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Full-waveform inversion (FWI) is a powerful methodology employed in estimating subsurface physical parameters. FWI is classically formulated as a data-fitting problem based on minimizing the squared $$l_2$$ -norm of the difference between the observed data and modeled data (i.e., the data residuals or errors). The FWI based on $$l_2$$ -norm (hereafter classical FWI) stems from the assumption that the errors follow a Gaussian probability distribution. However, since, in most cases the error is non-Gaussian for non-linear problems and therefore, the classical FWI tends to fail. In this work, we consider the Tsallis q-deformation of the exponential function, which is widely used in the field of statistical physics, to derive an alternative misfit function to mitigate the FWI sensitivity to spurious measurements (outliers). In this regard, we formulated the FWI from a q-generalization of the Laplace probability distribution (or q-Laplace distribution). Application on a typical Brazilian pre-salt velocity model with an acoustic synthetic noisy-data is presented, in which we compare the performance of our proposal against the classical FWI. In addition, we compare also our methodology with the robust misfit function based on $$l_1$$ -norm, which is based on the Laplace distribution. The results show that our proposal outperforms classical FWI, and also the robust FWI based on $$l_1$$ -norm. The results also reveal that our proposal accelerates the data inversion process when the data is polluted by outliers.

In this brief, we investigate the robust state estimation of nonlinear systems with heavy-tailed noise. Based on the fact that the multivariate Laplace (ML) distribution has heavier tails than Gaussian distribution but is only determined by its mean and covariance, we utilize the ML distribution to model the heavy-tailed measurement noise. Based on the Gaussian mixture form of ML distribution, the system state and the covariance of ML distribution are inferred by variational Bayesian method, and then a closed recursive robust nonlinear Kalman filter is obtained with the nonlinear integrations calculated by numerical integration methods. Simulation results demonstrate that the proposed algorithm is insensitive to its free parameters and can obtain better estimation accuracy than related robust algorithms.

In recent years, variable selection based on penalty likelihood methods has aroused great concern. Based on the Gibbs sampling algorithm of asymmetric Laplace distribution, this paper considers the quantile regression with adaptive Lasso and Lasso penalty from a Bayesian point of view. Under the non-Bayesian and Bayesian framework, several regularization quantile regression methods are systematically compared for error terms with different distributions and heteroscedasticity. Under the error term of asymmetric Laplace distribution, statistical simulation results show that the Bayesian regularized quantile regression is superior to other distributions in all quantiles. And based on the asymmetric Laplace distribution, the Bayesian regularized quantile regression approach performs better than the non-Bayesian approach in parameter estimation and prediction. Through real data analyses, we also confirm the above conclusions.

We provide bibliographical connections and extensions of several representations of the classical Laplace distribution, discussed recently in the study of Ding and Blitzstein. Beyond presenting rel...

Recent empirical studies have shown that firm growth rate distribution is not Gaussian but closely follows a Laplace distribution. This robust feature of the growth rate distribution challenges existing models based on Gibrat’s model because it predicts a Gaussian distribution. First, we analyze more than 100,000 Japanese firms and empirically show that the Laplace shape can be observed for the Japanese firms. Then, by using the theory of stochastic processes, we theoretically show that the absence of jumps causes the discrepancy between Gibrat’s model and the Laplace shape. In particular, based on the Laplace shape and the law of proportionate effect, we show that the firm growth process is a jump process. In other words, firm growth cannot be explained by the consequence of many small shocks but is determined by a few large jumps. The widely observed Laplace distribution reflects this jump property of firm growth dynamics.

This paper proposes a Bayesian nonparametric interval regression model assuming the noise on the lower and upper bounds of interval data follows an asymmetric Laplace distribution. In order to address various uncertainties in real applications and make model training more convenient and efficient, the asymmetric Laplace distribution is represented as a scale mixture of Gaussian distribution, which is amenable to variational inference (VI for short) (Blei et al., 2016). Inspired by multi-task learning, the marginal quantile regression functions for lower and upper bounds relate each other by the correlation parameter of the coefficients in infinite feature space. This model can give point estimate of interval bounds or predicted occurrence probability for some events, which provides new potential for the problems which cannot be solved by previous interval regression models. In addition, the probability mass waste problem is analyzed and alleviated by a local correction strategy. Moreover, we modify the model to be fit for the interval data represented as center and radius. We verify above analysis results by numerical experiment on artificial and real dataset. Finally, the approximation ability of this model and that of the other existing methods are compared by applications on public data sets.

