Beyond Box-Cox: A diffusion-inspired functional framework for nonlinear demand and discrete choice modeling

This paper presents a new diffusion-inspired functional framework for modeling nonlinear demand and discrete choice, with applications in transportation and related fields. The framework goes beyond traditional transformations by combining a diffusion-like core function with upper and lower bound functions, linked through a transfer function. Key properties such as monotonicity and concavity or convexity are guaranteed by simple, verifiable conditions on the component functions. A key advantage of this approach is that it breaks the nonlinear structure into separate parts, making it easier to integrate socio-economic variables. This enables the model to capture non-linear shifts, damping, and scaling effects across different socio-economic groups. The framework is low-parametric, bounded, continuous, and modular, which makes it easy to estimate using standard software. Its flexibility and strengths are illustrated in two case studies: (i) a discrete choice model of transport mode selection, and (ii) a nonlinear logistic regression model of vehicle ownership. In both cases, the framework enhances model fit, facilitates better control over tail behavior, demonstrates how heterogeneity can be effectively integrated, and yields more precise and behaviorally plausible elasticity estimates. Controlled simulations further demonstrate the framework’s robustness across a broad range of nonlinear processes. Adjusting the individual sub-components leads to distinct functional behaviors, preventing convergence toward a single common shape. This diversity indicates that the framework avoids ”copy-cat” behavior or functional collapse. As a result, its flexible, bounded structure can be tailored or relaxed depending on the application, offering virtually limitless possibilities for adapting functions to address different problems.