We propose an alternative to $k$-nearest neighbors for functional data whereby the approximating neighboring curves are piecewise functions built from a functional sample. Using a locally defined distance function that satisfies stabilization criteria, we establish pointwise and global approximation results in function spaces when the number of data curves is large enough. We exploit this feature to develop the asymptotic theory when a finite number of curves is observed at time-points given by an i.i.d. sample whose cardinality increases up to infinity. We use these results to investigate the problem of estimating unobserved segments of a partially observed functional data sample as well as to study the problem of functional classification and outlier detection. For such problems, our methods are competitive with and sometimes superior to benchmark predictions in the field.

翻译：我们为功能性数据的近邻提出一个替代方案,将近邻曲线作为功能性数据的近邻。使用本地定义的、符合稳定标准的远距函数,当数据曲线数量足够大时,我们建立点法和全球近似结果。我们利用这一特征来发展无症状理论,当一个主要特征提高到无限度的样本在时间点上观察到一定数量的曲线时,我们用这些结果来调查估算部分观测到的功能性数据样本中未观测到的部分内容的问题,并研究功能分类和外部检测问题。对于这些问题,我们的方法具有竞争力,有时甚至优于实地的基准预测。