The generalization of Latin hypercube sampling Shields, Michael D.; Zhang, Jiaxin
Reliability engineering & system safety,
April 2016, 2016-04-00, 20160401, Letnik:
148
Journal Article
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Latin hypercube sampling (LHS) is generalized in terms of a spectrum of stratified sampling (SS) designs referred to as partially stratified sample (PSS) designs. True SS and LHS are shown to ...represent the extremes of the PSS spectrum. The variance of PSS estimates is derived along with some asymptotic properties. PSS designs are shown to reduce variance associated with variable interactions, whereas LHS reduces variance associated with main effects. Challenges associated with the use of PSS designs and their limitations are discussed. To overcome these challenges, the PSS method is coupled with a new method called Latinized stratified sampling (LSS) that produces sample sets that are simultaneously SS and LHS. The LSS method is equivalent to an Orthogonal Array based LHS under certain conditions but is easier to obtain. Utilizing an LSS on the subspaces of a PSS provides a sampling strategy that reduces variance associated with both main effects and variable interactions and can be designed specially to minimize variance for a given problem. Several high-dimensional numerical examples highlight the strengths and limitations of the method. The Latinized partially stratified sampling method is then applied to identify the best sample strategy for uncertainty quantification on a plate buckling problem.
•Latin hypercube sampling is generalized in terms of a spectrum of stratified sample designs.•The Partially Stratified Sampling (PSS) method is proposed as this generalization.•A new method, Latinized Stratified Sampling (LSS), is proposed that is both stratified and Latin.•Merging the PSS and LSS methods into a unified approach enables large reductions in variance.•The method reduces variance associated with both main effects and interaction effects.
Modern machine learning (ML) techniques, in conjunction with simulation-based methods, present remarkable potential for various scientific and engineering applications. Within the materials science ...field, these data-based methods can be used to build efficient structure–property linkages that can be further integrated within multi-scale simulations, or guide experiments in a materials discovery setting. However, a critical shortcoming of state-of-the-art ML techniques is their lack of reliable uncertainty/error estimates, which severely limits their use for materials or other engineering applications where data is often scarce and uncertainties are substantial. This paper presents methods for Bayesian learning of neural networks (NN) that allow consideration of both aleatoric uncertainties that account for the inherent stochasticity of the data-generating process, and epistemic uncertainties, which arise from consideration of limited amounts of data. In particular, algorithms based on approximate variational inference and (pseudo-)Bayesian model averaging achieve an appropriate trade-off between accuracy of the uncertainty estimates and accessible computational cost. The performance of these algorithms is first presented on simple 1D examples to illustrate their behavior in both extrapolation and interpolation settings. The approach is then applied for the prediction of homogenized and localized properties of a composite material. In this setting, data is generated from a finite element model, which permits a study of the behavior of the probabilistic learning algorithms under various amounts of aleatoric and epistemic uncertainties.
•Bayesian neural networks for data-driven materials modeling using small data.•Algorithms based on variational inference and probabilistic model averaging.•Example of surrogate modeling of an elastoplastic composite material.•Aleatoric uncertainties arise from random placement of fibers in microstructure.•Epistemic uncertainties are quantified and can be reduced by gathering more data.
•Quantifies the true epistemic uncertainty in probability model form and parameter values.•Retains a full multimodel probabilistic description of epistemic uncertainties created by small ...datasets.•Simultaneously propagates many (thousands or more) probability models.•Reduces the computational cost of multimodel uncertainty propagation by several orders of magnitude.
This paper addresses the problem of uncertainty quantification and propagation when data for characterizing probability distributions are scarce. We propose a methodology wherein the full uncertainty associated with probability model form and parameter estimation are retained and efficiently propagated. This is achieved by applying the information-theoretic multimodel inference method to identify plausible candidate probability densities and associated probabilities that each method is the best model in the Kullback-Leibler sense. The joint parameter densities for each plausible model are then estimated using Bayes’ rule. We then propagate this full set of probability models by estimating an optimal importance sampling density that is representative of all plausible models, propagating this density, and reweighting the samples according to each of the candidate probability models. This is in contrast with conventional methods that try to identify a single probability model that encapsulates the full uncertainty caused by lack of data and consequently underestimate uncertainty. The result is a complete probabilistic description of both aleatory and epistemic uncertainty achieved with several orders of magnitude reduction in computational cost. It is shown how the model can be updated to adaptively accommodate added data and added candidate probability models. The method is applied for uncertainty analysis of plate buckling strength where it is demonstrated how dataset size affects the confidence (or lack thereof) we can place in statistical estimates of response when data are lacking.
