \({\mathcal{E}}_{\beta ,\eta }\), and we can then proceed iteratively.

The gradients of \({\mathcal{E}}_{\beta ,\eta }\) in the directions of *W* and *H* are given by

and

As in Kim and Park (2008), we want to iteratively find the critical points of \({\mathcal{E}}_{\beta ,\eta }\), i.e. the solutions of

The above formulae suggest a multiplicative descent rule (which need not be *gradient descent*; see Lee and Seung (2001)). Fix \((W,H)\in \mathbb {R}_+^{T\times N}\times \mathbb {R}_+^{N\times L}\). Assume that

we can then decrease the value of \({\mathcal{E}}_{\beta ,\eta }\) by decreasing \(H_{n,\ell }\). Rewriting (14) as

or rather

since *W*, *H*, and *D* all have nonnegative entries, both sides of this equation are nonnegative. This in turn can be written as \({\chi _{n,\ell }^h(W,H)<1}\) where

Thus, another way to decrease \(H_{n,\ell }\) while still retaining nonnegativity is to multiply it by \(\chi _{n,\ell }^h(W,H)\). Reviewing these steps, we also see that if \(\frac{\partial {\mathcal{E}}_{\beta ,\eta }}{\partial H_{n,\ell }}<0\), we want to increase \(H_{n,\ell }\), and can again multiply by \(\chi _{n,\ell }^h(W,H)\). Finally, if \(\frac{\partial {\mathcal{E}}^\beta }{\partial H_{n,\ell }}=0\) (i.e., we have found a critical point) \(\chi _{n,\ell }^h(W,H)=1\), so multiplying \(H_{n,\ell }\) by \(\chi _{n,\ell }^h(W,H)\) leaves \(H_{n,\ell }\) unchanged.

The update rule for \(W_{t,n}\) is similar. To start, assume that

then we can decrease \({\mathcal{E}}_{\beta ,\eta }\) by decreasing \(W_{t,n}\). We can rewrite (15) as

We can again rewrite this as the comparison of two nonnegative quantities;

This in turn is equivalent to \(\chi _{t,n}^w(W,H)<1\) where

In other words, we can decrease \(W_{t,n}\) by multiplying by \(\chi _{t,n}^w(W,H)\). One can similarly see that if \(\frac{\partial {\mathcal{E}}^\beta }{\partial W_{t,n}}<0\), gradient descent again increases or decreases *W* with the same sign as multiplying by \(\chi _{t,n}^w(W,H)\).

Our proposed update rule for *W* and *H* is now

which is equivalent to (6).

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Karve, V., Yager, D., Abolhelm, M. *et al.* Seasonal Disorder in Urban Traffic Patterns: A Low Rank Analysis.
*J. Big Data Anal. Transp.* (2021). https://doi.org/10.1007/s42421-021-00033-4

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### Keywords

- Traffic
- Normalization
- Sparse nonnegative matrix Factorization