Heavy-tailed noise models concern data contaminated with outliers. If the presence of outliers is not considered in the assumed model, the estimation performance of important parameters such as the mean and variance deteriorates. In this work, a misspecified Cramér-Rao bound is derived to show the reduced estimation performance when assuming a Gaussian distribution, although some portion of the data is generated by a Gaussian with inflated variance. This provides insight into one heavy-tailed distribution; the assumption of a different heavy-tailed distribution, the Student’s t distribution, is also investigated. The Cramér-Rao Bound for joint estimation of the location, scale, and shape parameters of the Student’s t-distribution is also derived to quantify the difference in performance when the number of degrees of freedom is unknown. Analysis of the corresponding maximum likelihood estimators and practical implementations of those estimators using the Expectation Maximization algorithm reveals the misspecified estimation performance when the contaminated data is not perfectly modeled by the chosen heavy-tailed distribution. Each of the assumptions is tested on realistic data with labeled outliers to identify the more advantageous assumption between a mixture of Gaussians and a Student’s t distribution when the true distribution of measurements is not necessarily a specific heavy-tailed model.

In this article, the principles of robust estimation are applied to the standard basic time scale equation to obtain a new method of assigning weights to clocks. Specifically, the Student’s t-distribution is introduced as a new statistical model for an ensemble of clocks that are experiencing phase jumps, frequency jumps or anomalies in their measurement links. The proposed robust time scale is designed to mitigate the effects of these anomalies without necessarily identifying them, but through applying a method of robust estimation for the parameters of a Student’s t-distribution. The proposed time scale algorithm using the Student’s t-distribution (ATST) is shown to achieve comparable robustness to phase jumps, frequency jumps, and anomalies in the measurements with respect to the AT1 oracle time scale. The AT1 oracle is a special realization of the AT1 time scale which corrects all anomalies by having prior knowledge of their occurrences. The similar performance of ATST and AT1 oracle suggests that the ATST algorithm is efficient for obtaining robustness with no prior knowledge or detection of the occurrences of anomalies.

This work derives compact closed-form expressions of the misspecified Cramér–Rao bound and pseudo-true parameters of time-delay and Doppler for a high dynamics signal model. Those expressions are validated by analyzing the mean square error (MSE) of the misspecified maximum likelihood estimator. A noteworthy outcome of these MSE results is that, for some magnitudes of acceleration and signal-to-noise ratios, neglecting the acceleration is beneficial in the MSE sense. The variance performance improvement is obtained at the cost of a systematic error in the true parameter estimation. This can be seen as a specific case of the trade-off between bias and variance. Neglecting the acceleration can improve the Doppler estimation when the error induced on the misspecified model is less than the variance increase due to including an extra parameter to estimate. Then, for some non-zero acceleration magnitudes and short integration times, the Doppler estimation using a misspecified model outperforms a correctly specified model in the MSE sense.

The derivation of estimation lower bounds is paramount to designing and assessing the performance of new estimators. A lot of effort has been devoted to the range-velocity estimation problem, a fundamental stage on several applications, but very few works deal with acceleration, being a key aspect in high dynamics applications. Considering a generic band-limited signal formulation, we derive a new general compact form Cramér-Rao bound (CRB) expression for joint time-delay, Doppler stretch, and acceleration estimation. This generalizes and expands upon known delay/Doppler estimation CRB results for both wideband and narrowband signals. This new formulation, especially easy to use, is created based on baseband signal samples, making it valid for a variety of remote sensors. The new CRB expressions are illustrated and validated with representative GPS L1 C/A and Linear Frequency Modulated (LFM) chirp band-limited signals. The mean square error (MSE) of a misspecified estimator (conventional delay/Doppler) is compared with the derived bound. The comparison indicates that for some acceleration ranges the misspecified estimator outperforms a well specified estimator that accounts for acceleration.

