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In a recent paper, we introduced the concept of generalized partial-slice monogenic functions. The class of these functions includes both the theory of monogenic functions and of slice monogenic functions with values in a Clifford algebra. |
In this article we continue the research, carried out in \cite{zajac}, on computing the $*$-product of domains in $\CC^N$. Assuming that $0\in G\subset\CC^N$ is an arbitrary Runge domain and $0\in D\subset\CC^N$ is a bounded, smooth and linearly convex domain (or a non-decreasing union of such ones), we establish a ge... |
We prove that any null-homotopic special holomorphic vector bundle automorphisms of a rank 2 vector bundle E over a Stein space X can be written as a finite product of unipotent holomorphic vector bundle automorphism as well as a finite product of exponentials. |
We provide characterizations of Carleson measures on a certain class of bounded pseudoconvex domains. An example of a vanishing Carleson measure whose Berezin transform does not vanish on the boundary is given in the class of the Hartogs triangles $$\mathbb{H}_{k}:=\left\{ \left(z_{1},z_{2}\right)\in\mathbb{C}^{2}:\le... |
In this note, we study the approximation of singular plurifinely plurisubharmonic function $u$ defined on a plurifinely domain $\Omega$. Under some conditions, we prove that $u$ can be approximated by an increasing sequence of plurisubharmonic functions defined on Euclidean neighborhoods of $\Omega$. |
This article presents several findings regarding second and third-order differential subordination of the form: $$ <br>p(z)+\gamma_1 zp'(z)+\gamma_2 z^2p''(z)\prec h(z)\implies p(z)\prec e^z $$ and $$ <br>p(z)+\gamma_1 zp'(z)+\gamma_2 z^2p''(z)+\gamma_3 z^3p'''(z)\prec h(z)\implies p... |
We give two sufficient criteria for schlichtness of envelopes of holomorphy in terms of topology. These are weakened converses of results of Kerner and Royden. |
In this article, we study about the solutions of second order linear differential equations by considering several conditions on the coefficients of homogenous linear differential equation and its associated non-homogenous linear differential equation. |
Let $\psi$ be a conformal map of the unit disk $\mathbb{D}$ onto an unbounded domain and, for $\alpha >0$, let ${F_\alpha }=\left\{ {z \in \mathbb{D}:\left| {\psi \left( z \right)} \right| = \alpha } \right\}$. If ${H^p}\left( \mathbb{D} \right)$ denotes the classical Hardy space and $d_\mathbb{D} {\left( {0,{F_\al... |
We prove that the open unit ball $\mathbb{B}_n$ of $\mathbb{C}^n$ $(n\ge 2)$ admits a nonsingular holomorphic foliation $\mathcal F$ by closed complex hypersurfaces such that both the union of the complete leaves of $\mathcal F$ and the union of the incomplete leaves of $\mathcal F$ are dense subsets of $\mathbb{B}_n$.... |
Let $A$ and $A'$ be two circular annuli and let $\rho$ be a radial metric defined in the annulus $A'$. Consider the class $\mathcal H_\rho$ of $\rho-$harmonic mappings between $A$ and $A'$. |
We present a geometric proof of the theorem saying that holomorphic maps from Runge domains to affine algebraic varieties admit approximation by Nash maps. Next we generalize this theorem. |
We consider the Wiener--Hopf factorization problem for a matrix function that is completely defined by its first column: the succeeding columns are obtained from the first one by means of a finite group of permutations. The symmetry of this matrix function allows us to reduce the dimension of the problem. In particular... |
Let $\Omega_1$, $\Omega_2$ be two domains in $\mathbb{C}^n$ with Kobayashi metrics $k_{\Omega_i}$ and consider $f \in \mathcal{O}(\Omega_1,\Omega_2)$ a holomorphic mapping. Let $\mathfrak{F}_1$ and $\mathfrak{F}_2$ be a family of geodesics defined on $\Omega_1$ and $\Omega_2$ respectively, where a geodesic between $z$... |
In this paper we discuss equidistribution results for weighted Fekete sets in subsets of the plane. More precisely, we show that Fekete sets are maximally spread out relative to a rescaled version of the Beurling--Landau density, in the "droplet" corresponding to the given weight. |
We study the Dirichlet problem for the complex Monge-Ampère operator on a B-regular domain $\Omega$, allowing boundary data that is singular or unbounded. We introduce the concept of pluri-quasibounded functions on $\Omega$ and $\partial \Omega$, defined by the existence of plurisuperharmonic majorants that dominate t... |
In this article we will first prove a result about convergence in capacity. Using the achieved result we will obtain a general decompositon theorem for complex Monge-Ampere measues which will be used to prove a comparison principle for the complex Monge-Ampere operator. |
