Ratingsysteme sind ein zentraler Bestandteil der Kreditrisikomodellierung. Neben der Bonitätsbeurteilung auf der Ebene der Kreditnehmer und der Risikoquantifizierung auf der Ebene der Ratingklassen spielt dabei die Bildung der Ratingklassen eine wesentliche Rolle. Die Literatur zur Ratingklassenbildung setzt auf modellfreie, in gewisser Weise willkürliche Optimierungsverfahren. Ein Ziel der vorliegenden Arbeit ist es, stattdessen ein parametrisches statistisches Modell zur Bildung der Ratingklassen einzuführen. Ein geeignetes Modell ist im Bereich der Bruchpunktschätzung zu finden. Dieses Modell und die in der mathematischen Literatur vorgeschlagenen Parameter- und Intervallschätzer werden in der vorliegenden Arbeit dargestellt und gründlich diskutiert. Dabei wird Wert auf eine anwendungsnahe und anschauliche Formulierung der mathematisch-statistischen Sachverhalte gelegt. Anschließend wird die Methodik der Bruchpunktschätzung auf einen konkreten Datensatz angewendet und mit verschiedenen anderen Kriterien zur Ratingklassenbildung verglichen. Hier erweist sich die Bruchpunktschätzung als vorteilhaft. Aufbauend auf der empirischen Untersuchung wird abschließend weiterer Forschungsbedarf abgeleitet. Dazu werden insbesondere Konzepte für den Mehrklassenfall und für abhängige Daten entworfen. / Rating systems are a key component of credit risk modeling. In addition to scoring at borrowers’ level and risk quantification at the level of rating classes, the formation of the rating classes plays a fundamental role. The literature on rating classification uses in a way arbitrary optimization methods. Therefore, one aim of this contribution is to introduce a parametric statistical model to form the rating classes. A suitable model can be found in the area of split-point estimation. This model and the proposed parameter and interval estimators are presented and thoroughly discussed. Here, emphasis is placed on an application-oriented and intuitive formulation of the mathematical and statistical issues. Subsequently, the methodology of split-point estimation is applied to a specific data set and compared with several other criteria for rating classification. Here, split-point estimation proves to be advantageous. Finally, further research questions are derived on the basis of the empirical study. In particular, concepts for the case of more than two classes and for dependent data are sketched.

The purpose of this thesis was to explore the potential of near-infrared spectroscopy (NIRS) as an access channel by establishing reliable signal detection to verify the existence of signal differences associated with changes in activity. This thesis focused on using NIRS to measure brain activity from the prefrontal cortex during an auditory Go-No-Go task. A singular spectrum analysis change-point detection algorithm was applied to identify transition points where the NIRS signal properties varied from previous data points in the signal, indicating a change in brain activity. With this algorithm, latency values for change-points detected ranged from 6.44 s to 9.34 s. The averaged positive predictive values over all runs were modest (from 49.41% to 67.73%), with the corresponding negative predictive values being generally higher (48.66% to 78.80%). However, positive and negative predictive values up to 97.22% and 95.14%, respectively, were achieved for individual runs. No hemispheric differences were found.

The purpose of this thesis was to explore the potential of near-infrared spectroscopy (NIRS) as an access channel by establishing reliable signal detection to verify the existence of signal differences associated with changes in activity. This thesis focused on using NIRS to measure brain activity from the prefrontal cortex during an auditory Go-No-Go task. A singular spectrum analysis change-point detection algorithm was applied to identify transition points where the NIRS signal properties varied from previous data points in the signal, indicating a change in brain activity. With this algorithm, latency values for change-points detected ranged from 6.44 s to 9.34 s. The averaged positive predictive values over all runs were modest (from 49.41% to 67.73%), with the corresponding negative predictive values being generally higher (48.66% to 78.80%). However, positive and negative predictive values up to 97.22% and 95.14%, respectively, were achieved for individual runs. No hemispheric differences were found.

The problem of determining whether or not a theoretical model is an accurate representation of an empirically observed phenomenon is one of the most challenging in the empirical scientific investigation. The following study explores the problem of stochastic model validation. Special attention is devoted to the unusual two-peaked shape of the empirically observed distributions of the conditional on realised volatility financial returns. The application of statistical hypothesis testing and simulation techniques leads to the conclusion that the conditional on realised volatility returns are distributed with a specific previously undocumented distribution. The probability density that represents this distribution is derived, characterised and applied for validation of the financial model.

