![Deterministic SFA(1) 6
transform
observation latent variable
Slow Feature Analysis
Slow Feature
𝑥1,1:𝑇
𝑥2,1:𝑇
𝑥 𝑀,1:𝑇
𝑦1,1:𝑇
𝑦2,1:𝑇
𝑦 𝑁,1:𝑇
𝑋 = [𝑥1,1:𝑇; 𝑥2,1:𝑇 … ; 𝑥 𝑀,1:𝑇] 𝑌 = [𝑦1,1:𝑇; 𝑦2,1:𝑇 … ; 𝑦 𝑁,1:𝑇]
𝑌 = 𝑉 𝑇 𝑋 (𝑉: 𝑀 × 𝑁 𝑚𝑎𝑡𝑟𝑖𝑥)](https://image.slidesharecdn.com/probabilisticsfaforbehavioranalysis-151023033911-lva1-app6891/85/Probabilistic-sfa-for-behavior-analysis-7-320.jpg)
![Deterministic SFA(2) 7
Determnistic SFA problem is formulated such optimization problem
min
𝑉
tr[ 𝐘 𝐘T] 𝑠. 𝑡. 𝐘𝟏 = 𝟎, 𝐘𝐘T = 𝐈
constraints: zero mean, unit variance
decorreration
𝑌: 1𝑠𝑡 order time difference](https://image.slidesharecdn.com/probabilisticsfaforbehavioranalysis-151023033911-lva1-app6891/85/Probabilistic-sfa-for-behavior-analysis-8-320.jpg)
![Canonical Correration Analysis 21
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′ = arg max
𝑎′,𝑏′
𝐶𝑜𝑟[𝑢, 𝑣]](https://image.slidesharecdn.com/probabilisticsfaforbehavioranalysis-151023033911-lva1-app6891/85/Probabilistic-sfa-for-behavior-analysis-22-320.jpg)
論文紹介 Probabilistic sfa for behavior analysis
- 1. Zafeiriou, Lazaros, et al. Neural Networks and Learning Systems, IEEE Transactions on Probabilistic Slow Feature Analysis for Behavior Analysis Presenter : S5lab. Shuuji Mihara
- 2. Abstract 1 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.
- 3. Index 2 1. Introduction – What’s SFA? 2. Deterministic SFA 3. Probabilistic SFA 4. EM-SFA 5. EM-SFA with DTW 6. Experiments 7. Conclusion
- 4. Index 3 1. Introduction – What’s SFA? 2. Deterministic SFA 3. Probabilistic SFA 4. EM-SFA 5. EM-SFA with DTW 6. Experiments 7. Conclusion
- 5. Slow Feature Analysis (2002 Wiskott) Objective : Extract Slow Feature from Time series data . 4 transform observation latent variable Slow Feature Analysis Slow Feature 1
- 6. Index 5 1. Introduction – What’s SFA? 2. Deterministic SFA 3. Probabilistic SFA 4. EM-SFA 5. EM-SFA with DTW 6. Experiments 7. Conclusion
- 7. Deterministic SFA(1) 6 transform observation latent variable Slow Feature Analysis Slow Feature 𝑥1,1:𝑇 𝑥2,1:𝑇 𝑥 𝑀,1:𝑇 𝑦1,1:𝑇 𝑦2,1:𝑇 𝑦 𝑁,1:𝑇 𝑋 = [𝑥1,1:𝑇; 𝑥2,1:𝑇 … ; 𝑥 𝑀,1:𝑇] 𝑌 = [𝑦1,1:𝑇; 𝑦2,1:𝑇 … ; 𝑦 𝑁,1:𝑇] 𝑌 = 𝑉 𝑇 𝑋 (𝑉: 𝑀 × 𝑁 𝑚𝑎𝑡𝑟𝑖𝑥)
- 8. Deterministic SFA(2) 7 Determnistic SFA problem is formulated such optimization problem min 𝑉 tr[ 𝐘 𝐘T] 𝑠. 𝑡. 𝐘𝟏 = 𝟎, 𝐘𝐘T = 𝐈 constraints: zero mean, unit variance decorreration 𝑌: 1𝑠𝑡 order time difference
- 9. Index 8 1. Introduction – What’s SFA? 2. Deterministic SFA 3. Probabilistic SFA 4. EM-SFA 5. EM-SFA with DTW 6. Experiments 7. Conclusion
