Many scientific and engineering applications require fitting regression models that are nonlinear in the parameters. Advances in computer hardware and software in recent decades have made it easier to fit such models. Relative to fitting regression models that are linear in the parameters, however, fitting nonlinear regression models is more complicated. In particular, software like the $\texttt{nls}$ R function requires care in how the model is parameterized and how initial values are chosen for the maximum likelihood iterations. Often special diagnostics are needed to detect and suggest approaches for dealing with identifiability problems that can arise with such model fitting. When using Bayesian inference, there is the added complication of having to specify (often noninformative or weakly informative) prior distributions. Generally, the details for these tasks must be determined for each new nonlinear regression model. This paper provides a step-by-step procedure for specifying these details for any appropriate nonlinear regression model. Following the procedure will result in a numerically robust algorithm for fitting the nonlinear regression model. We illustrate the methods with three different nonlinear models that are used in the analysis of experimental fatigue data and we include two detailed numerical examples.

翻译：许多科学与工程应用需要拟合参数非线性的回归模型。近几十年来计算机硬件与软件的进步使得拟合此类模型更加便捷。然而，与线性参数回归模型相比，非线性回归模型的拟合更为复杂。具体而言，诸如 $\texttt{nls}$ R函数等软件要求谨慎处理模型的参数化方式以及最大似然迭代中初始值的选取。通常需要特殊诊断方法检测并建议处理此类模型拟合中可能出现的可辨识性问题。当采用贝叶斯推断时，还需额外指定（通常为非信息性或弱信息性）先验分布。一般而言，每个新型非线性回归模型都需要针对这些任务确定具体细节。本文为任意适用的非线性回归模型提供了逐步确定这些细节的流程。遵循该流程将得到用于拟合非线性回归模型的数值稳健算法。我们通过三个用于实验疲劳数据解析的不同非线性模型展示了该方法，并提供了两个详细的数值示例。