Deep Energy-Based NARX Models -0.3cm 0.990.4pt40pt · 2021. 7. 14. · Deep Energy-Based NARX...
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![Page 1: Deep Energy-Based NARX Models -0.3cm 0.990.4pt40pt · 2021. 7. 14. · Deep Energy-Based NARX Models Johannes N. Hendriks1, Fredrik K. Gustafsson2 Ant^onio H. Ribeiro2, Adrian G.](https://reader035.fdocuments.us/reader035/viewer/2022081622/613c65764c23507cb6355b3c/html5/thumbnails/1.jpg)
Deep Energy-Based NARX Models
Johannes N. Hendriks1, Fredrik K. Gustafsson2
Antonio H. Ribeiro2, Adrian G. Wills1,Thomas B. Schon2
1The University of Newcastle, Australia2Uppsala University, Sweden
Workshop on Nonlinear System Identification Benchmarks2021
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Motivation
Common performance criteria such as maximum-likelihood orprediction-error criteria usually involve assumptions about
uncertainty, be they explicit or implicit
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Nonlinear ARX model (Gaussian noise)
§ Data model:yt “ fθpyt´1, ut´1q ` et ,
where et „ N p0, σq.
§ fθ ù model structure.
§ Maximum likelihood estimator:
pθ “ arg minθ
Tÿ
t“1
}yt ´ fθpyt´1, ut´1q}2.
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Nonlinear ARX model (Gaussian noise)
§ Data model:yt “ fθpyt´1, ut´1q ` et ,
where et „ N p0, σq.§ fθ ù model structure.
§ Maximum likelihood estimator:
pθ “ arg minθ
Tÿ
t“1
}yt ´ fθpyt´1, ut´1q}2.
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Nonlinear ARX model (Gaussian noise)
§ Data model:yt “ fθpyt´1, ut´1q ` et ,
where et „ N p0, σq.§ fθ ù model structure.
§ Maximum likelihood estimator:
pθ “ arg minθ
Tÿ
t“1
}yt ´ fθpyt´1, ut´1q}2.
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Energy-based NARX models
§ Arbitrary distributions:
yt |pyt´1, ut´1q „ pθpyt |yt´1, ut´1q,
§ Energy-based model:
pθpyt |yt´1, ut´1q “egθpyt ,yt´1,ut´1q
ş
egθpγ,yt´1,ut´1q dγ,
Gustafsson, F.K., Danelljan, M., Bhat, G., and Schon,T.B. (2020). Energy-based models for deepprobabilistic regression. In Proceedings of the European Conference on Computer Vision (ECCV)
§ gθ ù Highly flexible structure: in our case a neural network.
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Energy-based NARX models
§ Arbitrary distributions:
yt |pyt´1, ut´1q „ pθpyt |yt´1, ut´1q,
§ Energy-based model:
pθpyt |yt´1, ut´1q “egθpyt ,yt´1,ut´1q
ş
egθpγ,yt´1,ut´1q dγ,
Gustafsson, F.K., Danelljan, M., Bhat, G., and Schon,T.B. (2020). Energy-based models for deepprobabilistic regression. In Proceedings of the European Conference on Computer Vision (ECCV)
§ gθ ù Highly flexible structure: in our case a neural network.
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Energy-based NARX models
§ Arbitrary distributions:
yt |pyt´1, ut´1q „ pθpyt |yt´1, ut´1q,
§ Energy-based model:
pθpyt |yt´1, ut´1q “egθpyt ,yt´1,ut´1q
ş
egθpγ,yt´1,ut´1q dγ,
Gustafsson, F.K., Danelljan, M., Bhat, G., and Schon,T.B. (2020). Energy-based models for deepprobabilistic regression. In Proceedings of the European Conference on Computer Vision (ECCV)
§ gθ ù Highly flexible structure: in our case a neural network.
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Model training
§ Maximum likelihood
pθ “ arg maxθ
Tÿ
i“1
´ log pθpyt | yt´1, ut´1q
“ arg minθ
Tÿ
t“1
ˆ
´gθpyt , xtq ` ln
ż
egθpγ,xtq dγ
˙
§ Noise contrastive estimation:Gutmann, M. and Hyvarinen, A. (2010). Noise-contrastive estimation: A new estimation principle forunnormalized statistical models. In Proceedings of the International Conference on Artificial Intelligenceand Statistics (AISTATS), 297–304
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Model training
§ Maximum likelihood
pθ “ arg maxθ
Tÿ
i“1
´ log pθpyt | yt´1, ut´1q
“ arg minθ
Tÿ
t“1
ˆ
´gθpyt , xtq ` ln
ż
egθpγ,xtq dγ
˙
§ Noise contrastive estimation:Gutmann, M. and Hyvarinen, A. (2010). Noise-contrastive estimation: A new estimation principle forunnormalized statistical models. In Proceedings of the International Conference on Artificial Intelligenceand Statistics (AISTATS), 297–304
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Model training
§ Maximum likelihood
pθ “ arg maxθ
Tÿ
i“1
´ log pθpyt | yt´1, ut´1q
“ arg minθ
Tÿ
t“1
ˆ
´gθpyt , xtq ` ln
ż
egθpγ,xtq dγ
˙
§ Noise contrastive estimation:Gutmann, M. and Hyvarinen, A. (2010). Noise-contrastive estimation: A new estimation principle forunnormalized statistical models. In Proceedings of the International Conference on Artificial Intelligenceand Statistics (AISTATS), 297–304
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Example 1: AR model
yt “ 0.95yt´1 ` et .
