The L\'evy distribution, alongside the Normal and Cauchy distributions, is one of the only three stable distributions whose density can be obtained in a closed form. However, there are only a few specific goodness-of-fit tests for the L\'evy distribution. In this paper, two novel classes of goodness-of-fit tests for the L\'evy distribution are proposed. Both tests are based on V-empirical Laplace transforms. New tests are scale free under the null hypothesis, which makes them suitable for testing the composite hypothesis. The finite sample and limiting properties of test statistics are obtained. In addition, a generalization of the recent Bhati-Kattumannil goodness-of-fit test to the L\'evy distribution is considered. For assessing the quality of novel and competitor tests, the local Bahadur efficiencies are computed, and a wide power study is conducted. Both criteria clearly demonstrate the quality of the new tests. The applicability of the novel tests is demonstrated with two real-data examples.
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The autologistic actor attribute model, or ALAAM, is the social influence counterpart of the better-known exponential-family random graph model (ERGM) for social selection. Extensive experience with ERGMs has shown that the problem of near-degeneracy which often occurs with simple models can be overcome by using "geometrically weighted" or "alternating" statistics. In the much more limited empirical applications of ALAAMs to date, the problem of near-degeneracy, although theoretically expected, appears to have been less of an issue. In this work I present a comprehensive survey of ALAAM applications, showing that this model has to date only been used with relatively small networks, in which near-degeneracy does not appear to be a problem. I show near-degeneracy does occur in simple ALAAM models of larger empirical networks, define some geometrically weighted ALAAM statistics analogous to those for ERGM, and demonstrate that models with these statistics do not suffer from near-degeneracy and hence can be estimated where they could not be with the simple statistics.
Sequential models, such as Recurrent Neural Networks and Neural Ordinary Differential Equations, have long suffered from slow training due to their inherent sequential nature. For many years this bottleneck has persisted, as many thought sequential models could not be parallelized. We challenge this long-held belief with our parallel algorithm that accelerates GPU evaluation of sequential models by up to 3 orders of magnitude faster without compromising output accuracy. The algorithm does not need any special structure in the sequential models' architecture, making it applicable to a wide range of architectures. Using our method, training sequential models can be more than 10 times faster than the common sequential method without any meaningful difference in the training results. Leveraging this accelerated training, we discovered the efficacy of the Gated Recurrent Unit in a long time series classification problem with 17k time samples. By overcoming the training bottleneck, our work serves as the first step to unlock the potential of non-linear sequential models for long sequence problems.
An unconventional approach is applied to solve the one-dimensional Burgers' equation. It is based on spline polynomial interpolations and Hopf-Cole transformation. Taylor expansion is used to approximate the exponential term in the transformation, then the analytical solution of the simplified equation is discretized to form a numerical scheme, involving various special functions. The derived scheme is explicit and adaptable for parallel computing. However, some types of boundary condition cannot be specified straightforwardly. Three test cases were employed to examine its accuracy, stability, and parallel scalability. In the aspect of accuracy, the schemes employed cubic and quintic spline interpolation performs equally well, managing to reduce the $\ell_{1}$, $\ell_{2}$ and $\ell_{\infty}$ error norms down to the order of $10^{-4}$. Due to the transformation, their stability condition $\nu \Delta t/\Delta x^2 > 0.02$ includes the viscosity/diffusion coefficient $\nu$. From the condition, the schemes can run at a large time step size $\Delta t$ even when grid spacing $\Delta x$ is small. These characteristics suggest that the method is more suitable for operational use than for research purposes.
In root finding and optimization, there are many cases where there is a closed set $A$ one does not the sequence constructed by one's favourite method will converge to A (here, we do not assume extra properties on $A$ such as being convex or connected). For example, if one wants to find roots, and one chooses initial points in the basin of attraction for 1 root $x^*$ (a fact which one may not know before hand), then one will always end up in that root. In this case, one would like to have a mechanism to avoid this point $z^*$ in the next runs of one's algorithm. In this paper, we propose a new method aiming to achieve this: we divide the cost function by an appropriate power of the distance function to $A$. This idea is inspired by how one would try to find all roots of a function in 1 variable. We first explain the heuristic for this method in the case where the minimum of the cost function is exactly 0, and then explain how to proceed if the minimum is non-zero (allowing both positive and negative values). The method is very suitable for iterative algorithms which have the descent property. We also propose, based on this, an algorithm to escape the basin of attraction of a component of positive dimension to reach another component. Along the way, we compare with main existing relevant methods in the current literature. We provide several examples to illustrate the usefulness of the new approach.
