Miracle-dz/UniFAR-FTUs huggingface dataset:
Random field representations for stochastic elliptic boundary value problems and statistical inverse problems

This paper presents new results allowing an unknown non-Gaussian positive matrix-valued random field to be identified through a stochastic elliptic boundary value problem, solving a statistical inverse problem. A new general class of non-Gaussian positive-definite matrix-valued random fields, adapted to the statistical inverse problems in high stochastic dimension for their experimental identification, is introduced and its properties are analysed. A minimal parameterisation of discretised random fields belonging to this general class is proposed. Using this parameterisation of the general class, a complete identification procedure is proposed. New results of the mathematical and numerical analyses of the parameterised stochastic elliptic boundary value problem are presented. The numerical solution of this parametric stochastic problem provides an explicit approximation of the application that maps the parameterised general class of random fields to the corresponding set of random solutions. This approximation can be used during the identification procedure in order to avoid the solution of multiple forward stochastic problems. Since the proposed general class of random fields possibly contains random fields which are not uniformly bounded, a particular mathematical analysis is developed and dedicated approximation methods are introduced. In order to obtain an algorithm for constructing the approximation of a very high-dimensional map, complexity reduction methods are introduced and are based on the use of sparse or low-rank approximation methods that exploit the tensor structure of the solution which results from the parameterisation of the general class of random fields.
Composite Gaussian process models for emulating expensive functions

A new type of nonstationary Gaussian process model is developed for approximating computationally expensive functions. The new model is a composite of two Gaussian processes, where the first one captures the smooth global trend and the second one models local details. The new predictor also incorporates a flexible variance model, which makes it more capable of approximating surfaces with varying volatility. Compared to the commonly used stationary Gaussian process model, the new predictor is numerically more stable and can more accurately approximate complex surfaces when the experimental design is sparse. In addition, the new model can also improve the prediction intervals by quantifying the change of local variability associated with the response. Advantages of the new predictor are demonstrated using several examples.
Penalized maximum likelihood estimation and effective dimension

This paper extends some prominent statistical results including \emph{Fisher Theorem and Wilks phenomenon} to the penalized maximum likelihood estimation with a quadratic penalization. It appears that sharp expansions for the penalized MLE \(\tilde{\thetav}_{G} \) and for the penalized maximum likelihood can be obtained without involving any asymptotic arguments, the results only rely on smoothness and regularity properties of the of the considered log-likelihood function. The error of estimation is specified in terms of the effective dimension \(p_G \) of the parameter set which can be much smaller than the true parameter dimension and even allows an infinite dimensional functional parameter. In the i.i. d. case, the Fisher expansion for the penalized MLE can be established under the constraint "\(p_G^{2}/n\) is small" while the remainder in the Wilks result is of order \(p_G^{3}/n \).
Significance Testing and Group Variable Selection

Let X; Z be r and s-dimensional covariates, respectively, used to model the response variable Y as Y = m(X;Z) + \sigma(X;Z)\epsilon. We develop an ANOVA-type test for the null hypothesis that Z has no influence on the regression function, based on residuals obtained from local polynomial fitting of the null model. Using p-values from this test, a group variable selection method based on multiple testing ideas is proposed. Simulations studies suggest that the proposed test procedure outperforms the generalized likelihood ratio test when the alternative is non-additive or there is heteroscedasticity. Additional simulation studies, with data generated from linear, non-linear and logistic regression, reveal that the proposed group variable selection procedure performs competitively against Group Lasso, and outperforms it in selecting groups having nonlinear effects. The proposed group variable selection procedure is illustrated on a real data set.
Second Rethinking of Network Pruning in the Adversarial Setting

It is well known that deep neural networks (DNNs) are vulnerable to adversarial attacks, which are implemented by adding crafted perturbations onto benign examples. Min-max robust optimization based adversarial training can provide a notion of security against adversarial attacks. However, adversarial robustness requires a significantly larger capacity of the network than that for the natural training with only benign examples. This paper proposes a framework of concurrent adversarial training and weight pruning that enables model compression while still preserving the adversarial robustness and essentially tackles the dilemma of adversarial training. Furthermore, this work studies two hypotheses about weight pruning in the conventional network pruning setting and finds that weight pruning is essential for reducing the network model size in the adversarial setting, i.e., training a small model from scratch even with inherited initialization from the large model cannot achieve both adversarial robustness and model compression.
Classification and clustering of sequencing data using a poisson model

