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52 nd IEEE Conference on Decision and Control Firenze, Italy, December 10-13, 2013 Study of the effective number of parameters in nonlinear identification benchmarks Anna Marconato, Maarten Schoukens, Yves Rolain and Johan Schoukens ELEC

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Study of the effective number

  • f parameters in nonlinear

identification benchmarks

Anna Marconato, Maarten Schoukens, Yves Rolain and Johan Schoukens ELEC – Vrije Universiteit Brussel, Belgium

52nd IEEE Conference on Decision and Control – Firenze, Italy, December 10-13, 2013

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Wiener-Hammerstein benchmark

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Effective number of parameters

 Property of the identified model  Degrees of freedom in the model parametrization

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𝒐eff vs. 𝒐𝜾

Number of parameters

 SVMs?  Regularization?

𝒐𝜾 𝒐eff

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 E

 Motivation example: FIR case  Linear / Nonlinear in the parameters  Comparison on WH benchmark

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Outline

𝒐eff vs. 𝒐𝜾

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Motivation: FIR example

𝑧 = 𝑕 𝑙𝑣 𝑢 − 𝑙

𝑒 𝑙=0

system response

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Motivation: FIR example

𝑧 = 𝑕 𝑙𝑣 𝑢 − 𝑙

𝑒 𝑙=0

𝑕 = 𝐿𝑈𝐿 −1𝐿𝑈𝑧

system response least squares solution

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Motivation: FIR example

?

least squares solution system response

𝑧 = 𝑕 𝑙𝑣 𝑢 − 𝑙

𝑒 𝑙=0

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Motivation: FIR example

𝑧 = 𝑕 𝑙𝑣 𝑢 − 𝑙

𝑒 𝑙=0

𝑕 = 𝐿𝑈𝐿 −1𝐿𝑈𝑧 = 𝑊Σ−1𝑉𝑈𝑧 = 𝑊𝜄 𝐿 = 𝑉Σ𝑊𝑈

SVD

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Motivation: FIR example

𝑕 = 𝑊 𝜄

𝑜𝜄 × 1 𝑜eff × 1

𝑕 = 𝑊𝜄

𝑜𝜄 × 1 𝑜θ × 1

truncated solution least squares solution system response

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Regressor matrix and 𝒐eff

LINEAR REGRESSION

𝜄 = 𝐿𝑈𝐿 −1𝐿𝑈𝑧 = 𝑊Σ−1𝑉𝑈𝑧

𝐿 = 𝑉Σ𝑊𝑈

SVD

𝑧 = 𝐿𝜄 Rank K 𝒐eff

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Jacobian matrix and 𝒐eff

NONLINEAR IN THE PARAMETERS

Δ𝜄 = 𝐾𝑈𝐾 −1𝐾𝑈𝑓 = 𝑊Σ−1𝑉𝑈𝑓 𝜄 𝑗+1 = 𝜄 𝑗 + Δ𝜄 Rank J 𝒐eff

𝐾 = 𝑉Σ𝑊𝑈

SVD

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WH results: comparison

𝒐𝜾 =134 RMSE = 5.6 mV

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WH results: singular values of J

threshold

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WH results: 𝒐eff

𝑜eff = 𝜏𝑗

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𝜏𝑗2 + 𝜇

𝑜𝜄 𝑗=1 REGULARIZATION (ridge regression)

𝜇 = 1 𝑜eff = 33

𝜏2

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WH results: comparison

134 64 69 29 33 52

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Conclusion

Effective number of parameters

 Measure of model complexity for a given dataset

More correct comparison of nonlinear models

 WH benchmark

Rank reduced estimation based on truncated SVD

 NL in the parameters: still open problem

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Thank you for your attention! Any questions?

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Study of the effective number

  • f parameters in nonlinear

identification benchmarks

Anna Marconato, Maarten Schoukens, Yves Rolain and Johan Schoukens ELEC – Vrije Universiteit Brussel, Belgium

52nd IEEE Conference on Decision and Control – Firenze, Italy, December 10-13, 2013

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Silverbox results: comparison

𝒐𝜾 = 23 RMSE = 0.34 mV

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Silverbox results: comparison