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

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.

💡 Research Summary

The paper proposes a comprehensive, data‑driven framework for constructing statistical models of physical laws that explicitly incorporates measurement uncertainty. The authors begin by representing the joint probability density function (PDF) of experimentally measured variables using a kernel density estimator (KDE). Unlike conventional KDE applications where the kernel bandwidth is chosen heuristically or via cross‑validation, the bandwidth σ is determined objectively from the scatter observed during a calibration phase of the experimental apparatus. This calibration‑derived σ directly encodes the physical noise characteristics of the measurement system, ensuring that the estimated PDF faithfully reflects the true experimental uncertainty.

From the estimated joint PDF f(x, y), the conditional density f(y|x) is obtained by normalisation, and the conditional expectation E