In recent years, advances in high throughput sequencing technology have led to a need for specialized methods for the analysis of digital gene expression data. While gene expression data measured on a microarray take on continuous values and can be modeled using the normal distribution, RNA sequencing data involve nonnegative counts and are more appropriately modeled using a discrete count distribution, such as the Poisson or the negative binomial. Consequently, analytic tools that assume a Gaussian distribution (such as classification methods based on linear discriminant analysis and clustering methods that use Euclidean distance) may not perform as well for sequencing data as methods that are based upon a more appropriate distribution. Here, we propose new approaches for performing classification and clustering of observations on the basis of sequencing data. Using a Poisson log linear model, we develop an analog of diagonal linear discriminant analysis that is appropriate for sequencing data. We also propose an approach for clustering sequencing data using a new dissimilarity measure that is based upon the Poisson model. We demonstrate the performances of these approaches in a simulation study, on three publicly available RNA sequencing data sets, and on a publicly available chromatin immunoprecipitation sequencing data set.
On the canonical form of scale mixtures of skew-normal distributions

The canonical form of scale mixtures of multivariate skew-normal distribution is defined, emphasizing its role in summarizing some key properties of this class of distributions. It is also shown that the canonical form corresponds to an affine invariant co-ordinate system as defined in Tyler \emph{et} al. (2009), and a method for obtaining the linear transform that converts a scale mixture of multivariate skew-normal distribution into a canonical form is presented. Related results, where the particular case of the multivariate skew $t$ distribution is considered in greater detail, are the general expression of the Mardia indices of multivariate skewness and kurtosis and the reduction of dimensionality in calculating the mode.
Randomized Dimension Reduction on Massive Data

Scalability of statistical estimators is of increasing importance in modern applications and dimension reduction is often used to extract relevant information from data. A variety of popular dimension reduction approaches can be framed as symmetric generalized eigendecomposition problems. In this paper we outline how taking into account the low rank structure assumption implicit in these dimension reduction approaches provides both computational and statistical advantages. We adapt recent randomized low-rank approximation algorithms to provide efficient solutions to three dimension reduction methods: Principal Component Analysis (PCA), Sliced Inverse Regression (SIR), and Localized Sliced Inverse Regression (LSIR). A key observation in this paper is that randomization serves a dual role, improving both computational and statistical performance. This point is highlighted in our experiments on real and simulated data.
Data Augmentation for Hierarchical Capture-recapture Models

Capture-recapture studies are widely used to obtain information about abundance (population size or density) of animal populations. A common design is that in which multiple distinct populations are sampled, and the research objective is modeling variation in population size $N_{s}; s=1,2,. ,S$ among the populations such as estimating a treatment effect or some other source of variation related to landscape structure. The problem is naturally resolved using hierarchical models. We provide a Bayesian formulation of such models using data augmentation which preserves the individual encounter histories in the model and, as such, is amenable to modeling individual effects. We formulate the model by conditioning on the total population size among all populations. In this case, the abundance model can be formulated as a multinomial model that allocates individuals among sites. MCMC is easily carried out by the introduction of a categorical individual effect, $g_{i}$, which partitions the total population size. The prior distribution for the latent variable $g$ is derived from the model assumed for the population sizes $N_{s}$.
Distributed parameter estimation of discrete hierarchical models via marginal likelihoods

We consider discrete graphical models Markov with respect to a graph $G$ and propose two distributed marginal methods to estimate the maximum likelihood estimate of the canonical parameter of the model. Both methods are based on a relaxation of the marginal likelihood obtained by considering the density of the variables represented by a vertex $v$ of $G$ and a neighborhood. The two methods differ by the size of the neighborhood of $v$. We show that the estimates are consistent and that those obtained with the larger neighborhood have smaller asymptotic variance than the ones obtained through the smaller neighborhood.
The mean Euler characteristic and excursion probability of Gaussian random fields with stationary increments

Let $X=\{X(t),t\in {\mathbb{R}}^N\}$ be a centered Gaussian random field with stationary increments and $X(0)=0$. For any compact rectangle $T\subset {\mathbb{R}}^N$ and $u\in {\mathbb{R}}$, denote by $A_u=\{t\in T:X(t)\geq u\}$ the excursion set. Under $X(\cdot)\in C^2({\mathbb{R}}^N)$ and certain regularity conditions, the mean Euler characteristic of $A_u$, denoted by ${\mathbb{E}}\{\varphi(A_u)\}$, is derived. By applying the Rice method, it is shown that, as $u\to\infty$, the excursion probability ${\mathbb{P}}\{\sup_{t\in T}X(t)\geq u\}$ can be approximated by ${\mathbb{E}}\{\varphi(A_u)\}$ such that the error is exponentially smaller than ${\mathbb{E}}\{\varphi(A_u)\}$. This verifies the expected Euler characteristic heuristic for a large class of Gaussian random fields with stationary increments.
Discussion of"Multiple Testing for Exploratory Research"by J. J. Goeman and A. Solari

