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.