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We introduce computationally efficient Monte Carlo methods for studying the statistics of stochastic differential equations in two distinct settings. In the first, we derive importance sampling methods for data assimilation when the noise in the model and observations are small. The methods are formulated in discrete time, where the "posterior" distribution we want to sample from can be analyzed in an accessible small noise expansion. We show that a "symmetrization" procedure akin to antithetic coupling can improve the order of accuracy of the sampling methods, which is illustrated with numerical examples. In the second setting, we develop "stochastic continuation" methods to estimate level sets for statistics of stochastic differential equations with respect to their parameters. We adapt Keller's Pseudo-Arclength continuation method to this setting using stochastic approximation, and generalized least squares regression. Furthermore, we show that the methods can be improved through the use of coupling methods to reduce the variance of the derivative estimates that are involved.

This dissertation is a study of methods to automatedly detect and produce approximations of eddy current differential coil defect signatures in terms of a summed collection of Gaussian functions (SoG). Datasets consisting of varying material, defect size, inspection frequency, and coil diameter were investigated. Dimensionally reduced representations of the defect responses were obtained utilizing common existing reduction methods and novel enhancements to them utilizing SoG Representations. Efficacy of the SoG enhanced representations were studied utilizing common Machine Learning (ML) interpretable classifier designs with the SoG representations indicating significant improvement of common analysis metrics.

In recent years, wind and other variable renewable energy sources have gained a rapidly increasing share of the global energy mix. In this context the greatest concern facing renewable energy sources like wind is the uncertainty in production volumes as their generation ability is inherently dependent on weather conditions. When providing forecasts for newly commissioned wind farms there is a limited amount of historical power production data, while the number of potential features from different weather forecast providers is vast. Bayesian regularization is therefore seen as a possible technique for reducing model overfitting problems that may arise. This thesis investigates Bayesian Neural Networks in one-hour and day-ahead forecasting of wind power generation. Initial results show that Bayesian Neural Networks display equivalent predictive performance to Neural Networks trained by Maximum Likelihood in both one-hour and day ahead forecasting. Models selected using maximum evidence were found to have statistically significant lower test error performance compared to those selected based on minimum test error. Further results show that the Bayesian Framework is able to identify irrelevant features through Automatic Relevance Determination, though not resulting in a statistically significant error reduction in predictiveperformance in one-hour ahead forecasting. In day-ahead forecasting removing irrelevant features based on Automatic Relevance Determination is found to yield statistically significant improvements in test error. / Under de senaste åren har vind och andra variabla förnybara energikällor fått en snabbtökande andel av den globala energiandelen. I detta sammanhang är den största oron förförnybara energikällors produktionsvolymer vindosäkerheten, eftersom kraftverkens generationsförmåga i sig är beroende av väderförhållandena. Vid prognoser för nybyggdavindkraftverk finns en begränsad mängd historisk kraftproduktionsdata, medan antaletpotentiella mätvärden från olika väderprognosleverantörer är stor. Bayesian regulariseringses därför som en möjlig metod för att minska problem med den överanpassning avmodellerna som kan uppstå.Denna avhandling undersöker Bayesianska Neurala Nätverk (BNN) för prognosticeringen timme och en dag framåt av vindkraftproduktion. Resultat visar att BNN gerekvivalent prediktiv prestanda jämfört med neurala nätverk bildade med användandeav Maximum-likelihood för prognoser för en timme och dagsprognoser. Modeller somvalts med användning av maximum evidence visade sig ha statistiskt signifikant lägretestfelprestanda jämfört med de som valts utifrån minimaltestfel. Ytterligare resultatvisar att ett Bayesianskt ramverk kan identifiera irrelevanta särdrag genom automatiskrelevansbestämning. För prognoser för en timme framåt resulterar detta emellertid intei en statistiskt signifikant felreduktion i prediktiv prestanda. För 1-dagarsprognoser, närvi avlägsnar irrelevanta funktioner baserade på automatisk relevans, fås dock statistisktsignifikanta förbättringar av testfel.

