Regression analysis estimates the relationships among variables which has been widely used in growth curves, and cross-validation as a model selection method assesses the generalization ability of regression models. Classical methods assume that the observation values of variables are precise numbers while in many cases data are imprecisely collected. So this paper explores the Chapman-Richards growth model which is one of the widely used growth models with imprecise observations under the framework of uncertainty theory. The least squares estimates of unknown parameters in this model are given. Moreover, cross-validation with imprecise observations is proposed. Furthermore, estimates of the expected value and variance of the uncertain error using residuals are given. In addition, ways to predict the value of response variable with new observed values of predictor variables are discussed. Finally, a numerical example illustrates our approach.

中文翻译：

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具有不精确观察的不确定 Chapman-Richards 增长模型的交叉验证

回归分析估计变量之间的关系，已广泛用于增长曲线，交叉验证作为模型选择方法评估回归模型的泛化能力。经典方法假设变量的观察值是精确的数字，而在许多情况下，数据的收集并不精确。因此本文在不确定性理论的框架下探讨了查普曼-理查兹增长模型，它是目前广泛使用的、观测不精确的增长模型之一。给出了该模型中未知参数的最小二乘估计。此外，提出了具有不精确观察的交叉验证。此外，还给出了使用残差对不确定误差的期望值和方差的估计。此外，讨论了用预测变量的新观测值预测响应变量值的方法。最后，一个数值例子说明了我们的方法。