Goeman and Solari [Statist. Sci. 26 (2011) 584–597] have addressed the interesting topic of multiple testing for exploratory research, and provided us with nice suggestions for exploratory analysis. They defined properties that an inferential procedure should have for exploratory analysis: the procedure should be mild, flexible and post hoc. Their inferential procedure gives a lower bound on the number of false hypotheses among the selected hypotheses, and moreover whenever possible identifies elementary hypotheses that are false. The need to estimate a lower bound on the number of false hypotheses arises in various applications, and the partial conjunction approach was developed for this purpose in Biometrics 64 (2008) 1215–1222 (see also Philos. Trans. R. Soc. Lond. Ser. A 367 (2009) 4255–4271 for more details). For example, in a combined analysis of several studies that examine the same problem, it is of interest to give a lower bound on the number of studies in which the finding was reproduced. I will first address the relation between the method of Goeman and Solari and the partial conjunction approach. Then I will discuss possible extensions and address the issue of exploration in more general settings, where the local test may not be defined in advance or where the candidate hypotheses may not be known to begin with. 1. RELATION TO THE TESTING OF PARTIAL CONJUNCTION HYPOTHESES Let H1, . ,Hn be the elementary hypotheses. The idea of giving a lower bound on the number of false elementary hypotheses (or equivalently an upper bound on the number of true elementary hypotheses) appears in [1], and is closely related to the tests of partial conjunction hypotheses. The partial conjunction null hypothesis Hu/n in [1] asks Ruth Heller is Senior Lecturer, Department of Statistics and Operations Research, Tel-Aviv University, Tel-Aviv, Israel e-mail: ruheller@post.tau. ac.il. This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in Statistical Science, 2011, Vol. 26, No. 4, 598–600. This reprint differs from the original in pagination and typographic detail. whether fewer than u of the elementary hypotheses are false, and the alternative hypothesis is that at least u of the elementary hypotheses are false. Testing whether Hu/n is false at a significance level α in order (i.e., for u = 1,2, . . .) results in a 1 − α confidence lower bound on the number of false elementary hypotheses: Theorem 1.1. Let pu/n be a partial conjunction p-value for testing Hu/n. Let umax =max{u :p i/n ≤ α ∀i = 1, . , u}. Then with 1 − α confidence, the true number of false hypotheses is in [umax, n]. Proof. Let k be the true number of false elementary hypotheses. If k = n, that is, all elementary hypotheses are false, there is nothing to prove. If k < n, Pr(k ≥ umax) = 1−Pr(k < umax) = 1−Pr(P (k+1)/n ≤ α)≥ 1−α.
Adaptive estimation in regression and complexity of approximation of random fields

In this thesis we study adaptive nonparametric regression with noise misspecification and the complexity of approximation of random fields in dependence of the dimension. First, we consider the problem of pointwise estimation in nonparametric regression with heteroscedastic additive Gaussian noise. We use the method of local approximation applying the Lepski method for selecting one estimate from the set of linear estimates obtained by the different degrees of localization. This approach is combined with the "propagation conditions" on the choice of critical values of the procedure, as suggested recently by Spokoiny and Vial [Ann.Stat., 2009]. The "propagation conditions" are relaxed for the model with misspecified covariance structure. We show that this procedure allows a misspecification of the covariance matrix with a relative error of order 1/ log(n), where n is the sample size. The quality of estimation is measured in terms of "oracle" risk bounds. We then turn to the approximation of d-parametric random fields of tensor product-type by means of n-term partial sums of the Karhunen-Lo\`eve expansion. The analysis is restricted to the average case setting. The quantity of interest is the information complexity describing the minimal number of terms in the partial sums, which guarantees an error not exceeding a given level. The behavior of this quantity when the dimension goes to infinity is the subject of our study.
Conditional simulation of max-stable processes

Since many environmental processes are spatial in extent, a single extreme event may affect several locations, and the spatial dependence must be taken into account in an appropriate way. This paper proposes a framework for conditional simulation of max-stable processes and gives closed forms for the regular conditional distributions of Brown--Resnick and Schlather processes. We test the method on simulated data and present applications to extreme rainfall around Zurich and extreme temperatures in Switzerland. The proposed framework provides accurate conditional simulations and can handle problems of realistic size. Copyright 2013, Oxford University Press.
Variable Selection with Exponential Weights and $l_0$-Penalization

