  With the ubiquity of sensors in the IoT era, statistical observations are
becoming increasingly available in the form of massive (multivariate)
time-series. Formulated as unsupervised anomaly detection tasks, an abundance
of applications like aviation safety management, the health monitoring of
complex infrastructures or fraud detection can now rely on such functional
data, acquired and stored with an ever finer granularity. The concept of
statistical depth, which reflects centrality of an arbitrary observation w.r.t.
a statistical population may play a crucial role in this regard, anomalies
corresponding to observations with 'small' depth. Supported by sound
theoretical and computational developments in the recent decades, it has proven
to be extremely useful, in particular in functional spaces. However, most
approaches documented in the literature consist in evaluating independently the
centrality of each point forming the time series and consequently exhibit a
certain insensitivity to possible shape changes. In this paper, we propose a
novel notion of functional depth based on the area of the convex hull of
sampled curves, capturing gradual departures from centrality, even beyond the
envelope of the data, in a natural fashion. We discuss practical relevance of
commonly imposed axioms on functional depths and investigate which of them are
satisfied by the notion of depth we promote here. Estimation and computational
issues are also addressed and various numerical experiments provide empirical
evidence of the relevance of the approach proposed.
