The purpose of this paper is to introduce a new distance metric learning algorithm for ordinal regression. Ordinal regression addresses the problem of predicting classes for which there is a natural ordering, but the real distances between classes are unknown. Since ordinal regression walks a fine line between standard regression and classification, it is a common pitfall to either apply a regression-like numerical treatment of variables or underrate the ordinal information applying nominal classification techniques. On a different note, distance metric learning is a discipline that has proven to be very useful when improving distance-based algorithms such as the nearest neighbors classifier. In addition, an appropriate distance can enhance the explainability of this model. In our study we propose an ordinal approach to learning a distance, called chain maximizing ordinal metric learning. It is based on the maximization of ordered sequences in local neighborhoods of the data. This approach takes into account all the ordinal information in the data without making use of any of the two extremes of classification or regression, and it is able to adapt to data for which the class separations are not clear. We also show how to extend the algorithm to learn in a non-linear setup using kernel functions. We have tested our algorithm on several ordinal regression problems, showing a high performance under the usual evaluation metrics in this domain. Results are verified through Bayesian non-parametric testing. Finally, we explore the capabilities of our algorithm in terms of explainability using the case-based reasoning approach. We show these capabilities empirically on two different datasets, experiencing significant improvements over the case-based reasoning with the traditional Euclidean nearest neighbors.

本文的目的是介绍一种新的用于序数回归的距离度量学习算法。序数回归解决了预测具有自然排序的类的问题，但类之间的实际距离是未知的。由于序数回归在标准回归和分类之间徘徊，因此应用类似回归的变量数值处理或应用名义分类技术低估序数信息是一个常见的陷阱。另一方面，距离度量学习是一门在改进基于距离的算法（例如最近邻分类器）时被证明非常有用的学科。此外，适当的距离可以增强该模型的可解释性。在我们的研究中，我们提出了一种学习距离的有序方法，称为链最大化序数度量学习. 它基于数据局部邻域中有序序列的最大化。这种方法考虑了数据中的所有有序信息，不使用分类或回归的两个极端中的任何一个，并且能够适应类分离不明确的数据。我们还展示了如何使用核函数扩展算法以在非线性设置中学习。我们已经在几个有序回归问题上测试了我们的算法，在该领域的常用评估指标下显示出高性能。结果通过贝叶斯非参数检验进行验证。最后，我们使用基于案例的推理方法探索我们算法在可解释性方面的能力。我们在两个不同的数据集上凭经验展示了这些能力，