In various fields, data recorded continuously during a time interval and curve data, such as spectral data, become common. These kinds of data can be interpreted as functional data. In this paper we have studied binary classification from only positive and unlabeled functional data (PU classification for functional data). Our first contribution is to present a simple classification algorithm for this problem. The key feature of the algorithm is that it is not required an estimation of the unknown class prior (or the constant probability that a positive object is labeled). It is worth noting that the idea of our method can be applied to kernel linear discriminant analysis for general data. Our second contribution is to prove that, under mild regularity conditions similar to those in a supervised context, the proposed algorithm can achieve perfect asymptotic classification in the context of PU classification. In fact, we show that the proposed algorithm works well not only in numerical experiments but also for real data examples. Moreover, as an important practical application, we have used the proposed algorithm to identify handball players at risk for anterior cruciate ligament (ACL) injury based on ground reaction force data.

中文翻译：

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仅根据阳性和未标记的功能数据进行分类

在各个领域中，在时间间隔内连续记录的数据和诸如光谱数据的曲线数据变得普遍。这些类型的数据可以解释为功能数据。在本文中，我们仅根据阳性和未标记的功能数据（针对功能数据的PU分类）研究了二进制分类。我们的第一个贡献是提出一种针对此问题的简单分类算法。该算法的关键特征是不需要先验未知类（或标记阳性对象的恒定概率）。值得注意的是，我们方法的思想可以应用于一般数据的核线性判别分析。我们的第二个贡献是证明，在与监督环境类似的温和规律性条件下，该算法在PU分类的背景下可以实现完美的渐近分类。实际上，我们证明了所提出的算法不仅在数值实验中而且在实际数据示例中均能很好地工作。此外，作为重要的实际应用，我们已使用提出的算法根据地面反作用力数据来识别处于前十字韧带（ACL）伤害风险的手球运动员。