The modelome of line curvature: Many nonlinear models approximated by a single bi-linear metamodel with verbal profiling

A generic mathematical phenomenon (line curvature) is described quantitatively and linguistically: a range of very different in form and representation models z=Fm(x)z=Fm(x), m=1,2,…,38m=1,2,…,38, each yielding smooth, but curved relationships z=f(x)z=f(x) with 0 or 1 inflection points, were collected from different fields of science, ranging from systems biology and statistics to trigonometry and psychophysics (Isaeva et al. (2011)). The behavioural repertoire of each of the models was realised by exhaustive statistically designed computer experiments, yielding a total of about 50,000 curves z=f(x)z=f(x), each recorded at 100 xx-values. A modelome of curvature was formed by this set of arched or sigmoid curves and was preprocessed and combined in a joint metamodel based on a bi-linear subspace analysis. To describe a total of 99.9% of the variability in the curves, 12 eigenvectors were needed. These 12 common curve descriptors were successfully related back to the original model input parameters in each of the individual models. Furthermore, to give verbal meaning to the per se meaningless axes in this 12-dimensional eigenvector space, a total of 64 curve images were selected by a statistical design, printed and submitted to descriptive sensory analysis, using a panel of ten trained judges. A quantitative map between the eigenvector space and the sensory space was successfully established and then used for predicting what the human descriptive profiling would be for each of the 50,000 curves. Thus, a first version of a complete “modelome” of the mathematical phenomenon “line curvature” has been established by multivariate metamodelling and described in terms of quantitative maps both to the original model parameters in the 38 individual models and to human verbal description of curve shapes.

► We construct a compact representation of line curvature by its bi-linear metamodel. ► We map a verbal profiling of curvature into its mathematical expressions. ► We show that it is possible, given a function type, to estimate curve parameters by its human profiling.