Laplace Distribution Research Articles

Fourier-synthesis approach for static charge-density reconstruction from theoretical structure factors of CaB6.

In a pilot study, electron-density (ED) and ED Laplacian distributions were reconstructed for the challenging case of CaB6 (Pearson symbol cP7) with conceptually fractional B-B bonds from quantum-chemically calculated structure-factor sets with resolutions 0.5 Å-1 ≤ [sin(θ)/λ]max ≤ 5.0 Å-1 by means of Fourier-synthesis techniques. Convergence of norm deviations of the distributions obtained with respect to the reference ones was obtained in the valence region of the unit cell. The QTAIM (quantum theory of atoms in molecules) atomic charges, and the ED and ED Laplacian values at the characteristic critical points of the Fourier-synthesized distributions have been analysed for each resolution and found to display a convergent behaviour with increasing resolution. The presented method(exponent) (ME) type of Fourier-synthesis approach can qualitatively reconstruct all characteristic chemical bonding features of the ED from valence-electron structure-factor sets with resolutions of about 1.2 Å-1 and beyond, and from all-electron structure-factor sets with resolutions of about 2.0 Å-1 and beyond. Application of the ME type of Fourier-synthesis approach for reconstruction of ED and ED Laplacian distributions at experimental resolution is proposed to complement the usual extrapolation to infinite resolution in Hansen-Coppens multipole model derived static ED distributions.

Flexible Bayesian quantile regression for nonlinear mixed effects models based on the generalized asymmetric Laplace distribution

We propose flexible Bayesian quantile regression for a class of parametric nonlinear mixed effects models for longitudinal data based on the generalized asymmetric Laplace distribution, which exhibits more flexibility in skewness, mode and tail behaviour than the frequently used asymmetric Laplace distribution in quantile regression. An efficient Markov chain Monte Carlo procedure based on the adaptive random walk Metropolis-within-Gibbs sampling algorithm is derived for posterior inference. We demonstrate through simulation studies and empirical analysis that the proposed method could provide more accurate parameter estimation and better model fit than the existing methods.

New highly efficient high‐breakdown estimator of multivariate scatter and location for elliptical distributions

AbstractHigh‐breakdown‐point estimators of multivariate location and shape matrices, such as the ‐estimator with smoothed hard rejection and the Rocke ‐estimator, are generally designed to have high efficiency for Gaussian data. However, many phenomena are non‐Gaussian, and these estimators can therefore have poor efficiency. This article proposes a new tunable ‐estimator, termed the ‐estimator, for the general class of symmetric elliptical distributions, a class containing many common families such as the multivariate Gaussian, ‐, Cauchy, Laplace, hyperbolic, and normal inverse Gaussian distributions. Across this class, the ‐estimator is shown to generally provide higher maximum efficiency than other leading high‐breakdown estimators while maintaining the maximum breakdown point. Furthermore, the ‐estimator is demonstrated to be distributionally robust, and its robustness to outliers is demonstrated to be on par with these leading estimators while also being more stable with respect to initial conditions. From a practical viewpoint, these properties make the ‐estimator broadly applicable for practitioners. These advantages are demonstrated with an example application—the minimum‐variance optimal allocation of financial portfolio investments.

Quantile regression for massive data set

Traditional statistical analysis is challenged by modern massive data sets, which have huge sample size and dimension. Quantile regression has become a popular alternative to least squares method for providing comprehensive description of the response distribution and robustness against heavy-tailed error distributions. On the other hand, non-smooth quantile loss poses a new challenge to massive data sets. To address the problem, we transform the non-differentiable quantile loss function into a convex quadratic loss function based on Expectation-maximization (EM) algorithm using an asymmetric Laplace distribution. Both simulations and real data application are conducted to illustrate the performance of the proposed methods.

L1-norm discriminant analysis via Bhattacharyya error bounds under Laplace distributions

L1-norm discriminant analysis has been proposed to enhance the robustness of classical LDA in the presence of outliers. This paper develops L1-norm discriminant analysis by exploring Bhattacharyya error bounds under Laplace distributions. Unlike some previous models, we assume that the samples of each class in the projected space follow Laplace distributions. It is the first time that the Bhattacharyya error bound is derived under Laplace distributions. It is interesting to note that this bound has a closed-form expression in a one-dimensional projected space. We relax the Bhattacharyya error bound to achieve another bound that can facilitate the design of a tractable model. Since the parameters of Laplace distributions are estimated by the maximum likelihood estimation, this yields the problem of estimating class centroids in our model. We employ a simple yet effective strategy to estimate class centroids in the original space. To address the small-sample-size problem, we transform the original model into a difference criterion by introducing additional parameters. We achieve an alternative representation of our model and design an effective algorithm to solve it. In addition, we also extend our model to its kernel version. The experiments on a series of data sets are done to demonstrate the effectiveness of our method in dealing with contaminated data.

