In censored regression, the outcomes are a mixture of known values (uncensored) and open intervals (censored), meaning that the outcome is either known with precision or is an unknown value above or below a known threshold. The use of censored data is widespread, and correctly modeling it is essential for many applications. Although the literature on censored regression is vast, deep learning approaches have been less frequently applied. This paper proposes three loss functions for training neural networks on censored data using gradient backpropagation: the tobit likelihood, the censored mean squared error, and the censored mean absolute error. We experimented with three variations in the tobit likelihood that arose from different ways of modeling the standard deviation variable: as a fixed value, a reparametrization, and an estimation using a separate neural network for heteroscedastic data. The tobit model yielded better results, but the other two losses are simpler to implement. Another central idea of our research was that data are often censored and truncated simultaneously. The proposed losses can handle simultaneous censoring and truncation at arbitrary values from above and below.

在审查回归中，结果是已知值（未经审查）和开区间（审查）的混合，这意味着结果要么是精确已知的，要么是高于或低于已知阈值的未知值。审查数据的使用很广泛，正确建模对于许多应用程序至关重要。尽管有关审查回归的文献非常多，但深度学习方法的应用较少。本文提出了使用梯度反向传播在删失数据上训练神经网络的三种损失函数：tobit 似然、删失均方误差和删失平均绝对误差。我们对 tobit 似然的三种变化进行了实验，这些变化是由标准差变量建模的不同方式引起的：作为固定值、重新参数化以及使用单独的神经网络对异方差数据进行估计。tobit 模型产生了更好的结果，但其他两个损失更容易实现。我们研究的另一个中心思想是数据经常被同时审查和截断。所提出的损失可以处理上方和下方任意值的同时审查和截断。