Uncertainty Quantification Research Papers

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- by Keith Hayes
- •
- 6
  Computer Science, Sensitivity Analysis, Uncertainty Quantification, CSIRO

For very long the deterministic view of Nature prevailed in Physics. Starting from the Illuminism there was the believe that Mechanics was completely described by Newton laws. If the initial conditions of a mechanical systems were... more

For very long the deterministic view of Nature prevailed in Physics. Starting from the Illuminism there was the believe that Mechanics was completely described by Newton laws. If the initial conditions of a mechanical systems were known, then its state could be completely determined in the future as well as in the past.This was precisely the viewpoint of Lagrange and it ruled from the 18th century to around the middle of the 20th century.

However this panglossian view of Nature was disputed, especially with the advent of Thermodynamics that challenged the believe in reversibility and started the study of complex random systems. Also the discovery of simple systems with chaotic behaviour, that despite being deterministic, showed a very complex behaviour, showed that mechanical problems are far from simple. The Quantum Theory, on the other hand, showed that the processes in Nature are not deterministic, but stochastic. This change of paradigm is not yet accepted as is well described in the book of Thomas Kuhn, that discusses sociological aspects of sciences.

The main ideas of probability were slow to develop. Perhaps because chance events were interpreted as the wish of the gods and hence were not believed to be random. The first theoretical works on probability were connected with games of chance and since the set of possibilities are finite the studies are combinatorial, but the big breakthrough is when the continuous case is studied.

Stochastic modelling is, of course, harder than deterministic modelling and the implementation of the model is more costly. Let us see this in a simple example. In the simplest deterministic continuous model the parameters of the model are constants. The equivalent of this in the stochastic case, is to transform the constants in random variables. For each realisation of the random variable its value is also a constant, but the constant may change with the realisation following a certain distribution of probability. That is, the most simple object of a stochastic model, the random variables, are functions, hence objects of infinite dimension. Also, in a coherent deterministic model one expects a solution if one fixes suitable conditions. In a stochastic model this is not the case. For each realisation of the stochastic data the parameters are chosen and a deterministic problem is solved. Then, with the set of solutions obtained, statistics are made and the main result of the problem
is the distribution of probability of the results. The important is to know among the several possible solutions how they are distributed. The values obtained from a single realisation are of no importance, the distribution of the values is the important result. The understanding of this fact was very slow to come for the manufacturing industry, for example, and only after long year of application of quality control in the manufacturing came the realisation that they dealt with a process stochastic in nature. Since them stochastic modelling is essential in Engineering. Today Reliability represents a new way of design and it takes into account the inevitable uncertainties of the processes.

The solution of a stochastic model can be decomposed in three steps: generation of the stochastic data following their distribution of probability, solution of a deterministic problem for each sample generated, and finally computation of statistics with the results (for example, to construct a histogram) until they show a certain persistence (do not change accordingly an error criterium). Histograms represent the approximation of the solution of a stochastic problem. When one wants an approximation of the solution of an stochastic problem, histogram is the best approximation. If they are hard to find, or costly to compute, one make do with mean and dispersion, or a certain number of moments. In some situation, as the ones described in Chapter 8, one computes only a certain probability since moments are too hard to compute.

This book intents to show the main ideas of Stochastic Modelling and Uncertainty Quantification and the main tool used is Functional Analysis, more specifically, Hilbert Spaces. With this tool very complex ideas, as Conditional Probabilities, can be developed in a systematic way.

In Chapter 1 are discussed the main ideas of Probability Theory in a hilbertian context. This chapter is designed as a reference, but the main concepts of random variables and random processes are developed. Chapter 2 discusses how to construct a stochastic model using the Maximum Entropy Principal. When the data is only independent random variables the generation is simple. But in the case one has dependent random variables or a random process the generation is harder and one needs more tools. The main ideas of generation is discussed here. Chapter 3 discusses the problem of approximation of a random variable. The following chapters deals with applications. Chapter 4 discusses linear systems with uncertainties; chapter 5 discusses nonlinear systems with uncertainties; chapter 6 deals with differential equations and the next two chapters, 7 and 8, deals with optimisation with uncertainties.