We provide a rather general and very simple to compute lower bound for the asymptotic convergence factor of compact subsets of the set of complex numbers with connected complement and finitely many connected components . |
We study a cut-off function lemma in projective spaces. We believe that this is well-known. |
It is of interest to know the sharp bounds of the Hankel determinant, Zalcman functionals, Fekete-Szeg$ \ddot{o} $ inequality as a part of coefficient problems for different classes of functions. Let $\mathcal{H}$ be the class of functions $ f $ which are holomorphic in the open unit disk $\mathbb{D}=\{z\in\mathbb{C}:... |
Adaptive importance sampling (AIS) algorithms are widely used to approximate expectations with respect to complicated target probability distributions. When the target has heavy tails, existing AIS algorithms can provide inconsistent estimators or exhibit slow convergence, as they often neglect the target's tail b... |
Hamiltonian Monte Carlo is a powerful algorithm for sampling from difficult-to-normalize posterior distributions. However, when the geometry of the posterior is unfavorable, it may take many expensive evaluations of the target distribution and its gradient to converge and mix. |
Order statistics arising from $m$ independent but not identically distributed random variables are typically constructed by arranging some $X_{1}, X_{2}, \ldots, X_{m}$, with $X_{i}$ having distribution function $F_{i}(x)$, in increasing order denoted as $X_{(1)} \leq X_{(2)} \leq \ldots \leq X_{(m)}$. In this case, $... |
Given samples (x_1,...,x_m) and (z_1,...,z_n) which we believe are independent realizations of random variables X and Z respectively, where we further believe that Z=X+Y with Y independent of X, the problem is to estimate the distribution of Y. We present a new method for doing this, involving simulation. Experiments ... |
The R package "sensobol" provides several functions to conduct variance-based uncertainty and sensitivity analysis, from the estimation of sensitivity indices to the visual representation of the results. It implements several state-of-the-art first and total-order estimators and allows the computation of up to... |
A novel solution to the smoothing problem for multi-object dynamical systems is proposed and evaluated. The systems of interest contain an unknown and varying number of dynamical objects that are partially observed under noisy and corrupted observations. |
This paper presents a new open source Python framework for causal discovery from observational data and domain background knowledge, aimed at causal graph and causal mechanism modeling. The 'cdt' package implements the end-to-end approach, recovering the direct dependencies (the skeleton of the causal graph) a... |
Monte Carlo sampling methods are the standard procedure for approximating complicated integrals of multidimensional posterior distributions in Bayesian inference. In this work, we focus on the class of Layered Adaptive Importance Sampling (LAIS) scheme, which is a family of adaptive importance samplers where Markov ch... |
This paper is devoted to the problem of sampling Gaussian fields in high dimension. Solutions exist for two specific structures of inverse covariance : sparse and circulant. |
Several probability distributions have been proposed in the literature, especially with the aim of obtaining models that are more flexible relative to the behaviors of the density and hazard rate functions. Recently, a new generalization of the Lindley distribution was proposed by Ghitany et al. (2013), called power L... |
Although various forms of linkage map construction software are widely available, there is a distinct lack of packages for use in the R statistical computing environment. This article introduces the ASMap linkage map construction R package which contains functions that use the efficient MSTmap algorithm for clustering... |
The R package CVEK introduces a suite of flexible machine learning models and robust hypothesis tests for learning the joint nonlinear effects of multiple covariates in limited samples. It implements the Cross-validated Ensemble of Kernels (CVEK)(Liu and Coull 2017), an ensemble-based kernel machine learning method th... |
I consider the problem of integrating a function $f$ over the $d$-dimensional unit cube. I describe a multilevel Monte Carlo method that estimates the integral with variance at most $\epsilon^{2}$ in $O(d+\ln(d)d_{t}\epsilon^{-2})$ time, for $\epsilon>0$, where $d_{t}$ is the truncation dimension of $f$. |
