7 research outputs found
Representation and Characterization of Non-Stationary Processes by Dilation Operators and Induced Shape Space Manifolds
We have introduce a new vision of stochastic processes through the geometry
induced by the dilation. The dilation matrices of a given processes are
obtained by a composition of rotations matrices, contain the measure
information in a condensed way. 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, it coincides with the
Naimark Dilation and only one rotation matrix is computed, when the process is
non-stationary, a set of rotation matrices are computed. In particular, the
periodicity of the correlation function that may appear in some classes of
signal is transmitted to the set of dilation matrices. These rotation matrices,
which can be arbitrarily close to each other depending on the sampling or the
rescaling of the signal are seen as a distinctive feature of the signal. In
order to study this sequence of matrices, and guided by the possibility to
rescale the signal, the correct geometrical framework to use with the
dilation's theoretic results is the space of curves on manifolds, that is the
set of all curve that lies on a base manifold. To give a complete sight about
the space of curve, a metric and the derived geodesic equation are provided.
The general results are adapted to the more specific case where the base
manifold is the Lie group of rotation matrices. The notion of the shape of a
curve can be formalized as the set of equivalence classes of curves given by
the quotient space of the space of curves and the increasing diffeomorphisms.
The metric in the space of curve naturally extent to the space of shapes and
enable comparison between shapes.Comment: 19 pages, draft pape
Géométrie et topologie des processus périodiquement corrélés induit par la dilation : Application à l'étude de la variabilité des épidémies pédiatriques saisonnières
Each year emergency department are faced with epidemics that affect their organisation and deteriorate the quality of the cares. The analyse of these outbreak is tough due to their huge variability. We aim to study these phenomenon and to bring out a new paradigm in the analysis of their behavior. With this aim in mind, we propose to tackle this problem through geometry and topology: the variability process being periodically correlated, the theory of dilation exhibit a set of matrices that carry all the information about this process. This set of matrices allow to map the process into a Lie group, defined as the set of all curves on a manifold. Thus, it is possible to compare stochastic processes using properties of Lie groups. Then, we consider the point cloud formed by the set of dilation matrices, to gain more intuitions about the underlying process. We proved a relation between the temporal aspect of the signal and the structure of the set of its dilation matrices. We used and developped persistent homology tools, and were able to classify non-stationary processes. Eventually, we implement these techniques directly on the process of arrivals to detect the trigger of the epidemics. Overall we established a complete and a coherent framework, both theoretical and practical.Chaque année lors de la période hivernale, des phénomènes épidémiques affectent l’organisation des services d’urgences pédiatriques et dégradent la qualité de la réponse fournie. Ces phénomènes présentent une forte variabilité qui rend leur analyse difficile. Nous nous proposons d’étudier cette volatilité pour apporter une vision nouvelle et éclairante sur le comportement de ces épidémies. Pour ce faire, nous avons adopté une vision géométrique et topologique originale directement issue d’une application de la théorie de la dilation: le processus de variabilité étant périodiquement corrélé, cette théorie fournit un ensemble de matrices dites de dilations qui portent toute l’information utile sur ce processus. Cet ensemble de matrices nous permet de représenter les processus stochastiques comme des éléments d’un groupe de Lie particulier, à savoir le groupe de Lie constitué de l’ensemble des courbes sur une variété. Il est alors possible de comparer des processus par ce biais. Pour avoir une perception plus intuitive du processus de variabilité, nous nous sommes ensuite concentrés sur le nuage de points formé par l’ensemble des matrices de dilations. En effet, nous souhaitons mettre en évidence une relation entre la forme temporelle d’un processus et l’organisation de ces matrices de dilations. Nous avons utilisé et développé des outils d’homologie persistante et avons établi un lien entre la désorganisation de ce nuage de points et le type de processus sous-jacents. Enfin nous avons appliqué ces méthodes directement sur le processus de variabilité pour pouvoir détecter le déclenchement de l’épidémie. Ainsi nous avons établi un cadre complet et cohérent, à la fois théorique et appliqué pour répondre à notre problématique
Geometry and topology of periodically correlated processes : Analysis of the variability of seasonal pediatric epidemics
Chaque année lors de la période hivernale, des phénomènes épidémiques affectent l’organisation des services d’urgences pédiatriques et dégradent la qualité de la réponse fournie. Ces phénomènes présentent une forte variabilité qui rend leur analyse difficile. Nous nous proposons d’étudier cette volatilité pour apporter une vision nouvelle et éclairante sur le comportement de ces épidémies. Pour ce faire, nous avons adopté une vision géométrique et topologique originale directement issue d’une application de la théorie de la dilation: le processus de variabilité étant périodiquement corrélé, cette théorie fournit un ensemble de matrices dites de dilations qui portent toute l’information utile sur ce processus. Cet ensemble de matrices nous permet de représenter les processus stochastiques comme des éléments d’un groupe de Lie particulier, à savoir le groupe de Lie constitué de l’ensemble des courbes sur une variété. Il est alors possible de comparer des processus par ce biais. Pour avoir une perception plus intuitive du processus de variabilité, nous nous sommes ensuite concentrés sur le nuage de points formé par l’ensemble des matrices de dilations. En effet, nous souhaitons mettre en évidence une relation entre la forme temporelle d’un processus et l’organisation de ces matrices de dilations. Nous avons utilisé et développé des outils d’homologie persistante et avons établi un lien entre la désorganisation de ce nuage de points et le type de processus sous-jacents. Enfin nous avons appliqué ces méthodes directement sur le processus de variabilité pour pouvoir détecter le déclenchement de l’épidémie. Ainsi nous avons établi un cadre complet et cohérent, à la fois théorique et appliqué pour répondre à notre problématique.Each year emergency department are faced with epidemics that affect their organisation and deteriorate the quality of the cares. The analyse of these outbreak is tough due to their huge variability. We aim to study these phenomenon and to bring out a new paradigm in the analysis of their behavior. With this aim in mind, we propose to tackle this problem through geometry and topology: the variability process being periodically correlated, the theory of dilation exhibit a set of matrices that carry all the information about this process. This set of matrices allow to map the process into a Lie group, defined as the set of all curves on a manifold. Thus, it is possible to compare stochastic processes using properties of Lie groups. Then, we consider the point cloud formed by the set of dilation matrices, to gain more intuitions about the underlying process. We proved a relation between the temporal aspect of the signal and the structure of the set of its dilation matrices. We used and developped persistent homology tools, and were able to classify non-stationary processes. Eventually, we implement these techniques directly on the process of arrivals to detect the trigger of the epidemics. Overall we established a complete and a coherent framework, both theoretical and practical
Information topological characterization for non-stationary random processes
International audienc
Information Topological Characterization of Periodically Correlated Processes by Dilation Operators
International audienceGiving process information through spectral considerations has been tackled for decades. We propose in this work 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
Improving Health Care Management Through Persistent Homology of Time-Varying Variability of Emergency Department Patient Flow
International audienceExcessive 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 (DFA) 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 (TDA) 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 bi-objective optimization problem to track the variations of this parameter