In recent years, more attention has been paid prominently to accelerated degradation testing in order to characterize accurate estimation of reliability properties for systems that are designed to work properly for years of even decades. %In this regard, degradation data from particular testing levels of the stress variable(s) are extrapolated with an appropriate statistical model to obtain estimates of lifetime quantiles at normal use levels. In this paper we propose optimal experimental designs for repeated measures accelerated degradation tests with competing failure modes that correspond to multiple response components. The observation time points are assumed to be fixed and known in advance. The marginal degradation paths are expressed using linear mixed effects models. The optimal design is obtained by minimizing the asymptotic variance of the estimator of some quantile of the failure time distribution at the normal use conditions. Numerical examples are introduced to ensure the robustness of the proposed optimal designs and compare their efficiency with standard experimental designs.
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This work focuses on the space-time reduced-order modeling (ROM) method for solving large-scale uncertainty quantification (UQ) problems with multiple random coefficients. In contrast with the traditional space ROM approach, which performs dimension reduction in the spatial dimension, the space-time ROM approach performs dimension reduction on both the spatial and temporal domains, and thus enables accurate approximate solutions at a low cost. We incorporate the space-time ROM strategy with various classical stochastic UQ propagation methods such as stochastic Galerkin and Monte Carlo. Numerical results demonstrate that our methodology has significant computational advantages compared to state-of-the-art ROM approaches. By testing the approximation errors, we show that there is no obvious loss of simulation accuracy for space-time ROM given its high computational efficiency.
Various nonparametric approaches for Bayesian spectral density estimation of stationary time series have been suggested in the literature, mostly based on the Whittle likelihood approximation. A generalization of this approximation has been proposed in Kirch et al. who prove posterior consistency for spectral density estimation in combination with the Bernstein-Dirichlet process prior for Gaussian time series. In this paper, we will extend the posterior consistency result to non-Gaussian time series by employing a general consistency theorem of Shalizi for dependent data and misspecified models. As a special case, posterior consistency for the spectral density under the Whittle likelihood as proposed by Choudhuri, Ghosal and Roy is also extended to non-Gaussian time series. Small sample properties of this approach are illustrated with several examples of non-Gaussian time series.
In this paper we analyze, for a model of linear regression with gaussian covariates, the performance of a Bayesian estimator given by the mean of a log-concave posterior distribution with gaussian prior, in the high-dimensional limit where the number of samples and the covariates' dimension are large and proportional. Although the high-dimensional analysis of Bayesian estimators has been previously studied for Bayesian-optimal linear regression where the correct posterior is used for inference, much less is known when there is a mismatch. Here we consider a model in which the responses are corrupted by gaussian noise and are known to be generated as linear combinations of the covariates, but the distributions of the ground-truth regression coefficients and of the noise are unknown. This regression task can be rephrased as a statistical mechanics model known as the Gardner spin glass, an analogy which we exploit. Using a leave-one-out approach we characterize the mean-square error for the regression coefficients. We also derive the log-normalizing constant of the posterior. Similar models have been studied by Shcherbina and Tirozzi and by Talagrand, but our arguments are much more straightforward. An interesting consequence of our analysis is that in the quadratic loss case, the performance of the Bayesian estimator is independent of a global "temperature" hyperparameter and matches the ridge estimator: sampling and optimizing are equally good.
There has been substantial interest in developing Markov chain Monte Carlo algorithms based on piecewise-deterministic Markov processes. However existing algorithms can only be used if the target distribution of interest is differentiable everywhere. The key to adapting these algorithms so that they can sample from to densities with discontinuities is defining appropriate dynamics for the process when it hits a discontinuity. We present a simple condition for the transition of the process at a discontinuity which can be used to extend any existing sampler for smooth densities, and give specific choices for this transition which work with popular algorithms such as the Bouncy Particle Sampler, the Coordinate Sampler and the Zig-Zag Process. Our theoretical results extend and make rigorous arguments that have been presented previously, for instance constructing samplers for continuous densities restricted to a bounded domain, and we present a version of the Zig-Zag Process that can work in such a scenario. Our novel approach to deriving the invariant distribution of a piecewise-deterministic Markov process with boundaries may be of independent interest.
The rapid recent progress in machine learning (ML) has raised a number of scientific questions that challenge the longstanding dogma of the field. One of the most important riddles is the good empirical generalization of overparameterized models. Overparameterized models are excessively complex with respect to the size of the training dataset, which results in them perfectly fitting (i.e., interpolating) the training data, which is usually noisy. Such interpolation of noisy data is traditionally associated with detrimental overfitting, and yet a wide range of interpolating models -- from simple linear models to deep neural networks -- have recently been observed to generalize extremely well on fresh test data. Indeed, the recently discovered double descent phenomenon has revealed that highly overparameterized models often improve over the best underparameterized model in test performance. Understanding learning in this overparameterized regime requires new theory and foundational empirical studies, even for the simplest case of the linear model. The underpinnings of this understanding have been laid in very recent analyses of overparameterized linear regression and related statistical learning tasks, which resulted in precise analytic characterizations of double descent. This paper provides a succinct overview of this emerging theory of overparameterized ML (henceforth abbreviated as TOPML) that explains these recent findings through a statistical signal processing perspective. We emphasize the unique aspects that define the TOPML research area as a subfield of modern ML theory and outline interesting open questions that remain.