We examine the behaviour of the pseudo-marginal random walk Metropolis algorithm, where evaluations of the target density for the accept/reject probability are estimated rather than computed precisely. Under relatively general conditions on the target distribution, we obtain limiting formulae for the acceptance rate a... |
The EM algorithm is a method for finding the maximum likelihood estimate of a model in the presence of missing data. Unfortunately, EM does not produce a parameter covariance matrix for standard errors. |
In multivariate or spatial extremes, inference for max-stable processes observed at a large collection of locations is among the most challenging problems in computational statistics, and current approaches typically rely on less expensive composite likelihoods constructed from small subsets of data. In this work, we ... |
Random Forests [Breiman:2001] (RF) are a fully non-parametric statistical method requiring no distributional assumptions on covariate relation to the response. RF are a robust, nonlinear technique that optimizes predictive accuracy by fitting an ensemble of trees to stabilize model estimates. |
Estimating the spectral characteristics of a nonstationary random process is an important but challenging task, which can be facilitated by exploiting structural properties of the process. In certain applications, the observed processes are underspread, i.e., their time and frequency correlations exhibit a reasonably ... |
Constructing approximations that can accurately mimic the behavior of complex models at reduced computational costs is an important aspect of uncertainty quantification. Despite their flexibility and efficiency, classical surrogate models such as Kriging or polynomial chaos expansions tend to struggle with highly non-... |
In this article, we present a recently released R package for Bayesian calibration. Many industrial fields are facing unfeasible or costly field experiments. |
We desribe how we extend R with an external C++ code library by using the Rcpp package. Our working example uses the recent machine learning library and application 'Corels' providing optimal yet easily interpretable rule lists <<a href="https://arxiv.org/abs/1704.01701" data-arxiv-id="1704.01701" class="li... |
Visualization of high-dimensional data is counter-intuitive using conventional graphs. Parallel coordinates are proposed as an alternative to explore multivariate data more effectively. |
We present theoretical and practical properties of the affine-invariant ensemble sampler Markov chain Monte Carlo method. In high dimensions the affine-invariant ensemble sampler shows unusual and undesirable properties. |
A goal of systems biology is to understand the dynamics of intracellular systems. Stochastic chemical kinetic models are often utilized to accurately capture the stochastic nature of these systems due to low numbers of molecules. |
We introduce a new goodness-of-fit test for regular vine (R-vine) copula models. R-vine copulas are a very flexible class of multivariate copulas based on a pair-copula construction (PCC). |
We investigate the problem of sampling from posterior distributions with intractable normalizing constants in Bayesian inference. Our solution is a new generative modeling approach based on optimal transport (OT) that learns a deterministic map from a reference distribution to the target posterior through constrained ... |
Approximating the recent phylogeny of $N$ phased haplotypes at a set of variants along the genome is a core problem in modern population genomics and central to performing genome-wide screens for association, selection, introgression, and other signals. The Li & Stephens (LS) model provides a simple yet powerful h... |
Gibbs samplers are preeminent Markov chain Monte Carlo algorithms used in computational physics and statistical computing. Yet, their most fundamental properties, such as relations between convergence characteristics of their various versions, are not well understood. |
Digital nets are among the most successful methods to construct low-discrepancy point sets for quasi-Monte Carlo integration. Their quality is traditionally assessed by a measure called the $t$-value. |
Sample selection models represent a common methodology for correcting bias induced by data missing not at random. These models are not empirically identifiable without exclusion restrictions. |
Gaussian latent variable models are a key class of Bayesian hierarchical models with applications in many fields. Performing Bayesian inference on such models can be challenging as Markov chain Monte Carlo algorithms struggle with the geometry of the resulting posterior distribution and can be prohibitively slow. |