In this paper, we consider the finite mixture of quantile regression model from a Bayesian perspective by assuming the errors have the asymmetric Laplace distribution (ALD), and develop the Gibbs s...

We develop a random partition procedure based on a Dirichlet process prior with Laplace distribution. Gibbs sampling of a Laplace mixture of linear mixed regressions with a Dirichlet process is implemented as a random partition model when the number of clusters is unknown. Our approach provides simultaneous partitioning and parameter estimation with the computation of classification probabilities, unlike its counterparts. A full Gibbs-sampling algorithm is developed for an efficient Markov chain Monte Carlo posterior computation. The proposed method is illustrated with simulated data and one real data of the energy efficiency of Tsanas and Xifara (Energy and Buildings, 49, 560-567, 2012).

This article is a comparative study between the parametric asymptotic lower confidence limits and bootstrap lower confidence limits for the basic quantile based process capability indices based on the unified super-structure CNp⁢(u,v){C_{N_{p}}(u,v)} when the distribution of the quality characteristic follows an asymmetric non-normal distribution. We illustrate this method when the distribution of the quality characteristic is a member of the family of Esscher-transformed Laplace models introduced by S. George and D. George [11]. We obtain the bias corrected and accelerated (BCa) bootstrap confidence intervals of CNp⁢(u,v){C_{N_{p}}(u,v)}, which provide lower confidence intervals with coverage probability nearer to the nominal value compared to the asymptotic confidence intervals. We conclude that for asymmetric and peaked processes, the BCa confidence interval is a better alternative compared to the usual confidence intervals under the assumption that the quality characteristic follows a Gaussian type distribution. Numerical examples are given based on some real data.

Multiresolution models such as the wavelet-domain hidden Markov tree (HMT) model provide a powerful approach for image modeling and processing because it captures the key features of the wavelet coefficients of real-world data. It is observed that the Laplace distribution is peakier in the center and has heavier tails compared with the Gaussian distribution. Thus we propose a new HMT model based on the two-state, zero-mean Laplace mixture model (LMM), the LMM-HMT, which provides significantly potential for characterizing real-world textures. By using the HMT segmentation framework, we develop LMM-HMT based segmentation methods for image textures and dynamic textures. The experimental results demonstrate the effectiveness of the introduced model and segmentation methods.

The analysis of Tobit model with non-normal error distribution is extended to the case of asymmetric Laplace distribution (ALD). Since the ALD probability density function is known to be continuous but not differentiable, the usual mode-finding algorithms such as maximum likelihood can be difficult and result in the inconsistent parameter estimates. Various Markov chain Monte Carlo algorithms including probability integral transformation, griddy Gibbs, random walk Metropolis-Hastings, and tailored randomized block Metropolis-Hastings (TaRB-MH) are applied and compared. Results from a simulation study suggest that TaRB-MH is the best performing algorithm. Using a survey dataset on the wage earnings of Thai male workers to compare the Tobit model with normal and ALD errors through the model marginal likelihood and deviance information criterion, the results reveal that the model with the ALD error is preferred.

We propose a new family of linear mixed-effects models based on the generalized Laplace distribution. Special cases include the classical normal mixed-effects model, models with Laplace random effects and errors, and models where Laplace and normal variates interchange their roles as random effects and errors. By using a scale-mixture representation of the generalized Laplace, we develop a maximum likelihood estimation approach based on Gaussian quadrature. For model selection, we propose likelihood ratio testing and we account for the situation in which the null hypothesis is at the boundary of the parameter space. In a simulation study, we investigate the finite sample properties of our proposed estimator and compare its performance to other flexible linear mixed-effects specifications. In two real data examples, we demonstrate the flexibility of our proposed model to solve applied problems commonly encountered in clustered data analysis. The newly proposed methods discussed in this paper are implemented in the R package nlmm.

Speech recognition accuracy degrades in presence of additive noise, especially when recognizer's training data is clean. Several methods have been proposed to compensate effects of noise on recognition accuracy. Among these methods, Missing Feature Techniques (MFT) have shown promising results. Two different MF approaches have been introduced in literature: ''Model-Based'' and ''Feature-Based'' approaches. In the first category, the state distribution calculations should be changed and also some modifications are required to cope with filter bank features. But, in the second category, compensated representations of corrupted signals are reconstructed prior to recognition, and conventional recognizers, using MFCC features, are then used. In ''Feature-Based'' MFT, spectral vectors of speech signal frames are conventionally modeled by a Gaussian distribution (GD) and according to estimated parameters of the models, missed parts of speech representation are reconstructed. In this paper, we consider some researches that suggest multivariate Laplace distribution (MLD) to be a proper distribution for modeling speech signal. Here, we examine this idea in modeling log spectral representation of speech frames, and show that MLD acts better than Gaussian distribution. Moreover, We apply the Maximum Likelihood (ML) estimation of missing elements conditioned on observed values with respect to MLD and prove that the estimation equations are simple and tractable and by using this estimation in reconstruction of missing features, we gain better phoneme recognition accuracy against using ''GD'', in noisy conditions. In SNR values blew 10dB in the cases of all of the noises, MLD improves the recognition accuracy more than 4% in most of the cases.

