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On the validity of using the delta method for calculating the uncertainty of the predictions from an overparameterized model

Magnus Malmström, Isaac Skog, Daniel Axehill, Fredrik Gustafsson

Abstract

The uncertainty in the prediction calculated using the delta method for an overparameterized (parametric) black-box model is shown to be larger or equal to the uncertainty in the prediction of a canonical (minimal) model. Equality holds if the additional parameters of the overparameterized model do not add flexibility to the model. As a conclusion, for an overparameterized black-box model, the calculated uncertainty in the prediction by the delta method is not underestimated. The results are shown analytically and are validated in a simulation experiment where the relationship between the normalized traction force and the wheel slip of a car is modelled using e.g., a neural network

On the validity of using the delta method for calculating the uncertainty of the predictions from an overparameterized model

Abstract

The uncertainty in the prediction calculated using the delta method for an overparameterized (parametric) black-box model is shown to be larger or equal to the uncertainty in the prediction of a canonical (minimal) model. Equality holds if the additional parameters of the overparameterized model do not add flexibility to the model. As a conclusion, for an overparameterized black-box model, the calculated uncertainty in the prediction by the delta method is not underestimated. The results are shown analytically and are validated in a simulation experiment where the relationship between the normalized traction force and the wheel slip of a car is modelled using e.g., a neural network
Paper Structure (15 sections, 36 equations, 1 figure)

This paper contains 15 sections, 36 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Signal model and likelihood
  4. Neural network model structure
  5. True system
  6. Parameter covariance
  7. Error propagation using the delta method
  8. Overparametized models
  9. Models with redundant parameters of Category 1
  10. Models with redundant parameters of Category 2
  11. Nonlinear models
  12. Numerical examples
  13. Models with redundant parameters of Category 1
  14. Models with redundant parameters of Category 2
  15. Conclusion

Figures (1)

  • Figure 1: Calculated uncertainty in the prediction for a canonical and an overparameterized model using the delta method. In (a) the model is nonlinear and the overparameterized model has redundant parameters of Category 1. While in (b) the model is linear in its parameters and the overparameterization has redundant parameters of Category 2 which add flexibility to the model. In (c) the model is a two-layer with an increasing number of nodes in the hidden layer, i.e., it is nonlinear and the overparameterization with redundant parameters of Category 2. The simulation data is generated by \ref{['eq:magic']}.