of parameters in nonlinear identification benchmarks Anna - PowerPoint PPT Presentation

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 – Vrije Universiteit Brussel, Belgium

Wiener-Hammerstein benchmark 2

𝒐 eff vs. 𝒐 𝜾 𝒐 𝜾 Number of parameters   SVMs?  Regularization? 𝒐eff Effective number of parameters   Property of the identified model  Degrees of freedom in the model parametrization 3

Outline 𝒐eff vs. 𝒐 𝜾  E  Motivation example: FIR case  Linear / Nonlinear in the parameters  Comparison on WH benchmark 4

Motivation: FIR example 𝑒 𝑧 = 𝑕 𝑙 𝑣 𝑢 − 𝑙 𝑙=0 system response 5

Motivation: FIR example 𝑒 𝑧 = 𝑕 𝑙 𝑣 𝑢 − 𝑙 𝑙=0 system response least squares solution = 𝐿 𝑈 𝐿 −1 𝐿 𝑈 𝑧 𝑕 6

Motivation: FIR example 𝑒 𝑧 = 𝑕 𝑙 𝑣 𝑢 − 𝑙 𝑙=0 system response ? least squares solution 7

Motivation: FIR example 𝑒 𝑧 = 𝑕 𝑙 𝑣 𝑢 − 𝑙 𝑙=0 = 𝐿 𝑈 𝐿 −1 𝐿 𝑈 𝑧 = 𝑊Σ −1 𝑉 𝑈 𝑧 𝑕 = 𝑊𝜄 SVD 𝐿 = 𝑉Σ𝑊 𝑈 8

Motivation: FIR example 𝑕 = 𝑊𝜄 system response truncated solution 𝑜 𝜄 × 1 𝑜 θ × 1 𝜄 least squares solution 𝑕 = 𝑊 𝑜 𝜄 × 1 𝑜eff × 1 9

Regressor matrix and 𝒐 eff 𝑧 = 𝐿𝜄 LINEAR REGRESSION = 𝐿 𝑈 𝐿 −1 𝐿 𝑈 𝑧 = 𝑊Σ −1 𝑉 𝑈 𝑧 𝜄 SVD 𝐿 = 𝑉Σ𝑊 𝑈 𝒐eff Rank K 10

Jacobian matrix and 𝒐 eff Δ𝜄 = 𝐾 𝑈 𝐾 −1 𝐾 𝑈 𝑓 = 𝑊Σ −1 𝑉 𝑈 𝑓 NONLINEAR IN THE PARAMETERS 𝑗+1 = 𝜄 𝑗 + Δ𝜄 𝜄 SVD 𝐾 = 𝑉Σ𝑊 𝑈 𝒐eff Rank J 11

WH results: comparison RMSE = 5.6 mV 𝒐 𝜾 = 134 12

WH results: singular values of J threshold 13

WH results: 𝒐 eff 𝑜 𝜄 2 𝜏 𝑗 REGULARIZATION 𝑜eff = 𝜏 𝑗2 + 𝜇 (ridge regression) 𝑗=1 𝜇 = 1 𝜏 2 𝑜eff = 33 14

WH results: comparison 29 134 64 69 33 52 15

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 16

Thank you for your attention! Any questions? 17

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 – Vrije Universiteit Brussel, Belgium

Silverbox results: comparison RMSE = 0.34 mV 𝒐 𝜾 = 23 19

Silverbox results: comparison 20