Parsimonious Linear Fingerprinting for Time Series
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1 Parsimonious Linear Fingerprinting for Time Series Lei Li, B. Aditya Prakash, Christos Faloutsos School of Computer Science Carnegie Mellon University VLDB L. Li, 2010 VLDB2010, 36 th International Conference on Very Large Data Bases
2 Motivation Answering similarity queries in Time Series Databases SELECT * FROM TSDB WHERE data LIKE TSDB VLDB 2010 L. Li,
3 Motivation Similar motions VLDB 2010 L. Li,
4 Motivation Automatic labeling of human motion sequences VLDB 2010 L. Li,
5 Motivation Summarization / Compression VLDB 2010 L. Li,
6 Outline Motivation Proposed Method: Intuition & Example Experiments & Results PLiF: Insight Details Conclusion VLDB 2010 L. Li,
7 Intuition: Goals G1 G2 G3 G4 Good features/similarity function Good compression Ability to forecast Scalability VLDB 2010 L. Li,
8 Intuition: Goals G1 G2 G3 G4 Good features/similarity function (1a) lag independent (1b) frequency proximity (1c) grouping harmonics Good compression Ability to forecast Scalability VLDB 2010 L. Li,
9 Example: synthetic signals Equations (a) (b) (c) (d) (e) sin(2πt/100) cos(2πt/100) sin(2πt/98 + π/6) sin(2πt/110) + 0.2sin(2πt/30) cos(2πt/110) + 0.2sin(2πt/30 + π/4) VLDB 2010 L. Li,
10 Intuition (1a) Equations (a) sin(2πt/100) Time shift (b) (c) (d) (e) cos(2πt/100) sin(2πt/98 + π/6) sin(2πt/110) + 0.2sin(2πt/30) cos(2πt/110) + 0.2sin(2πt/30 + π/4) e.g. left-foot-start walking v.s. right-foot-start walking VLDB 2010 L. Li,
11 Intuition (1b) Equations (a) sin(2πt/100) (b) cos(2πt/100) (c) sin(2πt/98 + π/6) nearby frequency & time shift (d) (e) sin(2πt/110) + 0.2sin(2πt/30) cos(2πt/110) + 0.2sin(2πt/30 + π/4) e.g. running v.s. fast running VLDB 2010 L. Li,
12 Intuition (1c) Equations (a) (b) (c) (d) (e) sin(2πt/100) cos(2πt/100) sin(2πt/98 + π/6) sin(2πt/110) + 0.2sin(2πt/30) cos(2πt/110) + 0.2sin(2πt/30 + π/4) groups of harmonics ~ human voices VLDB 2010 L. Li,
13 : only two numbers to represent each! Proposed PLiF VLDB 2010 L. Li,
14 Intuition: how it works find hidden variable/pattern HV1 HV2 HV3 500 VLDB 2010 L. Li, 2010 f=1/100 f=1/110 f=1/30 14
15 Intuition: how it works find hidden variable/pattern HV2 = HV2 HV3 HV2 HV3 VLDB 2010 L. Li, 2010 f=1/110 f=1/30 Co-occur 15
16 HV1 HV VLDB 2010 L. Li,
17 HV1 HV VLDB 2010 L. Li,
18 Why it works? / How to interpret? Group of harmonics 1/110 & 1/30 harmonics. 1/100 + VLDB 2010 L. Li, 2010 Proposed PLiF 18 -
19 Basic Idea pattern/harmonics 1/110 & 1/30 running projection to harmonics (aka. frequency) walking VLDB 2010 L. Li, 2010 pattern/harmonics 1/100 19
20 Why not SVD/PCA? + Confused! VLDB 2010 L. Li, 2010 PCA no clear grouping PLiF 20 -
21 Outline Motivation Proposed Method: Intuition & Example Experiments & Results PLiF: Insight Details Conclusion VLDB 2010 L. Li,
22 Experiment: Goals to Verify G1 G2 G3 G4 Good features (low dimensional) Good compression Ability to forecast Scalability VLDB 2010 L. Li,
23 Experiments Datasets: Mocap 49 * BGP: 10 * 103k Chlorine:166 * 4k VLDB 2010 L. Li,
24 Result Visualization Mocap PLiF first two fingerprints With PLiF, now able to visualize very high dimensional time sequences VLDB 2010 L. Li,
