We propose new estimation methods for the factor loading matrix in modeling multivariate volatility processes. The key step of the methods is based on the weighted scatter estimators, which does not involve optimizing any objective function and was embedded with robust estimation properties. The method can therefore be easily applied to high-dimensional systems without running into computational problems. The estimation is proved to be consistent and the asymptotic distribution is derived. We compare the performance with other estimation methods and demonstrate its superiority when using both simulated data as well as real-world case studies. / Statistics

This work describes a framework for simultaneous estimation and modeling (SEAM) of dynamic systems using non-Gaussian belief fusion by first presenting the relevant fundamental formulations, then building upon these formulations incrementally towards a more general and ubiquitous framework. Multi-Gaussian belief fusion (MBF) is introduced as a natural and effective method of fusing non-Gaussian probability distribution functions (PDFs) in arbitrary dimensions efficiently and with no loss of accuracy. Construction of some multi-Gaussian structures for potential use in MBF is addressed. Furthermore, recursive Bayesian estimation (RBE) is developed for linearized systems with uncertainty in model parameters, and a rudimentary motion model correction stage is introduced. A subsequent improvement to motion model correction for arbitrarily non-Gaussian belief is developed, followed by application to observation models. Finally, SEAM is generalized to fully nonlinear and non-Gaussian systems. Several parametric studies were performed on simulated experiments in order to assess the various dependencies of the SEAM framework and validate its effectiveness in both estimation and modeling. The results of these studies show that SEAM is capable of improving estimation when uncertainty is present in motion and observation models as compared to existing methods. Furthermore, uncertainty in model parameters is consistently reduced as these parameters are updated throughout the estimation process. SEAM and its constituents have potential uses in robotics, target tracking and localization, state estimation, and more. / Doctor of Philosophy / The simultaneous estimation and modeling (SEAM) framework and its constituents described in this dissertation aim to improve estimation of signals where significant uncertainty would normally introduce error. Such signals could be electrical (e.g. voltages, currents, etc.), mechanical (e.g. accelerations, forces, etc.), or the like. Estimation is accomplished by addressing the problem probabilistically through information fusion. The proposed techniques not only improve state estimation, but also effectively "learn" about the system of interest in order to further refine estimation. Potential uses of such methods could be found in search-and-rescue robotics, robust control algorithms, and the like. The proposed framework is well-suited for any context where traditional estimation methods have difficulty handling heightened uncertainty.

Robust nonparametric spectral estimation includes generating an accurate estimate of the Power Spectral Density (PSD) for a given set of data while trying to minimize the bias due to data outliers. Robust nonparametric spectral estimation is applied in the domain of electrical communications and digital signal processing when a PSD estimate of the electromagnetic spectrum is desired (often for the goal of signal detection), and when the spectrum is also contaminated by Impulsive Noise (IN). Power Line Communication (PLC) is an example of a communication environment where IN is a concern because power lines were not designed with the intent to transmit communication signals. There are many different noise models used to statistically model different types of IN, but one popular model that has been used for PLC and various other applications is called the Middleton Class A model, and this model is extensively used in this thesis. The performances of two different nonparametric spectral estimation methods are analyzed in IN: the Welch method and the multitaper method. These estimators work well under the common assumption that the receiver noise is characterized by Additive White Gaussian Noise (AWGN). However, the performance degrades for both of these estimators when they are used for signal detection in IN environments. In this thesis basic robust estimation theory is used to modify the Welch and multitaper methods in order to increase their robustness, and it is shown that the signal detection capabilities in IN is improved when using the modified robust estimators. / Master of Science / One application of blind spectral estimation is blind signal detection. Unlike a car radio, where the radio is specifically designed to receive AM and PM radio waves, sometimes it is useful for a radio to be able to detect the presence of transmitted signals whose characteristics are not known ahead of time. Cognitive radio is one application where this capability is useful. Often signal detection is inhibited by Additive White Gaussian Noise (AWGN). This is analogous to trying to hear a friend speak (signal detection) in a room full of people talking (background AWGN). However, some noise environments are more impulsive in nature. Using the previous analogy, the background noise could be loud banging caused by machinery; the noise will not be as constant as the chatter of the crowd, but it will be much louder. When power lines are used as a medium for electromagnetic communication (instead of just sending power), it is called Power Line Communication (PLC), and PLC is a good example of a system where the noise environment is impulsive. In this thesis, methods used for blind spectral estimation are modified to work reliably (or robustly) for impulsive noise environments.