Title: Robust Monitoring Procedures for Dependent Data Author: Ondřej Chochola Department: Department of Probability and Mathematical Statistics Supervisor: Prof. RNDr. Marie Hušková, DrSc. Supervisor's e-mail address: huskova@karlin.mff.cuni.cz Abstract: In the thesis we focus on sequential monitoring procedures. We extend some known results towards more robust methods. The robustness of the procedures with respect to outliers and heavy-tailed observations is introduced via use of M-estimation instead of classical least squares estimation. Another extension is towards dependent and multivariate data. It is assumed that the observations are weakly dependent, more specifically they fulfil strong mixing condition. For several models, the appropriate test statistics are proposed and their asymptotic properties are studied both under the null hypothesis of no change as well as under the alternatives, in order to derive proper critical values and show consistency of the tests. We also introduce retrospective change-point procedures, that allow one to verify in a robust way the stability of the historical data, which is needed for the sequential monitoring. Finite sample properties of the tests need to be also examined. This is done in a simulation study and by application on some real data in the capital asset...

Cette thèse s’inscrit dans le cadre du projet « SCALA » financé par l’ANR à travers le programme ANR-11-SECU-0005. Son objectif consiste à surveiller des systèmes de contrôle et d’acquisition de données (SCADA) contre des attaques cyber-physiques. Il s'agit de résoudre un problème de détection-localisation séquentielle de signaux transitoires dans des systèmes stochastiques et dynamiques en présence d'états inconnus et de bruits aléatoires. La solution proposée s'appuie sur une approche par redondance analytique composée de deux étapes : la génération de résidus, puis leur évaluation. Les résidus sont générés de deux façons distinctes, avec le filtre de Kalman ou par projection sur l’espace de parité. Ils sont ensuite évalués par des méthodes d’analyse séquentielle de rupture selon de nouveaux critères d’optimalité adaptés à la surveillance des systèmes à sécurité critique. Il s'agit donc de minimiser la pire probabilité de détection manquée sous la contrainte de niveaux acceptables pour la pire probabilité de fausse alarme et la pire probabilité de fausse localisation. Pour la tâche de détection, le problème d’optimisation est résolu dans deux cas : les paramètres du signal transitoire sont complètement connus ou seulement partiellement connus. Les propriétés statistiques des tests sous-optimaux obtenus sont analysées. Des résultats préliminaires pour la tâche de localisation sont également proposés. Les algorithmes développés sont appliqués à la détection et à la localisation d'actes malveillants dans un réseau d’eau potable / This PhD thesis is registered in the framework of the project “SCALA” which received financial support through the program ANR-11-SECU-0005. Its ultimate objective involves the on-line monitoring of Supervisory Control And Data Acquisition (SCADA) systems against cyber-physical attacks. The problem is formulated as the sequential detection and isolation of transient signals in stochastic-dynamical systems in the presence of unknown system states and random noises. It is solved by using the analytical redundancy approach consisting of two steps: residual generation and residual evaluation. The residuals are firstly generated by both Kalman filter and parity space approaches. They are then evaluated by using sequential analysis techniques taking into account certain criteria of optimality. However, these classical criteria are not adequate for the surveillance of safety-critical infrastructures. For such applications, it is suggested to minimize the worst-case probability of missed detection subject to acceptable levels on the worst-case probability of false alarm and false isolation. For the detection task, the optimization problem is formulated and solved in both scenarios: exactly and partially known parameters. The sub-optimal tests are obtained and their statistical properties are investigated. Preliminary results for the isolation task are also obtained. The proposed algorithms are applied to the detection and isolation of malicious attacks on a simple SCADA water network

Ce travail développe des techniques non-supervisées de détection et de localisation en ligne de ruptures dans les signaux enregistrés dans un environnement bruité. Ces techniques reposent sur l'association d'une approche algébrique avec la TVE. L'approche algébrique permet d'appréhender aisément les ruptures en les caractérisant en termes de distributions de Dirac retardées et leurs dérivées dont la manipulation est facile via le calcul opérationnel. Cette caractérisation algébrique, permettant d'exprimer explicitement les instants d'occurrences des ruptures, est complétée par une interprétation probabiliste en termes d'extrêmes : une rupture est un évènement rare dont l'amplitude associée est relativement grande. Ces évènements sont modélisés dans le cadre de la TVE, par une distribution de Pareto Généralisée. Plusieurs modèles hybrides sont proposés dans ce travail pour décrire à la fois le comportement moyen (bruit) et les comportements extrêmes (les ruptures) du signal après un traitement algébrique. Des algorithmes entièrement non-supervisés sont développés pour l'évaluation de ces modèles hybrides, contrairement aux techniques classiques utilisées pour les problèmes d'estimation en question qui sont heuristiques et manuelles. Les algorithmes de détection de ruptures développés dans cette thèse ont été validés sur des données générées, puis appliqués sur des données réelles provenant de différents phénomènes, où les informations à extraire sont traduites par l'apparition de ruptures. / This work develops non supervised techniques for on-line detection and location of change-points in noisy recorded signals. These techniques are based on the combination of an algebraic approach with the Extreme Value Theory (EVT). The algebraic approach offers an easy identification of the change-points. It characterizes them in terms of delayed Dirac distributions and their derivatives which are easily handled via operational calculus. This algebraic characterization, giving rise to an explicit expression of the change-points locations, is completed with a probabilistic interpretation in terms of extremes: a change point is seen as a rare and extreme event. Based on EVT, these events are modeled by a Generalized Pareto Distribution.Several hybrid multi-components models are proposed in this work, modeling at the same time the mean behavior (noise) and the extremes ones (change-points) of the signal after an algebraic processing. Non supervised algorithms are proposed to evaluate these hybrid models, avoiding the problems encountered with classical estimation methods which are graphical ad hoc ones. The change-points detection algorithms developed in this thesis are validated on generated data and then applied on real data, stemming from different phenomenons, where change-points represent the information to be extracted.

