Detecting a change in the structure of a time series is a classical statistical problem. Here we consider a short memory causal linear process $X_i=\sum_{j=0}^\infty a_j\xi_{i-j}$, $i=1,\cdots,n$, where the innovations $\xi_i$ are independent and identically distributed and the coefficients $a_j$ are summable. The goal is to detect the existence of an unobserved time at which there is a change in the marginal distribution of the $X_i$'s. Our model allows us to simultaneously detect changes in the coefficients and changes in location and/or scale of the innovations. Under very simple moment and summability conditions, we investigate the asymptotic behaviour of the sequential empirical process based on the $X_i$'s both with and without a change-point, and show that two proposed test statistics are consistent. In order to find appropriate critical values for the test statistics, we then prove the validity of the moving block bootstrap for the sequential empirical process under both the hypothesis and the alternative, again under simple conditions. Finally, the performance of the proposed test statistics is demonstrated through Monte Carlo simulations.

The change point problem is a problem where a process changes regimes because a parameter changes at a point in time called the change point. The objective of this problem is to estimate the change point and each of the parameters of the stochastic process. In this thesis, we examine the change point problem for two classes of stochastic processes. First, we consider the volatility change point problem for stochastic diffusion processes driven by Brownian motions. Then, we consider the drift change point problem for Ornstein-Uhlenbeck processes driven by _-stable Levy motions. In each problem, we establish the consistency of the estimators, determine asymptotic behavior for the changing parameters, and finally, we perform simulation studies to computationally assess the convergence of parameters. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2020. / FAU Electronic Theses and Dissertations Collection

Change point detection (CPD) is the method of detecting sudden changes in timeseries, and its importance is great concerning network traffic. With increased knowledge of occurring changes in data logs due to updates in networking equipment,a deeper understanding is allowed for interactions between the updates and theoperational resource usage. In a data log that reflects the amount of network traffic, there are large variations in the time series because of reasons such as connectioncount or external changes to the system. To circumvent these unwanted variationchanges and assort the deliberate variation changes is a challenge. In this thesis, we utilize data logs retrieved from a network equipment vendor to detect changes, then compare the detected changes to when firmware/signature updates were applied, configuration changes were made, etc. with the goal to achieve a deeper understanding of any interaction between firmware/signature/configuration changes and operational resource usage. Challenges in the data quality and data processing are addressed through data manipulation to counteract anomalies and unwanted variation, as well as experimentation with parameters to achieve the most ideal settings. Results are produced through experiments to test the accuracy of the various change pointdetection methods, and for investigation of various parameter settings. Through trial and error, a satisfactory configuration is achieved and used in large scale log detection experiments. The results from the experiments conclude that additional information about how changes in variation arises is required to derive the desired understanding.

The thesis aims at change-point estimation in gradual change models. Methods avail- able in literature are reviewed and modified for point-of-stabilisation (PoSt) context, present e.g. in drug continuous manufacturing. We describe in detail the estimation in the linear PoSt model and we extend the methods to quadratic and Emax model. We describe construction of confidence intervals for the change-point, discuss their interpre- tation and show how they can be used in practice. We also address the situation when the assumption of homoscedasticity is not fulfilled. Next, we run simulations to calculate the coverage of confidence intervals for the change-point in discussed models using asymp- totic results and bootstrap with different parameter combinations. We also inspect the simulated distribution of derived estimators with finite sample. In the last chapter, we discuss the situation when the model for the data is incorrectly specified and we calculate the coverage of confidence intervals using simulations. 1

