In this paper we face the problem of representation of functional data with the tools of algebraic topology. We represent functions by means of merge trees and this representation is compared with that offered by persistence diagrams. We show that these two tree structures, although not equivalent, are both invariant under homeomorphic re-parametrizations of the functions they represent, thus allowing for a statistical analysis which is indifferent to functional misalignment. We employ a novel metric for merge trees and we prove a few theoretical results related to its specific implementation when merge trees represent functions. To showcase the good properties of our topological approach to functional data analysis, we first go through a few examples using data generated {\em in silico} and employed to illustrate and compare the different representations provided by merge trees and persistence diagrams, and then we test it on the Aneurisk65 dataset replicating, from our different perspective, the supervised classification analysis which contributed to make this dataset a benchmark for methods dealing with misaligned functional data.

翻译：在本文中,我们面临功能数据与代数表学工具的表述问题。我们通过合并树木的方式代表功能,而这种表述则与持久性图表相比。我们显示,这两种树结构虽然不等同,但都是在对其所代表的功能进行自成体系的重新校正的情况下无差别的,从而可以进行统计分析,而这种分析与功能对称无关。我们为合并树木采用了一种新颖的衡量标准,在合并树木代表功能时,我们证明了与其具体实施有关的一些理论结果。为了展示我们功能数据分析的表层学方法的良好特性,我们首先通过几个例子,利用在硅} 中生成的数据,用来说明和比较合并树木和持久性图表所提供的不同表述,然后我们从不同角度对Anerisk65数据集进行测试,从我们的不同角度对Anerisk65数据集进行复制,通过监督的分类分析使这一数据成为处理功能数据不吻合的方法的基准。