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L'imagerie hyperspectrale (HSI) repose sur le fait que, pour un matériau donné, la quantité de rayonnement émis varie avec la longueur d'onde. Les capteurs HSI mesurent donc le rayonnement des matériaux au sein de chaque pixel pour un très grand nombre de bandes spectrales contiguës et fournissent des images contenant des informations à la fois spatiale et spectrale. Les méthodes classiques de détection adaptative supposent généralement que le fond est gaussien à vecteur moyenne nul ou connu. Cependant, quand le vecteur moyen est inconnu, comme c'est le cas pour l'image hyperspectrale, il doit être inclus dans le processus de détection. Nous proposons dans ce travail d'étendre les méthodes classiques de détection pour lesquelles la matrice de covariance et le vecteur de moyenne sont tous deux inconnus.Cependant, la distribution statistique multivariée des pixels de l'environnement peut s'éloigner de l'hypothèse gaussienne classiquement utilisée. La classe des distributions elliptiques a été déjà popularisée pour la caractérisation de fond pour l’HSI. Bien que ces modèles non gaussiens aient déjà été exploités dans la modélisation du fond et dans la conception de détecteurs, l'estimation des paramètres (matrice de covariance, vecteur moyenne) est encore généralement effectuée en utilisant des estimateurs conventionnels gaussiens. Dans ce contexte, nous analysons de méthodes d’estimation robuste plus appropriées à ces distributions non-gaussiennes : les M-estimateurs. Ces méthodes de détection couplées à ces nouveaux estimateurs permettent d'une part, d'améliorer les performances de détection dans un environment non-gaussien mais d'autre part de garder les mêmes performances que celles des détecteurs conventionnels dans un environnement gaussien. Elles fournissent ainsi un cadre unifié pour la détection de cibles et la détection d'anomalies pour la HSI. / Hyperspectral imaging (HSI) extends from the fact that for any given material, the amount of emitted radiation varies with wavelength. HSI sensors measure the radiance of the materials within each pixel area at a very large number of contiguous spectral bands and provide image data containing both spatial and spectral information. Classical adaptive detection schemes assume that the background is zero-mean Gaussian or with known mean vector that can be exploited. However, when the mean vector is unknown, as it is the case for hyperspectral imaging, it has to be included in the detection process. We propose in this work an extension of classical detection methods when both covariance matrix and mean vector are unknown.However, the actual multivariate distribution of the background pixels may differ from the generally used Gaussian hypothesis. The class of elliptical distributions has already been popularized for background characterization in HSI. Although these non-Gaussian models have been exploited for background modeling and detection schemes, the parameters estimation (covariance matrix, mean vector) is usually performed using classical Gaussian-based estimators. We analyze here some robust estimation procedures (M-estimators of location and scale) more suitable when non-Gaussian distributions are assumed. Jointly used with M-estimators, these new detectors allow to enhance the target detection performance in non-Gaussian environment while keeping the same performance than the classical detectors in Gaussian environment. Therefore, they provide a unified framework for target detection and anomaly detection in HSI.

Rocket combustors and other high-performance chemical propulsion systems are prone to combustion instability. Recent simulations of rocket combustors using detailed chemical kinetics show that the constant pressure assumption used in classical treatments may be suspect due to high rates of heat release. This study is a exploration on the effects of these extraordinary rates of heat addition on the local pressure field, and interactions between the heat release and an acoustic field. <br> <br>The full problem is decomposed into simpler unit problems focused on the particular interactions of physical phenomena involved in combustion instability. The overall strategy consists of analyzing fundamental problems with simplified scenarios and then build up the complexity by adding more phenomena to the analysis. Seven unit problems are proposed in this study. <br> <br>The first unit problem consists of the pressure response to an unsteady heat release source in an unconfined one-dimensional domain. An analytical model based on the acoustic wave equation with planar symmetry and an unsteady heat source is derived and then compared against results from highly-resolved numerical simulations. Two different heat release profiles, one a Gaussian spatial distribution with a step temporal profile, and the other a Gaussian spatial distribution with a Gaussian temporal distribution, are used to model the heat source. The analytical solutions predict two different regimes in the pressure response depending on the Helmholtz number, which is defined as the ratio of the acoustic time over the duration of the heat release pulse. A critical Helmholtz number is found to dictate the pressure response regime. For compact cases, in the subcritical regime, the amplitude of the pressure pulse remains constant in space. For noncompact cases, above the critical Helmholtz number, the pressure pulse reaches a maximum at the center