arXiv subjects

W. John Braun

Publications and source records attributed to W. John Braun.

10 recordsLinked to original sources

arXiv2026

Bias Reduction for Local Polynomial Derivative Estimation

Local polynomial smoothing is commonly used in non-parametric regression, but local linear derivative estimation still has a bias of order $O(h^2)$. This paper proposes an iterative data sharpening method to reduce the bias of derivative estimates while retaining the simplicity of local linear fitting. The method is based on two expectation operators: $L_0$, acting on the regression function, and $L_1$, acting on the first-order derivative. By repeatedly applying the residual operator $R=I-L_0$, a series of sharpened derivative estimates can be constructed. After $l$ sharpening steps, the bias order can be reduced from $O(h^2)$ to $O(h^{2l+2})$. For the Gaussian kernel, all sharpening coefficients equal 1, giving a simple closed-form single-bandwidth expression. Simulation experiments on three smooth test functions show that this method can significantly reduce the estimation bias while revealing a bias-variance trade-off.

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arXiv2026

Nonparametric Estimation under General Nonlinear ODE Constraints: A Comparison with Parametric ODE-Fitting Methods

Many physical, biological, and epidemiological processes are governed by ordinary differential equations (ODEs) that are nonlinear in the state variable, including logistic population growth, chemical reaction kinetics, and epidemiological compartment models. We develop a differential equation-constrained local polynomial regression (DE-constrained LPR) framework for the general first-order ODE constraint g'(x) = F(x, g(x)), where F may be any Lipschitz continuous function, extending prior work restricted to exponential and linear ODE structures. Because F is generally nonlinear in g, the Taylor coefficients of the DE1-k estimator cannot be written in closed form; instead they are obtained by successive symbolic differentiation of F, and the estimator is computed by nonlinear least squares, requiring only a single local parameter at each evaluation point regardless of polynomial degree k. We derive the asymptotic conditional bias and variance of the DE1-k estimator, propose an AIMSE-optimal bandwidth that exploits the ODE structure to avoid direct estimation of high-order derivatives, and evaluate the method in a simulation study based on logistic growth, benchmarking against the parameter cascading method of Ramsay et al. (2007) (PCODE) and classical local linear regression. The DE-constrained estimator consistently outperforms local linear regression and is competitive with PCODE even though it estimates no structural parameter of the ODE; a sensitivity analysis across growth rates shows DE-constrained estimation becomes more accurate and more robust than PCODE as the curve steepens and PCODE's parameter estimation grows less stable. These results position DE-constrained LPR as a practical nonparametric alternative to parametric ODE-fitting methods when structural parameters are difficult to identify reliably.

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arXiv2026

Local Quasi-Linear Models: Kernel Differential Equation Regression and Fire Data Analysis

We introduce the local quasi-linear (LQL) model, a differential equation-constrained local polynomial regression framework for the general first-order linear ordinary differential equation (ODE) $g'(x)=a(x)g(x)+b(x)$, extending prior work on differential equation-constrained local polynomial regression (DE-constrained LPR) for the exponential growth model. We derive closed-form DE-constrained local polynomial (DE1-$k$) estimators for arbitrary Taylor degree $k$, establish their asymptotic conditional bias and variance, and propose two approaches for estimating $a(x)$ and $b(x)$ when they are unknown. A simulation study across two structurally different ODEs shows that DE1-$k$ estimation with automatic degree selection reduces both estimation error and ODE-consistency error relative to unconstrained local linear regression. We then apply the framework to the firebrand burning-rate experiment of Albini (1979), modelling the density-loss curve of wind-driven firebrands with a physically motivated forced-convection ODE; the DE-constrained estimator outperforms local linear regression in the sparsest species-diameter groups, where physical structure is most valuable in compensating for scarce data. We further examine the robustness of the LQL model to misspecification against a local quasi-exponential alternative. Together, these results extend the DE-constrained regression paradigm to a broad class of physically motivated linear models and provide a practical estimation tool for fire science and other application areas where mechanistic knowledge is available but data are sparse.

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arXiv2026

Local Quasi-Exponential Growth Models: Kernel Differential Equation Regression and Mouse Tumor Growth Data

Local polynomial regression faces several challenges when dealing with sparse data. The difficulty in capturing local features of the underlying function can lead to a possible misrepresentation of the true relationship. Furthermore, with limited data points in local neighborhoods, the variance of estimators can increase significantly. Local polynomial regression also requires a substantial amount of data to produce good models, making it less efficient for sparse datasets. This paper employs a differential equation-constrained regression approach, introduced by \citet{ding2014estimation}, for local quasi-exponential growth models. By incorporating first-order differential equations, this method extends the sparse design capacity of local polynomial regression while reducing bias and variance. We discuss the asymptotic biases and variances of kernel estimators using first-degree Taylor polynomials. Model comparisons are conducted using mouse tumor growth data, along with simulation studies that include tumor growth with different sparse designs, and simulated quasi-exponential growth with varying levels of noise and growth rates.

