Representation and Characterization of Nonstationary Processes by Dilation Operators and Induced Shape Space Manifolds.

We proposed in this work the introduction of a new vision of stochastic processes through geometry induced by dilation. The dilation matrices of a given process are obtained by a composition of rotation matrices built in with respect to partial correlation coefficients. Particularly interesting is the fact that the obtention of dilation matrices is regardless of the stationarity of the underlying process. When the process is stationary, only one dilation matrix is obtained and it corresponds therefore to Naimark dilation. When the process is nonstationary, a set of dilation matrices is obtained. They correspond to Kolmogorov decomposition. In this work, the nonstationary class of periodically correlated processes was of interest. The underlying periodicity of correlation coefficients is then transmitted to the set of dilation matrices. Because this set lives on the Lie group of rotation matrices, we can see them as points of a closed curve on the Lie group. Geometrical aspects can then be investigated through the shape of the obtained curves, and to give a complete insight into the space of curves, a metric and the derived geodesic equations are provided. The general results are adapted to the more specific case where the base manifold is the Lie group of rotation matrices, and because the metric in the space of curve naturally extends to the space of shapes; this enables a comparison between curves' shapes and allows then the classification of random processes' measures.

Improving Health Care Management Through Persistent Homology of Time-Varying Variability of Emergency Department Patient Flow.

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.