LogLogModeller

LogLogModeller is a statistical framework and software package designed to model relationships that exhibit power-law or scale-invariant behavior by operating in log-log space. The central idea is to transform both the dependent and independent variables using logarithms and then fit a linear model to estimate the scaling exponent, with the intercept capturing the baseline level in log units.

The framework supports estimation approaches such as ordinary least squares on log-transformed data, as well as weighted, robust, or orthogonal regression to address measurement error and heteroskedasticity. It provides options for handling zeros or small values through pseudo-counts or offset terms, and it offers diagnostic tools including residual analysis on the log scale, R-squared in log space, information criteria, and bootstrap confidence intervals. It also supports back-transformation to the original scale and interpretation of the exponent as a multiplicative effect.

Software design and availability: implemented as a library for major languages, with a straightforward API for data input, model specification, fitting, prediction, and plotting. It includes utilities for log-binning and cross-validation, as well as model comparison against alternatives such as log-normal or stretched exponential forms. The package emphasizes transparent handling of zeros, missing data, and scale issues to minimize bias in estimation.

Applications and limitations: log-log modeling is widely used in physics, biology, ecology, and economics for exploring scaling laws and power-law distributions. Limitations include sensitivity to zero and negative values, potential bias from log transformation under heavy measurement error, and the requirement that the relationship be well approximated by a monotone power-law in the tested range.

See also: Power law, Log transformation, Linear regression, Scaling law.