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arXiv2026

Differential Equation-Constrained Exponential-Type Local Polynomial Regression Under Model Misspecification

The issue of model misspecification is critical, yet it is often regarded as unavoidable in applied statistical modeling. Model misspecification can be mitigated by incorporating informative features and strengthening model formulations, such as through the integration of domain knowledge or structural constraints. In this paper, we propose a regression framework constrained by differential equations, which leverages first-order differential equations and adapts local polynomial regression techniques. Specifically, we focus on the local exponential growth model, characterized by an exponential-type differential equation. For this model, we examine the asymptotic biases and variances of kernel estimators constructed using Taylor polynomials of varying degrees. To evaluate model robustness, we conduct simulation studies comparing different estimators under two misspecification scenarios varying the levels of misspecification.

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arXiv2025

Differential Equation-Constrained Local Regression for Data with Sparse Design

Local polynomial regression of order one or higher often performs poorly in areas with sparse data. In contrast, local constant regression tends to be more robust in these regions, although it is generally the least accurate approach, especially near the boundaries of the data. Incorporating information from differential equations, which may approximately or exactly hold, is one way of extending the sparse design capacity of local constant regression while reducing bias and variance. A nonparametric regression method that exploits first-order differential equations is studied in this paper and applied to noisy mouse tumour growth data. Asymptotic biases and variances of kernel estimators using Taylor polynomials with different degrees are discussed. Model comparison is performed for different estimators through simulation studies under various scenarios that simulate exponential-type growth.

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arXiv2025

On Data Sharpening in Nonparametric Autoregressive Models

Data sharpening has been shown to reduce bias in nonparametric regression and density estimation. Its performance on nonlinear first order autoregressive models is studied theoretically and numerically in this paper. Although the asymptotic properties of data sharpening are not as favourable in the presence of serial dependence as in bivariate regression with independent responses, it is still found to reduce bias under mild conditions on the autoregression function. Numerical comparisons with the bias reduction method of Cheng et al. (2018) indicate that data sharpening is competitive in this setting.

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arXiv2025

The importance of exploration: Modelling site-constant foraging

Foraging site constancy, or repeated return to the same foraging location, is a foraging strategy used by many species to decrease uncertainty, but it is often unclear exactly how the foraging site is identified. Here we focus on the context where food harvesting is first preceded by a separate exploration period. Foraging then consists of three distinct steps: (1) exploration of the landscape (site-generation), (2) selection of a foraging site (site-selection), and (3) exploitation (harvesting) through repeated trips between the foraging site and "home base". This type of foraging has received scant attention by modellers, leading to two main knowledge gaps. First, little is known about how organisms implement steps (1) and (2). Second, it is not known how the choice of implementation method affects the outcomes of step (3). We investigate these two gaps using an agent-based model, with bumble bees as our model organism, foraging in a patchy resource landscape of crop, wildflower, and empty cells. We tested two different site-generation methods (random and circular foray exploration behaviour) and four different site-selection methods (random and optimizing based on distance from the nest, local wildflower density, or net rate of energy return) on a variety of outcomes from the site-constant harvesting step. We find that site-selection method has a high impact on crop pollination services as well as the percent of crop resources collected, while site-generation method has a high impact on the percent of time spent harvesting and the total trip time. We also find that some of the patterns we identify can be used to infer how a given real organism is identifying a foraging site. Our results underscore the importance of explicitly considering exploratory behaviour to better understand the ecological consequences of foraging dynamics.

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arXiv2024

Quick and Simple Kernel Differential Equation Regression Estimators for Data with Sparse Design

Local polynomial regression of order at least one often performs poorly in regions of sparse data. Local constant regression is exceptional in this regard, though it is the least accurate method in general, especially at the boundaries of the data. Incorporating information from differential equations which may approximately or exactly hold is one way of extending the sparse design capacity of local constant regression while reducing bias and variance. A nonparametric regression method that exploits first order differential equations is introduced in this paper and applied to noisy mouse tumour growth data. Asymptotic biases and variances of kernel estimators using Taylor polynomials with different degrees are discussed. Model comparison is performed for different estimators through simulation studies under various scenarios which simulate exponential-type growth.

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arXiv2024

Anisotropic local constant smoothing for change-point regression function estimation

Understanding forest fire spread in any region of Canada is critical to promoting forest health, and protecting human life and infrastructure. Quantifying fire spread from noisy images, where regions of a fire are separated by change-point boundaries, is critical to faithfully estimating fire spread rates. In this research, we develop a statistically consistent smooth estimator that allows us to denoise fire spread imagery from micro-fire experiments. We develop an anisotropic smoothing method for change-point data that uses estimates of the underlying data generating process to inform smoothing. We show that the anisotropic local constant regression estimator is consistent with convergence rate $O\left(n^{-1/{(q+2)}}\right)$. We demonstrate its effectiveness on simulated one- and two-dimensional change-point data and fire spread imagery from micro-fire experiments.

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