# A general approach to statistical modeling of physical laws: nonparametric regression

## TL;DR

This paper introduces a nonparametric regression method based on kernel estimators for modeling physical laws from experimental data, demonstrated through chaotic time series prediction.

## Contribution

It presents a general approach to extract physical laws from data using kernel-based nonparametric regression and introduces a new predictor cost function for model selection.

## Key findings

- Effective modeling of chaotic data using nonparametric regression
- New predictor cost function for optimal data selection
- Accurate future value prediction in noisy chaotic systems

## Abstract

Statistical modeling of experimental physical laws is based on the probability density function of measured variables. It is expressed by experimental data via a kernel estimator. The kernel is determined objectively by the scattering of data during calibration of experimental setup. A physical law, which relates measured variables, is optimally extracted from experimental data by the conditional average estimator. It is derived directly from the kernel estimator and corresponds to a general nonparametric regression. The proposed method is demonstrated by the modeling of a return map of noisy chaotic data. In this example, the nonparametric regression is used to predict a future value of chaotic time series from the present one. The mean predictor error is used in the definition of predictor quality, while the redundancy is expressed by the mean square distance between data points. Both statistics are used in a new definition of predictor cost function. From the minimum of the predictor cost function, a proper number of data in the model is estimated.