Model Selection Based on the Modulus of Continuity

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10.90.8train errortest errorBICSEBAIC0.70.60.50.40.30.20.100 5 10 15 20 25 30 35 40 45(a)10.9test errortrain errorMC0.80.70.60.50.40.30.20.100 5 10 15 20 25 30 35 40 45(b)Fig. 12. Performance Predictions of model selection methods in the case of combined function: (a) represents theestimated error in the L 2 sense versus the number of nodes using the AIC, BIC, and SEB methods and (b) representsthe estimated error in the L 1 sense versus the number of nodes using the MC method.26
5 ConclusionWe have suggested a new method of model selection in regression problems based on themodulus of continuity. The risk function bounds are investigated from the view point of themodulus of continuity for both the target and estimation functions. The final form of the riskfunction bounds incorporate the information of learned results such as the sum of absoluteweights as well as the structural information such as the number of nodes in the regressionmodel. To verify the validity of the suggested bound, the model selection of regression modelsusing the trigonometric polynomials in function approximation problem was performed. Asa result, the suggested method showed the better performance for various types of targetfunctions from the view points of risk and node ratios compared to other model selectionmethods. We also demonstrated that the risk prediction using the MC method was able tocatch the trend of test errors. This was mainly due to the fact that the MC based methodwas performed using the risk function bounds incorporating both the information of learnedresults as well as the structural information of the estimation network, the main criteriaof other model selection methods. Furthermore, the suggested MC method can be easilyextended to various types of regression models with nonlinear kernel functions which havesome smoothness constraints.27
Page 2: the trade-off between the under-fit
Page 10 and 11: Therefore, the upper bound of risk
Page 12 and 13: similar complexity while the fourth
Page 14 and 15: To see the risk prediction of the m
Page 16 and 17: 32.521.510.50AIC BIC SEB MC(a)32.52
Page 24 and 25: 0.80.60.40.20 0.025 0.05 0.1(a)0.80
Page 28 and 29: References1. Akaike, H.: Informatio
Page 30: Now, let us consider the target fun

Model Selection Based on the Modulus of Continuity

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