Hyperspectral Denoising Using Asymmetric Noise Modeling Deep Image Prior

Deep image prior (DIP) is a powerful technique for image restoration that leverages an untrained network as a handcrafted prior. DIP can also be used for hyperspectral image (HSI) denoising tasks and has achieved impressive performance. Recent works further incorporate different regularization terms to enhance the performance of DIP and successfully show notable improvements. However, most DIP-based methods for HSI denoising rarely consider the distribution of complicated HSI mixed noise. In this paper, we propose the asymmetric Laplace noise modeling deep image prior (ALDIP) for HSI mixed noise removal. Based on the observation that real-world HSI noise exhibits heavy-tailed and asymmetric properties, we model the HSI noise of each band using an asymmetric Laplace distribution. Furthermore, in order to fully exploit the spatial–spectral correlation, we propose ALDIP-SSTV, which combines ALDIP with a spatial–spectral total variation (SSTV) term to preserve more spatial–spectral information. Experiments on both synthetic data and real-world data demonstrate that ALDIP and ALDIP-SSTV outperform state-of-the-art HSI denoising methods.

THE GGE BIPLOT ON RCIM MODEL FOR ASSESSING THE GENOTYPE-ENVIRONMENT INTERACTION WITH SIMULATING OUTLIERS: ROBUSTNESS IN R-SQUARED PROCRUSTES

The genotype by environment interaction (GEI) analysis was usually done by Additive Main Effects and Multiplicative Interaction (AMMI) model with Biplot features, and recently there was a Row Column Interaction Model (RCIM) alternatively. In the Biplot of genotype (G) and genotype by environment (GE) interactions, known as the GGE Biplot, the main effect of environment (E) was deleted, while the main effect of G and the interaction effect of GE is kept and combined. Subsequently, continuing our recent research of the robustness of the GGE Biplot in RCIM models, this paper aims to develop the GGE Biplot by RCIM model to analyze the GEI with outlying observations. We used the RCIM model with Asymptotic Laplace Distribution (ALD) that was applied on the simulated data with scattered and single environment outliers to evaluate the robustness of the GGE Biplot. In addition, the robustness was evaluated using the R-squared statistic of the Procrustes analysis. It is shown that the GGE Biplot of RCIM with the ALD family function provides better robustness than the Gaussian. A noticeable superiority of the GGE Biplot with RCIM ALD appeared as the percentage of single environment outliers reach the number of rows of the data matrix.

High Dimensional Mode Hunting Using Pettiest Components Analysis.

Principal components analysis has been used to reduce the dimensionality of datasets for a long time. In this paper, we will demonstrate that in mode detection the components of smallest variance, the pettiest components, are more important. We prove that for a multivariate normal or Laplace distribution, we obtain boxes of optimal volume by implementing "pettiest component analysis," in the sense that their volume is minimal over all possible boxes with the same number of dimensions and fixed probability. This reduction in volume produces an information gain that is measured using active information. We illustrate our results with a simulation and a search for modal patterns of digitized images of hand-written numbers using the famous MNIST database; in both cases pettiest components work better than their competitors. In fact, we show that modes obtained with pettiest components generate better written digits for MNIST than principal components.

Private Sampling with Identifiable Cheaters

In this paper we study verifiable sampling from probability distributions in the context of multi-party computation. This has various applications in randomized algorithms performed collaboratively by parties not trusting each other. One example is differentially private machine learning where noise should be drawn, typically from a Laplace or Gaussian distribution, and it is desirable that no party can bias this process. In particular, we propose algorithms to draw random numbers from uniform, Laplace, Gaussian and arbitrary probability distributions, and to verify honest execution of the protocols through zero-knowledge proofs. We propose protocols that result in one party knowing the drawn number and protocols that deliver the drawn random number as a shared secret.

A uniform-Laplace mixture distribution

We introduce a new stochastic model incorporating a mixture of uniform and Laplace distributions. We present basic theoretical properties of this model and discuss related computational issues of parameter estimation via expectation–maximization computational schemes. We check the performance of the estimation algorithm on synthetic data and provide a data example illustrating modeling potential of this novel methodology. A related model involving a mixture of uniform and exponential distributions is studied as well along the same lines.