Description
Uncertainty Quantification (UQ) is a relatively new research area which describes the methods and approaches used to supply quantitative descriptions of the effects of uncertainty, variability and errors in simulation problems and models. It is rapidly becoming a field of increasing importance, with many real-world applications within statistics, mathematics, probability and engineering, but also within the natural sciences. Literature on the topic has up until now been largely based on polynomial chaos, which raises difficulties when considering different types of approximation and does not lead to a unified presentation of the methods. Moreover, this description does not consider either deterministic problems or infinite dimensional ones.
This book gives a unified, practical and comprehensive presentation of the main techniques used for the characterization of the effect of uncertainty on numerical models and on their exploitation in numerical problems. In particular, applications to linear and nonlinear systems of equations, differential equations, optimization and reliability are presented. Applications of stochastic methods to deal with deterministic numerical problems are also discussed.
Matlab® illustrates the implementation of these methods and makes the book suitable as a textbook and for self-study.
Contents
1. Elements of Probability Theory and Stochastic Processes. 2. Maximum Entropy and Information.
3. Representation of Random Variables.
4. Linear Algebraic Equations Under Uncertainty.
5. Nonlinear Algebraic Equations Involving Random Parameters. 6. Differential Equations Under Uncertainty.
7. Optimization Under Uncertainty.
8. Reliability-Based Optimization

ISBN: 978178540058

Through our project, we want to investigate the effect of policy uncertainty on the unemployment level and we expect to find a positive causal effect. An interesting debate has risen recently since many scholars argued that the... more

This paper deals with the applications of stochastic spectral methods for structural topology optimization in the presence of uncertainties. A non-intrusive polynomial chaos expansion is integrated into a topology optimization algorithm... more

ABSTRACT: The present paper examines how statistics of extremes can be used to enhance the assessment and performance prediction of monitored highway bridges. This is achieved by proposing an approach to obtain a monitoring-based... more

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- by William Oberkampf and +1
  Christopher Roy
- •
- 4
  Engineering, Uncertainty Quantification, Scientific Computing, Mathematical Sciences

A stochastic model of the dynamic behavior of sawn timber beams of Ar- gentinean Eucalyptus grandis is herein presented. The aim of this work is to study the influence of the timber knots in the dynamical response of timber beams. The... more

This chapter is devoted to introducing the theories of interval algebra to people who are interested in applying the interval methods to uncertainty analysis in science and engineering. In view of this purpose, we shall introduce the key... more

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- by Hend Dawood
- •
- 82
  Mathematics, Applied Mathematics, Ring Theory, Field Theory

Quantitative fire risk analysis aims at providing an assessment of fire safety on a scientific basis and taking relevant uncertainties into account in a rational quantitative manner. Under a probabilistic approach, performance measures... more

The present study was undertaken to develop and validate a simple, sensitive, accurate, precise and reproducible UV spectrophotometric method for cefuroxime axetil using methanol as solvent. In this method the simple UV spectrum of... more

ABSTRACT The Romanesque “tribune” of the abbey church of Cuxa was probably dismantled in the sixteenth century. No description or representation of this work of art, at the time of its existence, is to be found. It is known today... more

This half-day tutorial on Belief function (random sets) for the working scientist was presented on July 9th 2016 at the latest International Joint Conference on Artificial Intelligence (IJCAI-16). The tutorial is very comprehensive (468... more

Is there room enough in all creation for another 'Empty Universe Theory'? How should we view the realm in which we exist? Are the natures of matter and energy, their compositions and relationships with each other the fundamental key to... more

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- by Louis C Taylor
- •
- 19
  Mathematics, Physics, Metaphysics, Metaphilosophy

The principal aim of this book is to introduce to the widest possible audience an original view of belief calculus and uncertainty theory. In this geometric approach to uncertainty, uncertainty measures can be seen as points of a suitably... more

The principal aim of this book is to introduce to the widest possible audience an original view of belief calculus and uncertainty theory. In this geometric approach to uncertainty, uncertainty measures can be seen as points of a suitably complex geometric space, and manipulated in that space, for example, combined or conditioned.