We attempt to trace the history and development of Markov chain Monte Carlo (MCMC) from its early inception in the late 1940s through its use today. We see how the earlier stages of Monte Carlo (MC, not MCMC) research have led to the algorithms currently in use. |
Approximate Bayesian computation (ABC) has gained popularity over the past few years for the analysis of complex models arising in population genetic, epidemiology and system biology. Sequential Monte Carlo (SMC) approaches have become work horses in ABC. |
As a statistical tool to assist formative assessments in educational settings, diagnostic classification models (DCMs) have been increasingly used to provide diagnostic information regarding examinees' attributes. DCMs often adopt a dichotomous division such as the mastery and non-mastery of attributes to express ... |
Polynomial chaos is a powerful technique for propagating uncertainty through ordinary and partial differential equations. Random variables are expanded in terms of orthogonal polynomials and differential equations are derived for the expansion coefficients. |
The iotools package provides a set of tools for Input/Output (I/O) intensive datasets processing in R (R Core Team, 2014). Efficent parsing methods are included which minimize copying and avoid the use of intermediate string representations whenever possible. |
In this note we describe experiments on an implementation of two methods proposed in the literature for computing regions that correspond to a notion of order statistics for multidimensional data. Our implementation, which works for any dimension greater than one, is the only that we know of to be publicly available. |
A first proposal of a sparse and cellwise robust PCA method is presented. Robustness to single outlying cells in the data matrix is achieved by substituting the squared loss function for the approximation error by a robust version. |
Ising and Potts models are an important class of discrete probability distributions which originated from Statistical Physics and since then have found applications in several disciplines. Simulation from these models is a well known challenging problem. |
Multistate Markov models are a canonical parametric approach for data modeling of observed or latent stochastic processes supported on a finite state space. Continuous-time Markov processes describe data that are observed irregularly over time, as is often the case in longitudinal medical data, for example. |
Minimum distance estimation methodology based on an empirical distribution function has been popular due to its desirable properties including robustness. Even though the statistical literature is awash with the research on the minimum distance estimation, the most of it is confined to the theoretical findings: only f... |
An off-line handwritten alphabetical character recognition system using multilayer feed forward neural network is described in the paper. A new method, called, diagonal based feature extraction is introduced for extracting the features of the handwritten alphabets. |
We describe and analyze a variance reduction approach for Monte Carlo (MC) sampling that accelerates the estimation of statistics of computationally expensive simulation models using an ensemble of models with lower cost. These lower cost models --- which are typically lower fidelity with unknown statistics --- are us... |
In this article, a new nonparametric and robust method of forecasting hierarchical functional time series is presented. The method is compared with Hyndman and Shang's method with respect to their unbiasedness, effectiveness, robustness, and computational complexity. |
Recent work incorporating geometric ideas in Markov chain Monte Carlo is reviewed in order to highlight these advances and their possible application in a range of domains beyond Statistics. A full exposition of Markov chains and their use in Monte Carlo simulation for Statistical inference and molecular dynamics is p... |
Statistical Taylor expansion replaces the input precise variables in a conventional Taylor expansion with random variables each with known distribution, to calculate the result mean and deviation. It is based on the uncorrelated uncertainty assumption: Each input variable is measured independently with fine enough sta... |
We propose a very fast approximate Markov Chain Monte Carlo (MCMC) sampling framework that is applicable to a large class of sparse Bayesian inference problems, where the computational cost per iteration in several models is of order $O(ns)$, where $n$ is the sample size, and $s$ the underlying sparsity of the model. ... |
