Laplace Distribution Research Articles

Low-level turbulence risk assessment and visualization using temporal rate of change of headwind of an aircraft

In this study, we focused on the temporal rate of change of headwind, which is one of the recorded parameters in the aircraft’s in-flight quick access recorder data. We selected the Laplace distribution and utilized the scaling parameter to construct a low-level turbulence risk assessment model. Using this model, we calculated the risk of low-level turbulence occurrence at five airports in Japan based on the month, time of day, and wind speed. We visualized how the geographical conditions at each airport influenced risk in relation to airport wind speeds. We developed a low-level turbulence visualization site linked to weather conditions using these results to enable pilots to easily verify low-level turbulence risk and incorporate this information into their flight routines. These findings are anticipated to significantly enhance aircraft safety.

Deep Variational Network Toward Blind Image Restoration.

Blind image restoration (IR) is a common yet challenging problem in computer vision. Classical model-based methods and recent deep learning (DL)-based methods represent two different methodologies for this problem, each with their own merits and drawbacks. In this paper, we propose a novel blind image restoration method, aiming to integrate both the advantages of them. Specifically, we construct a general Bayesian generative model for the blind IR, which explicitly depicts the degradation process. In this proposed model, a pixel-wise non-i.i.d. Gaussian distribution is employed to fit the image noise. It is with more flexibility than the simple i.i.d. Gaussian or Laplacian distributions as adopted in most of conventional methods, so as to handle more complicated noise types contained in the image degradation. To solve the model, we design a variational inference algorithm where all the expected posteriori distributions are parameterized as deep neural networks to increase their model capability. Notably, such an inference algorithm induces a unified framework to jointly deal with the tasks of degradation estimation and image restoration. Further, the degradation information estimated in the former task is utilized to guide the latter IR process. Experiments on two typical blind IR tasks, namely image denoising and super-resolution, demonstrate that the proposed method achieves superior performance over current state-of-the-arts.

The probability density function of the surface electromyogram and its dependence on contraction force in the vastus lateralis

IntroductionThe probability density function (PDF) of the surface electromyogram (sEMG) depends on contraction force. This dependence, however, has so far been investigated by having the subject generate force at a few fixed percentages of MVC. Here, we examined how the shape of the sEMG PDF changes with contraction force when this force was gradually increased from zero.MethodsVoluntary surface EMG signals were recorded from the vastus lateralis of healthy subjects as force was increased in a continuous manner vs. in a step-wise fashion. The sEMG filling process was examined by measuring the EMG filling factor, computed from the non-central moments of the rectified sEMG signal.Results(1) In 84% of the subjects, as contraction force increased from 0 to 10% MVC, the sEMG PDF shape oscillated back and forth between the semi-degenerate and the Gaussian distribution.(2) The PDF–force relation varied greatly among subjects for forces between 0 and ~ 10% MVC, but this variability was largely reduced for forces above 10% MVC.(3) The pooled analysis showed that, as contraction force gradually increased, the sEMG PDF evolved rapidly from the semi-degenerate towards the Laplacian distribution from 0 to 5% MVC, and then more slowly from the Laplacian towards the Gaussian distribution for higher forces.ConclusionsThe study demonstrated that the dependence of the sEMG PDF shape on contraction force can only be reliably assessed by gradually increasing force from zero, and not by performing a few constant-force contractions. The study also showed that the PDF–force relation differed greatly among individuals for contraction forces below 10% MVC, but this variability was largely reduced when force increased above 10% MVC.

Interval-valued linear regression model with an asymmetric Laplace distribution

Interval-valued linear regression model with an asymmetric Laplace distribution

Asymmetric Laplace scale mixtures for the distribution of cryptocurrency returns

AbstractRecent studies about cryptocurrency returns show that their distribution can be highly-peaked, skewed, and heavy-tailed, with a large excess kurtosis. To accommodate all these peculiarities, we propose the asymmetric Laplace scale mixture (ALSM) family of distributions. Each member of the family is obtained by dividing the scale parameter of the conditional asymmetric Laplace (AL) distribution by a convenient mixing random variable taking values on all or part of the positive real line and whose distribution depends on a parameter vector $$\varvec{\theta }$$ θ providing greater flexibility to the resulting ALSM. Advantageously concerning the AL distribution, our family members allow for a wider range of values for skewness and kurtosis. For illustrative purposes, we consider different mixing distributions; they give rise to ALSMs having a closed-form probability density function where the AL distribution is obtained as a special case under a convenient choice of $$\varvec{\theta }$$ θ . We examine some properties of our ALSMs such as hierarchical and stochastic representations and moments of practical interest. We describe an EM algorithm to obtain maximum likelihood estimates of the parameters for all the considered ALSMs. We fit these models to the returns of two cryptocurrencies, considering several classical distributions for comparison. The analysis shows how our models represent a valid alternative to the considered competitors in terms of AIC, BIC, and likelihood-ratio tests.

