Regression Functions as Fields on Covariate Space: Evolution Equations, the Hessian, and Identification
Abstract
For each time t, the regression function mt(x)=E[Yt| Xt=x] of an outcome on contemporaneous covariates can be regarded as a field on covariate space. When the covariates of units move as a diffusion and each unit's outcome has its own dynamics, we show that mt satisfies a second-order evolution equation whose diffusion term is 1/2tr(Σ∇2 mt), the Hessian of the regression function weighted by the covariance Σ of covariate increments, whose transport term is the drift of the time-reversed covariate process, and whose source term is the conditional mean rate of change of the outcome; this yields a Feynman-Kac representation along reversed covariate paths. We relate this Hessian to quadratic regression (under Gaussian design the best quadratic approximation has Hessian E[∇2 m(X)]) and to kernel smoothing bias. For bistable replicator dynamics on covariate space, travelling fronts are exactly logistic in (x,t), so one logistic regression identifies the diffusion-reaction ratio and the invading strategy. We then ask whether spread in covariate space reflects fluctuating unit characteristics or imitation among similar units. Imitation acts through exactly the same operator as covariate fluctuation, so the two are observationally equivalent in the regression field; panel data separate them because they identify Σ directly. The two cannot be separated on a travelling front, and we propose a split-sample instrumental-variables test that works on seeded, non-front profiles, with size and power evaluated in agent-based experiments.
Related
- New Bernstein functions and subordination representations for time-nonlocal evolution equations
- Conformalized Quantile Regression and Minimax Limits of Fixed-Score Calibration under Known Covariate Shift
- Flexible covariate adjustments in regression discontinuity designs
- Covariate-dependent Nonparametric g-modeling for regression via infinite Mixture-of-Expertizing class
- Optimal Neural Estimation for High-Dimensional Nonparametric Regression under Multisource Covariate Shift
BibTeX
@misc{tanaka2026regression,
title = {{Regression Functions as Fields on Covariate Space: Evolution Equations, the Hessian, and Identification}},
author = {Nobuaki Tanaka},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37500},
eprint = {2610.37500},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37500},
}
RIS
TY - UNPB
TI - Regression Functions as Fields on Covariate Space: Evolution Equations, the Hessian, and Identification
AU - Nobuaki Tanaka
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.37500
UR - https://arxiv.org/abs/2610.37500
AB - For each time t, the regression function m_t(x)=E[Y_t| X_t=x] of an outcome on contemporaneous covariates can be regarded as a field on covariate space. When the covariates of units move as a diffusion and each unit's outcome has its own dynamics, we show that m_t satisfies a second-order evolution equation whose diffusion term is 1/2tr(Σ∇^2 m_t), the Hessian of the regression function weighted by the covariance Σ of covariate increments, whose transport term is the drift of the time-reversed covariate process, and whose source term is the conditional mean rate of change of the outcome; this yields a Feynman-Kac representation along reversed covariate paths. We relate this Hessian to quadratic regression (under Gaussian design the best quadratic approximation has Hessian E[∇^2 m(X)]) and to kernel smoothing bias. For bistable replicator dynamics on covariate space, travelling fronts are exactly logistic in (x,t), so one logistic regression identifies the diffusion-reaction ratio and the invading strategy. We then ask whether spread in covariate space reflects fluctuating unit characteristics or imitation among similar units. Imitation acts through exactly the same operator as covariate fluctuation, so the two are observationally equivalent in the regression field; panel data separate them because they identify Σ directly. The two cannot be separated on a travelling front, and we propose a split-sample instrumental-variables test that works on seeded, non-front profiles, with size and power evaluated in agent-based experiments.
ER -