Nonlinear regression models play a crucial role in signal processing and multi-sensor applications. Traditionally, performance bounds for these models assume independent Gaussian observations. In practice, the Gaussian assumption fails in multi-sensor systems if some proportion of sensors are corrupted by non-Gaussian noise and outliers. In this context, we extend the Misspecified Cramér-Rao Bound (MCRB) framework to the contaminated Gaussian noise model, where observations are generated from a mixture of nominal Gaussian noise and occasional outliers. Building on previous work with Complex Elliptically Symmetric noise models, we derive analytical MCRB expressions under the mismatched Gaussian assumption and study the asymptotic behavior of the corresponding Misspecified Maximum Likelihood Estimator (MMLE). To demonstrate practical relevance, we apply the theory to joint time-delay and Doppler estimation in GPS signals under contamination. Numerical simulations confirm that the MMLE root mean squared error converges to the theoretical MCRB, which aligns with the classical Gaussian CRB.

This work provides a comparative study of the complexity and performance for a range of different types of robust estimators. The interest of this analysis is to find the preferred robust estimator that can define the system time for a swarm of satellites. The Student’s t-distribution is used as a model for the noise corrupting the measurements. The ideal performance of an unbiased estimator for a fixed number of degrees of freedom is known in the form of the Cram´er-Rao Bound (CRB). In this article, two examples of a robust Mestimator and an approximation of the Maximum Likelihood Estimator (MLE) resulting from an Expectation-Maximization algorithm are each tested with respect to the performance bounds. Each estimator is also compared with the Gaussian MLE under Gaussian noise, to identify any losses in efficiency under Gaussian conditions. The complexity of the algorithms is also studied by comparing the time until convergence in the iterative update of the robust estimators.

Robust estimation methods are useful in mitigating the impact of anomalies in clock data. Such anomalous clock data is assumed to be well modeled by a Student’s t-distribution. This paper derives a lower bound on the performance of the misspecified Gaussian model using the theory of the Misspecified Cram´er-Rao bound (MCRB). The results of these derivations are verified by analyzing the Mean Square Error (MSE) of the misspecified Gaussian Maximum Likelihood Estimator (MLE) when using data generated by the Student’s t-distribution. The derived MCRB indicates a constraint on the MSE when assuming a Gaussian distribution. The MLE for the mean of the Student’s t-distribution is obtained with an Expectation maximization algorithm and is shown to obtain a lower MSE than the MCRB and hence, the misspecified estimator. This indicates an improvement in performance if anomalous clock data is appropriately accounted for in the statistical model.

The computation of a common reference time for a swarm of nanosatellites is restricted by the quality and availability of the timing measurements made with inter-satellite links. The presence of anomalies or absence of communication links is demonstrated to harm the stability of the time scale. The Least Squares (LS) estimator is introduced as a method of preprocessing measurement noise by using all available clock comparisons in the swarm. This estimator also provides filtered measurements when inter-satellite links are missing as long as each satellite maintains at least one link with another. Anomaly detection and removing corrupted satellite links are shown to be compatible with the LS estimator to mitigate the impact of anomalous measurements. When a satellite becomes completely isolated for some period of time, a correction at the beginning and the end of the isolation period are both detailed. The correction is simple and just requires resetting the weights of missing clocks and clocks being reintroduced. Continuity is shown to be maintained when a large portion of clocks are removed and later reintroduced at the same time.

This paper introduces a new statistical model for clock phases assuming a multivariate Gaussian distribution for the clock phase deviations from a common time scale. This model allows us to derive a maximum likelihood estimator for the clock phases, which is consistent with the current methods of computing a common time scale for a collection of clocks. Detailing a statistical model of the clock phases, which assumes a Gaussian distribution allows us to find the MLE for each clock’s phase deviation from a common time scale. For verification, the MLE for the clock phases is shown to be consistent with the result of the existing basic time scale equation. The statistical distribution of the frequency states resulting from this statistical model is Gaussian over a window of past time instants. This property can be used to design a new time scale based on the maximum likelihood estimator of frequency and frequency variances that are alternatives to the exponential filters designed for AT1. With the appropriate number of past frequency samples, this MLE has identical performance to the optimal AT1 algorithm in a nominal context. The statistical distribution of the frequency when the clock suffers a phase jump anomaly is then identified as a Student’s t-distribution. The Student’s t-distribution models the statistics of datasets contaminated by outliers, leading to the derivation of a different MLE that is robust to those outliers. The time scale using the robust MLE provides estimates of each clock’s frequency and frequency variance that are unaffected by phase jump anomalies and improves the long-term frequency stability when each clock in the ensemble experiences phase jump anomalies within some window of time.