<p>This doctoral thesis is comprised of four papers that all relate to the subject of Time Series Analysis.</p><p>The first paper of the thesis considers point estimation in a nonnegative, hence non-Gaussian, AR(1) model. The parameter estimation is carried out using a type of extreme value estimators (EVEs). A novel estimation method based on the EVEs is presented. The theoretical analysis is complemented with Monte Carlo simulation results and the paper is concluded by an empirical example.</p><p>The second paper extends the model of the first paper of the thesis and considers semiparametric, robust point estimation in a nonlinear nonnegative autoregression. The nonnegative AR(1) model of the first paper is extended in three important ways: First, we allow the errors to be serially correlated. Second, we allow for heteroskedasticity of unknown form. Third, we allow for a multi-variable mapping of previous observations. Once more, the EVEs used for parameter estimation are shown to be strongly consistent under very general conditions. The theoretical analysis is complemented with extensive Monte Carlo simulation studies that illustrate the asymptotic theory and indicate reasonable small sample properties of the proposed estimators.</p><p>In the third paper we construct a simple nonnegative time series model for realized volatility, use the results of the second paper to estimate the proposed model on S&P 500 monthly realized volatilities, and then use the estimated model to make one-month-ahead forecasts. The out-of-sample performance of the proposed model is evaluated against a number of standard models. Various tests and accuracy measures are utilized to evaluate the forecast performances. It is found that forecasts from the nonnegative model perform exceptionally well under the mean absolute error and the mean absolute percentage error forecast accuracy measures.</p><p>In the fourth and last paper of the thesis we construct a multivariate extension of the popular Diebold-Mariano test. Under the null hypothesis of equal predictive accuracy of three or more forecasting models, the proposed test statistic has an asymptotic Chi-squared distribution. To explore whether the behavior of the test in moderate-sized samples can be improved, we also provide a finite-sample correction. A small-scale Monte Carlo study indicates that the proposed test has reasonable size properties in large samples and that it benefits noticeably from the finite-sample correction, even in quite large samples. The paper is concluded by an empirical example that illustrates the practical use of the two tests.</p>

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.

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This doctoral thesis is comprised of four papers that all relate to the subject of Time Series Analysis. The first paper of the thesis considers point estimation in a nonnegative, hence non-Gaussian, AR(1) model. The parameter estimation is carried out using a type of extreme value estimators (EVEs). A novel estimation method based on the EVEs is presented. The theoretical analysis is complemented with Monte Carlo simulation results and the paper is concluded by an empirical example. The second paper extends the model of the first paper of the thesis and considers semiparametric, robust point estimation in a nonlinear nonnegative autoregression. The nonnegative AR(1) model of the first paper is extended in three important ways: First, we allow the errors to be serially correlated. Second, we allow for heteroskedasticity of unknown form. Third, we allow for a multi-variable mapping of previous observations. Once more, the EVEs used for parameter estimation are shown to be strongly consistent under very general conditions. The theoretical analysis is complemented with extensive Monte Carlo simulation studies that illustrate the asymptotic theory and indicate reasonable small sample properties of the proposed estimators. In the third paper we construct a simple nonnegative time series model for realized volatility, use the results of the second paper to estimate the proposed model on S&amp;P 500 monthly realized volatilities, and then use the estimated model to make one-month-ahead forecasts. The out-of-sample performance of the proposed model is evaluated against a number of standard models. Various tests and accuracy measures are utilized to evaluate the forecast performances. It is found that forecasts from the nonnegative model perform exceptionally well under the mean absolute error and the mean absolute percentage error forecast accuracy measures. In the fourth and last paper of the thesis we construct a multivariate extension of the popular Diebold-Mariano test. Under the null hypothesis of equal predictive accuracy of three or more forecasting models, the proposed test statistic has an asymptotic Chi-squared distribution. To explore whether the behavior of the test in moderate-sized samples can be improved, we also provide a finite-sample correction. A small-scale Monte Carlo study indicates that the proposed test has reasonable size properties in large samples and that it benefits noticeably from the finite-sample correction, even in quite large samples. The paper is concluded by an empirical example that illustrates the practical use of the two tests.