In the chapters in Part I, Theories of Uncertainty, the author offers an extensive recapitulation of the state of the art in the mathematics of uncertainty. This part of the book contains the most comprehensive summary to date of the whole of belief theory, with Chap. 4 outlining for the first time, and in a logical order, all the steps of the reasoning chain associated with modelling uncertainty using belief functions, in an attempt to provide a self-contained manual for the working scientist. In addition, the book proposes in Chap. 5 what is possibly the most detailed compendium available of all theories of uncertainty. Part II, The Geometry of Uncertainty, is the core of this book, as it introduces the author’s own geometric approach to uncertainty theory, starting with the geometry of belief functions: Chap. 7 studies the geometry of the space of belief functions, or belief space, both in terms of a simplex and in terms of its recursive bundle structure; Chap. 8 extends the analysis to Dempster’s rule of combination, introducing the notion of a conditional subspace and outlining a simple geometric construction for Dempster’s sum; Chap. 9 delves into the combinatorial properties of plausibility and commonality functions, as equivalent representations of the evidence carried by a belief function; then Chap. 10 starts extending the applicability of the geometric approach to other uncertainty measures, focusing in particular on possibility measures (consonant belief functions) and the related notion of a consistent belief function. The chapters in Part III, Geometric Interplays, are concerned with the interplay of uncertainty measures of different kinds, and the geometry of their relationship, with a particular focus on the approximation problem. Part IV, Geometric Reasoning, examines the application of the geometric approach to the various elements of the reasoning chain illustrated in Chap. 4, in particular conditioning and decision making. Part V concludes the book by outlining a future, complete statistical theory of random sets, future extensions of the geometric approach, and identifying high-impact applications to climate change, machine learning and artificial intelligence.

The book is suitable for researchers in artificial intelligence, statistics, and applied science engaged with theories of uncertainty. The book is supported with the most comprehensive bibliography on belief and uncertainty theory.

After over 50 years of spaceflight, the calculation of the necessary margin on the thermal protection system of an entering spacecraft remains a largely ad-hoc (in a tactical sense) process governed by engineering judgment. Over the past... more

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- by Matteo Diez and +2
  Daniele Peri
  Yusuke Tahara
- •
- 10
  Engineering, Ocean Engineering, Naval Engineering, Naval Architecture

Landslide inventory maps (LIMs) show where landslides have occurred in an area, and provide information useful to different types of landslide studies, including susceptibility and hazard modelling and validation, risk assessment ,... more

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- by Michele Santangelo and +1
  Fausto Guzzetti
- •
- 7
  Landslides, Uncertainty Quantification, Aerial Photographic Interpretation, Mapping

With the extensive use of rating systems in the web, and their significance in decision making process by users, the need for more accurate aggregation methods has emerged. The Naïve aggregation method, using the simple mean, is not... more

In this chapter the basic principles of two methodologies for uncertainty quantification (UQ) are discussed, namely the polynomial chaos method and the collocation method. UQ deals with the propagation of uncertainties through complex... more

The single most important statement that can be made with regard to the logical status of human and physical geographical reasoning is that it belongs to the class of non-monotonic reasoning. In other words, geographical reasoning is... more

Uncertainty quantification was conducted using non – intrusive form of Generalised Polynomial Chaos methods for square inline tube bundles. The input was treated as uncertain. Uncertainties were considered in Reynolds number of the flow... more

Stochastic spectral methods are efficient techniques for uncertainty quantification. Recently they have shown excellent performance in the statistical analysis of integrated circuits. In stochastic spectral methods, one needs to determine... more

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- by Ian G Main
- •
- 17
  Seismic Hazard, Forecasting, Seismology, Multidisciplinary

We begin by constructing the algebra of classical intervals and prove that it is a nondistributive abelian semiring. Next, we formalize the notion of interval dependency, along with discussing the algebras of two alternate theories of... more

The validation of a high-fidelity uncertainty quantification of a high-speed catamaran is presented, with focus on irregular wave analysis and approximation methods used in design optimization studies, namely a deterministic regular wave... more

