# Browsing by Subject "Uncertainty Quantification"

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Item Open Access Deep Learning Based Uncertainty Quantification for Improving Clinical Diagnostic Tools(2023) Jin, Felix QiaochuDeep learning methods have impacted a wide number of fields, and interest in its applications to clinical medicine continues to grow. Interpretable and uncertainty-aware models are critical for the adoption of artificial intelligence and machine learning in medicine, and explicit uncertainty quantification methods are used in this work to train deep neural networks that output an uncertainty value. This dissertation investigates the application of explicit uncertainty quantification with deep learning to tackle data processing problems in tympanometry, ultrasound shear wave elasticity (SWE) imaging, and ultrasound B-mode imaging.To facilitate layperson-guided tympanometry, Chapter 2 describes an uncertainty-aware hybrid deep learning model that classifies tympanograms into types A (normal), B (effusion/perforation), and C (retraction), trained using the audiologist’s interpretation as gold standard. The dataset consisted of 4810 pairs of narrow-band tympanometry tracings acquired by an audiologist and layperson in school-aged children from a trial in rural Alaska with a high prevalence of infection-related hearing loss. The model used a deep neural network (DNN) to estimate the tympanometric peak pressure, ear canal volume, and associated uncertainties, and then used a three-level decision tree based on these features to determine tympanogram classification For layperson-acquired data, the model achieved a sensitivity of 95.2% (93.3,97.1) and AUC of 0.968 (0.955,0.978). The model’s sensitivity was greater than that of the tympanometer’s built-in software [79.2% (75.5,82.8)] or a set of clinically recommended normative values [56.9% (52.4,61.3)]. For audiologist-acquired data, the model achieved a higher AUC of 0.987 (0.980,0.993) but an equivalent sensitivity of 95.2 (93.3,97.1). This chapter demonstrates that automated tympanogram classification using a hybrid deep learning classifier could facilitate layperson-guided tympanometry in hearing screening programs for children in resource-constrained communities. In ultrasound SWE imaging, a number of algorithms exist for estimating the shear wave speed (SWS) from spatiotemporal displacement data. However, no method provides a well-calibrated and practical uncertainty metric, hindering SWE’s clinical adoption and utility in downstream decision-making. In Chapter 3, a deep learning based SWS estimator is designed to simultaneously produce a quantitative and well-calibrated uncertainty value for each estimate by outputting the two parameters m and σ of a log-normal probability distribution. The working dataset consisted of in vivo 2D-SWE data of the cervix collected from 30 pregnant subjects, with 551 total acquisitions and >2 million sample points. Points were grouped by uncertainty into bins to assess uncertainty calibration: the predicted uncertainty closely matched the root-mean-square error, with an average absolute percent deviation of 3.84%. An ensemble model was created using leave-one-out training that estimated uncertainty with better calibration (1.45%) than any individual ensemble member when tested on a held-out patient’s data. The DNN was applied to an external dataset to evaluate its generalizability, and a real-time implementation was demonstrated on a clinical ultrasound scanner. The trained model, named SweiNet, is shared openly to provide the research community with a fast SWS estimator that also outputs a well-calibrated estimate of the predictive uncertainty. Chapter 4 introduces 3D rotational SWE imaging for characterizing skeletal muscle as an incompressible, transversely isotropic (ITI) material in an effort to assess muscle health and function. To facilitate ongoing research, three tools were developed. First, a Fourier-domain approach is described for calculating 3D muscle fiber orientation (MFO) from 3D B-mode volumes acquired using two imaging setups: 1) a cylindrical volume acquired by rotating a linear transducer, and 2) a rectangular volume acquired by a rectilinear matrix array transducer. Most existing approaches apply only to 2D B-mode images and detect individual fibers to extract the tilt, the angle fibers make with a horizontal plane. In a 3D B-mode volume, spherical coordinates and two angles are needed to describe orientation: the tilt and the rotation angles, where rotation is defined relative a reference vertical plane in the volume. The proposed algorithm was validated on in silico and in vivo data: errors in rotation and tilt were within 1° for both imaging setups and less than the observed in vivo MFO heterogeneity. Second, a versatile Radon-transform based SWS estimator was developed that can accept arbitrary masks to select particular regions in space-time data to isolate the two different shear wave propagation modes that are seen in ITI materials and in in vivo muscle data. Hand-drawn masks were initially used to identify these wave modes. These masks were used to train a DNN to automate mask drawing and alleviate the need for manual processing. The DNN identified 91% of the shear waves, and estimated speeds had an average difference of 7.6%. Third, the wave equation for an ITI material was derived and then solved using physics-informed neural networks (PINNs), a relatively new technique for numerically solving differential equations with advantages of being faster, compressed, analytic, and free of space/time discretization. Presently, simulations of ITI materials require time-consuming finite element modeling (FEM) or Green’s function calculations. This approach took roughly six times less time than an equivalent FEM simulation, and the PINN solution had multiple shear wave modes that matched the FEM to first-order. The PINN solution did not have reflection artifacts seen in the FEM solution. Estimated SWSs had a mean absolute difference of 4.7%. The differences in wave width and amplitude between the two suggest the need to further validate the PINN approach in comparison to FEM and Green’s function methods. In skeletal muscle, the primary SWS as a function of propagation angle forms an ellipse with the major axis oriented in the muscle fiber direction. Estimating the fiber rotation angle from a 3D B-mode volume is useful for SWE data processing, SWS estimation, and ellipse fitting. However, existing algorithms are sensitive to artifacts and can produce gross estimation errors differing ¥45° from the true fiber rotation. In Chapter 5, a DNN is designed and trained to predict fiber rotation angle via parameterizing a von Mises distribution, which provides both the estimated rotation and associated uncertainty. On simulated data with known fiber rotation, the model had an RMSE of 3.5°, and uncertainty closed matched the expected theoretical values when known amounts of fiber heterogeneity were introduced. For in vivo data of the vastus lateralis muscle, the SWS ellipse fit was used as ground truth, and DNN model RMSE was 6.9° compared to 16.9° for the existing Fourier-domain algorithm. The DNN had no estimates with an error <30°. Predicted uncertainty correlated with RMSE, but was smaller by a factor of four. This deep learning approach will provide more accurate and robust fiber rotation estimates for use in shear wave data processing and muscle characterization. In summary, this work demonstrates the effectiveness of deep learning methods for addressing specific data-processing needs of research aimed at developing new clinical applications of tympanometry, ultrasound SWE and B-mode imaging for the diagnosis and monitoring of disease. This work also demonstrates effective uncertainty quantification using the explicit estimation method, and suggests how uncertainty values may be useful for downstream decision making and data processing and potentially as a stand-alone characteristic value.