•Shows that optimal stratified designs are non-uniform for nonlinear systems.•Illustrates that the benefits of an optimally non-uniform sample design can be substantial compared to space-filling ...designs.•Proposes an adaptive approach that mitigates the practical challenges of achieving sample design optimality.•The method is applied to modeling shear localization in amorphous solids with stochastic initial conditions.
This paper compares space-filling and importance sampling (IS)-based Monte Carlo sample designs with those derived for optimality in the error of stratified statistical estimators. Space-filling designs are shown to be optimal for systems whose response depends linearly on the input random variables. They are, however, shown to be far from optimal when the system is nonlinear. To achieve optimality, it is shown that samples should be placed densely in regions of large variation (sparsely in regions of small variation). This notion is shown to be subtly, but importantly, different from other non-space-filling designs, particularly IS. To achieve near-optimal sample designs, the adaptive Gradient Enhanced Refined Stratified Sampling (GE-RSS) is proposed that sequentially refines the probability space in accordance with stratified sampling. The space is refined according to the estimated local variance of the system computed from gradients using a surrogate model. The method significantly reduces the error in stratified Monte Carlo estimators for strongly nonlinear systems, outperforms both space-filling methods and IS-based methods, and is simple to implement. Numerical examples on strongly nonlinear systems illustrate the improvement over space-filling and IS designs. The method is applied to study the probability of shear band formation in a bulk metallic glass.
•A new model for non-stationary and non-Gaussian stochastic processes is presented.•The model improves the ITAM by upgrading directly the autocorrelation function.•KL-ITAM improves the ...accuracy/efficiency of non-Gaussian stochastic process modeling.•Utilizes the K–L expansion for simulation of general non-Gaussian random processes.
A method is proposed for modeling non-Gaussian and non-stationary random processes using the Karhunen–Loève expansion and translation process theory that builds upon an existing family of procedures called the Iterative Translation Approximation Method (ITAM). The new method improves the ITAM by iterating directly on the non-stationary autocorrelation function. The existing ITAM requires estimation of the evolutionary spectrum from the autocorrelation function for which no unique relation exists. Consequently, computationally expensive estimates or simplifying assumptions/approximations reduced the ITAM performance for non-stationary processes. The proposed method improves the accuracy of the resulting process while maintaining computational efficiency. Several examples are provided.
We describe, for the first time, the use of a mobile device platform for remote direct observation of inhaler use and technique. The research programme commenced with a rapid systematic review of ...mobile device (or videophone) use for direct observation of therapy (MDOT). Ten studies (mainly pilots) were identified involving patients with tuberculosis, sickle cell disease and Alzheimer's disease. New studies are ongoing (ClinicalTrials.gov website) in TB, stroke, sickle cell disease, HIV and opioid dependence. Having identified no prior use of MDOT in inhaler monitoring, we implemented a feasibility study in 12 healthy volunteer children (2-12 years; 8 females and 4 males) over a period of 14 days, with twice daily video upload of their 'dummy' inhaler use. Two children uploaded 100% of the requested videos, with only one child having an inhaler upload rate of <75%. The quality of uploaded videos was generally good (only 1.7% of unacceptable quality for evaluation). The final aspect of the research was a pilot study using MDOT (6 weeks) in 22 children with difficult to treat asthma. Healthcare professionals evaluated inhaler technique using uploaded videos and provided telephone instruction on improving inhaler use. The main outcomes were assessed at week 12 post initiation of MDOT. By week 5, all children still engaging in MDOT (n = 18) were judged to have effective inhaler technique. Spirometry values did not vary to a significantly significant degree between baseline and 12 weeks (P>0.05), however, mean fraction of exhaled nitric oxide (FeNO) values normalised (mean 38.7 to 19.3ppm) and mean Asthma Control Test values improved (13.1 to mean 17.8). Feedback from participants was positive. Overall the findings open up a new paradigm in device independent (can be used for any type of inhaler device) monitoring, providing a platform for evaluating / improving inhaler use at home.
•A fully probabilistic multimodel approach for imprecise GSA is proposed.•This approach employs Bayesian multimodel inference to quantify uncertainties in model inputs.•The multimodel inference is ...combined with an importance reweighting scheme for estimation of imprecise Sobol indices.•The methodology can be used to assess confidence in computed sensitivity indices and inform testing and data collection efforts.
