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

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Presentation on theme: "Arizona State University1 Fast Mining of a Network of Coevolving Time Series Wei FanHanghang TongPing JiYongjie Cai."— Presentation transcript:

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

2 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

3 Arizona State University Contextual Network  Embedded with contextual information (Networks) 3 (a) A Simplified Sensor Network. 0.3 0.6 0.4 0.6 0.5 0.1 0.4 0.2 (b) Measured Temperature Time Series. The time series are inter-connected with each other by its embedded network.

4 Arizona State University Contextual Network (cont.)  Appear in many applications, e.g., 4 (a) Water Quality Monitoring 0.9 0.5 0.7 0.3 0.1 0.2 0.1 0.5 (b) Motion Capture(c) Epilepsy Signaling Contextual Network Coevolving Time Series

5 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 5 0.3 0.6 0.4 0.6 0.5 0. 1 0.4 0.2 +

6 Arizona State University Singular Value Decomposition (SVD) 6 Coevolving time seriesMatrix representation X t1t1 t2t2 t7t7 t17t17 21.5…1.81.61.5…3.23.63.8… 11…1.210.9…3.13.33.4… 1.71.5…1.71.61.5…3.43.83.9… 21…4.95.76…2.11.71.6… 0.70.6…55.55.8…1.31.11.2… 1.30.4…4.24.95.4…2.733.3… 1.80.8…4.65.45.8…33.43.7… t8t8 t9t9 t 18 t 19 … … … TS 1 TS 2 TS 3 TS 4 TS 5 TS 6 TS 7 Morning rush hours 1 5 9 13 17 21 Time Traffic Volume Afternoon rush hours

7 Arizona State University SVD (cont.)  Singular vectors for correlation detection 7 ≈ × × …1.81.61.5…3.23.63.8… …1.210.9…3.13.33.4… …1.71.61.5…3.43.83.9… …4.95.76…2.11.71.6… …55.55.8…1.31.11.2… …4.24.95.4…2.733.3… …4.65.45.8…33.43.7… 370 011.5 0.270.52 0.20.45 0.270.51 0.47-0.32 0.41-0.39 0.44-0.08 0.49-0.04 …0.260.290.31 …0.180.19 … …-0.15-0.22-0.26-0.27…0.290.340.36… 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

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

9 Arizona State University Step 1. Encode Correlation Among Time Series 9 coevolving time seriestime series matrix X X ≈ U×Z coefficient matrix …1.81.61.5…3.23.63.8… …1.210.9…3.13.33.4… …1.71.61.5…3.43.83.9… …4.95.76…2.11.71.6… …55.55.8…1.31.11.2… …4.24.95.4…2.733.3… …4.65.45.8…33.43.7… ≈ TS 1 TS 2 TS 3 TS 4 TS 5 TS 6 TS 7 × 2.4-0.74 1.99-0.8 2.36-0.72 0.933.26 0.53.28 1.462.26 1.752.36 …1.071.111.12…1.491.641.71… …1.231.461.58…0.190.130.14… 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 1 5 9 13 17 21 Time Traffic Volume Afternoon rush hours AR MR

10 Arizona State University Step 2. Encode Contextual Information 10 contextual informationcontextual matrix S S ≈ U×V ≈ 0.37 0.04-0.020.21 -0.14 0.280.310.2 × 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 7 2.4-0.74 1.99-0.8 2.36-0.72 0.933.26 0.53.28 1.462.26 1.752.36 TS 1TS 2TS 3TS 4TS 5TS 6TS 7 P1P1 P2P2 111-0.12-0.30.3 111-0.12-0.30.3 111-0.12-0.30.3 -0.12 110.7 -0.3 110.7 0.3 0.7 11 0.3 0.7 11 TS 1TS 2TS 3TS 4TS 5TS 6TS 7 coefficient matrix

11 Arizona State University Step 3. Encode Temporal Smoothness 11 0.740.57…1.240.920.64 0.20.06…0.270.240.12 Z:Z: AR MR

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

13 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

14 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

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

16 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

17 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

18 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

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

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

21 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

22 Arizona State University Thank you! 22 0.3 0.6 0.4 0.6 0.5 0.1 0.4 0.2 Yongjie Cai Source code: http://ycai.ws.gc.cuny.eduhttp://ycai.ws.gc.cuny.edu Q & A

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

24 Arizona State University Social Recommendation  Social network - Correlations from social connections to personal behaviors  User-item rating matrix factorization  Social network matrix factorization 24 … 1. 8 1. 6 1. 5 … 3. 2 3. 6 3. 8 … … 1. 2 1 0. 9 … 3. 1 3. 3 3. 4 … … 1. 7 1. 6 1. 5 … 3. 4 3. 8 3. 9 … … 4. 9 5. 7 6… 2. 1 1. 7 1. 6 … …5 5. 5 5. 8 … 1. 3 1. 1 1. 2 … … 4. 2 4. 9 5. 4 … 2. 7 3 3. 3 … … 4. 6 5. 4 5. 8 …3 3. 4 3. 7 … ≈ 2.4-0.74 1.99-0.8 2.36-0.72 0.933.26 0.53.28 1.462.26 1.752.36 …1.071.111.12…1.491.641.71… …1.231.461.58…0.190.130.14… × U Z ≈ × 0.37 0.04-0.020.21 -0.14 0.280.310.2 2.4 - 0.74 1.99-0.8 2.36 - 0.72 0.933.26 0.53.28 1.462.26 1.752.36 111-0.1-0.30.3 111-0.1-0.30.3 111-0.1-0.30.3 -0.1 110.7 -0.3 110.7 0.3 0.7 11 0.3 0.7 11 U V X S

25 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

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