Cette thèse s’intéresse à la construction d’une échelle de temps autonome et robuste aux erreurs d’horloge pour des essaims de satellites. Prévue pour une utilisation dans un essaim de nanosatellites, cette nouvelle échelle de temps appelée ATST (Autonomous Time Scale using the Student’s T-distribution) peut traiter les anomalies dues aux imperfections des horloges et aux liens inter-satellites manquants dans un environnement hostile. Plus précisément, les types d’anomalies traités incluent les sauts de phase, les sauts de fréquence, un bruit de mesure élevé dans certains liens et d’éventuelles données manquantes. En calculant la moyenne pondérée des résidus issus de l’équation de l’échelle de temps de base (BTSE), la contribution des satellites avec des mesures anormales est réduite pour la génération de l’échelle de temps. Les poids attribués à chaque horloge sont basés sur l’hypothèse que les résidus de l’ensemble suivent une loi de Student, ce qui permet d’utiliser des méthodes d’estimation robustes à la présence d’éléments aberrants. La performance de l’algorithme ATST est équivalente à celle de l’algorithme AT1 oracle, qui est une version de l’échelle de temps AT1 avec la capacité de détecter parfaitement toutes les anomalies dans des données simulées. Bien que l’algorithme n’ait pas de méthode de détection explicite, l’algorithme ATST affiche toujours un niveau de robustesse comparable à celui d’un détecteur parfait. Cependant, l’algorithme ATST est concu pour un essaim avec de nombreuses horloges de types homogènes et est limité par une complexité numérique élevée. De plus, les anomalies sont toutes traitées de la même manière sans distinction entre les différents types d’anomalies. Malgré ces limitations identifiées, ce nouvel algorithme ATST représente une contribution prometteuse dans le domaine des échelles de temps grâce a la robustesse atteinte. Une méthode de traitement des horloges ajoutées ou retirées de l’ensemble des horloges disponibles est également proposée dans cette thèse en conjonction avec la méthode ATST. La méthode obtenue préserve la continuité de phase et de fréquence de l’échelle de temps en attribuant un poids nul aux horloges concernées lorsque le nombre total d’horloges est modifié. Un estimateur des moindres carrés (Least Squares, LS) est présenté pour montrer comment les mesures des liens inter-satellites peuvent être traitées en amont pour réduire le bruit de mesure et en même temps remplacer les mesures manquantes. L’estimateur LS peut être utilisé avec une méthode de détection qui élimine les mesures anormales. Il peut alors remplacer les mesures supprimées par les estimations correspondantes. Cette thèse étudie également les performances de l’estimateur du maximum de vraisemblance (MLE) pour les paramètres des lois de probabilités à queues lourdes, plus précisément pour la loi de Student et pour un mélange de lois gaussiennes. Les améliorations obtenues en supposant que ces lois sont effectivement à queues lourdes par rapport à l’hypothèse de la loi gaussienne sont démontrées avec les bornes de Cramér-Rao mal-spécifiées (MCRB). Les expressions obtenues des MCRB pour une loi de Student et un mélange de lois gaussiennes confirment que les lois à queues lourdes sont meilleures pour l’estimation de la moyenne en présence de valeurs aberrantes. Elles permettent également de montrer que l’estimation des paramètres des lois à queues lourdes nécessite au moins 25 horloges pour obtenir une erreur d’estimation proche de la MCRB correspondante, c’est-à-dire que l’estimateur atteigne son efficacité asymptotique. Des propositions de pistes de recherche futures incluent le traitement des limitations de l’algorithme ATST concernant les types et le nombre d’horloges. Une nouvelle définition des pondérations des résidus issue d’une méthode d’apprentissage statistique utilisant des données d’apprentissage est envisageable grâce `a l’utilisation des résidus de l’échelle de temps de base BTSE. Une autre piste de recherche est le traitement des anomalies transitoires qui pose actuellement problème pour l’algorithme ATST. Un traitement de ce type d’erreurs pourrait être envisagé avec un algorithme d’apprentissage statistique ou avec un estimateur robuste de la fréquence des horloges sur une fenêtre de données passées. Mots clés: Estimation robuste, échelles de temps, détection des anomalies, bornes de Cramér-Rao mal-spécifiées.