Giving process information through spectral considerations has been tackled for decades. We propose, in this paper, a new way of dealing with such an objective by giving hidden information topology of the spectral measure of non-stationary and periodically correlated processes. We used first the Kolmogorov decomposition which is a natural extension of the Naimark operator theory to obtain a sequence of rotation matrices called the dilation matrices. These matrices carry all the spectral information of the process and belong to SO(n) or SU(n) with respect to, respectively, the real or complex nature of the periodically correlated processes studied. In order to give a topological interpretation of the positioning of these matrices on the space of rotation matrices, we have applied a persistent homology technique and next exposed fundamental attributes. We showed that different types of periodically correlated processes are endowed with a point cloud structure that can be easily discriminated by topological and information features.

Excessive admissions at the emergency department (ED) is a phenomenon very closely linked to the propagation of viruses. It is a cause of overcrowding for EDs and a public health problem. The aim of this work is to give EDs' leaders more time for decision making during this period. Based on the admissions time series associated with specific clinical diagnoses, we will first perform a detrended fluctuation analysis to obtain the corresponding variability time series. Next, we will embed this time series on a manifold to obtain a point cloud representation and use topological data analysis through persistent homology technic to propose two early real-time indicators. One is the early indicator of abnormal arrivals at the ED whereas the second gives the information on the time index of the maximum number of arrivals. The performance of the detectors is parameter dependent and it can evolve each year. That is why we also propose to solve a biobjective optimization problem to track the variations of this parameter.