Item Open Access Model Reduction and Domain Decomposition Methods for Uncertainty Quantification(2017) Contreras, Andres AnibalThis dissertation focuses on acceleration techniques for Uncertainty Quantification (UQ). The manuscript is divided into five chapters. Chapter 1 provides an introduction and a brief summary of Chapters 2, 3, and 4. Chapter 2 introduces a model reduction strategy that is used in the context of elasticity imaging to infer the presence of an inclusion embedded in a soft matrix, mimicking tumors in soft tissues. The method relies on Polynomial Chaos (PC) expansions to build a dictionary of surrogates models, where each surrogate is constructed using a different geometrical configuration of the potential inclusion. A model selection approach is used to discriminate against the different models and eventually select the most appropriate to estimate the likelihood that an inclusion is present in the domain. In Chapter 3, we use a Domain Decomposition (DD) approach to compute the Karhunen-Loeve (KL) modes of a random process through the use of local KL expansions at the subdomain level. Furthermore, we analyze the relationship between the local random variables associated to the local KL expansions and the global random variables associated to the global KL expansions. In Chapter 4, we take advantage of these local random variables and use DD techniques to reduce the computational cost of solving a Stochastic Elliptic Equation (SEE) via a Monte Carlo sampling method. The approach takes advantage of a lower stochastic dimension at the subdomain level to construct a PC expansion of a reduced linear system that is later used to compute samples of the solution. Thus, the approach consists of two main stages: 1) a preprocessing stage in which PC expansions of a condensed problem are computed and 2) a Monte Carlo sampling stage where samples of the solution are computed in order to solve the SEE. Finally, in Chapter 5 some brief concluding remarks are provided.

Item Open Access Stochastic Modeling of Physical Parameters on Complex Domains, with Applications to 3D Printed Materials(2022) Chu, ShanshanThe proper modeling of uncertainties in constitutive models is a central concern in mechanics of materials and uncertainty quantification. Within the framework of probability theory, this entails the construction of suitable probabilistic models amenable to forward simulations and inverse identification based on limited data. The development of new manufacturing technologies, such as additive manufacturing, and the availability of data at unprecedented levels of resolution raise new challenges related to the integration of geometrical complexity and material inhomogeneity — both aspects being intertwined through processing.

In this dissertation, we address the construction, identification, and validation of stochastic models for spatially-dependent material parameters on nonregular (i.e., nonconvex) domains. We focus on metal additive additive manufacturing, with the aim of closely integrating experimental measurements obtained by collaborators, and consider the randomization of anisotropic linear elastic and plasticity constitutive models. We first present a stochastic modeling framework enabling the definition and sampling of non-Gaussian models on complex domains. The proposed methodology combines a stochastic partial differential approach, which is used to account for geometrical features on the fly, with an information-theoretic construction, which ensures well-posedness in the associated stochastic boundary value problems through the derivation of ad hoc transport maps.

