{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,7]],"date-time":"2025-11-07T09:03:29Z","timestamp":1762506209134,"version":"build-2065373602"},"reference-count":37,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2013,11,6]],"date-time":"2013-11-06T00:00:00Z","timestamp":1383696000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>We formulate a multi-matrices factorization model (MMF) for the missing sensor data estimation problem. The estimation problem is adequately transformed into a matrix completion one. With MMF, an n-by-t real matrix, R, is adopted to represent the data collected by mobile sensors from n areas at the time, T1, T2, ... , Tt, where the entry, Rij, is the aggregate value of the data collected in the ith area at Tj . We propose to approximate R by seeking a family of d-by-n probabilistic spatial feature matrices, U(1), U(2), ... , U(t), and a probabilistic temporal feature matrix, V E Rdxt, where Rj \u2248 UT(j)Tj . We also present a solution algorithm to the proposed model. We evaluate MMF with synthetic data and a real-world sensor dataset extensively. Experimental results demonstrate that our approach outperforms the state-of-the-art comparison algorithms.<\/jats:p>","DOI":"10.3390\/s131115172","type":"journal-article","created":{"date-parts":[[2013,11,6]],"date-time":"2013-11-06T11:41:50Z","timestamp":1383738110000},"page":"15172-15186","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":17,"title":["Multi-Matrices Factorization with Application to Missing Sensor Data Imputation"],"prefix":"10.3390","volume":"13","author":[{"given":"Xiao-Yu","family":"Huang","sequence":"first","affiliation":[{"name":"Software Institute, Sun Yat-Sen University, Guangzhou 510275, China"},{"name":"School of Economics and Commerce, South China University of Technology, Guangzhou 510006, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wubin","family":"Li","sequence":"additional","affiliation":[{"name":"Department of Computing Science, Ume\u00e5 University, SE-901 87 Ume\u00e5, Sweden"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kang","family":"Chen","sequence":"additional","affiliation":[{"name":"Academy of Guangdong Telecom Co.Ltd, Guangzhou 510630, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xian-Hong","family":"Xiang","sequence":"additional","affiliation":[{"name":"Department of Interventional Radiology, the First Affiliated Hospital of Sun Yat-Sen University, Guangzhou 510080, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Rong","family":"Pan","sequence":"additional","affiliation":[{"name":"Software Institute, Sun Yat-Sen University, Guangzhou 510275, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lei","family":"Li","sequence":"additional","affiliation":[{"name":"Software Institute, Sun Yat-Sen University, Guangzhou 510275, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wen-Xue","family":"Cai","sequence":"additional","affiliation":[{"name":"School of Economics and Commerce, South China University of Technology, Guangzhou 510006, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2013,11,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1483","DOI":"10.1016\/j.neucom.2008.11.026","article-title":"K nearest neighbours with mutual information for simultaneous classification and missing data imputation","volume":"72","author":"Verleysen","year":"2009","journal-title":"Neurocomputation"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"931","DOI":"10.1061\/(ASCE)0733-947X(2005)131:12(931)","article-title":"Multiple imputation scheme for overcoming the missing values and variability issues in ITS data","volume":"131","author":"Ni","year":"2005","journal-title":"J. Transport. Eng."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"132","DOI":"10.3141\/1836-17","article-title":"Exploring imputation techniques for missing data in transportation management systems","volume":"1836","author":"Smith","year":"2003","journal-title":"Transport. Res. Record. J. Transport. Res. Board"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Qu, L., Zhang, Y., Hu, J., Jia, L., and Li, L. (2008, January 4\u20136). A BPCA Based Missing Value Imputing Method for Traffic Flow Volume Data. Eindhoven, The Neatherlands.","DOI":"10.1109\/IVS.2008.4621153"},{"key":"ref_5","unstructured":"Jiang, N., and Gruenwald, L. (2007). Advances in Databases: Concepts, Systems and Applications, Springer."},{"key":"ref_6","unstructured":"Netflix Prize. Avaiable online: http:\/\/www.netflixprize.com."},{"key":"ref_7","first-page":"1257","article-title":"Probabilistic matrix factorization","volume":"20","author":"Salakhutdinov","year":"2008","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_8","unstructured":"Koren, Y. Factorization Meets the Neighborhood: A Multifaceted Collaborative Filtering Model. Las Vegas, NV, USA."},{"key":"ref_9","first-page":"556","article-title":"Algorithms for non-negative matrix factorization","volume":"13","author":"Seung","year":"2001","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_10","unstructured":"Srebro, N. (2004). Learning with Matrix Factorizations. [Ph.D Thesis, Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology]."},{"key":"ref_11","first-page":"1329","article-title":"Maximum-margin matrix factorization","volume":"17","author":"Srebro","year":"2005","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_12","unstructured":"Independent and identically distributed random variables. Avaiable online: http:\/\/en.wikipedia.org\/wiki\/ndependent_and_identically_distributed_random_variables."