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Zafeiriou, Lazaros, et al.
Neural Networks and Learning Systems, IEEE Transactions on
Probabilistic Slow Feature Analysis
for Behavior Analysis
Presenter : S5lab. Shuuji Mihara
Abstract1
This Paper propose a number of extensions in both
deterministic and the probabilistic SFA optimization
framework. Particularly about EM-SFA.
This paper shed further light on the relation of the two
sequence EM-SFA and CCA(Canonical Correlation
Analysis).
The proposed EM-SFA with DTW(Dynamic Time
Warping) algorithms were applied for facial behavior
analysis, demonstrating their usefulness for this task.
Index2
1. Introduction – What’s SFA?
2. Deterministic SFA
3. Probabilistic SFA
4. EM-SFA
5. EM-SFA with DTW
6. Experiments
7. Conclusion
Index3
1. Introduction – What’s SFA?
2. Deterministic SFA
3. Probabilistic SFA
4. EM-SFA
5. EM-SFA with DTW
6. Experiments
7. Conclusion
Slow Feature Analysis (2002 Wiskott)
Objective : Extract Slow Feature from Time series data .
4
transform
observation latent variable
Slow Feature Analysis
Slow Feature
1
Index5
1. Introduction – What’s SFA?
2. Deterministic SFA
3. Probabilistic SFA
4. EM-SFA
5. EM-SFA with DTW
6. Experiments
7. Conclusion
Deterministic SFA(1)6
transform
observation latent variable
Slow Feature Analysis
Slow Feature
𝑥1,1:𝑇
𝑥2,1:𝑇
𝑥𝑀,1:𝑇
𝑦1,1:𝑇
𝑦2,1:𝑇
𝑦𝑁,1:𝑇
𝑋 = [𝑥1,1:𝑇; 𝑥2,1:𝑇 … ; 𝑥𝑀,1:𝑇] 𝑌 = [𝑦1,1:𝑇; 𝑦2,1:𝑇 … ; 𝑦𝑁,1:𝑇]
𝑌 = 𝑉𝑇𝑋 (𝑉:𝑀 × 𝑁 𝑚𝑎𝑡𝑟𝑖𝑥)
Deterministic SFA(2)7
Determnistic SFA problem is formulated such optimization problem
min𝑉
tr[ 𝐘 𝐘T] 𝑠. 𝑡. 𝐘𝟏 = 𝟎, 𝐘𝐘T = 𝐈
constraints: zero mean, unit variance
decorreration
𝑌: 1𝑠𝑡 order time difference
Index8
1. Introduction – What’s SFA?
2. Deterministic SFA
3. Probabilistic SFA
4. EM-SFA
5. EM-SFA with DTW
6. Experiments
7. Conclusion
Probabilistic Slow Feature Analysis(pSFA) 9
Observation model
𝒙𝑡 = 𝑾−1𝒚𝑡 +𝒘𝑡
𝒚𝑡: latent variable 𝒙𝑡 ∶ observed data
𝝀 ∶ dependency of 𝒚𝒕−𝟏
𝑾−𝟏: observation matrix
𝒗𝑡 , 𝒘𝑡: noise(Gaussian)
𝒚𝑡 = 𝝀𝒚𝑡−1 + 𝒗𝒕
System model
𝑣𝑡~𝑁 0, Σ𝑤𝑡~𝑁(0, 𝜎𝑥
2𝐼)
