We address the problem of estimating topological features from data in high dimensional Euclidean spaces under the manifold assumption. Our approach is based on the computation of persistent homology of the space of data points endowed with a sample metric known as Fermat distance. We prove that such metric space converges almost surely to the manifold itself endowed with an intrinsic metric that accounts for both the geometry of the manifold and the density that produces the sample. This fact implies the convergence of the associated persistence diagrams. The use of this intrinsic distance when computing persistent homology presents advantageous properties such as robustness to the presence of outliers in the input data and less sensitiveness to the particular embedding of the underlying manifold in the ambient space. We use these ideas to propose and implement a method for pattern recognition and anomaly detection in time series, which is evaluated in applications to real data.

The economic cost of roll refurbishment in the steel-making industry is considerable. In a cold rolling mill, wear and damage of rolls disrupt the industrial environment, so it is critical to predict the remaining useful life early and change the roll without causing disruption to the manufacturing process. However, since cold rolling is a complex process affected by multiple variables which are operated in adverse conditions, it is very challenging to mathematically analyse the roll wear and failure. For this reason, in the present paper, a data-driven solution is proposed to predict the correct time for changing individual rolls. To develop an accurate predictive model, several datasets containing high-resolution production data and roll refurbishment data collected from a UK based steel plant have been acquired and processed in a way that the roll wear is modelled as a Remaining Useful Life (RUL) problem, where the number of coils that a roll is able to process is viewed as the remaining cycles. Then hybrid deep learning models are used to predict the Remaining Useful Life of rolls used in steel making. This novel data-driven approach achieves high prediction accuracy and has been validated on a real-world dataset. The proposed approach not only helps avoiding early failure but also can serve as a critical step towards the design of an optimal, automated maintenance schedule for the roll management.