The validation of a high-fidelity uncertainty quantification of a high-speed catamaran is presented, with focus on irregular wave analysis and approximation methods used in design optimization studies, namely a deterministic regular wave approximation, and a stochastic regular wave UQ model. The approach includes a priori CFD simulations, followed by the ex post facto EFD campaign. The validation variables are the wave elevation, x-force, heave and pitch motions, vertical acceleration of the bridge and vertical velocity of the flight deck. Time series values are addressed as primary variables, whereas mean-crossing amplitudes, height, and period are indicated as secondary variables. The subseries and bootstrap methods are applied in order to estimate the validation values and 95% confidence intervals for expected value (EV), standard deviation (SD), mode, and quantile function. Additionally, validation values and confidence intervals for time series EV and SD are evaluated by time series theory, based on the sample variance and size, and the autocovariance function. The deterministic regular wave model assesses the EV of the x-force, whereas the stochastic regular wave UQ focuses on SSAs of pitch, acceleration, and velocity, as relevant merit factors for design optimization. On average, all variables are validated, but the validation uncertainty is quite large, due to limited number of inlet wave components of CFD and short run length for both CFD and EFD. For the evaluation of confidence intervals, the subseries method is found on average more conservative for EV and less conservative for SD, compared to time series theory. Primary variables are validated with an average 5% error and 37% confidence interval. Secondary variables are validated with an average 21% error and 38% confidence interval. The regular wave model is validated with nearly 9% error and 15% confidence interval, whereas the UQ model is validated with an average 1% error and 30% confidence interval.

This study is motivated by the sharp contrast between physical and probabilistic models of civil engineering. The current practice focuses on physical models while probabilistic ones are relatively underdeveloped. This unbalance can even... more

This study is motivated by the sharp contrast between physical and probabilistic models of civil engineering. The current practice focuses on physical models while probabilistic ones are relatively underdeveloped. This unbalance can even hinder advances in physical models. Moreover, this rarely acknowledged asymmetry creates the illusion that our deterministic models accurately capture all or at least the main aspects of reality. Thus, this thesis aspires to subtly adjust this imbalance by focusing on probabilistic models.

The main contribution is that it explores often neglected or oversimplified aspects of probabilistic analysis in civil engineering. These distinct but related issues are: (i ) selection of an appropriate distribution type; (ii ) effect of statistical uncertainty; (iii ) measurement uncertainty; (iv) long-term trends; and (v) dependence structure. These are demonstrated through analyzing extreme ground snow loads, although they are inevitably present for most random variables. Snow action, which has recently led to numerous structural failures in Central Europe, is treated as a vehicle of illustration to give a sharp focus to the study.

Methods developed in mathematical statistics, probability theory, information theory, and structural reliability are applied to tackle these issues. Fully statistical analysis of snow extremes is undertaken in conjunction with structural reliability analysis. The popular civil engineering approaches are compared with more advanced techniques that are able to capture the neglected effects in the former approaches. Snow water equivalent data of the Carpathian Region from more than 600 meteorological stations are analyzed, thus the results are representative for lowlands, for highlands, and for mountains as well. Furthermore, extensive parametric analyses are performed to extend the investigations to random variables other than ground snow intensity.

Based on the analysis of the above-listed issues the following main conclusions are drawn:

(i) It is demonstrated that mountains and highlands are better represented by Weibull, while lowlands by Fréchet distribution. In comparison, the currently standardized Gumbel model recommended in Eurocode often appreciably underestimates the snow maxima of lowlands and thus leads to overestimation of structural reliability.

(ii) It is shown that current practice, which neglects statistical uncertainties (parameter estimation and model selection uncertainties), can yield to even multiple orders of magnitude underestimation of failure probability. Bayesian posterior predictive distribution and Bayesian model averaging is proposed to account for parameter estimation and model selection uncertainties, respectively.

(iii) Statistical and interval based approaches are used to explore the effect of prevalently neglected measurement uncertainty on structural reliability. It is demonstrated that such simplification can lead to an order of magnitude underestimation of failure probability. If sufficient data are available to infer the probabilistic model of measurement uncertainty, then the statistical approach is recommended. Otherwise, the interval approach is advocated.

(iv) Using non-stationary extreme value analysis, statistically significant decreasing trend is found in annual ground snow extremes for most of the Carpathian Region. For some locations the effect of the trend on structural reliability is practically significant. This change is favorable from safety point of view as it increases reliability. Hence, revision of current regulations due to long-term trends is not needed from safety reasons, though it might be desirable from economic considerations. However, record lengths are insufficient to draw strong conclusions and to include trends in predicting extreme values with return periods of hundreds of years.

(v) Study of the widely used Gauss (normal, or Gaussian) copula assumption of time-continuous stochastic processes is performed. It reveals that Gauss copula can four times underestimate or even ten times overestimate time-variant failure probabilities obtained by other adopted copulas (t, Gumbel, rotated Gumbel, and rotated Clayton). Model averaging is proposed as a viable approach to rigorously account for copula function uncertainty.

