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Path-State Modeling for Time Series Anomaly Detection

Path-State Modeling for Time Series Anomaly Detection. Matt Mahoney. Outline. Review of time series anomaly detection Gecko Compression Path modeling Piecewise linear approximation of path Fast testing using state Experimental results on NASA valve data.

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Path-State Modeling for Time Series Anomaly Detection

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  1. Path-State Modeling for Time Series Anomaly Detection Matt Mahoney

  2. Outline • Review of time series anomaly detection • Gecko • Compression • Path modeling • Piecewise linear approximation of path • Fast testing using state • Experimental results on NASA valve data

  3. Normal Marotta Fuel Valve Solenoid Current (Used on Space Shuttle) Abnormal (poppet partially blocked) Problem: How to Detect Anomalies in Time Series Data

  4. Goal • Reduce human workload in specifying “normal” model • Editable rule based model (in SCL) • Real time testing (1K-10K samples per second)

  5. Manual Method • Identify features (zero crossings, peaks…) • Specify correct behavior using SCL rules

  6. Gecko (Stan Salvador) • Identify model states (parabolic segments) • Multiple training series are averaged by dynamic time warping • Classify points (x,dx,d2x) using RIPPER • Construct linear state machine • Pass/fail test result

  7. Compression Model Normal, uncompressed Abnormal, uncompressed Normal, compressed Abnormal, compressed Normal 1 Normal 2 Normal 1 or 2 Abnormal

  8. TEK Compression Anomaly Scores

  9. Goal Evaluation

  10. Problem with Gecko/RIPPER: State Machine May Underconstrain Model Training Segment 1: x = 0, dx = 0 Segment 2: 0 < x < 1, dx = 1 Test Segment 1: x = 0, dx = 0 Segment 2: 0 < x < 1, dx = 3 dx > 0.5 State 1 State 2 Accept

  11. Path Model dx Test Path (d2 = 4) 3 2 Training Path (scaled to unit cube) 1 x 1 2 3

  12. Path Model Example Training Training Normal Too steep Too low dx d2x x Anomaly Score

  13. Example TEK Results Anomaly Score TEK 0 TEK 1 TEK 10 TEK 11 TEK 12 (Training) (Normal)

  14. Problems with Path Modeling • Testing is slow, O(n2) • Compares n test points to n training points each • Model is complex (stores n points)

  15. Proposed Solution • Piecewise linear approximation of path • Editable (k segments, k << n) • Faster testing, O(kn) • State machine model (nearest segment) • Fast testing, O(n) (same as Gecko) • Local minima problem (same as Gecko)

  16. Piecewise Approximation Algorithm • Repeat n – k times • Remove vertex with lowest cost = dh2 • Run time is O(n log n) using doubly linked heap h d

  17. Test k: compare to all segments Nearest segment: 0-19 x dx Anomaly Score TEK0 training TEK3 near normal TEK12 stuck poppet TEK16 late release

  18. Paths (not segmented) TEK 16 TEK 12 x TEK 0 TEK 3 d2x dx

  19. TEK 0 approximation with k = 20 segments

  20. Test 2: compare only to current and next segment (fails) TEK 0 training TEK 3 OK TEK 12 local minima TEK 16 local minima

  21. Test 4 segments (previous, current, next 2) succeeds Training OK Skips past minimum Transitions back

  22. Test 4 fails with k = 50 Training OK Not complete Delayed completion

  23. Test 5 (previous, current, next 2, and one random segment) succeeds

  24. Path Fitting (optimal if no sharp bends) • Repeat n – k times • Remove lowest cost vertex (cost = dh2) • Move adjacent vertices by h/4 toward removed vertex

  25. Vertex Removal vs. Path Fitting • TEK 0 self anomaly scores • Path fitting better for k > 50 • Vertex removal better for k < 50 Vertex removal Path fitting K Maximum Total Maximum Total 200 0.000008 0.000656 0.000005 0.000350 100 0.000057 0.005802 0.000019 0.003903 50 0.000345 0.027968 0.000542 0.025327 20 0.010298 0.601229 0.015872 0.961845

  26. Path Modeling vs. Gecko • Data: Voltage Test 1 at 14V, 16V, 18V... to 32V • 10 x 20K points • 31 sets of 1-3 training files • Gecko • Transition threshold = 3 • Error threshold = 10 or 20 • Results: pass at 10 (P), pass at 20 (P/F) or fail • Path Modeling • Filter delay 2 x 50 samples per dimension • k = 50 segments • Test 5 (last, current, next 2, and random) • Results: maximum and total anomaly score

  27. Typical Results Test file + = Train Maximum Total Gecko V37898 V14 T21 R00s.txt 0.041018 58.254755 V37898 V16 T21 R00s.txt 0.021778 43.696323 V37898 V18 T21 R00s.txt 0.006596 26.814669 V37898 V20 T21 R00s.txt + 0.000913 0.705107 P V37898 V22 T21 R00s.txt 0.008819 48.095410 P/F V37898 V24 T21 R00s.txt 0.006635 23.487464 P V37898 V26 T21 R00s.txt + 0.000361 0.593473 P V37898 V28 T21 R00s.txt 0.009032 48.236476 V37898 V30 T21 R00s.txt 0.033475 194.134671 V37898 V32 T21 R00s.txt 0.076193 448.467580

  28. Gecko Summary (Stan) • Gecko • 1 training file: correct behavior • 10 self: 10 P (100% correct) • 90 others: 3 P/F, 87 F (97-100% correct) • 2-3 training files: some generalization • 26 self: 23 P, 3 F (14V, 14V, 16V) (88% correct) • 14V is too different from the others • 22 “between”: 8 P, 6 P/F, 8 F (36-63% correct) • 162 others: 1 P/F, 161 F (99-100% correct)

  29. Path Model Summary • Anomaly score proportional to training-test difference (correct) • Multiple training sets: no generalization (expected)

  30. Run Time Performance • Tested on data set 1 (218 x 20K points) • 50 training files = 106 samples • 168 test files = 3.36 x 106 samples • 750 MHz Duron, tsad4.cpp, g++ -O 2.95.2 • Read and filter 106 points: 23 sec • Approximate to k = 100 segments: 30 sec. • Test k: 162 sec (500 ns per point per segment)

  31. Summary

  32. Future Work • Test path modeling with other data sets • UCR archive, http://www.cs.ucr.edu/~eamonn/TSDMA/ • Power load profiles, http://www.delelect.com/pdfs/Del-Res.txt • Test with multiple dimensions • Generalization?

  33. Thank You Further Reading http://cs.fit.edu/~mmahoney/nasa/

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