A parametric approach to estimating and forecasting Value-at-Risk (VaR) and expected shortfall (ES) for a heteroscedastic financial return series is proposed. The well-known GJR-GARCH form models the volatility process, capturing the leverage effect. To capture potential skewness and heavy tails, the model assumes an asymmetric Laplace form as the conditional distribution of the series. Furthermore, dynamics in higher moments are modeled by allowing the shape parameter in this distribution to be time-varying. Estimation is via an adaptive Markov chain Monte Carlo (MCMC) sampling scheme, employing the Metropolis-Hastings (MH) algorithm with a mixture of Gaussian proposal distributions. A simulation study highlights accurate estimation and improved inference compared to a single-Gaussian-proposal MH method. The model is illustrated by applying it to four international stock market indices and two exchange rates, generating one-step-ahead forecasts of VaR and ES. Standard and non-standard tests are applied to these forecasts, and the finding is that the proposed model performs favourably compared to some popular competitors: in particular it is the only conservative model of risk over the period studied, which includes the recent global financial crisis.

The Riemannian geometry of the space Pm, of m × m symmetric positive definite matrices, has provided effective tools to the fields of medical imaging, computer vision and radar signal processing. Still, an open challenge remains, which consists of extending these tools to correctly handle the presence of outliers (or abnormal data), arising from excessive noise or faulty measurements. The present paper tackles this challenge by introducing new probability distributions, called Riemannian Laplace distributions on the space Pm. First, it shows that these distributions provide a statistical foundation for the concept of the Riemannian median, which offers improved robustness in dealing with outliers (in comparison to the more popular concept of the Riemannian center of mass). Second, it describes an original expectation-maximization algorithm, for estimating mixtures of Riemannian Laplace distributions. This algorithm is applied to the problem of texture classification, in computer vision, which is considered in the presence of outliers. It is shown to give significantly better performance with respect to other recently-proposed approaches.

Constructing a flexible parametric classes of probability distributions appeared as a quite popular approach in Bayesian analysis for the last few decades. This study is planned in the same direction for two component Mixture of Laplace probability distribution considering heterogeneous population. We have considered censored and complete sample environments and derived the closed form expressions for the Bayes estimators and their posterior risks. In addition we have worked out complete sample expressions for the Maximum Likelihood (ML) estimates along with their Posterior Risk and constructed components of the information matrix. To examine the performance of these estimators we have evaluated their properties for different sample sizes, censoring rates, proportions of the component of mixture and a variety of loss functions. To highlight the practical significance we have included an illustrative application example based on a real-life mixture data.

In today's hybrid video coding, Rate-Distortion Optimization (RDO) plays a critical role. It aims at minimizing the distortion under a constraint on the rate. Currently, the most popular RDO algorithm for one-pass coding is the one recommended in the H.264/AVC reference software. It, or HR-lambda for convenience, is actually a kind of universal method which performs the optimization only according to the quantization process while ignoring the properties of input sequences. Intuitively, it is not efficient all the time and an adaptive scheme should be better. Therefore, a new algorithm Lap- lambda is presented in this paper. Based on the Laplace distribution of transformed residuals, the proposed Lap-lambda is able to adaptively optimize the input sequences so that the overall coding efficiency is improved. Cases which cannot be well captured by the proposed models are considered via escape methods. Comprehensive simulations verify that compared with HR-lambda , Lap-lambda shows a much better or similar performance in all scenarios. Particularly, significant gains of 1.79 dB and 1.60 dB in PSNR are obtained for slow sequences and B-frames, respectively.

In this article, we consider a multivariate Laplace distribution. When its skewness is zero, the distribution becomes a member of the elliptical family of distributions. We provide a test with its asymptotic null and nonnull distributions, for testing that the skewness is zero. Characteristics of the Laplace distribution such as mean, covariance matrix, third and fourth cumulants, and moments are given. Mardia's real-valued measures of skewness β1p and kurtosis β2p are defined in terms of cumulants, and an inequality between the skewness and kurtosis—namely, β2p ≥ p 2 + β1p , where p is the dimension of the random vector—is given. When p = 1, this reduces to the well-known inequality in the univariate case.