We study how to verify specific frequency distributions when we observe a stream of $N$ data items taken from a universe of $n$ distinct items. We introduce the \emph{relative Fr\'echet distance} to compare two frequency functions in a homogeneous manner. We consider two streaming models: insertions only and sliding windows. We present a Tester for a certain class of functions, which decides if $f $ is close to $g$ or if $f$ is far from $g$ with high probability, when $f$ is given and $g$ is defined by a stream. If $f$ is uniform we show a space $\Omega(n)$ lower bound. If $f$ decreases fast enough, we then only use space $O(\log^2 n\cdot \log\log n)$. The analysis relies on the Spacesaving algorithm \cite{MAE2005,Z22} and on sampling the stream.
In this study, we propose a staging area for ingesting new superconductors' experimental data in SuperCon that is machine-collected from scientific articles. Our objective is to enhance the efficiency of updating SuperCon while maintaining or enhancing the data quality. We present a semi-automatic staging area driven by a workflow combining automatic and manual processes on the extracted database. An anomaly detection automatic process aims to pre-screen the collected data. Users can then manually correct any errors through a user interface tailored to simplify the data verification on the original PDF documents. Additionally, when a record is corrected, its raw data is collected and utilised to improve machine learning models as training data. Evaluation experiments demonstrate that our staging area significantly improves curation quality. We compare the interface with the traditional manual approach of reading PDF documents and recording information in an Excel document. Using the interface boosts the precision and recall by 6% and 50%, respectively to an average increase of 40% in F1-score.
For large Reynolds number flows, it is typically necessary to perform simulations that are under-resolved with respect to the underlying flow physics. For nodal discontinuous spectral element approximations of these under-resolved flows, the collocation projection of the nonlinear flux can introduce aliasing errors which can result in numerical instabilities. In Dzanic and Witherden (J. Comput. Phys., 468, 2022), an entropy-based adaptive filtering approach was introduced as a robust, parameter-free shock-capturing method for discontinuous spectral element methods. This work explores the ability of entropy filtering for mitigating aliasing-driven instabilities in the simulation of under-resolved turbulent flows through high-order implicit large eddy simulations of a NACA0021 airfoil in deep stall at a Reynolds number of 270,000. It was observed that entropy filtering can adequately mitigate aliasing-driven instabilities without degrading the accuracy of the underlying high-order scheme on par with standard anti-aliasing methods such as over-integration, albeit with marginally worse performance at higher approximation orders.
The Lasso is a method for high-dimensional regression, which is now commonly used when the number of covariates $p$ is of the same order or larger than the number of observations $n$. Classical asymptotic normality theory does not apply to this model due to two fundamental reasons: $(1)$ The regularized risk is non-smooth; $(2)$ The distance between the estimator $\widehat{\boldsymbol{\theta}}$ and the true parameters vector $\boldsymbol{\theta}^*$ cannot be neglected. As a consequence, standard perturbative arguments that are the traditional basis for asymptotic normality fail. On the other hand, the Lasso estimator can be precisely characterized in the regime in which both $n$ and $p$ are large and $n/p$ is of order one. This characterization was first obtained in the case of Gaussian designs with i.i.d. covariates: here we generalize it to Gaussian correlated designs with non-singular covariance structure. This is expressed in terms of a simpler ``fixed-design'' model. We establish non-asymptotic bounds on the distance between the distribution of various quantities in the two models, which hold uniformly over signals $\boldsymbol{\theta}^*$ in a suitable sparsity class and over values of the regularization parameter. As an application, we study the distribution of the debiased Lasso and show that a degrees-of-freedom correction is necessary for computing valid confidence intervals.
For the extended skew-normal distribution, which represents an extension of the normal (or Gaussian) distribution, we focus on the properties of the log-likelihood function and derived quantities in the the bivariate case. Specifically, we derive explicit expressions for the score function and the information matrix, in the observed and the expected form; these do not appear to have been examined before in the literature. Corresponding computing code in R language is provided, which implements the formal expressions.
We propose an approach to compute inner and outer-approximations of the sets of values satisfying constraints expressed as arbitrarily quantified formulas. Such formulas arise for instance when specifying important problems in control such as robustness, motion planning or controllers comparison. We propose an interval-based method which allows for tractable but tight approximations. We demonstrate its applicability through a series of examples and benchmarks using a prototype implementation.
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