Lately, spread spectrum systems are being increasingly used for commercial wireless communications because of their ability to reject various types of interference. This ability allows them to be used in multiple access systems. Direct sequence and frequency hopping systems have been the primary spread spectrum techniques used in practice. One technique which has not received much attention until recently is time hopping. In time hopping, a symbol is transmitted at a random position within the symbol period using a pulse width which is much smaller than the symbol period. Ultra-wideband (UWB) technology is a radar technology which shows promise for an relatively simple implementation of a time hopping system. This thesis looks at the error probability performance of a UWB time hopping multiple access system. Previous work has led to an estimate of the performance using a Gaussian approximation similar to that used for direct sequence systems. Through the use of a fast simulation technique, it will be shown that in certain situations, the Gaussian approximation fails to accurately predict the performance. A numerical analysis which uses characteristic functions is developed and shown to correctly predict the system’s performance under a wide range of situations. This numerical analysis also contributes to the understanding of the system. / Master of Science

Ziel dieser Arbeit ist theoretische Eigenschaften verschiedener Bootstrap Methoden zu untersuchen. Als Ergebnis führen wir die Konvergenzraten des Bootstrap-Verfahrens ein, die sich auf die Differenz zwischen der tatsächlichen Verteilung einer Statistik und der Resampling-Näherung beziehen. In dieser Arbeit analysieren wir die Verteilung der l2-Norm der Summe unabhängiger Vektoren, des Summen Maximums in hoher Dimension, des Wasserstein-Abstands zwischen empirischen Messungen und Wassestein-Barycenters. Um die Bootstrap-Konvergenz zu beweisen, verwenden wir die Gaussche Approximations technik. Das bedeutet dass man in der betrachteten Statistik eine Summe unabhängiger Vektoren finden muss, so dass Bootstrap eine erneute Abtastung dieser Summe ergibt. Ferner kann diese Summe durch Gaussche Verteilung angenähert und mit der Neuabtastung Verteilung als Differenz zwischen Kovarianzmatrizen verglichen werden. Im Allgemeinen scheint es sehr schwierig zu sein, eine solche Summe unabhängiger Vektoren aufzudecken, da einige Statistiken (zum Beispiel MLE) keine explizite Gleichung haben und möglicherweise unendlich dimensional sind. Um mit dieser Schwierigkeit fertig zu werden, verwenden wir einige neuartige Ergebnisse aus der statistischen Lerntheorie. Darüber hinaus wenden wir Bootstrap bei Methoden zur Erkennung von Änderungspunkten an. Im parametrischen Fall analysieren wir den statischen Likelihood Ratio Test (LRT). Seine hohen Werte zeigen Änderungen der Parameter Verteilung in der Datensequenz an. Das Maximum von LRT hat eine unbekannte Verteilung und kann mit Bootstrap kalibriert werden. Wir zeigen die Konvergenzraten zur realen maximalen LRT-Verteilung. In nicht parametrischen Fällen verwenden wir anstelle von LRT den Wasserstein-Abstand zwischen empirischen Messungen. Wir testen die Genauigkeit von Methoden zur Erkennung von Änderungspunkten anhand von synthetischen Zeitreihen und Elektrokardiographiedaten. Letzteres zeigt einige Vorteile des nicht parametrischen Ansatzes gegenüber komplexen Modellen und LRT. / The objective of this thesis is to explore theoretical properties of various bootstrap methods. We introduce the convergence rates of the bootstrap procedure which corresponds to the difference between real distribution of some statistic and its resampling approximation. In this work we analyze the distribution of Euclidean norm of independent vectors sum, maximum of sum in high dimension, Wasserstein distance between empirical measures, Wassestein barycenters. In order to prove bootstrap convergence we involve Gaussian approximation technique which means that one has to find a sum of independent vectors in the considered statistic such that bootstrap yields a resampling of this sum. Further this sum may be approximated by Gaussian distribution and compared with the resampling distribution as a difference between variance matrices. In general it appears to be very difficult to reveal such a sum of independent vectors because some statistics (for example, MLE) don't have an explicit equation and may be infinite-dimensional. In order to handle this difficulty we involve some novel results from statistical learning theory, which provide a finite sample quadratic approximation of the Likelihood and suitable MLE representation. In the last chapter we consider the MLE of Wasserstein barycenters model. The regularised barycenters model has bounded derivatives and satisfies the necessary conditions of quadratic approximation. Furthermore, we apply bootstrap in change point detection methods. In the parametric case we analyse the Likelihood Ratio Test (LRT) statistic. Its high values indicate changes of parametric distribution in the data sequence. The maximum of LRT has a complex distribution but its quantiles may be calibrated by means of bootstrap. We show the convergence rates of the bootstrap quantiles to the real quantiles of LRT distribution. In non-parametric case instead of LRT we use Wasserstein distance between empirical measures. We test the accuracy of change point detection methods on synthetic time series and electrocardiography (ECG) data. Experiments with ECG illustrate advantages of the non-parametric approach versus complex parametric models and LRT.