Figure: Gaussian error et
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Example 1: AR model
yt “ 0.95yt´1 ` et .
Figure: Gaussian mixture error et
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Example 1: AR model
yt “ 0.95yt´1 ` et .
Figure: Cauchy error et
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Example 1: AR model
yt “ 0.95yt´1 ` et .
Figure: Gaussian error et with conditional variance
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Example 2: ARX model
yt “ 1.5yt´1 ´ 0.7yt´2 ` ut´1 ` 0.5ut´2 ` et ,
(a) Sequence (b) t=56
Figure: Estimates of pθpyt |xtq for a validation data.
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Example 3: nonlinear model
Model:
y˚t “
´
0.8´ 0.5e´y˚2t´1
¯
y˚t´1 ´
´
0.3` 0.9e´y˚2t´1
¯
y˚t´2
` ut´1 ` 0.2ut´2 ` 0.1ut´1ut´2 ` vt ,
yt “yt ` wt
Process and output error:
vt „N p0, σ2v q
wt „N p0, σ2v q Figure: System only with
process noise. Input inblue and output in red.
Chen, S., Billings, S.A., and Grant, P.M. (1990). Non-Linear System Identification Using Neural Networks.International Journal of Control, 51(6), 1191–1214.
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Example 3: nonlinear model
Table: Simulated nonlinear MSE on the validation set for the fullyconnected network (FCN) NARX model and EB-NARX model
N “ 100 N “ 250 N “ 500FCN EB-NARX FCN EB-NARX FCN EB-NARX
σ “ 0.1 0.122 0.099 0.069 0.070 0.057 0.054σ “ 0.3 0.398 0.390 0.353 0.354 0.289 0.308σ “ 0.5 0.860 0.869 0.809 0.822 0.754 0.779
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Example 3: nonlinear model
(a) Sequence (b) t=56
Figure: Estimates of pθpyt |xtq for a validation data.
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Example 4: Coupled Electric Drives
Figure: Illustration of the CE8 coupled electric drives system
Wigren, T. and Schoukens, M. (2017). Coupled electric drives data set and reference models. Technical Report.Uppsala Universitet, 2017
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Example 4: Coupled Electric Drives
Figure: pθpyt |xtq sequence
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Example 4: Coupled Electric Drives
Figure: t “ 40
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Example 4: Coupled Electric Drives
Figure: t “ 57
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Example 4: Coupled Electric Drives
Figure: t “ 60
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Conclusion
§ Energy based NARX learns the full conditional distributionrather than the point estimate.
§ The current paper only considers one-step-ahead predictionsand not multi-step-ahead predictions.
§ Propagate MAP point estimates vs probabilities.
§ Studying energy-based models for nonlinear ARMAX, outputerror and other types of models that can handle more generalnoise types.
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Conclusion
§ Energy based NARX learns the full conditional distributionrather than the point estimate.
§ The current paper only considers one-step-ahead predictionsand not multi-step-ahead predictions.
§ Propagate MAP point estimates vs probabilities.
§ Studying energy-based models for nonlinear ARMAX, outputerror and other types of models that can handle more generalnoise types.
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Conclusion
§ Energy based NARX learns the full conditional distributionrather than the point estimate.
§ The current paper only considers one-step-ahead predictionsand not multi-step-ahead predictions.
§ Propagate MAP point estimates vs probabilities.
§ Studying energy-based models for nonlinear ARMAX, outputerror and other types of models that can handle more generalnoise types.
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Conclusion
§ Energy based NARX learns the full conditional distributionrather than the point estimate.
§ The current paper only considers one-step-ahead predictionsand not multi-step-ahead predictions.
§ Propagate MAP point estimates vs probabilities.
§ Studying energy-based models for nonlinear ARMAX, outputerror and other types of models that can handle more generalnoise types.
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Thank you!
To appear in the 19th IFAC Symposium in System Identification.
Paper: https://arxiv.org/abs/2012.04136
Code: https://github.com/jnh277/ebm arx
Contact:[email protected]@[email protected]@[email protected]
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Thank you!
To appear in the 19th IFAC Symposium in System Identification.
Paper: https://arxiv.org/abs/2012.04136
Code: https://github.com/jnh277/ebm arx
Contact:[email protected]@[email protected]@[email protected]
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Thank you!
To appear in the 19th IFAC Symposium in System Identification.
Paper: https://arxiv.org/abs/2012.04136
Code: https://github.com/jnh277/ebm arx
Contact:[email protected]@[email protected]@[email protected]
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