H-methods in applied sciences

The author has developed a framework for mathematical modelling within applied sciences. It is characteristic for data from 'nature and industry' that they have reduced rank for inference. It means that full rank solutions normally do not give satisfactory solutions. The basic idea of H-methods is to build up the mathematical model in steps by using weighing schemes. Each weighing scheme produces a score and/or a loading vector that are expected to perform a certain task. Optimisation procedures are used to obtain 'the best' solution at each step. At each step, the optimisation is concerned with finding a balance between the estimation task and the prediction task. The name H-methods has been chosen because of close analogy with the Heisenberg uncertainty inequality. A similar situation is present in modelling data. The mathematical modelling stops, when the prediction aspect of the model cannot be improved. H-methods have been applied to wide range of fields within applied sciences. In each case, the H-methods provide with superior solutions compared to the traditional ones. A background for the H-methods is presented. The H-principle of mathematical modelling is explained. It is shown how the principle leads to well-defined optimisation procedures. This is illustrated in the case of linear regression. The H-methods have been applied in different areas: general linear models, nonlinear models, multi-block methods, path modelling, multi-way data analysis, growth models, dynamic models and pattern recognition.