Global Sensitivity Analysis (GSA) aims to understand the relative importance of uncertain input variables to model response. Conventional GSA involves calculating sensitivity (Sobol’) indices for a model with known model parameter distributions. However, model parameters are affected by aleatory and epistemic uncertainty, with the latter often caused by lack of data. We propose a new framework to quantify uncertainty in probability model-form and model parameters resulting from small datasets and integrate these uncertainties into Sobol’ index estimates. First, the process establishes, through Bayesian multimodel inference, a set of candidate probability models and their associated probabilities. Imprecise Sobol’ indices are calculated from these probability models using an importance sampling reweighting approach. This results in probabilistic Sobol’ indices, whose distribution characterizes uncertainty in the sensitivity resulting from small dataset size. The imprecise Sobol’ indices thus provide a measure of confidence in the sensitivity estimate and, moreover, can be used to inform data collection efforts targeted to minimize the impact of uncertainties. Through an example studying the parameters of a Timoshenko beam, we show that these probabilistic Sobol’ indices converge to the true/deterministic Sobol’ indices as the dataset size increases and hence, distribution-form uncertainty reduces. The approach is then applied to assess the sensitivity of the out-of-plane properties of an E-glass fiber composite material to its constituent properties. This second example illustrates the approach for an important class of materials with wide-ranging applications when data may be lacking for some input parameters.
Transfer learning enables the transfer of knowledge gained while learning to perform one task (source) to a related but different task (target), hence addressing the expense of data acquisition and ...labelling, potential computational power limitations and dataset distribution mismatches. We propose a new transfer learning framework for task-specific learning (functional regression in partial differential equations) under conditional shift based on the deep operator network (DeepONet). Task-specific operator learning is accomplished by fine-tuning task-specific layers of the target DeepONet using a hybrid loss function that allows for the matching of individual target samples while also preserving the global properties of the conditional distribution of the target data. Inspired by conditional embedding operator theory, we minimize the statistical distance between labelled target data and the surrogate prediction on unlabelled target data by embedding conditional distributions onto a reproducing kernel Hilbert space. We demonstrate the advantages of our approach for various transfer learning scenarios involving nonlinear partial differential equations under diverse conditions due to shifts in the geometric domain and model dynamics. Our transfer learning framework enables fast and efficient learning of heterogeneous tasks despite considerable differences between the source and target domains.A promising area for deep learning is in modelling complex physical processes described by partial differential equations (PDEs), which is computationally expensive for conventional approaches. An operator learning approach called DeepONet was recently introduced to tackle PDE-related problems, and in new work, this approach is extended with transfer learning, which transfers knowledge obtained from learning to perform one task to a related but different task.
This paper outlines a methodology for Bayesian multimodel uncertainty quantification (UQ) and propagation and presents an investigation into the effect of prior probabilities on the resulting ...uncertainties. The UQ methodology is adapted from the information-theoretic method previously presented by the authors (Zhang and Shields, 2018) to a fully Bayesian construction that enables greater flexibility in quantifying uncertainty in probability model form. Being Bayesian in nature and rooted in UQ from small datasets, prior probabilities in both probability model form and model parameters are shown to have a significant impact on quantified uncertainties and, consequently, on the uncertainties propagated through a physics-based model. These effects are specifically investigated for a simplified plate buckling problem with uncertainties in material properties derived from a small number of experiments using noninformative priors and priors derived from past studies of varying appropriateness. It is illustrated that prior probabilities can have a significant impact on multimodel UQ for small datasets and inappropriate (but seemingly reasonable) priors may even have lingering effects that bias probabilities even for large datasets. When applied to uncertainty propagation, this may result in probability bounds on response quantities that do not include the true probabilities.
This paper introduces the 3rd-order Spectral Representation Method for simulation of non-stationary and non-Gaussian stochastic processes. The proposed method extends the classical 2nd-order Spectral ...Representation Method to expand the stochastic process from an evolutionary bispectrum and an evolutionary power spectrum, thus matching the process completely up to third-order. A Proper Orthogonal Decomposition (POD) approach is further proposed to enable an efficient FFT-based implementation that reduces computational cost significantly. Two examples are presented, including the simulation of a fully non-stationary seismic ground motion process, highlighting the accuracy and efficacy of the proposed method.
•3rd-order SRM is presented to simulate non-stationary and non-Gaussian processes.•Ensemble properties of the samples match the target moments up to third-order.•A proper orthogonal decomposition approach using fast Fourier transform is presented.•The method is applied to simulate non-stationary and non-Gaussian random processes.•Computational savings and statistical properties are illustrated.
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