Cyberattacks against critical infrastructures has been a growing problem for the past couple of years. These infrastructures are a particularly desirable target for adversaries, due to their vital importance in society. For instance, a stop in the operation of a critical infrastructure could result in a crippling effect on a nation's economy, security or public health. The reason behind this increase is that critical infrastructures have become more complex, often being integrated with a large network of various cyber components. It is through these cyber components that an adversary is able to access the system and conduct their attacks. In this thesis, we consider methods which can be used as a first line of defence against such attacks for Cyber-Physical Systems (CPS). Specifically, we start by studying how information leaks about a system's dynamics helps an adversary to generate attacks that are difficult to detect. In many cases, such attacks can be detrimental to a CPS since they can drive the system to a breaking point without being detected by the operator that is tasked to secure the system. We show that an adversary can use small amounts of data procured from information leaks to generate these undetectable attacks. In particular, we provide the minimal amount of information that is needed in order to keep the attack hidden even if the operator tries to probe the system for attacks.  We design defence mechanisms against such information leaks using the Hammersley-Chapman-Robbins lower bound. With it, we study how information leakage could be mitigated through corruption of the data by injection of measurement noise. Specifically, we investigate how information about structured input sequences, which we call events, can be obtained through the output of a dynamical system and how this leakage depends on the system dynamics. For example, it is shown that a system with fast dynamical modes tends to disclose more information about an event compared to a system with slower modes. However, a slower system leaks information over a longer time horizon, which means that an adversary who starts to collect information long after the event has occured might still be able to estimate it. Additionally, we show how sensor placements can affect the information leak. These results are then used to aid the operator to detect privacy vulnerabilities in the design of a CPS. Based on the Hammersley-Chapman-Robbins lower bound, we provide additional defensive mechanisms that can be deployed by an operator online to minimize information leakage. For instance, we propose a method to modify the structured inputs in order to maximize the usage of the existing noise in the system. This mechanism allows us to explicitly deal with the privacy-utility trade-off, which is of interest when optimal control problems are considered. Finally, we show how the adversary's certainty of the event increases as a function of the number of samples they collect. For instance, we provide sufficient conditions for when their estimation variance starts to converge to its final value. This information can be used by an operator to estimate when possible attacks from an adversary could occur, and change the CPS before that, rendering the adversary's collected information useless. / De senaste åren har cyberanfall mot kritiska infrastructurer varit ett växande problem. Dessa infrastrukturer är särskilt utsatta för cyberanfall, eftersom de uppfyller en nödvändig function för att ett samhälle ska fungera. Detta gör dem till önskvärda mål för en anfallare. Om en kritisk infrastruktur stoppas från att uppfylla sin funktion, då kan det medföra förödande konsekvenser för exempelvis en nations ekonomi, säkerhet eller folkhälsa. Anledningen till att mängden av attacker har ökat beror på att kritiska infrastrukturer har blivit alltmer komplexa eftersom de numera ingår i stora nätverk dör olika typer av cyberkomponenter ingår. Det är just genom dessa cyberkomponenter som en anfallare kan få tillgång till systemet och iscensätta cyberanfall. I denna avhandling utvecklar vi metoder som kan användas som en första försvarslinje mot cyberanfall på cyberfysiska system (CPS). Vi med att undersöka hur informationsläckor om systemdynamiken kan hjälpa en anfallare att skapa svårupptäckta attacker. Oftast är sådana attacker förödande för CPS, eftersom en anfallare kan tvinga systemet till en bristningsgräns utan att bli upptäcka av operatör vars uppgift är att säkerställa systemets fortsatta funktion. Vi bevisar att en anfallare kan använda relativt små mängder av data för att generera dessa svårupptäckta attacker. Mer specifikt så härleder ett uttryck för den minsta mängd information som krävs för att ett anfall ska vara svårupptäckt, även för fall då en operatör tar till sig metoder för att undersöka om systemet är under attack. I avhandlingen konstruerar vi försvarsmetoder mot informationsläcker genom Hammersley-Chapman-Robbins olikhet. Med denna olikhet kan vi studera hur informationsläckan kan dämpas genom att injicera brus i datan. Specifikt så undersöker vi hur mycket information om strukturerade insignaler, vilket vi kallar för händelser, till ett dynamiskt system som en anfallare kan extrahera utifrån dess utsignaler. Dessutom kollar vi på hur denna informationsmängd beror på systemdynamiken. Exempelvis så visar vi att ett system med snabb dynamik läcker mer information jämfört med ett långsammare system. Däremot smetas informationen ut över ett längre tidsintervall för långsammare system, vilket leder till att anfallare som börjar tjuvlyssna på ett system långt efter att händelsen har skett kan fortfarande uppskatta den. Dessutom så visar vi jur sensorplaceringen i ett CPS påverkar infromationsläckan. Dessa reultat kan användas för att bistå en operatör att analysera sekretessen i ett CPS. Vi använder även Hammersley-Chapman-Robbins olikhet för att utveckla försvarslösningar mot informationsläckor som kan användas \textit{online}. Vi föreslår modifieringar till den strukturella insignalen så att systemets befintliga brus utnyttjas bättre för att gömma händelsen. Om operatören har andra mål den försöker uppfylla med styrningen så kan denna metod användas för att styra avvängingen mellan sekretess och operatorns andra mål. Slutligen så visar vi hur en anfallares uppskattning av händelsen förbättras som en funktion av mängden data får tag på. Operatorn kan använda informationen för att ta reda på när anfallaren kan tänka sig vara redo att anfalla systemet, och därefter ändra systemet innan detta sker, vilket gör att anfallarens information inte längre är användbar. / <p>QC 20210820</p>

<div>Model-based clustering, with its flexibility and solid statistical foundations, is an important tool for unsupervised learning, and has numerous applications in a variety of fields. This dissertation focuses on nonparametric Bayesian approaches to model-based clustering under structural restrictions. These are additional constraints on the model that embody prior knowledge, either to regularize the model structure to encourage interpretability and parsimony or to encourage statistical sharing through underlying tree or network structure.</div><div><br></div><div>The first part in the dissertation focuses on the most commonly used model-based clustering models, mixture models. Current approaches typically model the parameters of the mixture components as independent variables, which can lead to overfitting that produces poorly separated clusters, and can also be sensitive to model misspecification. To address this problem, we propose a novel Bayesian mixture model with the structural restriction being that the clusters repel each other.The repulsion is induced by the generalized Matérn type-III repulsive point process. We derive an efficient Markov chain Monte Carlo (MCMC) algorithm for posterior inference, and demonstrate its utility on a number of synthetic and real-world problems. <br></div><div><br></div><div>The second part of the dissertation focuses on clustering populations with a hierarchical dependency structure that can be described by a tree. A classic example of such problems, which is also the focus of our work, is the phylogenetic tree with nodes often representing biological species. The structure of this problem refers to the hierarchical structure of the populations. Clustering of the populations in this problem is equivalent to identify branches in the tree where the populations at the parent and child node have significantly different distributions. We construct a nonparametric Bayesian model based on hierarchical Pitman-Yor and Poisson processes to exploit this, and develop an efficient particle MCMC algorithm to address this problem. We illustrate the efficacy of our proposed approach on both synthetic and real-world problems.</div>

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Segmented regression is a type of nonlinear regression that allows differing functional forms to be fit over different ranges of the explanatory variable. This paper considers the simple segmented regression case of two linear segments that are constrained to meet, often called the linear-linear model. Parameter estimation in the case where the joinpoint between the regimes is unknown can be tricky. Using a simulation study, four estimators for the parameters of the linear-linear model are evaluated. The bias and mean squared error of the estimators are considered under differing parameter combinations and sample sizes. Parameters estimated in the model are the location of the change-point, the slope and intercept of the first segment, the change in slope from the first segment to the second, and the variance over both segments.

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