constraints
𝝀𝒏𝟐 + 𝝈𝒏
𝟐 = 𝟏
Probabilistic Slow Feature Analysis(pSFA) 10
Slow Feature
𝝀𝒏𝟐 + 𝝈𝒏
𝟐 = 𝟏𝜆𝑛 large
𝜎𝑛 small
𝜆𝑛 small
𝜎𝑛 large𝒚𝑡 = 𝝀𝒚𝑡−1 + 𝒗𝒕
Index11
1. Introduction – What’s SFA?
2. Deterministic SFA
3. Probabilistic SFA
4. EM-SFA
5. EM-SFA with DTW
6. Experiments
7. Conclusion
EM-SFA(1)12
Previous method
Kalman smoother
ML
estimate Λ,𝑊−1
estimate 𝒚
Proposed method
Kalman smoother
Learning Sufficient Statistics
✕ Can’t estimate 𝜎𝑥2
EM algorithm
estimate Λ,𝑊−1, 𝜎𝑥2
Update
Sufficient Statistics
Kalman Smoother (1)13
State Space Model
𝒙𝑘 = 𝐴𝑘−1𝒙𝑘−1 + 𝜼𝒚𝑘 = 𝐻𝑘𝒙𝑘 + 𝝐
⇔𝒙𝑘 ~ 𝑁(𝐴𝑘−1𝒙𝑘−1, 𝑄𝑘−1)
𝒚𝑘 ~ 𝑁(𝑦𝑘𝒙𝑘−1, Σ𝑘)
Kalman Smoother (2)14
Estimate by Kalman Smoother
14
t,1x
t,2x
t,3x
t,1y
t,2y
t,3y
noise 𝜎𝑥2
ESTIMATION
Slow Feature
Probabilistic Slow Feature Analysis(pSFA) 15
Observation model
𝒙𝑡 = 𝑉𝒚𝑡 +𝒘𝑡
𝒚𝑡: latent variable 𝒙𝑡 ∶ observed data
𝚲 ∶ dependency of 𝒚𝒕−𝟏
𝑽 : observation matrix
𝒗𝑡 , 𝒘𝑡: noise(Gaussian)
𝒚𝑡 = 𝚲𝒚𝑡−1 + 𝒗𝒕
System model
𝑣𝑡~𝑁 0, Σ𝑤𝑡~𝑁(0, 𝜎𝑥
2𝐼)
constraints
𝝀𝒏𝟐 + 𝝈𝒏
𝟐 = 𝟏
Inference In SFA16
Parameter
Sufficient Statistics for EM
Kalman Smoother
EM-SFA Algorithm17
Index18
1. Introduction – What’s SFA?
2. Deterministic SFA
3. Probabilistic SFA
4. EM-SFA
5. EM-SFA with DTW
6. Experiments
7. Conclusion
Time Alignment19
Dynamic Time Warping(DTW)20
dynamic time warping (DTW) is an algorithm for
measuring similarity between two temporal sequences
which may vary in time or speed.
Canonical Correration Analysis21
canonical-correlation analysis (CCA) is a way of making
sense of cross-covariance matrices.
𝑢 = 𝑎′𝑥 𝑣 = 𝑏′𝑦
𝑥 = [𝑥1, 𝑥2, … ] y = [𝑦1, 𝑦2, … ]
multivariate data
univariate
𝑎′𝑏′ = argmax𝑎′,𝑏′
𝐶𝑜𝑟[𝑢, 𝑣]
EM-SFA with DTW22
The proposed EM-SFA is more suitable for aligning time
series, since it incorporates temporal constraints (via the
first-order Markov prior), while CCA incorporates a fully
connected MRF prior over the latent space
EM-SFA for Two Sequences23
The Complete joint likelihood distribution
log𝑃 𝑋1, 𝑋2, 𝑌 𝜃)
= log𝑃 𝑦1 0, Σ1) +
𝑡=2
𝑇
log𝑃 𝑦𝑡 𝑦𝑡−1, Λ)
+
𝑡=1
𝑇
log𝑃 𝑥𝑡1 𝑦𝑡 , 𝑉1, 𝜎𝑥,1
2 )+
𝑡=1
𝑇
log𝑃 𝑥𝑡2 𝑦𝑡, 𝑉2, 𝜎𝑥,2
2 )
EM-SFA with DTW24
Index25
1. Introduction – What’s SFA?
2. Deterministic SFA
3. Probabilistic SFA
4. EM-SFA
5. EM-SFA with DTW
6. Experiments
7. Conclusion
Experiment A –Synthetic Data26
27Action Unit(AU)
Experiment B –Real data1Unsupervised AU Temporal Phase Segmentation 28
Experiment B –Real data2Temporal Alignment CTW VS EM-SFA with DTW 29
Experiment B –Real data3Conflict Detection 30
Experiment B –Real data3Conflict Detection 31
Index32
1. Introduction – What’s SFA?
2. Deterministic SFA
3. Probabilistic SFA
4. EM-SFA
5. EM-SFA with DTW
6. Experiments
7. Conclusion
Conclusion33
This Paper propose a number of extensions in both
deterministic and the probabilistic SFA optimization
framework. Particularly about EM-SFA.
This paper shed further light on the relation of the two
sequence EM-SFA and CCA(Canonical Correlation
Analysis).
The proposed EM-SFA with DTW(Dynamic Time
Warping) algorithms were applied for facial behavior
analysis, demonstrating their usefulness for this task.
State Space Model(1)34
State Space Model
𝑥2 𝑥𝑇𝑥1
𝑦1 𝑦2 𝑦𝑇
latent variable
observed variable
sys-eq
obs-eq