Estimating nonlinear regression errors without doing regression

A method for estimating nonlinear regression errors and their distributions without performing regression is presented. Assuming continuity of the modeling function the variance is given in terms of conditional probabilities extracted from the data. For N data points the computational demand is N. Comparing the predicted residual errors with those derived from a linear model assumption provides a signal for nonlinearity. The method is successfully illustrated with data generated by the Ikeda and Lorenz maps augmented with noise. As a by-product the embedding dimensions of these maps are also extracted. This note contains derivations of the formalism and elaborations of the results presented in C. Peterson, ”Determining dependency structures and estimating nonlinear regression errors without doing regression”, International Journal of Modern Physics 6, 611-616 (1995). The latter should be used for citations.