# Band depths based on multiple time instances

## TL;DR

This paper introduces a novel concept of data depth for vector-valued functions using bands formed by convex hulls at multiple time points, bridging traditional and functional data analysis.

## Contribution

It proposes a new depth measure based on multiple time-instance bands, extending existing methods to vector-valued functions and providing a continuum from traditional to full functional bands.

## Key findings

- Defines m-bands using convex hulls at multiple time points
- Introduces a new depth concept for vector-valued functions
- Connects traditional and functional data analysis methods

## Abstract

Bands of vector-valued functions $f:T\mapsto\mathbb{R}^d$ are defined by considering convex hulls generated by their values concatenated at $m$ different values of the argument. The obtained $m$-bands are families of functions, ranging from the conventional band in case the time points are individually considered (for $m=1$) to the convex hull in the functional space if the number $m$ of simultaneously considered time points becomes large enough to fill the whole time domain. These bands give rise to a depth concept that is new both for real-valued and vector-valued functions.