Acoustic beamforming using maximum likelihood criteria with Laplacian model of the desired signal

Acoustic beamforming is a technique used in audio engineering and signal processing to filter sound waves in the spatial domain. Usually, an array of microphones is used to capture sound from different directions. These array signals are then combined to reinforce each other in the desired direction while cancelling the noise or interference from other directions. An important application of acoustic beamforming is the cocktail party scenario where multiple people speak simultaneously in a noisy room. To capture the speech of only the desired speaker, acoustic beamforming is used. Usually, in such cases, the target speech is modelled as a zero-mean complex Gaussian distributed random variable. However, the target speech coefficients are sparse in the time-frequency domain. Hence, in our work, we model the speech coefficients using a zero-mean circular Laplacian distribution. After modelling the target speech, we formulate a beamformer based on the maximum likelihood criteria. We add a distortionless constraint to the proposed beamformer to further improve performance. The final solution of the proposed beamformer encourages sparsity, indicating that it models the target speech better than the complex Gaussian distribution. Simulations show the effectiveness of the proposed beamformer in capturing the target speech and rejecting interfering speakers.

Bounded diffusing diffusivities: Brownian yet non-Gaussian diffusion

Brownian yet non-Gaussian diffusion has been recently reported in a huge number of biological and soft matter systems. Meanwhile, an archetypal theoretical model called ‘diffusing diffusivities’ is proposed to interpret it. Based on this spirit of diffusing diffusivities, we extend the original diffusing diffusivities (with the unbounded exponential distribution) to the case that the diffusivity is constructed by a bounded stochastic process, i.e., corresponding diffusivities (with certain upper and lower bounds) obeying bounded power-law distribution. We demonstrate that Brownian yet non-Gaussian diffusion can be reproduced by this bounded diffusing diffusivities, via numerical simulations and analytic derivations. Specifically, the short-time distribution of displacement, as the indicator of the Brownian yet non-Gaussian diffusion, is derived analytically by means of superstatistical approach. This short-time distribution is distinct from the Laplace distribution that appears in the original model. The long-time Gaussian displacement distribution is obtained by utilizing the subordination concept. The bounded diffusing diffusivity here may be beneficial to further understanding the diffusive process of particles in complex and inhomogeneous environments. Our work enriches the diffusing diffusivity family and sheds new light on the concept of the Brownian yet non-Gaussian diffusion under stochastic process.

Robust multi-target regression with improved stochastic configuration networks and its applications

To improve the accuracy and robustness of stochastic configuration networks (SCNs) for resolving multi-target regression tasks, this paper proposes a robust modeling approach based on improved stochastic configuration networks. A parallel implementation of SCN models is designed to incrementally generate the hidden nodes, which enhances the diversity of hidden layer mapping through information superposition and spanning connection. We employ an elastic net regularization model to sparsely constrain the model parameters to characterize the correlation among multiple targets. Then, the mixture Laplace distributions are used as the prior distribution of each target modeling error, and the output weights of the SCN model are re-evaluated by maximizing a posteriori estimation to enhance model’s robustness with respect to some uncertainties presented in training samples. The modelling performance of the proposed solution is tested on six standard datasets and the historical data of a municipal solid waste incineration process. The experimental results show that the proposed modeling technique has advantages in terms of both the prediction accuracy and the robustness.

SA-SatMVS: Slope Feature-Aware and Across-Scale Information Integration for Large-Scale Earth Terrain Multi-View Stereo

