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arXiv:2610.37500 · v2

Regression Functions as Fields on Covariate Space: Evolution Equations, the Hessian, and Identification

Nobuaki Tanaka
math.STcs.LG
首次提交 2026-10-28 00:24
版本更新 2026-10-28 11:10
雷达观测 2026-10-30 09:53
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摘要

For each time $t$, the regression function $m_t(x)=E[Y_t\mid X_t=x]$ 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 $\frac12\mathrm{tr}(Σ\nabla^2 m_t)$, 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 $E[\nabla^2 m(X)]$) 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.

作者备注：13 pages, 2 figures. Code for the symbolic checks and simulations is included as ancillary files

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摘要变化

For each time $t$, the regression function $m_t(x)=E[Y_t\mid X_t=x]$ of an outcome on contemporaneous covariates can be regarded as a field on covariate space. We study how this field evolves whenWhen the covariates of individual units move as a diffusion and the outcome attached to each unit changes's accordingoutcome tohas its own dynamics. Under mild regularity, we show that $m_t$ satisfies a second-order evolution equation whose diffusion term is $\frac12\mathrm{tr}(Σ\nabla^2 m_t)$, the trace of the Hessian of the regression function weighted by the instantaneous covariance $Σ$ of the 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. The equation is a regression-theoretic reading of a classical transport law, and; itthis yields a Feynman-Kac representation of $m_t$ along reversed covariate paths. We connectrelate thethis Hessian appearing in the equation to the Hessian estimated by quadratic regression (under Gaussian design the best quadratic approximation has Hessian $E[\nabla^2 m(X)]$ exactly) and to thekernel leadingsmoothing bias of kernel smoothers. For binary outcomes governed by bistable replicator dynamics with imitation among units with similar covariates, the closed equation is ofon Nagumocovariate typespace, its planar travelling fronts are exactly logistic in $(x,t)$, and aso singleone logistic regression ofidentifies the outcome ondiffusion-reaction covariatesratio and timethe identifiesinvading strategy. We then ask whether spread in covariate space reflects fluctuating unit characteristics or imitation among similar units. Imitation acts through exactly the ratiosame ofoperator diffusionas tocovariate reactionfluctuation, andso the directiontwo ofare invasion.observationally Weequivalent discussin identificationthe underregression field; panel versusdata repeatedseparate cross-sectionalthem designsbecause andthey aidentify $Σ$ directly. The two-stage estimationcannot strategybe separated on a travelling front, and we illustratepropose thea resultssplit-sample instrumental-variables test that works on seeded, non-front profiles, with simulationssize and power evaluated in agent-based experiments.

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v2 · 2026-10-28 11:10 · 最新观测版本

Regression Functions as Fields on Covariate Space: Evolution Equations, the Hessian, and Identification

For each time $t$, the regression function $m_t(x)=E[Y_t\mid X_t=x]$ 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 $\frac12\mathrm{tr}(Σ\nabla^2 m_t)$, 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 $E[\nabla^2 m(X)]$) 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.

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v1 · 2026-10-28 00:24

Regression Functions as Fields on Covariate Space: Evolution Equations, the Hessian, and Identification

For each time $t$, the regression function $m_t(x)=E[Y_t\mid X_t=x]$ of an outcome on contemporaneous covariates can be regarded as a field on covariate space. We study how this field evolves when the covariates of individual units move as a diffusion and the outcome attached to each unit changes according to its own dynamics. Under mild regularity we show that $m_t$ satisfies a second-order evolution equation whose diffusion term is $\frac12\mathrm{tr}(Σ\nabla^2 m_t)$, the trace of the Hessian of the regression function weighted by the instantaneous covariance $Σ$ of the 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. The equation is a regression-theoretic reading of a classical transport law, and it yields a Feynman-Kac representation of $m_t$ along reversed covariate paths. We connect the Hessian appearing in the equation to the Hessian estimated by quadratic regression (under Gaussian design the best quadratic approximation has Hessian $E[\nabla^2 m(X)]$ exactly) and to the leading bias of kernel smoothers. For binary outcomes governed by bistable replicator dynamics with imitation among units with similar covariates, the closed equation is of Nagumo type, its planar travelling fronts are exactly logistic in $(x,t)$, and a single logistic regression of the outcome on covariates and time identifies the ratio of diffusion to reaction and the direction of invasion. We discuss identification under panel versus repeated cross-sectional designs and a two-stage estimation strategy, and we illustrate the results with simulations.

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