Hierarchical nonlinear, multivariate, and spatially-dependent time-frequency functional methods

Notions of time and frequency are important constituents of most scientific inquiries, providing complimentary information. In an era of "big data," methodology for analyzing functional and/or image data is increasingly important. This dissertation develops methodology at the cross-section of time-frequency analysis and functional data and consists of three distinct, but related, contributions. First, we propose nonparametric methodology for nonlinear multivariate time-frequency functional data. In particular, we consider polynomial nonlinear functional data models that accommodate higher dimensional functional covariates, including time-frequency images, along with their interactions. The necessary dimension reduction for model estimation proceeds through carefully chosen basis expansions (empirical orthogonal functions) and feature-extraction stochastic search variable selection (SSVS). Properties of the methodology are examined through an extensive simulation study. Finally, we illustrate the approach through an application that attempts to characterize spawning behavior of shovelnose sturgeon in terms of high-density depth and temperature profiles. The second contribution proposes model-based time-frequency estimation through Bayesian lattice filter time-varying autoregressive models. In this context, we take a fully Bayesian approach and allow both the autoregressive coefficients and innovation variance to vary over time. Importantly, our model is estimated within the partial autocorrelation domain (i.e., through the partial autocorrelation coefficients). Additionally, all of the full conditional distributions required for our algorithm are of standard form and thus can be easily implemented using a Gibbs sampler. Further, as a by-product of the lattice filter recursions, our approach avoids undesirable matrix inversions. As such, estimation is computationally efficient and stable. We conduct a comprehensive simulation study that compares our method with other competing methods and find that, in most cases, our approach performs superior in terms of average squared error between the estimated and true time-varying spectral density. Lastly, we demonstrate our methodology through several real case studies. The final project of the dissertation develops models that accommodate spatially dependent functional responses with spatially dependent image predictors. The methodology is motivated by a soil science study that seeks to model spatially correlated water content functionals as a function of electro-conductivity images. The water content curves are measured at different locations within the study field and at various depths, whereas the electro-conductivity images are spatially referenced images of wavelength by depth. Estimation is facilitated by taking a Bayesian approach, where the necessary dimension reduction for model implementation proceeds using basis function expansions along with SSVS. Finally, the methodology is illustrated through an application to our motivating data.Includes bibliographical references (pages 122-133

Doctor of Philosophy

dissertationFunctional magnetic resonance imaging (fMRI) measures the change of oxygen consumption level in the blood vessels of the human brain, hence indirectly detecting the neuronal activity. Resting-state fMRI (rs-fMRI) is used to identify the intrinsic functional patterns of the brain when there is no external stimulus. Accurate estimation of intrinsic activity is important for understanding the functional organization and dynamics of the brain, as well as differences in the functional networks of patients with mental disorders. This dissertation aims to robustly estimate the functional connectivities and networks of the human brain using rs-fMRI data of multiple subjects. We use Markov random field (MRF), an undirected graphical model to represent the statistical dependency among the functional network variables. Graphical models describe multivariate probability distributions that can be factorized and represented by a graph. By defining the nodes and the edges along with their weights according to our assumptions, we build soft constraints into the graph structure as prior information. We explore various approximate optimization methods including variational Bayesian, graph cuts, and Markov chain Monte Carlo sampling (MCMC). We develop the random field models to solve three related problems. In the first problem, the goal is to detect the pairwise connectivity between gray matter voxels in a rs-fMRI dataset of the single subject. We define a six-dimensional graph to represent our prior information that two voxels are more likely to be connected if their spatial neighbors are connected. The posterior mean of the connectivity variables are estimated by variational inference, also known as mean field theory in statistical physics. The proposed method proves to outperform the standard spatial smoothing and is able to detect finer patterns of brain activity. Our second work aims to identify multiple functional systems. We define a Potts model, a special case of MRF, on the network label variables, and define von Mises-Fisher distribution on the normalized fMRI signal. The inference is significantly more difficult than the binary classification in the previous problem. We use MCMC to draw samples from the posterior distribution of network labels. In the third application, we extend the graphical model to the multiple subject scenario. By building a graph including the network labels of both a group map and the subject label maps, we define a hierarchical model that has richer structure than the flat single-subject model, and captures the shared patterns as well as the variation among the subjects. All three solutions are data-driven Bayesian methods, which estimate model parameters from the data. The experiments show that by the regularization of MRF, the functional network maps we estimate are more accurate and more consistent across multiple sessions