Satellite multi-view stereo (MVS) is a fundamental task in large-scale Earth surface reconstruction. Recently, learning-based multi-view stereo methods have shown promising results in this field. However, these methods are mainly developed by transferring the general learning-based MVS framework to satellite imagery, which lacks consideration of the specific terrain features of the Earth’s surface and results in inadequate accuracy. In addition, mainstream learning-based methods mainly use equal height interval partition, which insufficiently utilizes the height hypothesis surface, resulting in inaccurate height estimation. To address these challenges, we propose an end-to-end terrain feature-aware height estimation network named SA-SatMVS for large-scale Earth surface multi-view stereo, which integrates information across different scales. Firstly, we transform the Sobel operator into slope feature-aware kernels to extract terrain features, and a dual encoder–decoder architecture with residual blocks is applied to incorporate slope information and geometric structural characteristics to guide the reconstruction process. Secondly, we introduce a pixel-wise unequal interval partition method using a Laplacian distribution based on the probability volume obtained from other scales, resulting in more accurate height hypotheses for height estimation. Thirdly, we apply an adaptive spatial feature extraction network to search for the optimal fusion method for feature maps at different scales. Extensive experiments on the WHU-TLC dataset also demonstrate that our proposed model achieves the best MAE metric of 1.875 and an RMSE metric of 3.785, which constitutes a state-of-the-art performance.

A New Multimodal Modification of the Skew Family of Distributions: Properties and Applications to Medical and Environmental Data

The skew distribution has the characteristic of appropriately modeling asymmetric unimodal data. However, in practice, there are several cases in which the data present more than one mode. In the literature, it is possible to find a large number of authors who have studied extensions based on the skew distribution to model this type of data. In this article, a new family is introduced, consisting of a multimodal modification to the family of skew distributions. Using the methodology of the weighted version of a function, we perform the product of the density function of a family of skew distributions with a polynomial of degree 4, thus obtaining a more flexible model that allows modeling data sets, whose distribution contains at most three modes. The density function, some properties, moments, skewness coefficients, and kurtosis of this new family are presented. This study focuses on the particular cases of skew-normal and Laplace distributions, although it can be applied to any other distribution. A simulation study was carried out, to study the behavior of the model parameter estimates. Illustrations with real data, referring to medicine and environmental data, show the practical performance of the proposed model in the two particular cases presented.

Bayesian inference for Laplace distribution based on complete and censored samples with illustrations

In this paper, Bayesian estimates are derived for the location and scale parameters of the Laplace distribution based on complete, Type-I, and Type-II censored samples under different prior settings. Subsequently, Bayesian point and interval estimates, as well as the associated statistical inference, are discussed in detail. The developed methods are then applied to two real data sets for illustrative purposes. Moreover, a detailed Monte Carlo simulation study is carried out for evaluating the performance of the inferential methods developed here. Finally, we provide a brief discussion of the established results to demonstrate their practical utility and present some associated problems of further interest. Overall, this study fills an existing gap in the development of Bayesian inferential techniques for the parameters of the two-parameter Laplace distribution, making this research innovative and offering more investigative implications. It showcases the potential for broader methodological applications of Bayesian inference for complex real-world data sets, especially in scenarios involving different forms of censoring. This research provides a critical tool for statistical analysis in different fields such as engineering and finance, where the Laplace distribution is frequently adopted as a fundamental model.

Integration of the wind and solar power for the dynamic economic emission dispatch with the charging and discharging of plug-in electric vehicles

Wind and solar power are incorporated into the dynamic economic emission dispatch with the plug-in electric vehicles (DEED-PEV) in this paper, seeking to reduce the generation cost and pollutant emission significantly. A Laplacian non-dominated sorting genetic algorithm-II (LNSGA-II) and a constraint handling method (CHM) are proposed for the DEED-PEV. The LNSGA-II applies the Laplace distribution to both the crossover and mutation. The Laplace-based crossover makes each individual carry out unidirectional search towards its partner and enhances the information exchange between each pair of individuals. The Laplace-based mutation makes each individual carry out bi-directional search surrounding itself and adjust the searching range adaptively. Therefore, the LNSGA-II is able to explore and exploit the decision space of the DEED-PEV sufficiently, contributing to the reductions of the generation cost and pollutant emission in the objective space. For the variables of every individual, the CHM limits them within the feasible operating zones. The CHM also limits them within the dynamic boundaries. It further adjusts them according to the power balances. The power generation constraints and up/down ramp rate constraints can be always satisfied while the elimination of the total violation of the other constraints can be expedited. Therefore, the CHM can transform infeasible individuals to feasible ones rapidly. Four strategies are investigated for six PEV charging/discharging scenarios. The first three strategies consider no more than one kind of renewable energy generation while the fourth strategy involves both the wind power and solar power. Experimental results suggest that the fourth strategy is more economical and environment-friendly than the other three strategies. Furthermore, the LNSGA-II outperforms the other three competitor algorithms for six DEED-PEV problems in the light of hypervolume, coverage rate and spacing.

