28,367 research outputs found
Kernel-based stochastic collocation for the random two-phase Navier-Stokes equations
In this work, we apply stochastic collocation methods with radial kernel
basis functions for an uncertainty quantification of the random incompressible
two-phase Navier-Stokes equations. Our approach is non-intrusive and we use the
existing fluid dynamics solver NaSt3DGPF to solve the incompressible two-phase
Navier-Stokes equation for each given realization. We are able to empirically
show that the resulting kernel-based stochastic collocation is highly
competitive in this setting and even outperforms some other standard methods
Comparison of data-driven uncertainty quantification methods for a carbon dioxide storage benchmark scenario
A variety of methods is available to quantify uncertainties arising with\-in
the modeling of flow and transport in carbon dioxide storage, but there is a
lack of thorough comparisons. Usually, raw data from such storage sites can
hardly be described by theoretical statistical distributions since only very
limited data is available. Hence, exact information on distribution shapes for
all uncertain parameters is very rare in realistic applications. We discuss and
compare four different methods tested for data-driven uncertainty
quantification based on a benchmark scenario of carbon dioxide storage. In the
benchmark, for which we provide data and code, carbon dioxide is injected into
a saline aquifer modeled by the nonlinear capillarity-free fractional flow
formulation for two incompressible fluid phases, namely carbon dioxide and
brine. To cover different aspects of uncertainty quantification, we incorporate
various sources of uncertainty such as uncertainty of boundary conditions, of
conceptual model definitions and of material properties. We consider recent
versions of the following non-intrusive and intrusive uncertainty
quantification methods: arbitary polynomial chaos, spatially adaptive sparse
grids, kernel-based greedy interpolation and hybrid stochastic Galerkin. The
performance of each approach is demonstrated assessing expectation value and
standard deviation of the carbon dioxide saturation against a reference
statistic based on Monte Carlo sampling. We compare the convergence of all
methods reporting on accuracy with respect to the number of model runs and
resolution. Finally we offer suggestions about the methods' advantages and
disadvantages that can guide the modeler for uncertainty quantification in
carbon dioxide storage and beyond
Distribution-Free Uncertainty Quantification for Kernel Methods by Gradient Perturbations
We propose a data-driven approach to quantify the uncertainty of models
constructed by kernel methods. Our approach minimizes the needed distributional
assumptions, hence, instead of working with, for example, Gaussian processes or
exponential families, it only requires knowledge about some mild regularity of
the measurement noise, such as it is being symmetric or exchangeable. We show,
by building on recent results from finite-sample system identification, that by
perturbing the residuals in the gradient of the objective function, information
can be extracted about the amount of uncertainty our model has. Particularly,
we provide an algorithm to build exact, non-asymptotically guaranteed,
distribution-free confidence regions for ideal, noise-free representations of
the function we try to estimate. For the typical convex quadratic problems and
symmetric noises, the regions are star convex centered around a given nominal
estimate, and have efficient ellipsoidal outer approximations. Finally, we
illustrate the ideas on typical kernel methods, such as LS-SVC, KRR,
-SVR and kernelized LASSO.Comment: 18 figure
Explanation Uncertainty with Decision Boundary Awareness
Post-hoc explanation methods have become increasingly depended upon for
understanding black-box classifiers in high-stakes applications, precipitating
a need for reliable explanations. While numerous explanation methods have been
proposed, recent works have shown that many existing methods can be
inconsistent or unstable. In addition, high-performing classifiers are often
highly nonlinear and can exhibit complex behavior around the decision boundary,
leading to brittle or misleading local explanations. Therefore, there is an
impending need to quantify the uncertainty of such explanation methods in order
to understand when explanations are trustworthy. We introduce a novel
uncertainty quantification method parameterized by a Gaussian Process model,
which combines the uncertainty approximation of existing methods with a novel
geodesic-based similarity which captures the complexity of the target black-box
decision boundary. The proposed framework is highly flexible; it can be used
with any black-box classifier and feature attribution method to amortize
uncertainty estimates for explanations. We show theoretically that our proposed
geodesic-based kernel similarity increases with the complexity of the decision
boundary. Empirical results on multiple tabular and image datasets show that
our decision boundary-aware uncertainty estimate improves understanding of
explanations as compared to existing methods
Forecasting wireless demand with extreme values using feature embedding in Gaussian processes
Wireless traffic prediction is a fundamental enabler to proactive network optimisation in 5G and beyond. Forecasting extreme demand spikes and troughs is essential to avoiding outages and improving energy efficiency. However, current forecasting methods predominantly focus on overall forecast performance and/or do not offer probabilistic uncertainty quantification. Here, we design a feature embedding (FE) kernel for a Gaussian Process (GP) model to forecast traffic demand. The FE kernel enables us to trade-off overall forecast accuracy against peak-trough accuracy. Using real 4G base station data, we compare its performance against both conventional GPs, ARIMA models, as well as demonstrate the uncertainty quantification output. The advantage over neural network (e.g. CNN, LSTM) models is that the probabilistic forecast uncertainty can directly feed into decision processes in optimisation modules
DAUX: a Density-based Approach for Uncertainty eXplanations
Uncertainty quantification (UQ) is essential for creating trustworthy machine
learning models. Recent years have seen a steep rise in UQ methods that can
flag suspicious examples, however, it is often unclear what exactly these
methods identify. In this work, we propose an assumption-light method for
interpreting UQ models themselves. We introduce the confusion density matrix --
a kernel-based approximation of the misclassification density -- and use this
to categorize suspicious examples identified by a given UQ method into three
classes: out-of-distribution (OOD) examples, boundary (Bnd) examples, and
examples in regions of high in-distribution misclassification (IDM). Through
extensive experiments, we shed light on existing UQ methods and show that the
cause of the uncertainty differs across models. Additionally, we show how the
proposed framework can make use of the categorized examples to improve
predictive performance
Nonparametric Uncertainty Quantification for Stochastic Gradient Flows
This paper presents a nonparametric statistical modeling method for
quantifying uncertainty in stochastic gradient systems with isotropic
diffusion. The central idea is to apply the diffusion maps algorithm to a
training data set to produce a stochastic matrix whose generator is a discrete
approximation to the backward Kolmogorov operator of the underlying dynamics.
The eigenvectors of this stochastic matrix, which we will refer to as the
diffusion coordinates, are discrete approximations to the eigenfunctions of the
Kolmogorov operator and form an orthonormal basis for functions defined on the
data set. Using this basis, we consider the projection of three uncertainty
quantification (UQ) problems (prediction, filtering, and response) into the
diffusion coordinates. In these coordinates, the nonlinear prediction and
response problems reduce to solving systems of infinite-dimensional linear
ordinary differential equations. Similarly, the continuous-time nonlinear
filtering problem reduces to solving a system of infinite-dimensional linear
stochastic differential equations. Solving the UQ problems then reduces to
solving the corresponding truncated linear systems in finitely many diffusion
coordinates. By solving these systems we give a model-free algorithm for UQ on
gradient flow systems with isotropic diffusion. We numerically verify these
algorithms on a 1-dimensional linear gradient flow system where the analytic
solutions of the UQ problems are known. We also apply the algorithm to a
chaotically forced nonlinear gradient flow system which is known to be well
approximated as a stochastically forced gradient flow.Comment: Find the associated videos at: http://personal.psu.edu/thb11
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