In the context of a linear model with a sparse coefficient vector, exponential weights methods have been shown to be achieve oracle inequalities for prediction. We show that such methods also succeed at variable selection and estimation under the necessary identifiability condition on the design matrix, instead of much stronger assumptions required by other methods such as the Lasso or the Dantzig Selector. The same analysis yields consistency results for Bayesian methods and BIC-type variable selection under similar conditions.
Real-Time Semiparametric Regression

We develop algorithms for performing semiparametric regression analysis in real time, with data processed as it is collected and made immediately available via modern telecommunications technologies. Our definition of semiparametric regression is quite broad and includes, as special cases, generalized linear mixed models, generalized additive models, geostatistical models, wavelet nonparametric regression models and their various combinations. Fast updating of regression fits is achieved by couching semiparametric regression into a Bayesian hierarchical model or, equivalently, graphical model framework and employing online mean field variational ideas. An Internet site attached to this article, realtime-semiparametric-regression.net, illustrates the methodology for continually arriving stock market, real estate, and airline data. Flexible real-time analyses based on increasingly ubiquitous streaming data sources stand to benefit. This article has online supplementary material.
Testing the disjunction hypothesis using Voronoi diagrams with applications to genetics

Testing of the disjunction hypothesis is appropriate when each gene or location studied is associated with multiple $p$-values, each of which is of individual interest. This can occur when more than one aspect of an underlying process is measured. For example, cancer researchers may hope to detect genes that are both differentially expressed on a transcriptomic level and show evidence of copy number aberration. Currently used methods of $p$-value combination for this setting are overly conservative, resulting in very low power for detection. In this work, we introduce a method to test the disjunction hypothesis by using cumulative areas from the Voronoi diagram of two-dimensional vectors of $p$-values. Our method offers much improved power over existing methods, even in challenging situations, while maintaining appropriate error control. We apply the approach to data from two published studies: the first aims to detect periodic genes of the organism Schizosaccharomyces pombe, and the second aims to identify genes associated with prostate cancer.
Nonparametric testing for no-effect with functional responses and functional covariates

This paper studies the problem of nonparametric testing for the no-effect of a random functional covariate on a functional response. That means testing whether the conditional expectation of the response given the covariate is almost surely zero or not without imposing any model relating response and covariate. The response and the covariate take values in possibly different separable Hilbert spaces. Hence the situations with scalar response or covariate will be particular cases. Our test is based on the remark that checking the no-effect of the functional covariate is equivalent to checking the nullity of the conditional expectation of the response given a sufficiently rich set of projections of the covariate. Such projections could be on elements from finite-dimension subspaces of the Hilbert space where the covariate takes values. Then, the idea is to search a finite-dimension element of norm 1 that is, in some sense, the least favorable for the null hypothesis. With at hand such a least favorable direction, it remains to check the nullity of the conditional expectation of the functional response given the scalar product between the covariate and the selected direction. We follow these steps using a nearest neighbors (NN) smoothing approach. As a result, our test statistic is a quadratic form involving univariate NN smoothing and the asymptotic critical values are given by the standard normal law. The test is able to detect nonparametric alternatives, not only linear ones. The responses could be heteroscedastic with conditional variance of unknown form. The law of the covariate does not need to be known. An empirical study with simulated data is reported. The cases of functional response and functional or scalar covariate are considered. Our conclusion is that the test could be easily implemented and performs well in simulations and real data applications.
Fast dimension-reduced climate model calibration and the effect of data aggregation

How will the climate system respond to anthropogenic forcings? One approach to this question relies on climate model projections. Current climate projections are considerably uncertain. Characterizing and, if possible, reducing this uncertainty is an area of ongoing research. We consider the problem of making projections of the North Atlantic meridional overturning circulation (AMOC). Uncertainties about climate model parameters play a key role in uncertainties in AMOC projections. When the observational data and the climate model output are high-dimensional spatial data sets, the data are typically aggregated due to computational constraints. The effects of aggregation are unclear because statistically rigorous approaches for model parameter inference have been infeasible for high-resolution data. Here we develop a flexible and computationally efficient approach using principal components and basis expansions to study the effect of spatial data aggregation on parametric and projection uncertainties. Our Bayesian reduced-dimensional calibration approach allows us to study the effect of complicated error structures and data-model discrepancies on our ability to learn about climate model parameters from high-dimensional data. Considering high-dimensional spatial observations reduces the effect of deep uncertainty associated with prior specifications for the data-model discrepancy. Also, using the unaggregated data results in sharper projections based on our climate model. Our computationally efficient approach may be widely applicable to a variety of high-dimensional computer model calibration problems.