The spike-and-slab quantile LASSO for robust variable selection in cancer genomics studies.

Data irregularity in cancer genomics studies has been widely observed in the form of outliers and heavy-tailed distributions in the complex traits. In the past decade, robust variable selection methods have emerged as powerful alternatives to the nonrobust ones to identify important genes associated with heterogeneous disease traits and build superior predictive models. In this study, to keep the remarkable features of the quantile LASSO and fully Bayesian regularized quantile regression while overcoming their disadvantage in the analysis of high-dimensional genomics data, we propose the spike-and-slab quantile LASSO through a fully Bayesian spike-and-slab formulation under the robust likelihood by adopting the asymmetric Laplace distribution (ALD). The proposed robust method has inherited the prominent properties of selective shrinkage and self-adaptivity to the sparsity pattern from the spike-and-slab LASSO (Roc̆ková and George, J Am Stat Associat, 2018, 113(521): 431-444). Furthermore, the spike-and-slab quantile LASSO has a computational advantage to locate the posterior modes via soft-thresholding rule guided Expectation-Maximization (EM) steps in the coordinate descent framework, a phenomenon rarely observed for robust regularization with nondifferentiable loss functions. We have conducted comprehensive simulation studies with a variety of heavy-tailed errors in both homogeneous and heterogeneous model settings to demonstrate the superiority of the spike-and-slab quantile LASSO over its competing methods. The advantage of the proposed method has been further demonstrated in case studies of the lung adenocarcinomas (LUAD) and skin cutaneous melanoma (SKCM) data from The Cancer Genome Atlas (TCGA).

Absolute moments of the variance-gamma distribution

We obtain exact formulas for the absolute raw and central moments of the variance-gamma distribution, as infinite series involving the modified Bessel function of the second kind and the modified Lommel function of the first kind. When the skewness parameter is equal to zero (the symmetric variance-gamma distribution), the infinite series reduces to a single term. Moreover, for the case that the shape parameter is a half-integer (in our parameterisation of the variance-gamma distribution), we obtain a closed-form expression for the absolute moments in terms of confluent hypergeometric functions. As a consequence, we deduce new exact formulas for the absolute raw and central moments of the asymmetric Laplace distribution and the product of two correlated zero mean normal random variables, and more generally the sum of independent copies of such random variables.

SBL-LCGL: sparse Bayesian learning based on Laplace distribution for robust cone-beam x-ray luminescence computed tomography

Objective. To address the quality and accuracy issues in the distribution of nanophosphors (NPs) using Cone-beam x-ray luminescence computed tomography (CB-XLCT) by proposing a novel reconstruction strategy.Approach. This paper introduces a sparse Bayesian learning reconstruction method termed SBL-LCGL, which is grounded in the Lipschitz continuous gradient condition and the Laplace prior to overcome the ill-posed inverse problem inherent in CB-XLCT.Main results. The SBL-LCGL method has demonstrated its effectiveness in capturing the sparse features of NPs and mitigating the computational complexity associated with matrix inversion. Both numerical simulation andin vivoexperiments confirm that the method yields satisfactory imaging results regarding the position and shape of the targets.Significance. The advancements presented in this work are expected to enhance the clinical applicability of CB-XLCT, contributing to its broader adoption in medical imaging and diagnostics.

A quantile-regression approach to bivariate longitudinal joint modeling

Joint modeling of longitudinal outcomes and time-to-event data has become a major re- search interest in the last thirty years. A joint model is useful since it helps (i) to under- stand the evolution of the outcome(s) of interest over time, (ii) to understand the effects of the outcomes on the time of occurrence of some event(s) of interest (e.g. death/relapse), and (iii) to study the effects of the time-varying and time-invariant predictors on the longitudinal and time-to-event process. Traditional linear mixed models are routinely used for modeling the longitudinal process. However, for non-Gaussian/skewed outcomes it is more appealing to use quantile-regression models since such models do not assume any specific probability distribution for the outcomes. In this article, we present a bivariate quantile-regression approach for jointly modeling longitudinal process and time-to-event. In a Bayesian setting we consider an Asymmetric Laplace Distribution (ALD) for modeling different quantiles of the outcomes, and a semi-parametric Proportional Hazards (PH) model for time-to-event. Model parameters are estimated using Markov chain Monte Carlo (MCMC) algorithm, and we discuss the computational complexities through several simulation studies. Our numerical studies illustrate the usefulness of our model over the other traditional models. Journal of Statistical Research 2024, Vol. 58, No. 1, pp. 111-130.

