|1.||Introduction. Non-linearity in data.|
|2.||Explorative study of non-linearity.|
|4.||Polynomial surfaces in low dimension.|
|7.||Estimation in non-linear models.|
|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|
|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.