Arizona State University1 Fast Mining of a Network of Coevolving Time Series Wei FanHanghang TongPing JiYongjie Cai

Arizona State University Ubiquitous Coevolving Time Series 2 b) Chlorine concentration levels in water distribution network a) Temperature monitoring in a smart building Coevolving Time Series c) Marker tracking in motion capture d) Physiological signal in health care

Arizona State University Contextual Network  Embedded with contextual information (Networks) 3 (a) A Simplified Sensor Network (b) Measured Temperature Time Series. The time series are inter-connected with each other by its embedded network.

Arizona State University Contextual Network (cont.)  Appear in many applications, e.g., 4 (a) Water Quality Monitoring (b) Motion Capture(c) Epilepsy Signaling Contextual Network Coevolving Time Series

Arizona State University Problem Definition  NoT Missing Value Recovery Problem Given: NoT - a network of time series R = Recover: its missing parts indicated by the indicator W

Arizona State University Singular Value Decomposition (SVD) 6 Coevolving time seriesMatrix representation X t1t1 t2t2 t7t7 t17t … … … 11… … … … … … 21… … … … … … … … … … … … t8t8 t9t9 t 18 t 19 … … … TS 1 TS 2 TS 3 TS 4 TS 5 TS 6 TS 7 Morning rush hours Time Traffic Volume Afternoon rush hours

Arizona State University SVD (cont.)  Singular vectors for correlation detection 7 ≈ × × … … … … … … … … … … … … … … … … … … … … … … … … … … … MR: Morning rush hours AR: Afternoon rush hours AR MR + AR P1P1 P2P2 strength of P 1 strength of P 2 TS 1 TS 2 TS 3 TS 4 TS 5 TS 6 TS 7 Limitations: Contextual Information Temporal Smoothness P1P1 P2P2 U Σ Z X MR AR

Arizona State University Outline  Motivation  Dynamic Contextual Matrix Factorization  Experiments  Conclusion 8

Arizona State University Step 1. Encode Correlation Among Time Series 9 coevolving time seriestime series matrix X X ≈ U×Z coefficient matrix … … … … … … … … … … … … … … … … … … … … … ≈ TS 1 TS 2 TS 3 TS 4 TS 5 TS 6 TS 7 × … … … … … … P1P1 P2P2 TS 1 TS 2 TS 3 TS 4 TS 5 TS 6 TS 7 P1P1 P2P2 time series latent factor t7t7 t17t17 t8t8 t9t9 t 18 t 19 … … … indicator matrix Morning rush hours Time Traffic Volume Afternoon rush hours AR MR

Arizona State University Step 2. Encode Contextual Information 10 contextual informationcontextual matrix S S ≈ U×V ≈ × contextual latent factor TS 1 TS 2 TS 3 TS 4 TS 5 TS 6 TS 7 P1P1 P2P2 TS 1 TS 2 TS 3 TS 4 TS 5 TS 6 TS TS 1TS 2TS 3TS 4TS 5TS 6TS 7 P1P1 P2P TS 1TS 2TS 3TS 4TS 5TS 6TS 7 coefficient matrix

Arizona State University Step 3. Encode Temporal Smoothness … … Z:Z: AR MR

Arizona State University Put It All Together - DCMF #3 #2 #1 #2 ok #3 #1 12

Arizona State University Proposed Algorithm  Algorithm – Key idea: EM algorithm – Sketch: Group Z and V, group U to θ Alternatively update {Z,V} and θ(U) Forward and backward algorithms for Z 13  Properties – Converge to a local optimum – Time complexity: Linear in T #3#3 #3#3 S ≈U x V #2#2 #2#2 #1#1 #1#1 #2#2 #2#2 ok #3#3 #3#3 #1#1 #1#1

Arizona State University Relations with Existing Tools  Kalman filter and smoother  DynaMMo [Li+ KDD 2009]  SoRec [Ma+ CIKM 2008]  PMF [Salakhutdinov+ NIPS 2007]  SmoothSoRec, SmoothPMF 14 All special cases of our DCMF model

Arizona State University Outline  Motivation  Dynamic Contextual Matrix Factorization  Experiments  Conclusion 15

Arizona State University Experimental Evaluations  Parameter Sensitivity – how robust is our DCMF algorithm?  Effectiveness – how accurate is our DCMF algorithm in terms of recovering the missing values of the input time series?  Efficiency – how does our DCMF algorithm scale w.r.t. T ? 16

Arizona State University Parameter Sensitivity (a) Impact of l(b) Impact of λ 17  l: dimensionality of latent factors. – RMSE stabilizes after l reaches 15.  λ: weight to control the contribution of contextual network

Arizona State University Effectiveness Results Evaluation of missing value recovery. Lower is better 18 (a) Motes (n=54, T= 14400) (b) Chlorine (n=166, T= 4310) Ours

Arizona State University A Case Study – Running Instance (b) y-coordinate (c) z-coordinate 19 * (a) Marker positions* Partially missing: LWRB Closest in U: RWRB, RWRA, LWRA

Arizona State University Scalability  T: the length of time series 20

Arizona State University Conclusion  NoT: a network of coevolving time series  Main contributions – Model formulation multiple time series + contextual information +temporal smoothness – Algorithm EM algorithm Linear scalability in the length of time series – Empirical evaluation outperform all the existing competitors 21

Arizona State University Thank you! Yongjie Cai Source code: Q & A

Arizona State University Effectiveness Results (cont.)  Motion Capture Dataset 23

Arizona State University Social Recommendation  Social network - Correlations from social connections to personal behaviors  User-item rating matrix factorization  Social network matrix factorization 24 … … … … … … … … … … … … … … … … … … … … … ≈ … … … … … … × U Z ≈ × U V X S

Arizona State University  Find user latent factor shared by users’ social network S and the rating matrix X – Improve the recommendation accuracy Social Recommendation (cont.) 25 [Ma+ CIKM 2008] Lack of temporal smoothness