  With the rise of Industry 4.0, huge amounts of data are now generated that
are apt to be modelled as functional data. In this setting, standard profile
monitoring methods aim to assess the stability over time of a completely
observed functional quality characteristic. However, in some practical
situations, evaluating the process state in real-time, i.e., as the process is
running, could be of great interest to significantly improve the effectiveness
of monitoring. To this aim, we propose a new method, referred to as functional
real-time monitoring (FRTM), that is able to account for both phase and
amplitude variation through the following steps: (i) registration; (ii)
dimensionality reduction; (iii) monitoring of a partially observed functional
quality characteristic. An extensive Monte Carlo simulation study is performed
to quantify the performance of FRTM with respect to two competing methods.
Finally, an example is presented where the proposed method is used to monitor
batches from a penicillin production process in real-time.
