Gaussian process surrogates for failure detection: a Bayesian experimental design approach

An important task of uncertainty quantification is to identify {the probability of} undesired events, in particular, system failures, caused by various sources of uncertainties. In this work we consider the construction of Gaussian {process} surrogates for failure detection and failure probability estimation. In particular, we consider the situation that the underlying computer models are extremely expensive, and in this setting, determining the sampling points in the state space is of essential importance. We formulate the problem as an optimal experimental design for Bayesian inferences of the limit state (i.e., the failure boundary) and propose an efficient numerical scheme to solve the resulting optimization problem. In particular, the proposed limit-state inference method is capable of determining multiple sampling points at a time, and thus it is well suited for problems where multiple computer simulations can be performed in parallel. The accuracy and performance of the proposed method is demonstrated by both academic and practical examples

A historical view on the maximum entropy

How to find unknown distributions is questioned in many pieces of research. There are several ways to figure them out, but the main question is which acts more reasonably than others. In this paper, we focus on the maximum entropy principle as a suitable method of discovering the unknown distribution, which recommends some prior information based on the available data set. We explain its features by reviewing some papers. Furthermore, we recommend some articles to study around the generalized maximum entropy issue, which is more suitable when autocorrelation data exists. Then, we list the beneficial features of the maximum entropy as a result. Finally, some disadvantages of entropy are expressed to have a complete look at the maximum entropy principle, and we list its drawbacks as the final step

Fast high-dimensional Bayesian classification and clustering

We introduce a fast approach to classification and clustering applicable to high-dimensional continuous data, based on Bayesian mixture models for which explicit computations are available. This permits us to treat classification and clustering in a single framework, and allows calculation of unobserved class probability. The new classifier is robust to adding noise variables as a drawback of the built-in spike-and-slab structure of the proposed Bayesian model. The usefulness of classification using our method is shown on metabololomic example, and on the Iris data with and without noise variables. Agglomerative hierarchical clustering is used to construct a dendrogram based on the posterior probabilities of particular partitions, to provide a dendrogram with a probabilistic interpretation. An extension to variable selection is proposed which summarises the importance of variables for classification or clustering and has probabilistic interpretation. Having a simple model provides estimation of the model parameters using maximum likelihood and therefore yields a fully automatic algorithm. The new clustering method is applied to metabolomic, microarray, and image data and is studied using simulated data motivated by real datasets. The computational difficulties of the new approach are discussed, solutions for algorithm acceleration are proposed, and the written computer code is briefly analysed. Simulations shows that the quality of the estimated model parameters depends on the parametric distribution assumed for effects, but after fixing the model parameters to reasonable values, the distribution of the effects influences clustering very little. Simulations confirms that the clustering algorithm and the proposed variable selection method is reliable when the model assumptions are wrong. The new approach is compared with the popular Bayesian clustering alternative, MCLUST, fitted on the principal components using two loss functions in which our proposed approach is found to be more efficient in almost every situation

Auto-regressive model based polarimetric adaptive detection scheme part I: Theoretical derivation and performance analysis

This paper deals with the problem of target detection in coherent radar systems exploiting polarimetric diversity. We resort to a parametric approach and we model the disturbance affecting the data as a multi-channel autoregressive (AR) process. Following this model, a new polarimetric adaptive detector is derived, which aims at improving the target detection capability while relaxing the requirements on the training data size and the computational burden with respect to existing solutions. A complete theoretical characterization of the asymptotic performance of the derived detector is provided, using two different target fluctuation models. The effectiveness of the proposed approach is shown against simulated data, in comparison with alternative existing solutions

Statistical methods for scale-invariant and multifractal stochastic processes.

This thesis focuses on stochastic modeling, and statistical methods, in finance and in climate science. Two financial markets, short-term interest rates and electricity prices, are analyzed. We find that the evidence of mean reversion in short-term interest rates is week, while the “log-returns” of electricity prices have significant anti-correlations. More importantly, empirical analyses confirm the multifractal nature of these financial markets, and we propose multifractal models that incorporate the specific conditional mean reversion and level dependence. A second topic in the thesis is the analysis of regional (5◦ × 5◦ and 2◦ × 2◦ latitude- longitude) globally gridded surface temperature series for the time period 1900-2014, with respect to a linear trend and long-range dependence. We find statistically significant trends in most regions. However, we also demonstrate that the existence of a second scaling regime on decadal time scales will have an impact on trend detection. The last main result is an approximative maximum likelihood (ML) method for the log- normal multifractal random walk. It is shown that the ML method has applications beyond parameter estimation, and can for instance be used to compute various risk measures in financial markets