Structural and dynamic characterization of two-component waterproof mortars by 1H NMR, FT-IR, mechanical and SEM investigations

Advanced 1H NMR relaxometry is an important tool used for the characterization of a large range of materials like mortars. CPMG pulse sequence and saturation recovery methods were used for the measurement of four samples (MF, C166, P288, AQSE) of two-component waterproof mortars at 1, 3, 7, and 28 days after preparation. In general, four dynamics components were observed for the T2 Laplace distributions can be associated with 1H from bound water and with those from small, medium, and large pores. Due to particular characteristics, such as high elasticity the flexural tensile strength and compressive strength could not be measured for all samples. SEM images were recorded, and the reported composition was correlated to NMR parameters and FT-IR spectra. The hydrating behaviour of the two-component waterproofing mortar dried under natural conditions was observed for 28 days and then the hydration process was characterized using 2D 1H NMR T1-T2 correlation map.

An alternative stochastic model for linear portfolios

Stock market returns often tend to follow a non-normal probability distribution due to extreme losses in the tails. These cause fatter tails than normal and consequently heavy-tailed probability distributions are mostly used for modeling returns. In this work, we consider the generalized T (GT) distribution which can be heavy-tailed through its parameters and propose to use it in modeling the random stock returns. The GT distribution also contains the normal, Student t and generalized Laplace distributions as special or limiting cases of the shape parameters. The closed form expressions for the important risk measures are obtained. In case of a portfolio modeling, a multivariate extension of the distribution within the class of elliptical distributions is used. The tail mean-variance portfolio model based on the multivariate GT distribution is developed and optimal portfolio problem is solved. Risk measures for the random return of an asset whose density function is a mixture of the GT densities are obtained. The computability of all expressions derived is shown, and a real data application consisting of seven real stocks from the same sector is given.

A Gaussian-multivariate Laplacian mixture distribution based robust cubature Kalman filter

In this paper, the state estimation problems of nonlinear systems with outlier-corrupted measurements are investigated. First, to model the non-Gaussian noises caused by the randomly occurring measurement outliers, we propose a new Gaussian-multivariate Laplacian mixture (GMLM) distribution and analyze its distribution characteristics. Then, the measurement likelihood probability density function (PDF) is formulated as the GMLM distribution and further expressed as a hierarchical Gaussian expression. Furthermore, by variational Bayesian (VB) method, a robust cubature Kalman filter is derived (GMLMRCKF). Finally, simulation test and real date are utilized to evaluate the effectiveness of our GMLMRCKF, the results illustrate that the proposed filter has better estimation accuracy and consistence in the case of non-stationary heavy-tailed noises than the existing robust filters, i.e., it has almost the same performance as the standard CKF in the absence of outliers and better robust performance in the presence of unknown outliers.

A Bayesian Semi-parametric Quantile Regression Approach for Joint Modeling of Longitudinal Ordinal and Continuous Responses

Quantile regression (QR) models are one of the methods for longitudinal data analysis. When responses seemto be skew and asymmetric due to outliers and heavy-tails, QR models may work suitably. This paper developes the semi-parametric quantile regression model for analyzing longitudinal continuous and ordinal mixed responses. The latent variable model and some threshold parameters are used to perform the quantile regression model’s ordinal part. The error of the latent variable model has Asymmetric Laplace (AL) distribution. The error term’s distribution is assumed to be AL distribution to model the continuous responses. The correlations of longitudinal responses belong to the same individual and those of mixed continuous and ordinal responses are considered using a random-effects approach. The regression spline is used to approximate the non-parametric part of the model. The parameter estimation procedure is performed under aBayesian paradigm using the Gibbs sampling method. A simulation study is performed to demonstrate the proposed model’s performance where the relative biases, standard errors, and root of MSEs of estimated parameters are decreased in the semi- parametric QR joint model when the number of subjects is increased. In our application, it was found that the mother’s age and her child’s age have significant effects on reading ability, and antisocial behavior depends on the child’s gender.

Bayesian group bridge composite quantile regression

Bayesian regularized composite quantile regression (CQR) method with group bridge penalty is adopted to conduct covariate selection and estimation in CQR. MCMC algorithm was improved for posterior inference employing a scale mixture of normal of the asymmetric Laplace distribution (ALD). The suggested algorithm uses priors for the coefficients of regression, which are scale mixtures of multivariate uniform distributions with a particular Gamma distribution as a mixing distribution. Simulation results and analyses of real data show that the suggested MCMC sampler has excellent mixing feature and outperforms the current approaches in terms of prediction accuracy and model selection.

