Estimating nonlinear regression errors without doing regression

TL;DR

This paper introduces a novel method to estimate nonlinear regression errors and their distributions directly from data without performing regression, using conditional probabilities, which is computationally feasible and effective for complex systems.

Contribution

The paper presents a new approach to estimate nonlinear regression errors without regression, leveraging conditional probabilities and applied to chaotic maps with noise.

Findings

Successfully estimated errors for Ikeda and Lorenz maps

Detected nonlinearity by comparing residual errors

Extracted embedding dimensions of chaotic maps

Abstract

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 N2. 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.