Delay Embedding on Time Series - zzutk.github.io - Delay Embedding on Ti… · 9/22/2015 Zhifei...
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Delay Embedding on Time Series
9/22/2015 Zhifei Zhang 1
Zhifei Zhang
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9/22/2015 Zhifei Zhang 2
Outline
1. Why delay embedding?
2. How delay embedding works?
3. Modeling and classification
4. Experimental evaluation
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9/22/2015 Zhifei Zhang 3
Why delay embedding?
Time-varying signal (Online)
Real-time processing (Real-time)
Limited storage (Small memory)
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9/22/2015 Zhifei Zhang 4
Why delay embedding?
Online, real-timeand small memoryfails most existing works.
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9/22/2015 Zhifei Zhang 5
Why delay embedding?
t
f(t)1
0
-1
f(t)
f(t+1) 1
-1-1
0
0 1
p1
p2
1-D time series (time space)
2-D time series (embedding space)
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9/22/2015 Zhifei Zhang 6
Why delay embedding?
t
f(t)1
0
-1
f(t)
f(t+1) 1
-1-1
0
0 1
p1
p2
Online Real-timeSmall memory
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9/22/2015 Zhifei Zhang 7
Why delay embedding?
t
f(t)1
0
-1
f(t)
f(t+1) 1
-1-1
0
0 1
p1
p2
OnlineReal-timeSmall memory
[t, f(t) ][t+1, f(t+1)]
[ f(t), f(t+1) ]
O(1)
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9/22/2015 Zhifei Zhang 8
Why delay embedding?
t
f(t)1
0
-1
f(t)
f(t+1) 1
-1-1
0
0 1
p1
p2
OnlineReal-timeSmall memory
Discretization
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9/22/2015 Zhifei Zhang 9
How delay embedding works?
For different patterns, the trajectories are
distinguishable
• Frequency• Amplitude• Tendency
• Length• Phase• Baseline
For the same pattern, the trajectories are
invariant
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9/22/2015 Zhifei Zhang 10
How delay embedding works?Frequency Amplitude Tendency
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9/22/2015 Zhifei Zhang 11
How delay embedding works?
Attenuation in frequency, amplitude and both
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9/22/2015 Zhifei Zhang 12
How delay embedding works?
Some other ways to perform delay embedding
SignalDelay
EmbeddingDifferential Embedding
Derivative Delay Embedding
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9/22/2015 Zhifei Zhang 13
How delay embedding works?
Some other ways to perform delay embedding
SignalDelay
EmbeddingDifferential Embedding
Derivative Delay Embedding
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9/22/2015 Zhifei Zhang 14
Modeling and Classification
Persistent Homology Differential Equations
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9/22/2015 Zhifei Zhang 15
Modeling and Classification
Persistent Homology Differential Equations
• Periodical patterns• Time-consuming• Low fidelity
• Smooth trajectory• Predefined equations• Off-line
To achieve online processing, we may simply record and match trajectories
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9/22/2015 Zhifei Zhang 16
Modeling and Classification
t
f(t)1
0
-1
f(t)
f(t+1) 1
-1-1
0
0 1
p1
p2
Directed Graph
Markov Process
Geographic Distribution
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9/22/2015 Zhifei Zhang 17
Modeling and Classification
S1
S2
S3
S4
S5
Toy example of classification
Class 1Class 2Testing
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9/22/2015 Zhifei Zhang 18
Modeling and Classification
Training stream Testing stream
Class m
2 0 2 1 0 2 0
Buffer
DDE DDE
MGM 1
MGM m
MGM n
MGM
Ge
ogr
aph
icd
istr
ibu
tio
nM
arko
vp
roce
ssUpdate…P(St+1|St)…
Query
Comparesimilarity
DDE-MGM Scheme
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9/22/2015 Zhifei Zhang 19
Experimental Evaluation
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9/22/2015 Zhifei Zhang 20
Experimental Evaluation
Testing data
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9/22/2015 Zhifei Zhang 21
Experimental Evaluation
Example segments and geographic distribution
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9/22/2015 Zhifei Zhang 22
Experimental Evaluation
1
3
7
1313.3 12.711.6
10.9
0
2
4
6
8
10
12
14
10 30 50 80
Efficiency for Different Grid Size
Memory cost (KB) Max. Freq. (kHz)
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9/22/2015 Zhifei Zhang 23
Experimental Evaluation
43.3
29.1220.89
14.9 158
15
34
70
86
0
20
40
60
80
100
20 30 40 50 60
Classification Results for Different Grid Size
Error rate (%) Comp. time (10ms)
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9/22/2015 Zhifei Zhang 24
Experimental Evaluation
17.6 15.9 15.5 14.9 14.910
23
33
55
70
0
10
20
30
40
50
60
70
80
1 2 3 4 5
Classification Results for Different Testing Time
Error rate (%) Comp. time (10ms)
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9/22/2015 Zhifei Zhang 25
Experimental Evaluation
DDE-MGM VS. DTW-base Approach
Leave-20%-out cross validation × 100 iterations
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9/22/2015 Zhifei Zhang 26
Experimental Evaluation
3.33
5.56
0.84
7.6
0
1
2
3
4
5
6
7
8
DDE-MGM State-of-the-art
Classification Results for Different Testing Time
Error rate (%) Comp. time (sec)
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9/22/2015 Zhifei Zhang 27