Diese Arbeit befasst sich mit einem Multiplier-Bootstrap Verfahren für die Konstruktion von Likelihood-basierten Konfidenzbereichen in zwei verschiedenen Fällen. Im ersten Fall betrachten wir das Verfahren für ein einzelnes parametrisches Modell und im zweiten Fall erweitern wir die Methode, um Konfidenzbereiche für eine ganze Familie von parametrischen Modellen simultan zu schätzen. Theoretische Resultate zeigen die Validität der Bootstrap-Prozedur für eine potenziell begrenzte Anzahl an Beobachtungen, eine große Anzahl an betrachteten parametrischen Modellen, wachsende Parameterdimensionen und eine mögliche Misspezifizierung der parametrischen Annahmen. Im Falle eines einzelnen parametrischen Modells funktioniert die Bootstrap-Approximation, wenn die dritte Potenz der Parameterdimension ist kleiner als die Anzahl an Beobachtungen. Das Hauptresultat über die Validität des Bootstrap gilt unter der sogenannten Small-Modeling-Bias Bedingung auch im Falle, dass das parametrische Modell misspezifiert ist. Wenn das wahre Modell signifikant von der betrachteten parametrischen Familie abweicht, ist das Bootstrap Verfahren weiterhin anwendbar, aber es führt zu etwas konservativeren Schätzungen: die Konfidenzbereiche werden durch den Modellfehler vergrößert. Für die Konstruktion von simultanen Konfidenzbereichen entwickeln wir ein Multiplier-Bootstrap Verfahren um die Quantile der gemeinsamen Verteilung der Likelihood-Quotienten zu schätzen und eine Multiplizitätskorrektur der Konfidenzlevels vorzunehmen. Theoretische Ergebnisse zeigen die Validität des Verfahrens; die resultierende Approximationsfehler hängt von der Anzahl an betrachteten parametrischen Modellen logarithmisch. Hier betrachten wir auch wieder den Fall, dass die parametrischen Modelle misspezifiziert sind. Wenn die Misspezifikation signifikant ist, werden Bootstrap-generierten kritischen Werte größer als die wahren Werte sein und die Bootstrap-Konfidenzmengen sind konservativ. / The thesis studies a multiplier bootstrap procedure for construction of likelihood-based confidence sets in two cases. The first one focuses on a single parametric model, while the second case extends the construction to simultaneous confidence estimation for a collection of parametric models. Theoretical results justify the validity of the bootstrap procedure for a limited sample size, a large number of considered parametric models, growing parameters’ dimensions, and possible misspecification of the parametric assumptions. In the case of one parametric model the bootstrap approximation works if the cube of the parametric dimension is smaller than the sample size. The main result about bootstrap validity continues to apply even if the underlying parametric model is misspecified under a so-called small modelling bias condition. If the true model deviates significantly from the considered parametric family, the bootstrap procedure is still applicable but it becomes conservative: the size of the constructed confidence sets is increased by the modelling bias. For the problem of construction of simultaneous confidence sets we suggest a multiplier bootstrap procedure for estimating a joint distribution of the likelihood ratio statistics, and for adjustment of the confidence level for multiplicity. Theoretical results state the bootstrap validity; a number of parametric models enters a resulting approximation error logarithmically. Here we also consider the case when parametric models are misspecified. If the misspecification is significant, then the bootstrap critical values exceed the true ones and the bootstrap confidence set becomes conservative. The theoretical approach includes non-asymptotic square-root Wilks theorem, Gaussian approximation of Euclidean norm of a sum of independent vectors, comparison and anti-concentration bounds for Euclidean norm of Gaussian vectors. Numerical experiments for misspecified regression models nicely confirm our theoretical results.

