The inflection point is an important feature of sigmoidal height-diameter (H-D) models. It is often cited as one of the properties favoring sigmoidal model forms. However, there are very few studies analyzing the inflection points of H-D models. The goals of this study were to theoretically and empirically examine the behaviors of inflection points of six common H-D models with a regional dataset. The six models were the Wykoff (WYK), Schumacher (SCH), Curtis (CUR), Hossfeld IV (HOS), von Bertalanffy-Richards (VBR), and Gompertz (GPZ) models. The models were first fitted in their base forms with tree species as random effects and were then expanded to include functional traits and spatial distribution. The distributions of the estimated inflection points were similar between the two-parameter models WYK, SCH, and CUR, but were different between the three-parameter models HOS, VBR, and GPZ. GPZ produced some of the largest inflection points. HOS and VBR produced concave H-D curves without inflection points for 12.7% and 39.7% of the tree species. Evergreen species or decreasing shade tolerance resulted in larger inflection points. The trends in the estimated inflection points of HOS and VBR were entirely opposite across the landscape. Furthermore, HOS could produce concave H-D curves for portions of the landscape. Based on the studied behaviors, the choice between two-parameter models may not matter. We recommend comparing several three-parameter model forms for consistency in estimated inflection points before deciding on one. Believing sigmoidal models to have inflection points does not necessarily mean that they will produce fitted curves with one. Our study highlights the need to integrate analysis of inflection points into modeling H-D relationships.

中文翻译：

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高径模型拐点分析


拐点是 S 形高径 (H-D) 模型的一个重要特征。它经常被认为是有利于 sigmoidal 模型形式的属性之一。然而，分析H-D模型拐点的研究却很少。本研究的目标是通过区域数据集从理论上和实证上检验六种常见 H-D 模型的拐点行为。这六款车型分别是 Wykoff (WYK)、Schumacher (SCH)、Curtis (CUR)、Hossfeld IV (HOS)、von Bertalanffy-Richards (VBR) 和 Gompertz (GPZ) 车型。这些模型首先以树种作为随机效应拟合其基本形式，然后扩展到包括功能特征和空间分布。双参数模型 WYK、SCH 和 CUR 的估计拐点分布相似，但三参数模型 HOS、VBR 和 GPZ 的估计拐点分布不同。 GPZ 产生了一些最大的拐点。 HOS 和 VBR 为 12.7% 和 39.7% 的树种产生了没有拐点的凹形 H-D 曲线。常绿物种或耐荫性降低导致较大的拐点。居屋和可变居住面积的估计拐点趋势在整个景观中完全相反。此外，HOS 可以为部分景观产生凹形 H-D 曲线。根据所研究的行为，双参数模型之间的选择可能并不重要。我们建议在决定使用一种三参数模型形式之前，先比较几种三参数模型形式，以确保估计拐点的一致性。相信 S 形模型具有拐点并不一定意味着它们会生成具有拐点的拟合曲线。我们的研究强调需要将拐点分析整合到 HD 关系建模中。