**Non-linear methods**

*Topics*

1. | Introduction. Non-linearity in data. | |

2. | Explorative study of non-linearity. | |

3. | Gauss-Newton procedure. | |

4. | Polynomial surfaces in low dimension. | |

5. | Graphic analysis. | |

6. | Case study | |

7. | Estimation in non-linear models. | |

8. | Modelling strategies. | |

9. | Examples | |

10. | Importance of variables. | |

11. | Curvature. Degree of non-linearity. | |

12. | Linear and non-negativity constraints. | |

13. | Trend and non-linearity. | |

14. | Predictions in non-linear models | |

15. | Analysis of residuals. | |

16. | Comparisons of methods | |

17. | Confidence intervals | |

18. | Outlier detection | |

19. | Sensitivity analysis | |

20. | Case study | |

21. | Guidelines for presentation of results |

Industrial data often show sign of non-linearity. The challenge, when working with non-linearity in industrial data, is that the rank is typically rather low. Standard procedures that are implemented in the program packages usually break down. The H-method has been applied with success to handle non-linearity in industrial data. There are typically two situations that are important.

The first one is the situation, where there is not specified a given mathematical model. But linear analysis show a clear sign of curvature in data. In these cases efficient procedures have been developed that is based on finding score vectors such that a polynomial in score vectors provide with efficient predictions. In terms of the original variables it means finding a polynomial surface in the original variables. These procedures have shown their success when working with industrial data. The reason is that the linearity in industrial data often shows itself as a weak curvature that typically is adequately modelled by a second or third order polynomial function of appropriate score vectors.

The second one is where there is given a specific non-linear model that the data are expected to 'obey'. The Gauss-Newton procedure suggests that the solution should be found by successive linear approximation. By using the H-method appropriate low rank solutions for each linear approximation are derived. The procedures developed have been found successful both in finding solutions, where the Gauss-Newton method (and Marquard's regularisations) break down, and also in deriving better predictions from non-linear models than traditional methods.

See a short introduction to some of the ideas.

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