Nonlinear modelling of curvature by bi-linear metamodelling

The phenomenon of line curvature - that a smooth line z=f(x) deviates from being straight - is often observed in scientific data. Fitting nonlinear mathematical models to curves today requires slow iterative search processes prone to errors due to local optima. A new generic method, the direct look-up method, is presented for mathematical description of such curvature with less subjectivity and simpler parameter estimation. The new modelometric method is based on bi-linear metamodelling to emulate a whole set of potentially relevant nonlinear models capable of describing curvature. A comprehensive set of 38 nonlinear mathematical models was here collected from different scientific disciplines. Each model can generate a wide range of monotonous sigmoid or arched output shapes depending on their set of input parameter values. For each nonlinear model, its model phenome - its repertoire of possible output curves - was established once and for all by computer simulations statistically designed to fill the relevant model parameter space at a chosen resolution. This simulated curve set was compressed by Principal Component Analysis. The resulting set of 38 bi-linear metamodels emulates the input-output behaviour of the nonlinear models. Then, to parameterise new curves, the input data of each curve were fitted to all relevant nonlinear models via their metamodels. Models with good enough fit were listed as plausible, and their unknown parameter values were estimated from their closest known simulation design points. Thereby, the slow, iterative nonlinear curve fitting was replaced by a fast linear projection with a simple look-up quantification. The traditional problem of choosing initial values to avoid local optima was eliminated. The multivariate metamodelling allowed a wide set of nonlinear curvature descriptions to be handled.