A test based on the ratio of the sample mean absolute deviation and the sample standard deviation is proposed for testing the Laplace distribution hypothesis. The asymptotic null distribution for this test statistic is found to be normal. The use of Anderson–Darling test based on a data transformation is also discussed.

A generalized Laplace distribution using the Kumaraswamy distribution is introduced. Different structural properties of this new distribution are derived, including the moments, and the moment generating function. We discuss maximum likelihood estimation of the model parameters and obtain the observed and expected information matrix. A real data set is used to compare the new model with widely known distributions.

Gaussian models are frequently used for road profiles. However, these models are often only valid for short sections of the road. Here we present a comprehensive approach to describe various aspects of road surface/elevation by using extensions of Gaussian models arising from random gamma distributed variances. These random variances result in the Laplace distribution and thus we refer to the so defined models as Laplace models. The approach is shown to perform well in modelling road topography, road roughness and multi-valued responses of forces and bending moments containing transients. The different Laplace models are presented together with numerical examples and Matlab code for simulation.

This paper proposes a coding tree unit (CTU) level rate control for HEVC based on the Laplace distribution modeling of the transformed residuals. Firstly, we give a study on the relationship model among the optimal quantization step, the Laplace parameter and the Lagrange multiplier. Based on the relationship model, the quantization parameter for each CTU can be dynamically adjusted according to the distribution of the transformed residual. Secondly, a CTU level rate control scheme is proposed to achieve accurate rate control as well as high coding performance. Experimental results show that the proposed rate control scheme achieves better coding performance than the state-of-the-art rate control schemes for HEVC in terms of both objective and subjective quality.

This paper introduces a new non-Gaussian detection method for complex-valued synthetic aperture sonar (SAS) imagery. The detection method is based on a multivariate extension of the Laplace distribution derived using a scale mixture of Gaussian distributions. A goodness-of-fit test in the form of a likelihood ratio is then conducted on a sonar imagery data set consisting of high-frequency (HF) and broadband (BB) images coregistered over the same region on the seafloor showing the proposed model's applicability in sonar imagery. Detection based on testing the equality of parameters from two populations is then implemented on a database containing actual SAS images of the seafloor with synthetically generated targets inserted into the images and compared to a similar non-Gaussian technique. Detection performance in this paper is given in terms of receiver-operator characteristic (ROC) curve attributes, probability of detection, and average false alarm rate.

An approach to multi-channel blind de-convolution is developed, which uses an adaptive filter that performs blind source separation in the Fourier space. The approach keeps (during the learning process) the same permutation and provides appropriate scaling of components for all frequency bins in the frequency space. Experiments indicate that Generalized Laplace Distribution can be used effectively to blind de-convolution of convolution mixtures of sources in Fourier space compared to the conventional Laplacian and Gaussian function.

A random variable X is said to have the skew-Laplace distribution if its pdf is f(x) = 2g(x)G(λx), where g(·) and G(·), respectively, denote the pdf and the cdf of the Laplace distribution. This distribution – in spite of its simplicity – appears not to have been studied in detail. The only work that appears to give some details of this distribution is Gupta et. al [Random Operators and Stochastic Equations, Vol. 10 (2002), pp. 133–140], where expressions for the expectation, variance, skewness and the kurtosis of X are given. But these expressions appear to contain some errors. In this paper, we provide a comprehensive description of the mathematical properties of X. The properties derived include the k th moment, variance, skewness, kurtosis, moment generating function, characteristic function, cumulant generating function, the k th cumulant, hazard rate function, mean deviation about the mean, mean deviation about the median, Renyi entropy, Shannon's entropy, cumulative residual entropy and the asympto...

Based on a random sample from the Laplace population with unknown shape and scale parameters, one- and two-sided confidence bands on the entire cumulative distribution function and simultaneous confidence intervals for the interval probabilities under the distribution are constructed using Kolmogorov–Smirnov type statistics. Small sample and asymptotic percentiles of the relevant statistics are provided.

