ROC curves and cost curves are two popular ways of visualising classifier performance, finding appropriate thresholds according to the operating condition, and deriving useful aggregated measures such as the area under the ROC curve (AUC) or the area under the optimal cost curve. In this note we present some new findings and connections between ROC space and cost space, by using the expected loss over a range of operating conditions. In particular, we show that ROC curves can be transferred to cost space by means of a very natural way of understanding how thresholds should be chosen, by selecting the threshold such that the proportion of positive predictions equals the operating condition (either in the form of cost proportion or skew). We call these new curves {ROC Cost Curves}, and we demonstrate that the expected loss as measured by the area under these curves is linearly related to AUC. This opens up a series of new possibilities and clarifies the notion of cost curve and its relation to ROC analysis. In addition, we show that for a classifier that assigns the scores in an evenly-spaced way, these curves are equal to the Brier Curves. As a result, this establishes the first clear connection between AUC and the Brier score.

翻译：ROC 曲线和成本曲线是两种受欢迎的方式,通过可视化分类性能,根据运行条件找到适当的阈值,并得出有用的综合措施,如ROC 曲线下的区域或最佳成本曲线下的区域。在本说明中,我们通过使用一系列运行条件下的预期损失,介绍了ROC空间和成本空间之间的一些新发现和联系。我们特别表明,通过非常自然地理解如何选择阈值,将ROC 曲线转换到成本空间,选择的阈值,使正预测的比例等于运行条件(以成本比例或斯凯夫的形式)。我们称之为这些新曲线{RC 成本曲线 或最佳成本曲线下的区域。我们证明,这些曲线下区域测量的预期损失与ACUC有线性关系。这开启了一系列新的可能性,并澄清成本曲线的概念及其与ROC 分析的关系。此外,我们还表明,对于以平衡方式分配分数的分类器来说,这些正比值等于运行条件(以成本比例或斯凯夫为形式)。我们称之为这些新的曲线 {RC 确定Brier 之间的分数。