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

Problem: How to Detect Anomalies in Time Series Data

• Normal Marotta Fuel Valve Solenoid Current (Used on Space Shuttle)

• Abnormal (poppet partially blocked)

Goal

• Reduce human workload in specifying “normal” model

• Editable rule based model (in SCL)

• Real time testing (1K-10K samples per second)

Manual Method

• Identify features (zero crossings, peaks…)

• Specify correct behavior using SCL rulesRecorded Current Signature of a Known Good Valve

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

-0.2 0 0.2 0.4 0.6 0.8 1

Time (Seconds)

Val

ve C

urr

ent

1

2

3

4

1

2

3

4

Energized

De-Energized

Energizing

De-Energizing

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

Compression Model

Normal, uncompressed

Abnormal, uncompressed

Normal, compressed

Abnormal, compressed

Normal 1 Normal 2

Normal 1 or 2 Abnormal

TEK Compression Anomaly Scores

0

0.2

0.4

0.6

0.8

1

1.2

GZIP PAQ3 RK

Nor 0

Nor 1

Ab 0

Ab 1

Goal Evaluation

Manual Gecko Compres-sion

Reduce Workload

No Yes Yes

Real Time Yes Yes Possible

Editable model

Yes Yes No

Problem with Gecko/RIPPER: State Machine May Underconstrain Model

TrainingSegment 1: x = 0, dx = 0Segment 2: 0 < x < 1, dx = 1

TestSegment 1: x = 0, dx = 0Segment 2: 0 < x < 1, dx = 3

State 1 State 2

dx > 0.5

Accept

Path Model

dx

x1 2 3

1

2

3

Training Path (scaled to unit cube)

Test Path (d2 = 4)

Path Model ExampleTraining Training Normal Too steep Too low

x

dxd2x

Anomaly Score

Example TEK Results

TEK 0 TEK 1 TEK 10 TEK 11 TEK 12(Training) (Normal)

AnomalyScore

Problems with Path Modeling

• Testing is slow, O(n2)– Compares n test points to n training points

each

• Model is complex (stores n points)

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)

Piecewise Approximation Algorithm

• Repeat n – k times– Remove vertex with lowest cost = dh2

• Run time is O(n log n) using doubly linked heap

d

h

Test k: compare to all segments

TEK0 training TEK3 near normal TEK12 stuck poppet TEK16 late release

x

dx

AnomalyScore

Nearest segment: 0-19

Paths (not segmented)

TEK 16

TEK 0TEK 3

TEK 12

x

dxd2x

TEK 0 approximation with k = 20 segments

Test 2: compare only to current and next segment (fails)

TEK 0 training TEK 3 OK TEK 12 local minima TEK 16 local minima

Test 4 segments (previous, current, next 2) succeeds

Training OK Skips past minimum Transitions back

Test 4 fails with k = 50

Training OK Not complete Delayed completion

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

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

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

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

Typical Results

Test file + = Train Maximum Total GeckoV37898 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

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)

Path Model Summary

• Anomaly score proportional to training-test difference (correct)

• Multiple training sets: no generalization (expected)

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)

Summary

Path Model Gecko

Meets all goals Yes Yes

Output Numeric Pass/fail

Training speed O(n log n) O(n2) (DTW)

Test speed O(n) O(n)

Parameters Filter delay, number of segments

Transition and error thresholds

Local minima Yes Yes

Generalization No Some

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?

Thank You

Further Reading

http://cs.fit.edu/~mmahoney/nasa/