We introduce a general framework for regression in the errors-in-variables regime, allowing for full flexibility about the dimensionality of the data, observational error probability density types, the (nonlinear) model type and the avoidance of ad-hoc definitions of loss functions. In this framework, we introduce model fitting for partially unpaired data, i.e. for given data groups the pairing information of input and output is lost (semi-supervised). This is achieved by constructing mixture model densities, which directly model the loss of pairing information allowing inference. In a numerical simulation study linear and nonlinear model fits are illustrated as well as a real data study is presented based on life expectancy data from the world bank utilizing a multiple linear regression model. These results show that high quality model fitting is possible with partially unpaired data, which opens the possibility for new applications with unfortunate or deliberate loss of pairing information in data.

翻译：本文提出了一种适用于误差变量回归的通用框架，该框架充分考虑了数据的维度灵活性、观测误差概率密度类型、模型（非线性）类型，并避免了损失函数的临时定义。在此框架中，我们引入了针对部分未配对数据的模型拟合方法，即对于给定的数据组，输入与输出的配对信息已丢失（半监督情况）。这是通过构建混合模型密度实现的，该密度直接建模配对信息的缺失以支持推断。在数值模拟研究中，我们展示了线性和非线性模型的拟合效果，并基于世界银行的人均预期寿命数据，采用多元线性回归模型进行了实证研究。结果表明，即使数据存在部分未配对情况，仍可实现高质量的模型拟合，这为数据中因意外或故意丢失配对信息的新应用场景提供了可能。