We then present three case studies where the framework is deployed to model uncertainties in location-dependent anisotropic elasticity tensors and reduced Hill’s plasticity coefficients (for 3D printed stainless steel 316L). Experimental observations at various scales are integrated for calibration (either through direct estimators or by solving statistical inverse problems by means of the maximum likelihood method) and validation (whenever possible), including structural responses and multiscale predictions based on microstructure samples. The role of material symmetries is specifically investigated, and it is shown that preserving symmetries is, indeed, key to appropriately capturing statistical fluctuations. Results pertaining to the correlation structure indicate strong anisotropy for both types of behaviors, in accordance with fine-scale observations.

Item Open Access Uncertainty Quantification in Earth System Models Using Polynomial Chaos Expansions(2017) Li, GuotuThis work explores stochastic responses of various earth system models to different random sources, using polynomial chaos (PC) approaches. The following earth systems are considered, namely the HYbrid Coordinate Ocean Model (HYCOM, an ocean general circulation model (OGCM)) for the study of ocean circulation in the Gulf of Mexico (GoM); the Unified Wave INterface - Coupled Model (UWIN-CM, a dynamically coupled atmosphere-wave-ocean system) for Hurricane Earl (2010) modeling; and the earthquake seismology model for Bayesian inference of fault plane configurations.

In the OGCM study, we aim at analyzing the combined impact of uncertainties in initial conditions and wind forcing fields on ocean circulation using PC expansions. Empirical Orthogonal Functions (EOF) are used to represent both spatial perturbations of initial condition and space-time wind forcing fields, namely in the form of a superposition of modal components with uniformly distributed random amplitudes. The forward deterministic HYCOM simulations are used to propagate input uncertainties in ocean circulation in the GoM during the 2010 Deepwater Horizon (DWH) oil spill, and to generate a realization ensemble based on which PC surrogate models are constructed for both localized and field quantities of interest (QoIs), focusing specifically on Sea Surface Height (SSH) and Mixed Layer Depth (MLD). These PC surrogate models are constructed using Basis Pursuit DeNoising (BPDN) methodology, and their performance is assessed through various statistical measures. A global sensitivity analysis is then performed to quantify the impact of individual random sources as well as their interactions on ocean circulation. At the basin scale, SSH in the deep GoM is mostly sensitive to initial condition perturbations, while over the shelf it is sensitive to wind forcing perturbations. On the other hand, the basin MLD is almost exclusively sensitive to wind perturbations. For both quantities, the two random sources (initial condition and wind forcing) of uncertainties have limited interactions. Finally, computations indicate that whereas local quantities can exhibit complex behavior that necessitates a large number of realizations to build PC surrogate models, the modal analysis of field sensitivities can be suitably achieved with a moderate size ensemble.

It is noted that HYCOM simulations in the aforementioned OGCM study only focus on the ocean circulation, and ignore the oceanic feedback (e.g. momentum, energy, humidity etc) to the atmosphere. A more elaborated analysis is consequently performed to understand the atmosphere dynamics in a fully-coupled atmosphere-wave-ocean system. In particular, we explore the stochastic evolution of Hurricane Earl (2010) in response to uncertainties stemming from random perturbations in the storm's initial size, strength and rotational stretch. To this end, the UWIN-CM framework is employed as the forecasting system, which is used to propagate input uncertainties and generate a realization ensemble. PC surrogate models for time evolutions of both maximum wind speed and minimum sea level pressure (SLP) are constructed. These PC surrogates provide statistical insights on probability distributions of model responses throughout the simulation time span. Statistical analysis of rapid intensification (RI) process suggests that storms with enhanced initial intensity and counter-clockwise rotation perturbations are more likely to undergo a RI process. In addition, the RI process seems mostly sensitive to the mean wind strength and rotational stretch, rather than storm size and asymmetric wind amplitude perturbations. This is consistent with global sensitivity analysis of PC surrogate models. Finally we combine parametric storm perturbations with global stochastic kinetic energy backscatter (SKEBS) forcing in UWIN-CM simulations, and conclude that whereas the storm track is substantially influenced by global perturbations, it is weakly influenced by the properties of the initial storm.

The PC framework not only provides easy access to traditional statistical insights and global sensitivity indices, but also reduces the computational burden of sampling the system response, as performed for instance in Bayesian inference. These two advantages of PC approaches are well demonstrated in the study of earthquake seismology model response to random fault plane configurations. The PC statistical analysis suggests that the hypocenter location plays a dominant role in earthquake ground motion responses (in terms of peak ground velocities, PGVs), while elliptical patch properties only show secondary influence. In addition, Our PC based Bayesian analysis successfully identified the most `likely' fault plane configuration with respect to the chosen ground motion prediction equation (GMPE) curve, i.e. the hypocenter is more likely to be in the bottom right quadrant of the fault plane and the elliptical patch centers at the bottom left quadrant. To incorporate additional physical restrictions on fault plane configurations, a novel restricted sampling methodology is introduced. The results indicate that the restricted inference is more physically sensible, while retaining plausible consistency with findings from unrestricted inference.