Hamiltonian Monte Carlo (HMC) has been widely adopted in the statistics community because of its ability to sample high-dimensional distributions much more efficiently than other Metropolis-based methods. Despite this, HMC often performs sub-optimally on distributions with high correlations or marginal variances on mu... |
Existing guarantees for algorithms sampling from nonlogconcave measures on $\mathbb{R}^d$ are generally inexplicit or unscalable. Even for the class of measures with logdensities that have bounded Hessians and are strongly concave outside a Euclidean ball of radius $R$, no available theory is comprehensively satisfact... |
Gibbs random fields (GRF) are polymorphous statistical models that can be used to analyse different types of dependence, in particular for spatially correlated data. However, when those models are faced with the challenge of selecting a dependence structure from many, the use of standard model choice methods is hamper... |
We describe the vote package in R, which implements the plurality (or first-past-the-post), two-round runoff, score, approval and single transferable vote (STV) electoral systems, as well as methods for selecting the Condorcet winner and loser. We emphasize the STV system, which we have found to work well in practice ... |
We consider the approximation of expectations with respect to the distribution of a latent Markov process given noisy measurements. This is known as the smoothing problem and is often approached with particle and Markov chain Monte Carlo (MCMC) methods. |
The demand for high-performance computing (HPC) is ever-increasing for everyday statistical computing purposes. The downside is that we need to write specialized code for each HPC environment. |
State-space models (SSMs) are a powerful statistical tool for modelling time-varying systems via a latent state. In these models, the latent state is never directly observed. |
Record linkage is the process of matching together records from different data sources that belong to the same entity. Record linkage is increasingly being used by many organizations including statistical, health, government etc. to link administrative, survey, and other files to create a robust file for more comprehe... |
We consider the problem of local radioelectric property estimation from global electromagnetic scattering measurements. This challenging ill-posed high dimensional inverse problem can be explored by intensive computations of a parallel Maxwell solver on a petaflopic supercomputer. |
A new data-based smoothing parameter for circular kernel density (and its derivatives) estimation is proposed. Following the plug-in ideas, unknown quantities on an optimal smoothing parameter are replaced by suitable estimates. |
An intriguing new class of piecewise deterministic Markov processes (PDMPs) has recently been proposed as an alternative to Markov chain Monte Carlo (MCMC). In order to facilitate the application to a larger class of problems, we propose a new class of PDMPs termed Gibbs zig-zag samplers, which allow parameters to be ... |
In this paper, the parameter estimation of ARMA(p,q) model is given by approximate Bayesian computation algorithm. In order to improve the sampling efficiency of the algorithm, approximate Bayesian computation should select as many statistics as possible with parameter information in low dimension. |
Motivation: Elastic net regression is a form of penalized regression that lies between ridge and least absolute shrinkage and selection operator (LASSO) regression. The elastic net penalty is a powerful tool controlling the impact of correlated predictors and the overall complexity of generalized linear regression mod... |
This short note investigates convergence of adaptive MCMC algorithms, i.e.\ algorithms which modify the Markov chain update probabilities on the fly. We focus on the Containment condition introduced in \cite{roberts2007coupling}. |
We propose a novel use of a recent new computational tool for Bayesian inference, namely the Approximate Bayesian Computation (ABC) methodology. ABC is a way to handle models for which the likelihood function may be intractable or even unavailable and/or too costly to evaluate; in particular, we consider the problem o... |
This study describes a method to quantify potential gait changes in human subjects. Microsoft Kinect devices were used to provide and track coordinates of fifteen different joints of a subject over time. |
We consider Bayesian inference problems with computationally intensive likelihood functions. We propose a Gaussian process (GP) based method to approximate the joint distribution of the unknown parameters and the data. |