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1109\/MC.2009.263","article-title":"Matrix factorization techniques for rcommender systems","volume":"42","author":"Koren","year":"2009","journal-title":"Computer"},{"key":"ref_14","unstructured":"Xu, W., Liu, X., and Gong, Y. (August, January 28). Document Clustering Based on Non-Negative Matrix Factorization. Pisa, Italy."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"788","DOI":"10.1038\/44565","article-title":"Learning the parts of objects by non-negative matrix factorization","volume":"401","author":"Lee","year":"1999","journal-title":"Nature"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"4164","DOI":"10.1073\/pnas.0308531101","article-title":"Metagenes and molecular pattern discovery using matrix factorization","volume":"101","author":"Brunet","year":"2004","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"717","DOI":"10.1007\/s10208-009-9045-5","article-title":"Exact matrix completion via convex optimization","volume":"9","author":"Recht","year":"2009","journal-title":"Found. Comput. Math."},{"key":"ref_18","first-page":"1457","article-title":"Non-negative matrix factorization with sparseness constraints","volume":"5","author":"Hoyer","year":"2004","journal-title":"J. Mach. Learn. Res."},{"key":"ref_19","first-page":"19","article-title":"Online learning for matrix factorization and sparse coding","volume":"11","author":"Mairal","year":"2010","journal-title":"J. Mach. Learn. Res."},{"key":"ref_20","unstructured":"Ke, Q., and Kanade, T. (2005, January 20\u201326). Robust L1Norm Factorization in the Presence of Outliers and Missing Data by Alternative Convex Programming. San Diego, CA, USA."},{"key":"ref_21","unstructured":"Nati, N.S., and Jaakkola, T. (2003, January 21\u201324). Weighted Low-Rank Approximations. Washington, DC, USA."},{"key":"ref_22","unstructured":"Abernethy, J., Bach, F., Evgeniou, T., and Vert, J.P. (2006). Low-Rank Matrix Factorization with Attributes, Technical Report; N-24\/06\/MM."},{"key":"ref_23","unstructured":"Vapnik, V. (1998). Statistical Learning Theory, Wiley."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Rissanen, J. (2010). Minimum Description Length Principle, Springer.","DOI":"10.1007\/978-0-387-30164-8_540"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2053","DOI":"10.1109\/TIT.2010.2044061","article-title":"The power of convex relaxation: Near-optimal matrix completion","volume":"56","author":"Tao","year":"2010","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_26","unstructured":"Cover, T.M., and Thomas, J.A. (2012). Elements of Information Theory, John Wiley & Sons."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"489","DOI":"10.1109\/TIT.2005.862083","article-title":"Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequencyinformation","volume":"52","author":"Romberg","year":"2006","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1289","DOI":"10.1109\/TIT.2006.871582","article-title":"Compressed sensing","volume":"52","author":"Donoho","year":"2006","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_29","unstructured":"Bertsekas, D.P. (1999). Nonlinear Programming, Athena Scientific."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"611","DOI":"10.1111\/1467-9868.00196","article-title":"Probabilistic principal component analysis","volume":"61","author":"Tipping","year":"1999","journal-title":"J. Royal Stat. Soc. Ser. B Stat. Methodol."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"512","DOI":"10.1109\/TITS.2009.2026312","article-title":"PPCA-based missing data imputation for traffic flow volume: A systematical approach","volume":"10","author":"Qu","year":"2009","journal-title":"IEEE Trans. Intell. Transport. Syst."},{"key":"ref_32","unstructured":"Marlin, B. (2004). Collaborative Filtering: A Machine Learning Perspective. [Ph.D Thesis, University of Toronto]."},{"key":"ref_33","unstructured":"Nguyen, L.N., and Scherer, W.T. (2003). Imputation Techniques to Account for Missing Data in Support of Intelligent Transportation Systems Applications, University of Virginia. UVACTS-13-0-78."},{"key":"ref_34","unstructured":"Gold, D.L., Turner, S.M., Gajewski, B.J., and Spiegelman, C. (2001, January 7\u201311). Imputing Missing Values in Its Data Archives for Intervals under 5 Minutes. Washington, DC, USA."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Shuai, M., Xie, K., Pu, W., Song, G., and Ma, X. (2008, January 5\u20137). An Online Approach Based on Locally Weighted Learning for Real Time Traffic Flow Prediction. Irvine, CA, USA.","DOI":"10.1145\/1463434.1463490"},{"key":"ref_36","unstructured":"Zhuhai. Avaiable online: http:\/\/en.wikipedia.org\/wiki\/Zhuhai."},{"key":"ref_37","unstructured":"Floating Car Data. Avaiable online: http:\/\/en.wikipedia.org\/wiki\/Floating_car_data."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/13\/11\/15172\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:50:25Z","timestamp":1760219425000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/13\/11\/15172"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,11,6]]},"references-count":37,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2013,11]]}},"alternative-id":["s131115172"],"URL":"https:\/\/doi.org\/10.3390\/s131115172","relation":{},"ISSN":["1424-8220"],"issn-type":[{"type":"electronic","value":"1424-8220"}],"subject":[],"published":{"date-parts":[[2013,11,6]]}}}