Uncertainty Quantification

Uncertainty quantification (UQ) is concerned with including and characterising uncertainties in mathematical models. Major steps comprise proper description of system uncertainties, analysis and efficient quantification of uncertainties in predictions and design problems, and statistical inference on uncertain parameters starting from available measurements. Research in UQ addresses fundamental mathematical and statistical challenges, but has also wide applicability in areas such as engineering, environmental, physical and biological applications. This workshop focussed on mathematical challenges at the interface of applied mathematics, probability and statistics, numerical analysis, scientific computing and application domains. The workshop served to bring together experts from those disciplines in order to enhance their interaction, to exchange ideas and to develop new, powerful methods for UQ

An Adaptive and a Multilevel Adaptive Sparse Grid approach to address global uncertainty and sensitivity

In the field of heterogeneous catalysis, first- principle-based microkinetic modeling has been proven to be an essential tool to provide a deeper understanding of the microscopic interplay between reactions. It avoids the bias of being fitted to experimental data, which allows us to extract information about the materials’ properties that cannot be drawn from experimental data. Unfortunately, the catalytic models draw information from electronic structure theory (e.g. Density Functional Theory) which contains a sizable error due to intrinsic approximations to make the computational costs feasible. Although the errors are commonly accepted and known, this work will analyse how significant the impact of these errors can be on the model outcome. We first explain how these errors are propagated into a model outcome, e.g., turnover-frequency (TOF), and how significant the outcome is impacted. Secondly, we quantify the propagation of single errors by a local and global sensitivity analysis, including a discussion of their dis-/advantages for a catalytic model. The global approach requires the numerical quadrature of high dimensional integrals as the catalytic model often depends on multiple parameters. This, we tackle with a local and dimension-adaptive sg! (sg!) approach. sg!s have shown to be very useful for medium dimensional problems since their adaptivity feature allows for an accurate surrogate model with a modest number of points. Despite the models’ high dimensionality, the outcome is mostly dominated by a fraction of the input parameter, which implies a high refinement in only a fraction of the dimensions (dimension-adaptive). Additionally, the kinetic data shows characteristics of sharp transitions between "non-active" and "active" areas, which need a higher order of refinement (local-adaptive). The efficiency of the adaptive sg! is tested on different toy models and a realistic first principle model, including the Sensitivity Analysis. Results show that for catalytic models, a local derivative-based sensitivity analysis gives only limited information. However, the global approach can identify the important parameters and allows extracting information from more complex models in more detail. The Sparse Grid approach is useful for reducing the total number of points, but what if evaluating the point itself is very expensive? The second part of this work concentrates on solving high dimensional integrals for models whose evaluations are costly due to, e.g. being only implicitly given by a Monte Carlo model. The evaluation contains an error due to finite sampling. To lower the error, we would have to increase computational effort for a high number of samples. To tackle this problem, we extend the SG method with a multilevel approach to lower the cost. Unlike existing approaches, we will not use the telescoping sum but utilise the sparse grid’s intrinsically given hierarchical structure. We assume that not all the SG points need the same accuracy but that we can double the points’ variance and halve the drawn samples with every refinement step. We demonstrate the methodology on different toy models and a realistic kinetic Monte Carlo system for CO oxidation. Therefore, we compare the non- multilevel adaptive Sparse Grid (ASG) with the Multilevel Adaptive Sparse Grid (MLASG). Results show that ith the multilevel extension we can save up to two orders of magnitude without challenging the accuracy of the surrogate model compared to a non-mulitlevel SG.Auf dem Gebiet der heterogenen Katalyse hat sich die First-Principle-basierte mikrokinetische Model- lierung als wesentliches Werkzeug bewährt, um ein tieferes Verständnis der mikroskopischen Wech- selwirkung zwischen