Les travaux présentés dans ce mémoire de thèse concernent le développement et la mise en oeuvre d'un algorithme de filtrage particulaire pour le recalage de la navigation inertielle par mesures altimétriques. Le filtre développé, le MRPF (Mixture Regularized Particle Filter), s'appuie à la fois sur la modélisation de la densité a posteriori sous forme de mélange fini, sur le filtre particulaire régularisé ainsi que sur l'algorithme mean-shift clustering. Nous proposons également une extension du MRPF au filtre particulaire Rao-Blackwellisé appelée MRBPF (Mixture Rao-Blackwellized Particle Filter). L'objectif est de proposer un filtre adapté à la gestion des multimodalités dues aux ambiguïtés de terrain. L'utilisation des modèles de mélange fini permet d'introduire un algorithme d'échantillonnage d'importance afin de générer les particules dans les zones d'intérêt. Un second axe de recherche concerne la mise au point d'outils de contrôle d'intégrité de la solution particulaire. En nous appuyant sur la théorie de la détection de changement, nous proposons un algorithme de détection séquentielle de la divergence du filtre. Les performances du MRPF, MRBPF, et du test d'intégrité sont évaluées sur plusieurs scénarios de recalage altimétrique / This thesis deals with the development of a mixture particle filtering algorithm for inertial navigation update via radar-altimeter measurements. This particle filter, the so-called MRPF (Mixture Regularized Particle Filter), combines mixture modelling of the posterior density, the regularized particle filter and the mean-shift clustering algorithm. A version adapted to the Rao-Blackwellized particle filter, the MRBPF (Mixture Rao-Blackwellized Particle Filter), is also presented. The main goal is to design a filter well suited to multimodal densities caused by terrain amibiguity. The use of mixture models enables us to introduce an alternative importance sampling procedure aimed at proposing samples in the high likelihood regions of the state space. A second research axis is concerned with the development of particle filtering integrity monitoring tools. A novel particle filter divergence sequential detector, based on change detection theory, is presented. The performances of the MRPF, MRBPF and the divergence detector are reported on several terrain navigation scenarios

Change-point analysis is devoted to the detection and estimation of the time of structural changes within a data set of time-ordered observations. In this thesis, we estimate simultaneously multiple change-points by the least squares method and examine asymptotic properties of such estimators. Using argmin theorems, we prove weak and strong consistency under different moment conditions and investigate convergence in distribution. The identification of the limit variable allows us to derive an asymptotic confidence region for the unknown parameters. Based on a simulation study we evaluate these results.

This thesis addresses the optimization of the Variational Autoencoder-based Change Point Detection (VAE-CP) approach in time series analysis, a vital component in data-driven decision making. We evaluate the impact of various distance metrics and hyperparameters on the model’s performance using a systematic exploration and robustness testing on diverse real-world datasets. Findings show that the Dynamic Time Warping (DTW) distance metric significantly enhances the quality of the extracted latent variable space and improves change point detection. The research underscores the potential of the VAE-CP approach for more effective and robust handling of complex time series data, advancing the capabilities of change point detection techniques. / Denna uppsats behandlar optimeringen av en Variational Autoencoder-baserad Change Point Detection (VAE-CP)-metod i tidsserieanalys, en vital komponent i datadrivet beslutsfattande. Vi utvärderar inverkan av olika distansmått och hyperparametrar på modellens prestanda med hjälp av systematisk utforskning och robusthetstestning på diverse verkliga datamängder. Resultaten visar att distansmåttet Dynamic Time Warping (DTW) betydligt förbättrar kvaliteten på det extraherade latenta variabelutrymmet och förbättrar detektionen av brytpunkter (eng. change points). Forskningen understryker potentialen med VAE-CP-metoden för mer effektiv och robust hantering av komplexa tidsseriedata, vilket förbättrar förmågan hos tekniker för att upptäcka brytpunkter.