Single and double exponential models fitted to step length symmetry series are used to evaluate the timecourse of adaptation and de-adaptation in instrumented split-belt treadmill tasks. Whilst the nonlinear regression literature has developed substantially over time, the split-belt treadmill training literature has not been fully utilising the fruits of these developments. In this research area, the current methods of model fitting and evaluation have three significant limitations: (i) optimisation algorithms that are used for model fitting require a good initial guess for regression parameters; (ii) the coefficient of determination ($R^2$) is used for comparing and evaluating models, yet it is considered to be an inadequate measure of fit for nonlinear regression; and, (iii) inference is based on comparison of the confidence intervals for the regression parameters that are obtained under the untested assumption that the nonlinear model has a good linear approximation. In this research, we propose a transformed set of parameters with a common language interpretation that is relevant to split-belt treadmill training for both the single and double exponential models. We propose parameter bounds for the exponential models which allow the use of particle swarm optimisation for model fitting without an initial guess for the regression parameters. For model evaluation and comparison, we propose the use of residual plots and Akaike’s information criterion (AIC). A method for obtaining confidence intervals that does not require the assumption of a good linear approximation is also suggested. A set of MATLAB (MathWorks, Inc., Natick, MA, USA) functions developed in order to apply these methods are also presented. Single and double exponential models are fitted to both the group-averaged and participant step length symmetry series in an experimental dataset generating new insights into split-belt treadmill training. The proposed methods may be useful for research involving analysis of gait symmetry with instrumented split-belt treadmills. Moreover, the demonstration of the suggested statistical methods on an experimental dataset may help the uptake of these methods by a wider community of researchers that are interested in timecourse of motor training.

中文翻译：

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带状跑步机训练趋势的非线性回归

拟合步长对称序列的单指数模型和双指数模型用于评估仪表式皮带式跑步机任务中适应和去适应的时程。尽管非线性回归文献已经随着时间的推移而得到实质性发展，但皮带分离式跑步机训练文献尚未充分利用这些发展成果。在这一研究领域中，当前的模型拟合和评估方法具有三个重大局限性：（i）用于模型拟合的优化算法需要对回归参数进行良好的初步猜测；（ii）决定系数（${\mathrm{[R}}^2$）用于模型的比较和评估，但被认为是非线性回归拟合的不足量度；（iii）推断是基于对回归参数的置信区间的比较，该回归参数是在未经测试的假设下获得的，该假设是非线性模型具有良好的线性近似。在这项研究中，我们提出了一组具有通用语言解释的参数转换集，该参数与单指数和双指数模型的皮带式跑步机训练有关。我们为指数模型提出参数边界，该模型边界允许使用粒子群算法进行模型拟合，而无需对回归参数进行初步猜测。为了进行模型评估和比较，我们建议使用残差图和赤池信息准则（AIC）。还提出了一种无需置信线性良好的假设即可获得置信区间的方法。还介绍了为了应用这些方法而开发的一组MATLAB函数（MathWorks，Inc.，内蒂克，MA，美国）。将单指数和双指数模型拟合到实验数据集中的群体平均和参与者步长对称性序列，从而产生对皮带式跑步机训练的新见解。所提出的方法可能对使用仪器式皮带分离式跑步机进行步态对称性分析的研究有用。此外，在实验数据集上对建议的统计方法的论证可能有助于更广泛的对运动训练的时程感兴趣的研究人员采用这些方法。