Missing Value Estimation for Hierarchical Time Series: A Study of Hierarchical Web Traffic

Transcription

1 Missing Value Estimation for Hierarchical Time Series: A Study of Hierarchical Web Traffic Zitao Liu University of Pittsburgh ztliu@cs.pitt.edu November 16, 2015 This is the joint work with Chris Yan, Jimmy Yang and Milos Hauskrecht. Zitao Liu (University of Pittsburgh) ICDM 2015 November 16, / 26

2 Hierarchical Time Series U.S. Home Mail Answers Groups Flickr Tumblr Games Live Screen Mobile News More World Search Web My Yahoo Sign In Politics Mail Trending Now Yahoo! Homepage News Autos Sports Finance Dating Used Cars 2 Brock Lesnar 7 19 Kids and C 3 Clint Eastwood 8 Britney Spears 5 Meagan Good Sports Autos Homes 6 Iggy Azalea 4 Mega Millions jac New Cars Weather Daily live conce 1 Elin Nordegren 9 401(k) plans 10 Oil prices Lawyer: Exit wounds indicate Brown surrendered 17 Days To Kickoff Play Fantasy Foot Experts tasked by slain teen Michael Brown's family to examine hismail body say at least six bullets hit him. 'Was my child in pain?'» 1 5 of 50 Jobs Finance Shopping Beauty Brown family's autopsy Celeb there for ex-assistant U.S. Olympic champs booed Family nabs record gator Odd truth about tasty fish Health Live Events on Yahoo Food What is hierarchical time series(hts)? Yahoo! web pages are arranged Movies in certain Travel hierarchy and their daily page views become a hierarchical time series. Tech Aug 18 More Yahoo Sites Aug 19 Taylor Swift Liv Yes 5 p.m. ET 11:30 p.m. ET REMIND ME Yahoo en Español Zitao Liu (University of Pittsburgh) REMIND ME Aug 21 John Legend 12:10 a.m. ET REMIND ME ICDM 2015 Aug 24 Justin Timberlake 5 p.m. ET REMIND ME Only on Yahoo 4 ways make your eyes November 16,to /pop 26wit

3 Hierarchical Time Series Time series are organized in a hierarchical tree structure and they are consistent between hierarchy levels. U.S. News World Yahoo! Homepage Autos Politics Used Cars Parent Sports Mail New Cars Children Finance Consistent: Parent = Child 1 + Child 2 + Child Child n Zitao Liu (University of Pittsburgh) ICDM 2015 November 16, / 26

4 Motivation Why we care about modeling HTS? Resource management. User behaviors understanding. Advertisement pricing policy. Zitao Liu (University of Pittsburgh) ICDM 2015 November 16, / 26

5 Problem & Goal However, missing values occur: machine failures networking disturbances human mistakes Missing values will contaminate other time series through the hierarchy consequentially. Zitao Liu (University of Pittsburgh) ICDM 2015 November 16, / 26

6 Problem Parent x x x x Child 1 Child 2 Child 3 Child 4 Child 5 x x x x t 1 t 2 t 3 t 4 Time Accurately estimate the missing values. s.t Estimation is hierarchically consistent. Zitao Liu (University of Pittsburgh) ICDM 2015 November 16, / 26

7 HTSImpute In this work, we develop a new missing value estimation algorithm HTSImpute which utilizes the temporal dependence information within each individual time series (LOcal regression (LOESS)) exploits the intra-relations between different time series (Subspace Projection) guarantees hierarchical consistency (Hierarchical Consistency Projection) Zitao Liu (University of Pittsburgh) ICDM 2015 November 16, / 26

8 HTSImpute - LOESS Use LOESS to initially estimate the missing values. Value??? Time Advantages: nonparametric robust locally weighted Zitao Liu (University of Pittsburgh) ICDM 2015 November 16, / 26

9 HTSImpute - Subspace Projection Full rank matrix Low rank matrix Rank: n Rank: l Zitao Liu (University of Pittsburgh) ICDM 2015 November 16, / 26