of the heat source, and then decays in space converging to a lower far field amplitude. At the limits of very small and very large Helmholtz numbers, the heat release response tends to be a constant pressure process and a constant volume process, respectively. The parameters of the study are chosen to be representative of the extreme conditions in a rocket combustor. The analytical models for both heat source profiles closely match the simulations with a slight overprediction. The differences observed in the analytical solutions are due to neglecting mean flow property variations and the absence of loss mechanisms. The numerical simulations also reveal the presence of nonlinear effects such as weak shocks that cannot be captured by the linear acoustic wave equation. <br> <br>The second unit problem extends the analysis of the pressure response of an unsteady heat release source to an unconfined three-dimensional domain. An analytical model based on the spherical acoustic wave equation with an unsteady heat source is derived and then compared against results from highly-resolved three-dimensional numerical simulations. Two different heat release profiles, a three-dimensional Gaussian spherical distribution with either a step or a Gaussian temporal distribution, are used to model the heat source. Two different regimes in the pressure response depending on the Helmholtz number are found. This analysis also reveals that whereas for the one-dimensional case the pressure amplitude is constant over the distance, for the three-dimensional case it decays with the radial distance from the heat source. In addition, although for moderate heat release values the analytical solution is able to capture the dynamics of the fluid response, for large heat release values the nonlinear effects deviate the highly-resolved numerical solution from the analytical model. <br> <br>The third unit problem studies the pressure response of a fluctuating unsteady heat release source to an unconfined one-dimensional domain. An analytical model based on the acoustic wave equation with planar symmetry and an unsteady heat source is derived and then compared against results from highly-resolved numerical simulations. Two different heat release profiles, a flat spatial distribution with sinusoidal temporal profile and a Gaussian spatial distribution and sinusoidal temporal profile, are used to model the heat source. For both cases, the acoustically compact and noncompact regimes depending on the Helmholtz number are analyzed. While in the compact regime the amplitude of the pressure is constant over the distance, in the noncompact regime the amplitude of the pressure fluctuation is larger within the heat source area of application, and once outside the heat source decays to a far field pressure value. In addition, the analytical model does not capture the nonlinear effects present in the highly-resolved numerical simulations for large rates of heat release such as the ones present in rocket combustors.<br> <br>Finally, the last four unit problems focus on the interaction between unsteady heat release and the longitudinal acoustic modes of a combustor. The goal is to assess and quantify how pressure fluctuations due to unsteady heat release amplify a longitudinal acoustic mode. To investigate the nonlinear effects and the limitations based on the acoustic wave equation, the analytical models are compared against highly-resolved numerical simulations. The fourth unit problem consists of the pressure response to a moving rigid surface that generates a forced sinusoidal velocity fluctuation in a one-dimensional open-ended cavity. The fifth unit problem combines an analytical solution from the velocity harmonic fluctuation with an unsteady heat pulse with Gaussian spatial and temporal distribution developed in the first unit problem. The choice of an open-ended cavity simplifies the analysis and serves as a stepping stone to the sixth unit problem, which also includes the pressure reflections provoked by the acoustic boundaries of the duct. This sixth unit problem describes the establishment of a 1L acoustic longitudinal mode inside a closed duct using the harmonic velocity fluctuations from the fourth unit problem. A wall on the left end of the duct is only moved for one cycle at the 1L mode frequency to establish a 1L mode in the initially quiescent fluid. The last unit problem combines the analytical solution of the 1L mode acoustic field developed in the sixth unit problem with an unsteady heat pulse with Gaussian spatial and temporal distribution, and also accounts for pressure reflections. The derivation of the present analytical models includes the identification of relevant length and time scales that are condensed into the Helmholtz number, the phase shift between the longitudinal fluctuating pressure field and the heat source, and ratio of the fluctuating periods. The analytical solution is able to capture with an acceptable degree of accuracy the pressure trace of the numerical solution during the fist few cycles of the 1L mode, but it quickly deviates very significantly from the numerical solution due to wave steepening and the formation of weak shocks. Therefore, models based on the acoustic wave equation can provide a good understanding of the combustion instability behavior, but not accurately predict the evolution of the pressure fluctuations as the nonlinear effects play a major role in the combustion dynamics of liquid rocket engines.