25 Result Clustering Mocap PLiF first two fingerprints PLiF + thresholding Pred. walk run walking running Accuracy = 46/49 PCA + kmeans Pred. walk run Accuracy = 25/49 VLDB 2010 L. Li,
26 Result Clustering BGP data: PLiF + hierarchical clustering VLDB 2010 L. Li,
27 Intuition: Goals G1 G2 G3 G4 Good features/similarity function Good compression Ability to forecast Scalability VLDB 2010 L. Li,
28 Result - Compression Chlorine 166 * 4k error Storing only the PLiF features & sampling of hidden variables compression ratio VLDB 2010 L. Li, Ideal
29 Result - Compression Mocap: 93 * 300 error Storing only the PLiF features & sampling of hidden variables compression ratio Ideal VLDB 2010 L. Li,
30 Intuition: Goals later G1 G2 G3 G4 Good features/similarity function Good compression Ability to forecast Scalability VLDB 2010 L. Li,
31 wall clock time (s) CMU SCS wall clock time (s) Scalability Linear ~ sequence length sequence length sequence length VLDB 2010 L. Li,
32 PLiF Scalability Optimized algorithm Details later wall clock time SLOPE=1/3 PLiF-basic VLDB 2010 L. Li,
33 Intuition: Goals later G1 G2 G3 G4 Good features/similarity function Good compression Ability to forecast Scalability VLDB 2010 L. Li,
34 Outline Motivation Proposed Method: Intuition & Example Experiments & Results PLiF: Insight Details Conclusion VLDB 2010 L. Li,
35 Proposed Method: PLiF S1 S2 S3 S4 Learning Dynamics Finding Canonical Form Handling the Lag Grouping Harmonics VLDB 2010 L. Li,
36 Step 1. Learning Dynamics Use machine learning to find: Transition of Hidden Variables (HV): one timetick to other Mixing weights: HVs observed data Time series of hidden variables VLDB 2010 L. Li,
37 Underlying Model: Linear Dynamical Systems Details VLDB 2010 L. Li,
38 Dynamics/Transition in Hidden Variables transition matrix HV(t+1) HV(t) VLDB 2010 L. Li,
39 Mixing Weights + mixing/output matrix C VLDB 2010 L. Li,
40 Details Learning the Parameters Expectation-Maximization maximizing the expected log likelihood: Standard EM: expensive! Further speed optimization in our PLiF: matrix inversion using Woodbury matrix identity VLDB 2010 L. Li,
41 Step 2: Canonicalization But, hidden variables hard to interpret non-unique: many combinations are essentially the same Intuition: To make hidden variables compact and uniquely identified VLDB 2010 L. Li,
42 HV before f=1/110 f=1/100 f=1/30 Canonicalization adds Interpretability Harmonics frequency scaling (subtle) Time series of HV after canonicalization (real part) VLDB 2010 L. Li,
43 Step 2: Canonicalization Again, Estimating how each signal is composed of harmonics /patterns but, in complex space Mixing matrix (complex valued) VLDB 2010 L. Li,
44 Step 3:Handling Lag Intuition: Groups emerge.. reducing redundancy eliminating phase shift Conjugate! Mixing matrix (complex valued) VLDB 2010 L. Li,
45 Step 3:Handling Lag + Idea: only magnitude counts removing duplicates VLDB 2010 L. Li,
46 Step 3:Handling Lag + interpretability harmonics.1/100 harmonics 1/110 harmonics 1/30 VLDB 2010 L. Li,
47 Step 4:Grouping Harmonics + Intuition: Still a little redundancy harmonics.1/100 Think Minimum Description Length harmonics 1/110 harmonics 1/30 VLDB 2010 L. Li,
48 Step 4: Grouping Harmonics + Dimensional Reduction VLDB 2010 L. Li,
