Experimental Information and Statistical Modeling of Physical Laws

TL;DR

This paper introduces a statistical framework for modeling physical laws using kernel estimators, defining experimental information and redundancy, and proposing a model cost function to determine optimal data usage.

Contribution

It presents a novel approach combining kernel estimation with information theory to model physical laws and determine optimal data quantity for accurate modeling.

Findings

Redundancy increases with more experiments.

Experimental information converges to data complexity.

Conditional average serves as an effective nonparametric estimator.

Abstract

Statistical modeling of physical laws connects experiments with mathematical descriptions of natural phenomena. The modeling is based on the probability density of measured variables expressed by experimental data via a kernel estimator. As an objective kernel the scattering function determined by calibration of the instrument is introduced. This function provides for a new definition of experimental information and redundancy of experimentation in terms of information entropy. The redundancy increases with the number of experiments, while the experimental information converges to a value that describes the complexity of the data. The difference between the redundancy and the experimental information is proposed as the model cost function. From its minimum, a proper number of data in the model is estimated. As an optimal, nonparametric estimator of the relation between measured variables the conditional average extracted from the kernel estimator is proposed. The modeling is demonstrated on noisy chaotic data.