Based on these findings, practical recommendations are made for normal and safety critical structures. The effects and proposed approaches are illustrated through real life examples. Reliability of the more than 130 years old wrought iron structure of Eiffel-hall and a steel hall of Paks Nuclear Power Plant is analyzed. The tackled challenges are general and relevant for most random variables such as wind, traffic, and earthquake actions. Therefore, the findings can be applied for these as well and they can help to draft more consistent standards and to build safer structures.

A B S T R A C T Stochastic optimization methods, such as genetic algorithms, search for the global minimum of the misfit function within a given parameter range and do not require any calculation of the gradients of the misfit surfaces.... more

A B S T R A C T Stochastic optimization methods, such as genetic algorithms, search for the global minimum of the misfit function within a given parameter range and do not require any calculation of the gradients of the misfit surfaces. More importantly, these methods collect a series of models and associated likelihoods that can be used to estimate the posterior probability distribution. However, because genetic algorithms are not a Markov chain Monte Carlo method, the direct use of the genetic-algorithm-sampled models and their associated likelihoods produce a biased estimation of the posterior probability distribution. In contrast, Markov chain Monte Carlo methods, such as the Metropolis–Hastings and Gibbs sampler, provide accurate posterior probability distributions but at considerable computational cost. In this paper, we use a hybrid method that combines the speed of a genetic algorithm to find an optimal solution and the accuracy of a Gibbs sampler to obtain a reliable estimation of the posterior probability distributions. First, we test this method on an analytical function and show that the genetic algorithm method cannot recover the true probability distributions and that it tends to underestimate the true uncertainties. Conversely, combining the genetic algorithm optimization with a Gibbs sampler step enables us to recover the true posterior probability distributions. Then, we demonstrate the applicability of this hybrid method by performing one-dimensional elastic full-waveform inversions on synthetic and field data. We also discuss how an appropriate genetic algorithm implementation is essential to attenuate the " genetic drift " effect and to maximize the exploration of the model space. In fact, a wide and efficient exploration of the model space is important not only to avoid entrapment in local minima during the genetic algorithm optimization but also to ensure a reliable estimation of the posterior probability distributions in the subsequent Gibbs sampler step.

Previous studies of auditory expectation have focused on the expectedness perceived by listeners retrospectively in response to events. In contrast, this research examines predictive uncertainty—a property of listeners' prospective state... more

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- by Niels Chr. Hansen
- •
- 17
  Music, Information Theory, Auditory Perception, Music Cognition

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- by Reidar Bratvold
- •
- 2
  Decision Making, Uncertainty Quantification

Computer vision is an ever growing discipline whose ambitious goal is to enable machines with the intelligent visual skills humans and animals are provided by Nature, allowing them to interact effortlessly with complex, dynamic... more

Computer vision is an ever growing discipline whose ambitious goal is to enable machines with the intelligent
visual skills humans and animals are provided by Nature, allowing them to interact effortlessly
with complex, dynamic environments. Designing automated visual recognition and sensing systems
typically involves tackling a number of challenging tasks, and requires an impressive variety of sophisticated
mathematical tools. In most cases, the knowledge a machine has of its surroundings is at best
incomplete – missing data is a common problem, and visual cues are affected by imprecision. The need
for a coherent mathematical ‘language’ for the description of uncertain models and measurements then
naturally arises from the solution of computer vision problems.

The theory of evidence (sometimes referred to as ‘evidential reasoning’, ‘belief theory’ or ‘Dempster-
Shafer theory’) is, perhaps, one of the most successful approaches to uncertainty modelling, as arguably
the most straightforward and intuitive approaches to a generalized probability theory. Emerging in the
last Sixties from a profound criticism of the more classical Bayesian theory of inference and modelling
of uncertainty, it stimulated in the last decades an extensive discussion of the epistemic nature of both
subjective ‘degrees of beliefs’ and frequentist ‘chances’ or relative frequencies. More recently, a renewed
interest in belief functions, the mathematical generalization of probabilities which are the object of study
of the theory of evidence, has seen a blossoming of applications to a variety of fields of applied science.