Deconvolution technique can be utilized in the forward-looking radar (FLR). However, the forward-looking imaging performance degenerates greatly due to the effect of high-speed movement of the platform. In this article, an efficient Bayesian forward-looking superresolution imaging algorithm based on Doppler deconvolution in expanded beam space is proposed. First, the Doppler phase information caused by the high-speed platform is fully exploited and the Doppler matrix is integrated with the antenna pattern. The Doppler convolution model of the echo signal for forward-looking is derived in this article. Then, the Doppler phase information is adopted to perform the Doppler deconvolution. Moreover, an expanded beam space is constructed to enhance the sparsity of the imaging scene. The complex Gaussian distribution and the Laplace distribution have been used to model the distribution characteristics of noise and targets in the imaging scene, respectively. Finally, based on the Bayesian framework, the forward-looking imaging problem is converted into the convex optimization problem. The performance assessment based on simulated and experimental data, also in comparison to the conventional real beam, truncated singular value decomposition (TSVD), iterative adaptive approach (IAA) methods, has demonstrated the effectiveness of our proposed algorithm under high-speed platform scenarios.

Vibration monitoring has been a reliable source of information for machine fault diagnosis. Several methods are available for bearing fault diagnosis under constant speed condition. A fault diagnosis framework consisting of a novel pre-processing tool, named cumulative distribution sharpness (CDS) profiling is proposed for variable speed conditions. We first provide evidences suggesting that the bearing fault signals follow Laplace distribution. Under the influence of Gaussian noise, however, the sharpness of the distribution decreases. This is helpful in separating the periodic fault and noise regions in the time-domain vibration signal by calculating local CDS values. The proposed CDS profiling (CDSP) is thus obtained by sweeping a window over the vibration signal and estimating the sharpness of cumulative distribution of such windowed signal. For changing noise variance, the monotonous and continuous nature of CDS is ensured to obtain a profile that retains the fault periodicity. A short-time Fourier transform of the CDSP is then calculated, followed by multiple time-frequency curves extraction (MTFCE). Two important fault features - Prominence and Compliance - are proposed in this paper. Finally, a fuzzy inference system is used to obtain diagnosis percentages from the features. Thus, the proposed method can classify the faults into healthy, inner race and outer race faults. The results are then validated on experimental data with variable operating speed. We show that prominence of the fault characteristic frequency improves due to CDSP. The accuracy of the proposed method is found to be better than the benchmark method of MTFCE.

The paper introduces a Bayesian estimation method for quantile regression in univariate ordinal models. Two algorithms are presented that utilize the latent variable inferential framework of Albert and Chib (1993) and the normal-exponential mixture representation of the asymmetric Laplace distribution. Estimation utilizes Markov chain Monte Carlo simulation - either Gibbs sampling together with the Metropolis-Hastings algorithm or only Gibbs sampling. The algorithms are employed in two simulation studies and implemented in the analysis of problems in economics (educational attainment) and political economy (public opinion on extending "Bush Tax" cuts). Investigations into model comparison exemplify the practical utility of quantile ordinal models., 24 pages

As a generative model , probabilistic linear discriminant analysis (PLDA) has achieved good performance in supervised learning tasks. The model incorporates both within-individual and between-individual variation, and remaining unexplained data variation is assumed to follow Gaussian distribution . However, the assumption of Gaussian distribution makes the model sensitive to the presence of noise and outliers in training set. To address this issue, this paper proposes a robust probabilistic linear discriminant analysis model by assuming Laplace prior on the noise term. Instead of solving high-dimensional linear systems, we embed a Kronecker-decomposable component in the new model for tensor data, significantly reducing the size of problems. As the non-conjugacy of Laplace distribution complicates the calculation of the posteriors of latent variables, we express it to a hierarchical architecture using an Inverse Gamma distribution and then adopt variational expectation–maximization (EM) algorithm to learn model parameters. The reconstruction and classification experiments on several public databases show the superiority of the proposed model compared with the state-of-the-art LDA-based algorithms.

The probabilistic neural network (PNN) is an efficient approach that can compute nonlinear decision boundaries, widely used for classification. In this paper, the often used Gaussian distribution function is replaced by a new probability density function which provides a new variant of the PNN method. Most of the higher-dimensional data are statistically found to be not from the normal distribution, and hence, we have replaced it by the symmetric Laplace distribution. Further, the estimation of the smoothing parameter in the proposed PNN model is carried out with three different evolutionary algorithms, namely bat algorithm (BA), grey wolf optimizer (GWO), and whale optimization algorithm (WOA) with a novel fitness function. These different proposed PNN models with variable smoothing parameter estimation methods are tested on five different benchmark data sets. The performance of proposed three Laplace distribution-based variants of PNN incorporated with BA, GWO, and WOA are reported and compared with Gaussian-based variants of PNN and also other commonly used classifiers: the conventional PNN, extreme learning machine, and K-nearest neighbor in terms of measurement accuracy. The results demonstrate that the proposed approaches using evolutionary algorithms can provide as much as a ten percent increase in accuracy over the conventional PNN method.