10 HTSImpute - Hierarchical Consistency Projection [ ] 0 1 Summing matrix Ω? + Noise Y = Ω ˆL + ɛ Y where ˆL is the true estimate of all leaf time series. We define the hierarchical consistency projection operator using ordinary least square as follows: P HTS (Y, Ω) = ΩˆL = Ω(Ω Ω) 1 Ω Y Zitao Liu (University of Pittsburgh) ICDM 2015 November 16, / 26

11 HTSImpute - Idea Illustration Goal Start Zitao Liu (University of Pittsburgh) ICDM 2015 November 16, / 26

12 HTSImpute - Idea Illustration Goal Start Zitao Liu (University of Pittsburgh) ICDM 2015 November 16, / 26

13 HTSImpute - Idea Illustration Goal Subspace Projection Start Zitao Liu (University of Pittsburgh) ICDM 2015 November 16, / 26

14 HTSImpute - Idea Illustration Goal Subspace Projection Hierarchical Consistent Projection HTS Start Zitao Liu (University of Pittsburgh) ICDM 2015 November 16, / 26

15 HTSImpute - Idea Illustration Goal Subspace Projection Hierarchical Consistent Projection HTS Start Zitao Liu (University of Pittsburgh) ICDM 2015 November 16, / 26

16 HTSImpute - Idea Illustration Goal Subspace Projection Hierarchical Consistent Projection HTS Subspace HTS Start Zitao Liu (University of Pittsburgh) ICDM 2015 November 16, / 26

17 HTSImpute - Idea Illustration Subspace Projection Hierarchical Consistent Projection HTS HTS Goal Subspace HTS Start Zitao Liu (University of Pittsburgh) ICDM 2015 November 16, / 26

18 Experiments - Dataset 1 E (a) wide E2 E3 E4 E5 E6 E7 E8 (a) EMEA M1 F1 F2 F3 M2 M3 M4 M5 M6 M F4 F5 F6 M8 M (b) FP M10 M11 M12 (b) balance (c) deep (c) Media Figure 1: Synthetic data. Figure 2: Yahoo! web traffic data. Zitao Liu (University of Pittsburgh) ICDM 2015 November 16, / 26

19 Experiments - Metrics Avg-MAPE: measures the estimation accuracy. Estimated Value True Value True Value Avg-HCG: measures the hierarchical consistency. Estimated Parent Value Sum of Estimated Child Values Zitao Liu (University of Pittsburgh) ICDM 2015 November 16, / 26

20 Experiments - Baseline Regression Methods Local regression (LOESS) Subspace Methods Matrix Factorization (MF) Matrix Completion (MC) using softimpute weight Low Rank Approximation (wlra) Latent Variable Models probabilistic PCA (ppca) Zitao Liu (University of Pittsburgh) ICDM 2015 November 16, / 26

21 Experiments - Results Table 1: Avg-MAPE results on FP dataset. # MP (%) LOESS NMF KL NMF Euclidean MC ppca wlra HTSImpute Zitao Liu (University of Pittsburgh) ICDM 2015 November 16, / 26

22 Experiments - Results Table 2: Avg-HCG results on FP dataset (log 10 scale). # MP (%) LOESS NMF KL NMF Euclidean MC ppca wlra HTSImpute Zitao Liu (University of Pittsburgh) ICDM 2015 November 16, / 26

23 Conclusion In this work, we have presented a algorithm for HTS missing value estimation, specializing in taking advantage of temporal dependence information within each individual time series. utilizing intra-relations between different time series across the hierarchy. providing high satisfaction of the hierarchical consistency. Zitao Liu (University of Pittsburgh) ICDM 2015 November 16, / 26

24 Thank you November 16, 2015 Zitao Liu (University of Pittsburgh) ICDM 2015 November 16, / 26

25 HTSImpute - Hierarchical Consistency Projection =[ ] 0 1 Summing matrix Ω Zitao Liu (University of Pittsburgh) ICDM 2015 November 16, / 26

26 HTSImpute - Hierarchical Consistency Projection [ ] 0 1 Summing matrix Ω? + Noise Zitao Liu (University of Pittsburgh) ICDM 2015 November 16, / 26