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Το αντικείμενο της παρούσας διατριβής είναι η διερεύνηση μεθόδων στατιστικής συμπερασματολογίας για την προσαρμογή και έλεγχο της κατανομής γάμα και της αντίστροφης κανονικής (inverse Gaussian) κατανομής σε δεδομένα με θετική λοξότητα. Τα πρότυπα αυτά χρησιμοποιούνται ευρέως στην ανάλυση αξιοπιστίας και ελέγχου μακροβιότητας καθώς και σε άλλες εφαρμογές. Αρχικά γίνεται μια περιγραφή εναλλακτικών μεθόδων στατιστικής συμπερασματολογίας για τις διπαραμετρικές και τις τριπαραμετρικές οικογένειες κατανομών γάμα και αντίστροφης κανονικής. Στη συνέχεια διερευνάται η χρήση μεθόδων στατιστικής συμπερασματολογίας για την εκτίμηση των παραμέτρων της διπαραμετρικής γάμα κατανομής με χρήση της εμπειρικής ροπογεννήτριας συνάρτησης. Μέθοδοι εκτιμητικής, όπως είναι η μέθοδος των μικτών ροπών και των γενικευμένων ελαχίστων τετραγώνων, εφαρμόζονται και συγκρίνονται με την μέθοδο της μέγιστης πιθανοφάνειας μέσω πειραμάτων προσομοίωσης Monte Carlo. Επίσης, διερευνώνται έλεγχοι καλής προσαρμογής για τη διπαραμετρική γάμα κατανομή. Οι έλεγχοι αυτοί περιλαμβάνουν τους κλασικούς ελέγχους και έναν έλεγχο που χρησιμοποιεί την εμπειρική ροπογεννήτρια συνάρτηση. Με χρήση πειραμάτων προσομοίωσης Monte Carlo, γίνεται σύγκριση των ελέγχων ως προς το πραγματικό επίπεδο σημαντικότητας και την ισχύ έναντι άλλων λοξών προς τα δεξιά κατανομών. Στη συνέχεια εφαρμόζονται έλεγχοι καλής προσαρμογής γάμα κατανομών σε πραγματικά δεδομένα, τα οποία έχουν αναλυθεί νωρίτερα από άλλους ερευνητές. Για τον έλεγχο της τριπαραμετρικής γάμα κατανομής εφαρμόζεται μόνο ο έλεγχος με χρήση της εμπειρικής ροπογεννήτριας συνάρτησης, αφού δεν είναι γνωστοί κλασικοί έλεγχοι που χρησιμοποιούν την εμπειρική συνάρτηση κατανομής. Τέλος, γίνεται εκτίμηση ποσοστιαίων σημείων της αντίστροφης κανονικής κατανομής. Αρχικά, εκτιμώνται ποσοστιαία σημεία για την τριπαραμετρική κατανομή και στη συνέχεια εφαρμόζονται δύο μέθοδοι υπολογισμού ποσοστιαίων σημείων για την περίπτωση της διπαραμετρικής κατανομής. Η εκτίμηση των ποσοστιαίων σημείων σε κάθε οικογένεια κατανομών χρησιμοποιεί δύο μεθόδους ενδιάμεσης εκτίμησης των παραμέτρων της κατανομής. Οι μέθοδοι συγκρίνονται ως προς το μέσο τετραγωνικό σφάλμα και τη σχετική μεροληψία με τη βοήθεια πειραμάτων προσομοίωσης. / The subject of the present dissertation is the investigation of procedures of statistical inference for fitting and testing the gamma distribution and inverse Gaussian distribution, with data having positive skewness. These distributions are used widely in reliability analysis and lifetime models as well as in other applications. In the beginning, we describe alternative methods of statistical inference for the two and three-parameter families of gamma and inverse Gaussian distributions. Then, we examine methods of statistical inference in order to estimate the parameters of the two-parameter gamma distribution using the empirical moment generating function. Estimation procedures, like the method of mixed moments and the method of generalized least squares, are applied and compared with the method of maximum likelihood through Monte Carlo simulations. Also, we investigate goodness of fit tests for the two-parameter gamma distribution. These tests include the classical tests and a test based on the empirical moment generating function. Using Monte Carlo simulations, we compare the actual level of the tests and the power of the tests against skewed to the right distributions. We apply goodness of fit tests of gamma distributions to real life data, which have been examined earlier by other researchers. For the three-parameter gamma distribution we apply only one test using the empirical moment generating function since there are no classical tests using the empirical distribution function. Finally, we estimate quantiles of the inverse Gaussian distribution. We start estimating quantiles for the three-parameter distribution and then we apply two procedures which estimate quantiles for the two-parameter distribution. The estimates of the quantiles for each family of distributions use two procedures for estimating intermediary the parameters of the distribution. The procedures are compared with respect to the normalized mean square error and the relative bias using simulations.