Reaktionen zu ermöglichen. Leider basieren die katalytischen Modelle auf Informationen aus der elektronischen Strukturtheorie (z. B. Dichtefunktionaltheorie), die aufgrund in- trinsischer Näherungen einen beträchtlichen Fehler enthalten. In dieser Arbeit werden wir analysieren wie signifikant die Auswirkungen dieser Fehler auf das Modellergebnis sein können. Dazu erklären wir zunächst, wie diese Fehler in ein Modellergebnis, wie z. B. Turnover-Frequency (TOF), übertragen werden. Des Weiteren quantifizieren wir die Auswirkung einzelner Fehler mittels einer lokalen und globalen Sensitivitätsanalyse und erklären die Unterschiede beider Methoden. Der globale Sensitivitätsansatz erfordert das Lösen hochdimensionaler Integrale bzw. ein akkurates Ersatzmodel zum Auswerten, wofür wir einen lokalen und dimensions-adaptiven Sparse Grid-Ansatz benutzen. Sparse Grids (SG) haben sich für mitteldimensionale Probleme als sehr nützlich erwiesen, da ihre Adaptivitätsfunktion ein genaues Ersatzmodell mit einer kleinen Anzahl von Punkten er- möglicht. Trotz der hohen Dimensionalität der Modelle wird das Ergebnis meist von einem Bruchteil der Modellparameter dominiert, was eine hohe Verfeinerung in nur einem Bruchteil der Dimensionen erfordert (dimensionsadaptiv). Darüber hinaus zeigen die kinetischen Daten Charakteristiken scharfer Übergänge zwischen "nicht aktiven" und "aktiven" Bereichen, die eine höhere Verfeinerung (lokal- adaptiv) erfordern. Die Effizienz des adaptiven SG wird an verschiedenen Testmodellen und einem realistischen First-Principle-Modell, einschließlich der Sensitivitätsanalyse, getestet. Die Ergebnisse zeigen, dass für katalytische Modelle eine lokale Sensitivitätsanalyse auf Basis lokaler Ableitungen nur begrenzte Informationen liefert. Dagegen kann der globale Ansatz die wichtigen Parameter identi- fizieren und ermöglicht es, Informationen aus komplexeren Modellen detaillierter zu extrahieren. Der Sparse Grid-Ansatz reduziert die Gesamtzahl an Punkten, aber was ist, wenn die Auswertung eines Punktes schon sehr teuer ist? Deswegen konzentriert sich der zweite Teil dieser Arbeit auf die Lösung hochdimensionaler Integrale für Modelle, deren Auswertungen nur implizit, z.B. durch ein Monte-Carlo-Modell, gegeben ist. Wir erweitern die SG-Methode um einen mehrstufigen Ansatz, der die Kosten senken soll. Im Gegensatz zu bestehenden Ansätzen werden wir nicht die Teleskop- summe verwenden, sondern die intrinsisch gegebene hierarchische Struktur des SG ausnutzen. Jede Funktionsauswertung enthält einen Fehler, aufgrund einer begrenzten Probenmenge, aber nicht alle SG-Punkte benötigen die gleiche Genauigkeit. Deswegen können wir bei jedem Verfeinerungsschritt die Varianz der Punkte verdoppeln und somit die Menge der gezogenen Stichproben halbieren und die Kosten minimieren. Wir demonstrieren die Methodik an verschiedenen Testmodellen und einem realistischen kinetischen Monte-Carlo-Modell. Dabei vergleichen wir den reinen adaptiven Sparse Grid (ASG) Ansatz mit dem Multilevel Adaptive Sparse Grid (MLASG). Die Ergebnisse zeigen, dass wir mit der mehrstufigen Erweiterung im Vergleich zur ASG, bis zu zwei Größenordnungen an CPU (Central Processing Unit)- Zeit einsparen können, ohne die Genauigkeit des Ersatzmodells zu beeinflussen

Coupled Sequential Process-Performance Simulation and Multi-Attribute Optimization of Structural Components Considering Manufacturing Effects

Coupling of material, process, and performance models is an important step towards a fully integrated material-process-performance design of structural components. In this research, alternative approaches for introducing the effects of manufacturing and material microstructure in plasticity constitutive models are studied, and a cyberinfrastructure framework is developed for coupled process-performance simulation and optimization of energy absorbing components made of magnesium alloys. The resulting mixed boundary/initial value problem is solved using nonlinear finite element analysis whereas the optimization problem is decomposed into a hierarchical multilevel system and solved using the analytical target cascading methodology. The developed framework is demonstrated on process-performance optimization of a sheetormed, energy-absorbing component using both classical and microstructure-based plasticity models. Sheetorming responses such as springback, thinning, and rupture are modeled and used as manufacturing process attributes whereas weight, mean crush force, and maximum crush force are used as performance attributes. The simulation and optimization results show that the manufacturing effects can have a considerable impact on design of energy absorbing components as well as the optimum values of process and product design variables