This thesis studies the design and implementation of a linear minimum mean-square error (LMMSE) receiver in asynchronous bandlimited direct-sequence code-division multiple-access (DS-CDMA) systems that employ long-code pseudo-noise (PN) sequences and operate in multipath environments. The receiver is shown to be capable of multiple-access interference (MAI) suppression and multipath diversity combining without the knowledge of other users' signature sequences. It outperforms any other linear receiver by maximizing output signal-to-noise ratio (SNR) with the aid of a new chip filter which exploits the cyclostationarity of the received signal and combines all paths of the desired user that fall within its supported time span. This work is motivated by the shortcomings of existing LMMSE receivers which are either incompatible with long-code CDMA or constrained by limitations in the system model. The design methodology is based on the concept of linear/conjugate linear (LCL) filtering and satisfying the orthogonality conditions to achieve the LMMSE filter response. Moreover, the proposed LMMSE receiver addresses two drawbacks of the coherent Rake receiver, the industry's current solution for multipath reception. First, unlike the Rake receiver which uses the chip-matched filter (CMF) and treats interference as additive white Gaussian noise (AWGN), the LMMSE receiver suppresses interference by replacing the CMF with a new chip pulse filter. Second, in contrast to the Rake receiver which only processes a subset of strongest paths of the desired user, the LMMSE receiver harnesses the energy of all paths of the desired user that fall within its time support, at no additional complexity. The performance of the proposed LMMSE receiver is analyzed and compared with that of the coherent Rake receiver with probability of bit error, <i>Pe</i>, as the figure of merit. The analysis is based on the accurate improved Gaussian approximation (IGA) technique. Closed form conditional <i>Pe</i> expressions for both the LMMSE and Rake receivers are derived. Furthermore, it is shown that if quadriphase random spreading, moderate to large spreading factors, and pulses with small excess bandwidth are used, the widely-used standard Gaussian Approximation (SGA) technique becomes accurate even for low regions of <i>Pe</i>. Under the examined scenarios tailored towards current narrowband system settings, the LMMSE receiver achieves 60% gain in capacity (1. 8 dB in output SNR) over the selective Rake receiver. A third of the gain is due to interference suppression capability of the receiver while the rest is credited to its ability to collect the energy of the desired user diversified to many paths. Future wideband systems will yield an ever larger gain. Adaptive implementations of the LMMSE receiver are proposed to rid the receiver from dependence on the knowledge of multipath parameters. The adaptive receiver is based on a fractionally-spaced equalizer (FSE) whose taps are updated by an adaptive algorithm. Training-based, pilot-channel-aided (PCA), and blind algorithms are developed to make the receiver applicable to both forward and reverse links, with or without the presence of pilot signals. The blind algorithms are modified versions of the constant modulus algorithm (CMA) which has not been previously studied for long-code CDMA systems. Extensive simulation results are presented to illustrate the convergence behavior of the proposed algorithms and quantify their performance loss under various levels of MAI. Computational complexities of the algorithms are also discussed. These three criteria (performance loss, convergence rate, and computational complexity) determine the proper choice of an adaptive algorithm with respect to the requirements of the specific application in mind.

This thesis studies the design and implementation of a linear minimum mean-square error (LMMSE) receiver in asynchronous bandlimited direct-sequence code-division multiple-access (DS-CDMA) systems that employ long-code pseudo-noise (PN) sequences and operate in multipath environments. The receiver is shown to be capable of multiple-access interference (MAI) suppression and multipath diversity combining without the knowledge of other users' signature sequences. It outperforms any other linear receiver by maximizing output signal-to-noise ratio (SNR) with the aid of a new chip filter which exploits the cyclostationarity of the received signal and combines all paths of the desired user that fall within its supported time span. This work is motivated by the shortcomings of existing LMMSE receivers which are either incompatible with long-code CDMA or constrained by limitations in the system model. The design methodology is based on the concept of linear/conjugate linear (LCL) filtering and satisfying the orthogonality conditions to achieve the LMMSE filter response. Moreover, the proposed LMMSE receiver addresses two drawbacks of the coherent Rake receiver, the industry's current solution for multipath reception. First, unlike the Rake receiver which uses the chip-matched filter (CMF) and treats interference as additive white Gaussian noise (AWGN), the LMMSE receiver suppresses interference by replacing the CMF with a new chip pulse filter. Second, in contrast to the Rake receiver which only processes a subset of strongest paths of the desired user, the LMMSE receiver harnesses the energy of all paths of the desired user that fall within its time support, at no additional complexity. The performance of the proposed LMMSE receiver is analyzed and compared with that of the coherent Rake receiver with probability of bit error, <i>Pe</i>, as the figure of merit. The analysis is based on the accurate improved Gaussian approximation (IGA) technique. Closed form conditional <i>Pe</i> expressions for both the LMMSE and Rake receivers are derived. Furthermore, it is shown that if quadriphase random spreading, moderate to large spreading factors, and pulses with small excess bandwidth are used, the widely-used standard Gaussian Approximation (SGA) technique becomes accurate even for low regions of <i>Pe</i>. Under the examined scenarios tailored towards current narrowband system settings, the LMMSE receiver achieves 60% gain in capacity (1. 