MCMC algorithms such as Metropolis-Hastings algorithms are slowed down by the computation of complex target distributions as exemplified by huge datasets. We offer in this paper a useful generalisation of the Delayed Acceptance approach, devised to reduce the computational costs of such algorithms by a simple and univ... |
We introduce a novel approach to compositional data analysis based on $L^{\infty}$-normalization, addressing challenges posed by zero-rich high-throughput data. Traditional methods like Aitchison's transformations require excluding zeros, conflicting with the reality that omics datasets contain structural zeros th... |
Models with intractable normalizing functions arise frequently in statistics. Common examples of such models include exponential random graph models for social networks and Markov point processes for ecology and disease modeling. |
We present a sequential Monte Carlo sampler variant of the partial rejection control algorithm, and show that this variant can be considered as a sequential Monte Carlo sampler with a modified mutation kernel. We prove that the new sampler can reduce the variance of the incremental importance weights when compared wit... |
The R package sns implements Stochastic Newton Sampler (SNS), a Metropolis-Hastings Monte Carlo Markov Chain algorithm where the proposal density function is a multivariate Gaussian based on a local, second-order Taylor series expansion of log-density. The mean of the proposal function is the full Newton step in Newto... |
Given a set of data points sampled from some underlying space, there are two important challenges in geometric and topological data analysis when dealing with sampled data: reconstruction -- how to assemble discrete samples into global structures, and inference -- how to extract geometric and topological information fr... |
Solving semiparametric models can be computationally challenging because the dimension of parameter space may grow large with increasing sample size. Classical Newton's method becomes quite slow and unstable with intensive calculation of the large Hessian matrix and its inverse. |
This article introduces a novel methodology for the massive parallelization of projection-based depths, addressing the computational challenges of data depth in high-dimensional spaces. We propose an algorithmic framework based on Refined Random Search (RRS) and demonstrate significant speedup (up to 7,000 times faste... |
We propose a general-purpose approximation to the Ferguson-Klass algorithm for generating samples from Lévy processes without Gaussian components. We show that the proposed method is more than 1000 times faster than the standard Ferguson-Klass algorithm without a significant loss of precision. |
Boosting is a popular algorithm in supervised machine learning with wide applications in regression and classification problems. It combines weak learners, such as regression trees, to obtain accurate predictions. |
Simulations play a crucial role in the modern scientific process. Yet despite (or due to) this ubiquity, the Data Science community shares neither a comprehensive definition for a "high-quality" study nor a consolidated guide to designing one. |
In shared micromobility networks, such as bike-share and scooter-share networks, using trip data to accurately estimate demand in docked and dockless systems is critical to analyzing how the system is operating, such as identifying the number of dissatisfied users, operational costs, and equity in access, especially fo... |
The distribution function of the sum $Z$ of two standard normally distributed random variables $X$ and $Y$ is computed with the concept of copulas to model the dependency between $X$ and $Y$. By using implicit copulas such as the Gauss- or t-copula as well as Archimedean Copulas such as the Clayton-, Gumbel- or Frank-... |
Statistical testing is widespread and critical for a variety of scientific disciplines. The advent of machine learning and the increase of computing power has increased the interest in the analysis and statistical testing of multidimensional data. |
bayesian_pyhf is a Python package that allows for the parallel Bayesian and frequentist evaluation of multi-channel binned statistical models. The Python library pyhf is used to build such models according to the HistFactory framework and already includes many frequentist inference methodologies. |
Mixtures of matrix Gaussian distributions provide a probabilistic framework for clustering continuous matrix-variate data, which are becoming increasingly prevalent in various fields. Despite its widespread adoption and successful application, this approach suffers from over-parameterization issues, making it less sui... |
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