Optimal transport distances have found many applications in machine learning for their capacity to compare non-parametric probability distributions. Yet their algorithmic complexity generally prevents their direct use on large scale datasets. Among the possible strategies to alleviate this issue, practitioners can rely on computing estimates of these distances over subsets of data, {\em i.e.} minibatches. While computationally appealing, we highlight in this paper some limits of this strategy, arguing it can lead to undesirable smoothing effects. As an alternative, we suggest that the same minibatch strategy coupled with unbalanced optimal transport can yield more robust behavior. We discuss the associated theoretical properties, such as unbiased estimators, existence of gradients and concentration bounds. Our experimental study shows that in challenging problems associated to domain adaptation, the use of unbalanced optimal transport leads to significantly better results, competing with or surpassing recent baselines.
Interpretation of Deep Neural Networks (DNNs) training as an optimal control problem with nonlinear dynamical systems has received considerable attention recently, yet the algorithmic development remains relatively limited. In this work, we make an attempt along this line by reformulating the training procedure from the trajectory optimization perspective. We first show that most widely-used algorithms for training DNNs can be linked to the Differential Dynamic Programming (DDP), a celebrated second-order trajectory optimization algorithm rooted in the Approximate Dynamic Programming. In this vein, we propose a new variant of DDP that can accept batch optimization for training feedforward networks, while integrating naturally with the recent progress in curvature approximation. The resulting algorithm features layer-wise feedback policies which improve convergence rate and reduce sensitivity to hyper-parameter over existing methods. We show that the algorithm is competitive against state-ofthe-art first and second order methods. Our work opens up new avenues for principled algorithmic design built upon the optimal control theory.
Modeling multivariate time series has long been a subject that has attracted researchers from a diverse range of fields including economics, finance, and traffic. A basic assumption behind multivariate time series forecasting is that its variables depend on one another but, upon looking closely, it is fair to say that existing methods fail to fully exploit latent spatial dependencies between pairs of variables. In recent years, meanwhile, graph neural networks (GNNs) have shown high capability in handling relational dependencies. GNNs require well-defined graph structures for information propagation which means they cannot be applied directly for multivariate time series where the dependencies are not known in advance. In this paper, we propose a general graph neural network framework designed specifically for multivariate time series data. Our approach automatically extracts the uni-directed relations among variables through a graph learning module, into which external knowledge like variable attributes can be easily integrated. A novel mix-hop propagation layer and a dilated inception layer are further proposed to capture the spatial and temporal dependencies within the time series. The graph learning, graph convolution, and temporal convolution modules are jointly learned in an end-to-end framework. Experimental results show that our proposed model outperforms the state-of-the-art baseline methods on 3 of 4 benchmark datasets and achieves on-par performance with other approaches on two traffic datasets which provide extra structural information.
Accurate segmentation of the prostate from magnetic resonance (MR) images provides useful information for prostate cancer diagnosis and treatment. However, automated prostate segmentation from 3D MR images still faces several challenges. For instance, a lack of clear edge between the prostate and other anatomical structures makes it challenging to accurately extract the boundaries. The complex background texture and large variation in size, shape and intensity distribution of the prostate itself make segmentation even further complicated. With deep learning, especially convolutional neural networks (CNNs), emerging as commonly used methods for medical image segmentation, the difficulty in obtaining large number of annotated medical images for training CNNs has become much more pronounced that ever before. Since large-scale dataset is one of the critical components for the success of deep learning, lack of sufficient training data makes it difficult to fully train complex CNNs. To tackle the above challenges, in this paper, we propose a boundary-weighted domain adaptive neural network (BOWDA-Net). To make the network more sensitive to the boundaries during segmentation, a boundary-weighted segmentation loss (BWL) is proposed. Furthermore, an advanced boundary-weighted transfer leaning approach is introduced to address the problem of small medical imaging datasets. We evaluate our proposed model on the publicly available MICCAI 2012 Prostate MR Image Segmentation (PROMISE12) challenge dataset. Our experimental results demonstrate that the proposed model is more sensitive to boundary information and outperformed other state-of-the-art methods.
In this paper we introduce a covariance framework for the analysis of EEG and MEG data that takes into account observed temporal stationarity on small time scales and trial-to-trial variations. We formulate a model for the covariance matrix, which is a Kronecker product of three components that correspond to space, time and epochs/trials, and consider maximum likelihood estimation of the unknown parameter values. An iterative algorithm that finds approximations of the maximum likelihood estimates is proposed. We perform a simulation study to assess the performance of the estimator and investigate the influence of different assumptions about the covariance factors on the estimated covariance matrix and on its components. Apart from that, we illustrate our method on real EEG and MEG data sets. The proposed covariance model is applicable in a variety of cases where spontaneous EEG or MEG acts as source of noise and realistic noise covariance estimates are needed for accurate dipole localization, such as in evoked activity studies, or where the properties of spontaneous EEG or MEG are themselves the topic of interest, such as in combined EEG/fMRI experiments in which the correlation between EEG and fMRI signals is investigated.
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