- 10. Probabilistic Slow Feature Analysis (pSFA) 9 Observation model 𝒙 𝑡 = 𝑾−1 𝒚 𝑡 + 𝒘 𝑡 𝒚 𝑡: latent variable 𝒙 𝑡 ∶ observed data 𝝀 ∶ dependency of 𝒚 𝒕−𝟏 𝑾−𝟏: observation matrix 𝒗 𝑡, 𝒘 𝑡: noise(Gaussian) 𝒚 𝑡 = 𝝀𝒚 𝑡−1 + 𝒗 𝒕 System model 𝑣 𝑡~𝑁 0, Σ 𝑤𝑡~𝑁(0, 𝜎 𝑥 2 𝐼) constraints 𝝀 𝒏 𝟐 + 𝝈 𝒏 𝟐 = 𝟏
- 11. Probabilistic Slow Feature Analysis (pSFA) 10 Slow Feature 𝝀 𝒏 𝟐 + 𝝈 𝒏 𝟐 = 𝟏𝜆 𝑛 large 𝜎 𝑛 small 𝜆 𝑛 small 𝜎 𝑛 large 𝒚 𝑡 = 𝝀𝒚 𝑡−1 + 𝒗 𝒕
- 12. Index 11 1. Introduction – What’s SFA? 2. Deterministic SFA 3. Probabilistic SFA 4. EM-SFA 5. EM-SFA with DTW 6. Experiments 7. Conclusion
- 13. 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 𝜎 𝑥 2 EM algorithm estimate Λ, 𝑊−1 , 𝜎 𝑥 2 Update Sufficient Statistics
- 14. Kalman Smoother (1) 13 State Space Model 𝒙 𝑘 = 𝐴 𝑘−1 𝒙 𝑘−1 + 𝜼 𝒚 𝑘 = 𝐻 𝑘 𝒙 𝑘 + 𝝐 ⇔ 𝒙 𝑘 ~ 𝑁(𝐴 𝑘−1 𝒙 𝑘−1, 𝑄 𝑘−1) 𝒚 𝑘 ~ 𝑁(𝑦 𝑘 𝒙 𝑘−1, Σ 𝑘)
- 15. Kalman Smoother (2) 14 Estimate by Kalman Smoother 14 t,1x t,2x t,3x t,1y t,2y t,3y noise 𝜎𝑥 2 ESTIMATION Slow Feature
- 16. Probabilistic Slow Feature Analysis (pSFA) 15 Observation model 𝒙 𝑡 = 𝑉𝒚 𝑡 + 𝒘 𝑡 𝒚 𝑡: latent variable 𝒙 𝑡 ∶ observed data 𝚲 ∶ dependency of 𝒚 𝒕−𝟏 𝑽 : observation matrix 𝒗 𝑡, 𝒘 𝑡: noise(Gaussian) 𝒚 𝑡 = 𝚲𝒚 𝑡−1 + 𝒗 𝒕 System model 𝑣 𝑡~𝑁 0, Σ 𝑤𝑡~𝑁(0, 𝜎 𝑥 2 𝐼) constraints 𝝀 𝒏 𝟐 + 𝝈 𝒏 𝟐 = 𝟏
- 17. Inference In SFA 16 Parameter Sufficient Statistics for EM Kalman Smoother
- 19. Index 18 1. Introduction – What’s SFA? 2. Deterministic SFA 3. Probabilistic SFA 4. EM-SFA 5. EM-SFA with DTW 6. Experiments 7. Conclusion
- 21. 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.
- 22. Canonical Correration Analysis 21 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′ = arg max 𝑎′,𝑏′ 𝐶𝑜𝑟[𝑢, 𝑣]
- 23. EM-SFA with DTW 22 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
- 24. EM-SFA for Two Sequences 23 The Complete joint likelihood distribution log 𝑃 𝑋1, 𝑋2, 𝑌 𝜃) = log 𝑃 𝑦1 0, Σ1) + 𝑡=2 𝑇 log 𝑃 𝑦𝑡 𝑦𝑡−1, Λ) + 𝑡=1 𝑇 log 𝑃 𝑥 𝑡 1 𝑦𝑡, 𝑉1, 𝜎 𝑥,1 2 ) + 𝑡=1 𝑇 log 𝑃 𝑥 𝑡 2 𝑦𝑡, 𝑉2, 𝜎 𝑥,2 2 )
- 25. EM-SFA with DTW 24
- 26. Index 25 1. Introduction – What’s SFA? 2. Deterministic SFA 3. Probabilistic SFA 4. EM-SFA 5. EM-SFA with DTW 6. Experiments 7. Conclusion
- 27. Experiment A –Synthetic Data 26
- 29. Experiment B –Real data1 Unsupervised AU Temporal Phase Segmentation 28
- 30. Experiment B –Real data2 Temporal Alignment CTW VS EM-SFA with DTW 29
- 31. Experiment B –Real data3 Conflict Detection 30
- 32. Experiment B –Real data3 Conflict Detection 31
- 33. Index 32 1. Introduction – What’s SFA? 2. Deterministic SFA 3. Probabilistic SFA 4. EM-SFA 5. EM-SFA with DTW 6. Experiments 7. Conclusion
- 34. Conclusion 33 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.
- 35. State Space Model(1) 34 State Space Model 𝑥2 𝑥 𝑇𝑥1 𝑦1 𝑦2 𝑦 𝑇 latent variable observed variable sys-eq obs-eq