In this Book we are going to show how, indeed, the fruitful interaction of computer vision and evidential
reasoning is able stimulate a number of advances in both fields. From a methodological point of
view, novel theoretical advances concerning the geometric and algebraic properties of belief functions as
mathematical objects will be illustrated in some detail in Part II, with a focus on a perspective ‘geometric
approach’ to uncertainty and an algebraic solution of the issue of conflicting evidence. In Part III we will
illustrate how these new perspectives on the theory of belief functions arise from important computer vision
problems, such as articulated object tracking, data association and object pose estimation, to which
in turn the evidential formalism can give interesting new solutions. Finally, some initial steps towards
a generalization of the notion of total probability to belief functions will be taken, in the perspective of
endowing the theory of evidence with a complete battery of estimation and inference tools to the benefit
of scientists and practitioners.

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- by Fabio Cuzzolin
- •
- 38
  Probability Theory, Artificial Intelligence, Statistics, Bayesian

Flood management alternatives are often evaluated on the basis of flood parameters such as depth and velocity. As these parameters are uncertain, so is the evaluation of the alternatives. It is thus important to incorporate the... more

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- by Chen Liang and +1
  Sankaran Mahadevan
- •
- 7
  Sensitivity Analysis, Uncertainty Quantification, Optimization, Multidisciplinary Design Optimization

Prediction is the key objective of many machine learning applications. Accurate, reliable and robust predictions are essential for optimal and fair decisions by downstream components of artificial intelligence systems, especially in... more

Stringer-to-floor beam connections were reported as one of the most fatigue-prone details in riveted steel railway bridges. To detect stiffness degradation that results from the initiation and growth of fatigue cracks, an automated damage... more

Stringer-to-floor beam connections were reported as one of the most fatigue-prone details in riveted steel railway bridges. To detect stiffness degradation that results from the initiation and growth of fatigue cracks, an automated damage detection framework was proposed by the authors (Eftekhar Azam et al., 2019; Rageh et al., 2018). The proposed method relies on Proper Orthogonal Decomposition (POD) and Artificial Neural Networks (ANNs) to identify damage location and intensity under non-stationary, unknown train loads. Bridge computational models were used to simulate damage scenarios and for training the ANNs. Damage detection method efficiency and accuracy were shown to be significantly influenced by the level of modeling uncertainties (MUs). To investigate the applicability of the proposed framework to in-service bridges, a systematic analysis of the effect of MUs on the proposed POD-ANN framework was necessary. MU influence on the performance of the POD-ANN damage detection method was investigated and a new procedure for generating training data for ANNs was proposed. The procedure was based on synergizing Proper Orthogonal Modes (POMs) extracted from measured structural response and POMs calculated from the numerical model. The current study integrated numerical and field investigations. The main objective of the numerical investigation was to identify a robust damage feature independent of the level and location of assumed MUs. Results showed that Damage Location (DL) and Damage Intensity (DI) were detected with high accuracy for studied uncertainty cases; however, as expected, damage detection accuracy reduced as MU increased. A hybrid experimental-numerical approach was then implemented for the field investigation studies. This approach applied identified damage features from the numerical investigation to measurements from an in-service railway bridge to produce damage scenarios used to train the framework. MATLAB algorithms were developed that preprocessed field data and eliminated POM variations resulted from loading uncertainties. ANNs were trained and tested using the field strain estimated POMs from the hybrid approach and DL and DI results were obtained for the studied railway bridge under non-stationary, unknown train loads. These results show the promise of the POD-ANN method as a robust, real-time fatigue damage identification tool for steel railway bridges.

The Novikov–Furutsu (NF) theorem is a well-known mathematical tool, used in stochastic dynamics for correlation splitting, that is, for evaluating the mean value of the product of a random functional with a Gaussian argument multiplied by... more

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- by Marcos Chaos and +2
  Jeffrey Santner
  Tanvir Farouk
- •
- 8
  Mechanical Engineering, Chemical Engineering, Uncertainty Quantification, Boundary Conditions

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- by Reidar Bratvold and +1
  J. Eric Bickel
- •
- 9
  Risk, Decision Analysis, Oil and gas, Social Perception

The aim of this thesis is to solve a stochastic inverse problem, related to a model for voice production, using artificial neural networks (ANNs). Th- ree parameters of the model are considered uncertain and random variables are... more

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- by Julien Mauprivez and +1
  Rubens Sampaio
- •
- 4
  Inverse Problems, Uncertainty Quantification, Stochastic Modeling, Voice Recognition

Uncertainty Quantification

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