8 dB in output SNR) over the selective Rake receiver. A third of the gain is due to interference suppression capability of the receiver while the rest is credited to its ability to collect the energy of the desired user diversified to many paths. Future wideband systems will yield an ever larger gain. Adaptive implementations of the LMMSE receiver are proposed to rid the receiver from dependence on the knowledge of multipath parameters. The adaptive receiver is based on a fractionally-spaced equalizer (FSE) whose taps are updated by an adaptive algorithm. Training-based, pilot-channel-aided (PCA), and blind algorithms are developed to make the receiver applicable to both forward and reverse links, with or without the presence of pilot signals. The blind algorithms are modified versions of the constant modulus algorithm (CMA) which has not been previously studied for long-code CDMA systems. Extensive simulation results are presented to illustrate the convergence behavior of the proposed algorithms and quantify their performance loss under various levels of MAI. Computational complexities of the algorithms are also discussed. These three criteria (performance loss, convergence rate, and computational complexity) determine the proper choice of an adaptive algorithm with respect to the requirements of the specific application in mind.

<p>Recently, the use of cosmic ray muons in critical national security applications, e.g., nuclear nonproliferation and safeguards verification, has gained attention due to unique muon properties such as high energy and low attenuation even in very dense materials. Applications where muon tomography has been demonstrated include cargo screening for detection of special nuclear materials smuggling, source localization, material identification, determination of nuclear fuel debris location in nuclear reactors, etc. However, muon image reconstruction techniques are still limited in resolution mostly due to multiple Coulombscattering (MCS) within the target object. Improving and expanding muon tomography would require development of efficient & flexible physics-based algorithms to model the MCS process and accurately estimate the most probable trajectory of a muon as it traverses the target object. The present study introduces a novel algorithmic approach that utilizes Bayesian probability theory and a Gaussian approximation of MCS to estimate the most probable path of cosmic ray muons as they traverse uniform media.</p> <p>Using GEANT4, an investigation was conducted involving the trajectory of 10,000 muon particles that underwent bombardment from a point source parallel to the x-axis. The proposed algorithm was assessed through four types of simulations. In the first type, muons with energies of 1 GeV, 3 GeV, 10 GeV, and 100 GeV were utilized to evaluate the algorithms’ performance and accuracy. The second type of simulation involved the use of target cubes composed of different materials, including aluminum, iron, lead, and uranium. These simulations specifically focused on muons with an energy of 3 GeV. Next, the third type of simulation entailed employing target cubes with varying lengths, such as 10 cm, 20 cm, 40 cm, and 80 cm, specifically using muons with an energy of 3 GeV and a uranium target. Lastly, all the previous simulations were revised to accommodate a source of poly-energetic muons. This revision was undertaken to create a more realistic source scenario that aligns with the distribution of muon energies encountered in real-world situations.</p> <p>The results demonstrate significant improvements in precision and muon flux utilization when comparing different algorithms. The Generalized Muon Trajectory Estimation (GMTE) algorithm shows around 50% improvement in precision compared to currently used Straight Line Path (SLP) algorithm across all test scenarios. Additionally, GMTE algorithm exhibits around 38% improvement in precision compared to the extensively used Point of Closest Approach (PoCA) algorithm. Similarly for both mono and poly energetic source of muons, the GMTE algorithm shows 10%-35% increase in muon flux utilization for high Z materials and a 10%-15% increase for medium Z materials compared to the PoCA algorithm. Similarly, it demonstrates 6%-9% increase in muon flux utilization for both medium and high Z materials compared to the SLP algorithm across all test scenarios. These results highlight the enhanced performance and efficiency of GMTE algorithm in comparison to SLP and PoCA algorithms.</p> <p>Through these extensive simulations, our objective was to comprehensively evaluate the performance and effectiveness of the proposed algorithm across a range of variables, including energy levels, materials, and target geometries. The findings of our study demonstrate that the utilization of these algorithm enables improved resolution and reduced measurement time for cosmic ray muons when compared with current SLP and PoCA algorithm.</p>