Interval censoring arises frequently in life history data, as individuals are often only observed at a sequence of assessment times. This leads to a situation where we do not know when an event of interest occurs, only that it occurred somewhere between two assessment times. Here, the focus will be on methods of estimation for recurrent event data, current status data, and multistate data, subject to interval censoring. With recurrent event data, the focus is often on estimating the rate and mean functions. Nonparametric estimates are readily available, but are not smooth. Methods based on local likelihood and the assumption of a Poisson process are developed to obtain smooth estimates of the rate and mean functions without specifying a parametric form. Covariates and extra-Poisson variation are accommodated by using a pseudo-profile local likelihood. The methods are assessed by simulations and applied to a number of datasets, including data from a psoriatic arthritis clinic. Current status data is an extreme form of interval censoring that occurs when each individual is observed at only one assessment time. If current status data arise in clusters, this must be taken into account in order to obtain valid conclusions. Copulas offer a convenient framework for modelling the association separately from the margins. Estimating equations are developed for estimating marginal parameters as well as association parameters. Efficiency and robustness to the choice of copula are examined for first and second order estimating equations. The methods are applied to data from an orthopedic surgery study as well as data on joint damage in psoriatic arthritis. Multistate models can be used to characterize the progression of a disease as individuals move through different states. Considerable attention is given to a three-state model to characterize the development of a back condition known as spondylitis in psoriatic arthritis, along with the associated risk of mortality. Robust estimates of the state occupancy probabilities are derived based on a difference in distribution functions of the entry times. A five-state model which differentiates between left-side and right-side spondylitis is also considered, which allows us to characterize what effect spondylitis on one side of the body has on the development of spondylitis on the other side. Covariate effects are considered through multiplicative time homogeneous Markov models. The robust state occupancy probabilities are also applied to data on CMV infection in patients with HIV.

Interval censoring arises frequently in life history data, as individuals are often only observed at a sequence of assessment times. This leads to a situation where we do not know when an event of interest occurs, only that it occurred somewhere between two assessment times. Here, the focus will be on methods of estimation for recurrent event data, current status data, and multistate data, subject to interval censoring. With recurrent event data, the focus is often on estimating the rate and mean functions. Nonparametric estimates are readily available, but are not smooth. Methods based on local likelihood and the assumption of a Poisson process are developed to obtain smooth estimates of the rate and mean functions without specifying a parametric form. Covariates and extra-Poisson variation are accommodated by using a pseudo-profile local likelihood. The methods are assessed by simulations and applied to a number of datasets, including data from a psoriatic arthritis clinic. Current status data is an extreme form of interval censoring that occurs when each individual is observed at only one assessment time. If current status data arise in clusters, this must be taken into account in order to obtain valid conclusions. Copulas offer a convenient framework for modelling the association separately from the margins. Estimating equations are developed for estimating marginal parameters as well as association parameters. Efficiency and robustness to the choice of copula are examined for first and second order estimating equations. The methods are applied to data from an orthopedic surgery study as well as data on joint damage in psoriatic arthritis. Multistate models can be used to characterize the progression of a disease as individuals move through different states. Considerable attention is given to a three-state model to characterize the development of a back condition known as spondylitis in psoriatic arthritis, along with the associated risk of mortality. Robust estimates of the state occupancy probabilities are derived based on a difference in distribution functions of the entry times. A five-state model which differentiates between left-side and right-side spondylitis is also considered, which allows us to characterize what effect spondylitis on one side of the body has on the development of spondylitis on the other side. Covariate effects are considered through multiplicative time homogeneous Markov models. The robust state occupancy probabilities are also applied to data on CMV infection in patients with HIV.

In this thesis, robust set-valued estimation is studied and its application to in-flight alignment of strapdown inertial navigation systems (SINS) with large heading uncertainty is performed. It is known that the performance of the Kalman filter is vulnerable to modeling errors. One of the estimation methods, which are robust against modeling errors, is robust set-valued estimation. In this approach, the filter calculates the set of all possible states, which are consistent with uncertainty inputs satisfying an integral quadratic constraint (IQC) for given measured system outputs. In this thesis, robust set-valued filter with deterministic input is derived. In-flight alignment of SINS with Kalman filtering using external measurements is a widely used technique to eliminate the initial errors. However, if the initial errors are large then the performance of standard Kalman filtering technique is degraded due to modeling error caused by linearization process. To solve this problem, a novel linear norm-bounded uncertain error model is proposed where the remaining second orders terms due to linearization process are considered as norm-bounded uncertainty regarding only the heading error is large. Using the uncertain error model, the robust set-valued filter is applied to in-flight alignment problem. The comparison of the Kalman filter and the robust filter is done on a simulated trajectory and a real-time data. The simulation results show that the modeling errors can be compensated to some extent in Kalman filter by increasing the process noise covariance matrix. However, for very large initial heading errors, the proposed method outperforms the Kalman filter.