New approaches for analysing functional data - a focus on shape

Functional data - sets of measurement sequences arising from a generating source of continuous nature - has become pervasive in many applications, across many fields of research. The commonly adopted methodology for exploring such data treats the observed units as functions, with continuous functional structure. This introduces a nigh boundless range of new modes of variability in shape and structure, unique to only this type of data. Thus, methodology encompassing the structural variability of functional data has risen to the attention in functional data literature. In this thesis, we approach the shape features of functional data from three different angles, utilizing functional notions of statistical depth as well as metrics, sensitive to variations in structure. Our first approach demonstrates the advantages of data-driven methodology, that allows for the expert to utilize their contextual knowledge of the data. Thus, we provide a consistent framework, through statistical depth, for applying this ad hoc understanding to the analysis. In our second approach, we take a more universal outlook towards shape-encompassing methodology and introduce a new functional depth definition that utilizes some very recently discovered general notions of shape outlyingness. The third approach takes a step back from the context of depth and focuses on the more general concept of shape-sensitive metrics in functional spaces. We, in particular, introduce and study the properties of a new family of integrated metrics, that provide a notion of the overall local likeness (based on any pilot metric) of two curves.