49 Step 4: Grouping Harmonics + harmonics.1/100 Group of harmonics 1/110 & 1/30 VLDB 2010 L. Li,
50 Parsimonious Linear Fingerprinting PLiF Goals Goals steps G1 G2 Good features/similarity function (1a) lag independent (1b) frequency proximity (1c) grouping harmonics Good compression S1 PLiF alg. steps Learning Dynamics G3 Ability to forecast S2 Canonical Form G4 Scalability S3 S4 Handling Lag Grouping Harmonics VLDB 2010 L. Li,
51 Outline Motivation Proposed Method: Intuition & Example Experiments & Results PLiF: Insight Details Conclusion VLDB 2010 L. Li,
52 Conclusion Need for finding compact representation of time series data Intuition & Insights of PLiF Interpretation of PLiF & How it works Experiments on a diverse set of data It really works! It is fast & scalable. VLDB 2010 L. Li,
53 Parsimonious Linear Fingerprinting PLiF Goals Goals steps G1 G2 Good features/similarity function (1a) lag independent (1b) frequency proximity (1c) grouping harmonics Good compression S1 PLiF alg. steps Learning Dynamics G3 Ability to forecast S2 Canonical Form G4 Scalability S3 S4 Handling Lag Grouping Harmonics VLDB 2010 L. Li,
54 Question? Thanks! B. Aditya Prakash Lei Li Christos Faloutsos VLDB 2010 L. Li,
55 appendix BACKUP VLDB 2010 L. Li,
56 Why not Fourier (DFT)? 1. FT cannot do forecasting VLDB
57 Why not Fourier (DFT)? 1. FT cannot do forecasting VLDB
58 Why not Fourier (DFT)? FT spectrum 1. FT cannot do forecasting 2. No arbitrary frequency true freq. VLDB 2010 frequency 58
59 Why not Fourier (DFT)? 1. FT cannot do forecasting 2. No arbitrary frequency 3. nearby frequency treated differently, not suited for across signals freq.=5 freq.=5.1 VLDB
60 Details for Implementation Read this only if you want to implement it VLDB 2010 L. Li,
61 Details Modelling the data: Linear Dynamical Systems VLDB 2010 L. Li,
62 Details Linear Dynamical Systems: parameters name µ 0 initial state for hidden variable meaning & example e.g. initial position, velocity & acceleration A transition matrix how the states move forward, e.g. soccer flying in the air C Q 0 transmission/ projection/ output matrix Initial covariance hidden state observation, e.g. camera taking picture of the soccer Q transition covariance how precision is the soccer motion R transmission/ projection covariance i.e. observation noise; e.g. how accurate is the camera 62
63 Details Learning the Dynamics Expectation-Maximization maximizing the expected log likelihood VLDB 2010 L. Li,
64 Details Finding Canonical Form Intuition: find the canonical dynamics taking eigenvalue decomposition of the transition matrix A compensate C with C h is a projection of the data to the dynamics but... VLDB 2010 L. Li,
65 Details Lags and Harmonics group Handling the lag: Intuition: phase/shift should not matter step: eliminating duplicate conjugate in C h, taking magnitude, ==> C m Group harmonics taking SVD or PCA on C m resulting fingerprints H 1 VLDB 2010 L. Li,
66 3D VIEW OF HIDDEN VARIABLES VLDB 2010 L. Li,
67 frequency scaling (subtle) Example: parsimonious HV after canonicalization VLDB 2010 L. Li,
68 SPEEDUP OPTIMIZATION VLDB 2010 L. Li,
69 Scalability Speedup the computation of matrix inverse using Woodbury matrix identity VLDB 2010 L. Li,
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