Seasonal solar irradiance forecasting using artificial intelligence techniques with uncertainty analysis

Renewable integration in utility grid is crucial in the current energy scenario. Optimized utilization of renewable energy can minimize the energy consumption from the grid. This demands accurate forecasting of renewable contribution and planning. Most of the researches aim to find a suitable forecasting model in terms of accuracy and error metrics. However, the uncertainty and variability in these forecasts are also significant. This work combines point forecast with interval forecast to provide comprehensive information about the forecast uncertainty. In this work, solar irradiance forecasting is carried out using artificial intelligence (AI) techniques. Forecasting is done using seasonal auto-regressive moving average with exogenous factors (SARIMAX), support vector regression (SVR), long short term memory (LSTM) techniques and performance is evaluated. SVR model exhibited the best performance with R2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^2$$\\end{document} values of 0.97 and 0.96 for winter and summer respectively and 0.85 for monsoon and post-monsoon seasons. This is followed by forecast error distribution studies and uncertainty analysis. For this, SVR forecast error data is fitted using laplace distribution. Uncertainty study is carried out using confidence intervals and coverage rates. Excellent coverage rates are obtained for various confidence levels for all seasons, indicating the appropriate fitting of error distribution. For the narrow 85% confidence band, coverage rates of 89%, 95%, 90%, and 88% are obtained for winter, summer, monsoon and post-monsoon respectively. The work emphasizes the need for error-distribution studies, modeling of forecast errors and their application in providing reliable forecast intervals with the perspective of enhancing system reliability.

Laplacian eigenvalue distribution, diameter and domination number of trees

For a graph G with domination number γ, Hedetniemi, Jacobs and Trevisan (2016) proved that m G [ 0 , 1 ) ≤ γ , where m G [ 0 , 1 ) means the number of Laplacian eigenvalues of G in the interval [ 0 , 1 ) . Let T be a tree with diameter d. In this paper, we show that m T [ 0 , 1 ) ≥ ( d + 1 ) / 3 . All trees achieving the lower bound are completely characterized. Moreover, we prove that the domination number of a tree is ( d + 1 ) / 3 if and only if it has exactly ( d + 1 ) / 3 Laplacian eigenvalues less than one. As an application, it also provides a new type of tree, which shows the sharpness of the inequality due to Hedetniemi, Jacobs and Trevisan.

Laplace-Logistic Unit Distribution with Application in Dynamic and Regression Analysis

This manuscript presents a new two-parameter unit stochastic distribution, obtained by transforming the Laplace distribution, using a generalized logistic map, into a unit interval. The distribution thus obtained is named the Laplace-logistic unit (abbreviated LLU) distribution, and its basic stochastic properties are examined in detail. Also, the procedure for estimating parameters based on quantiles is provided, along with the asymptotic properties of the obtained estimates and the appropriate numerical simulation study. Finally, the application of the LLU distribution in dynamic and regression analysis of real-world data with accentuated “peaks” and “fat” tails is also discussed.

Detecting outliers in local ionospheric model for GNSS precise positioning

Global Navigation Satellite System fast precise positioning can be achieved with accurate ionospheric corrections computed from an adequate number of GNSS stations in a local region. In low-latitude regions, the presence of electron density gradients over short distances can lead to outliers in the map of ionospheric corrections and decrease its accuracy. In this study, we explored outlier detection in ionospheric correction mapping through statistical residuals during a four-month test in 2021. Our findings indicate that the residuals of the local ionospheric model conform to the Laplace distribution. To determine outliers, we use an empirical rule for the Laplace distribution, setting thresholds at μ ± 3b, μ ± 3.5b, and μ ± 5.8b for data retention rates of 95%, 97%, and 99.7%, respectively. Here, μ represents the location parameter, which corresponds to the median of data, and b is the scale parameter, calculated as the medium absolute deviation. We found that while removing outliers can improve model accuracy, it may result in unavailable prediction due to a lack of data in a spare network. For example, applying a μ ± 3.5b threshold for outlier removal led to approximately 2.5% of recording time having no ionospheric corrections map in low-latitude regions, however, the local model has the potential to improve its mean accuracy by up to 50% for both low and mid-latitudes. Therefore, choosing the appropriate percentile threshold depends on the network configuration and the desired accuracy. Removing erroneous satellite data to improve ionospheric accuracy brings positive impacts on precise positioning.