Laplacian eigenvalue distribution of a graph with given independence number

For a graph G, let α(G) be the independence number of G, let L(G) be the Laplacian matrix of G, and let mGI be the number of eigenvalues of L(G) in the interval I. Ahanjideh, Akbari, Fakharan and Trevisan proved that α(G)≤mG[0,n−α(G)] if G is an n-vertex connected graph. Choi, Moon and Park characterized graphs with α(G)=mG[0,n−α(G)] for α(G)=2 and α(G)=n−2. In this paper, we give a characterization for α(G)=3 and α(G)=n−3.

Kumaraswamy Esscher Transformed Laplace Distribution: Properties, Application and Extensions

In this article, we introduce a new generalized family of Esscher transformed Laplace distribution, namely the Kumaraswamy Esscher transformed Laplace distribution. We study the various properties of the distribution including the survival function, hazard rate function, cumulative hazard rate function and reverse hazard rate function. The parameters of the distribution are estimated using the maximum likelihood method of estimation. A real application of this distribution on breaking stress of carbon fibres is also considered. Further, we introduce and study the exponentiated and transmuted exponentiated Kumaraswamy Esscher transformed Laplace distributions.

A mixed effects changepoint quantile regression model for longitudinal data with application on COVID-19 data

IntroductionLongitudinal individual response profiles could exhibit a mixture of two or more phases of increase or decrease in trend throughout the follow-up period, with one or more unknown transition points (changepoints). The detection and estimation of these changepoints is crucial. Most of the proposed statistical methods for detecting and estimating changepoints in literature rely on distributional assumptions that may not hold. In this case, a good alternative is to use a robust approach; the quantile regression model. There are methods in the literature to deal with quantile regression models with a changepoint. These methods ignore the within-subject dependence of longitudinal data.MethodsWe propose a mixed effects quantile regression model with changepoints to account for dependence structure in the longitudinal data. Fixed effects parameters, in addition to the location of the changepoint, are estimated using the profile estimation method. The stochastic approximation EM algorithm is proposed to estimate the fixed effects parameters exploiting the link between an asymmetric Laplace distribution and the quantile regression. In addition, the location of the changepoint is estimated using the usual optimization methods.Results and discussionA simulation study shows that the proposed estimation and inferential procedures perform reasonably well in finite samples. The practical use of the proposed model is illustrated using COVID-19 data. The data focus on the effect of global economic and health factors on the monthly death rate due to COVID-19 from 1 April 2020 to 30th April 2021. the results show a positive effect on the monthly number of patients with COVID-19 in intensive care units (ICUs) for both 0.5th and 0.8th quantiles of new monthly deaths per million. The stringency index, hospital beds, and diabetes prevalence have no significant effect on both 0.5th and 0.8th quantiles of new monthly deaths per million.

A Bayesian Variable Selection Method for Spatial Autoregressive Quantile Models

In this paper, a Bayesian variable selection method for spatial autoregressive (SAR) quantile models is proposed on the basis of spike and slab prior for regression parameters. The SAR quantile models, which are more generalized than SAR models and quantile regression models, are specified by adopting the asymmetric Laplace distribution for the error term in the classical SAR models. The proposed approach could perform simultaneously robust parametric estimation and variable selection in the context of SAR quantile models. Bayesian statistical inferences are implemented by a detailed Markov chain Monte Carlo (MCMC) procedure that combines Gibbs samplers with a probability integral transformation (PIT) algorithm. In the end, empirical numerical examples including several simulation studies and a Boston housing price data analysis are employed to demonstrate the newly developed methodologies.

A Mixture Autoregressive Model Based on an Asymmetric Exponential Power Distribution

In nonlinear time series analysis, the mixture autoregressive model (MAR) is an effective statistical tool to capture the multimodality of data. However, the traditional methods usually need to assume that the error follows a specific distribution that is not adaptive to the dataset. This paper proposes a mixture autoregressive model via an asymmetric exponential power distribution, which includes normal distribution, skew-normal distribution, generalized error distribution, Laplace distribution, asymmetric Laplace distribution, and uniform distribution as special cases. Therefore, the proposed method can be seen as a generalization of some existing model, which can adapt to unknown error structures to improve prediction accuracy, even in the case of fat tail and asymmetry. In addition, an expectation-maximization algorithm is applied to implement the proposed optimization problem. The finite sample performance of the proposed approach is illustrated via some numerical simulations. Finally, we apply the proposed methodology to analyze the daily return series of the Hong Kong Hang Seng Index. The results indicate that the proposed method is more robust and adaptive to the error distributions than other existing methods.