This brief develops a robust multiple model strategy for nonlinear system identification with system output data corrupted by outliers. The nonlinear system is described as a global model that combines multiple local nonlinear state-space models (SSMs) identified at the prechosen working points. Industrial data contaminated with outliers is a common problem in practical processes, which imposes great challenges for nonlinear processes modeling. In order to handle the outliers, the robust observation model based on Laplace distribution, instead of the conventional Gaussian distribution, is used to model the outliers corrupted output data. The approach to estimate parameters of all local models simultaneously is derived using the expectation maximization (EM) algorithm, and a particle filter (PF) is introduced to numerically calculate the cost function (Q-function) in the EM algorithm. The effectiveness of the proposed method is verified through a numerical example and the practical two-link robotic manipulator.

When |X| k has a gamma distribution with parametersb −k andk −1,Jakuszenkow [1979] has, for the loss function $$(\hat \theta - \theta )^2 \theta ^{ - 2} $$ , considered the best multiple of ΣX 2 as an estimator ofb 2 and shown that it is Lehmann unbiased. In this paper, the best multiple of (Σ|X i |k)2/k is shown to be Lehmann unbiased, admissible and better than Jakuszenkow's estimator.

In this paper, a new model-based Bayesian compressive sensing technique for non-stationary images is proposed. Our algorithm is based on the recently addressed triplet Markov fields (TMF) model. The TMF model is well appropriate for non-stationary image processing, owing to the introduction of a third random field which reflects different non-stationarity of images. Furthermore, TMF can extract the interactions among the neighboring sites of an image in a more complete way than the classic hidden Markov models do. In this paper, the inter-scale dependencies between the wavelet coefficients is exploited explicitly in the proposed TMF model, which results in the wavelet domain TMF model. Our proposed model considers the intra- and inter-scale dependencies and the non-stationarity of images simultaneously. Also, we have developed our proposed algorithm for both Gaussian and non-Gaussian measurement noises, and we have modeled the non-Gaussianity of the noise via Laplace distribution. To approximate the posterior distributions of the hidden variables, we resort to a variational Bayesian expectation–maximization algorithm. Simulation results in both the optical and synthetic aperture radar images show that this model leads to an improvement over state-of-the-art algorithms in terms of the reconstruction error and the structural similarity.

In this paper, the grayscale image segmentation problem is investigated and a new mixture model with shifted asymmetric Laplace distribution component is proposed. Instead of the conventional Gaussian model, the shifted asymmetric Laplace distribution model is adopted to model the pixels. The spatial constraint on neighboring pixels is introduced into the proposed shifted asymmetric Laplace mixture model, which makes the model be robust to noise and outliers of the images. The unknown model parameters are estimated via the expectation-maximization (EM) algorithm, which can guarantee convergence to a local minimum. The experimental verification is performed on both synthesized images and images of real chip to prove the effectiveness of our image segmentation method.

This paper introduces a robust identification solution for the linear parameter varying Autoregressive Exogenous systems with outlier-contaminated outputs. The Laplace distribution with heavy tails and the expectation maximization algorithm are combined to build the robust system identification framework. To overcome the obstacles brought by the outliers, the Laplace distribution which can be decomposed into infinite Gaussian components, is applied to mathematically model the system noise. The problem of parameter estimation is solved using the expectation maximization algorithm, and the equations to infer the system model and noise parameters are simultaneously provided in the developed identification method. Finally, the verification tests performed on a numerical example and a mechanical unit are used to prove the validity of the developed identification method.

This paper proposes a sub-frame layer RDO algorithm to further improve coding performance of low-delay hierarchical prediction structure in High Efficiency Video Coding. In the proposed algorithm, each frame is segmented into two sub-frames according to motion intensity, and a sub-frame layer is taken as a basic optimizing unit. An inter-frame dependency model is proposed to quantify inter-frame dependency of a frame or a sub-frame. The proposed inter-frame dependency model is composed of two parts, Laplace distribution based direct inter-frame dependency model and distortion propagation chain. The direct inter-frame dependency model is used to evaluate dependency caused by direct reference, and the distortion propagation chain shows the relationship between direct reference and indirect reference. With quantified inter-frame dependency, Lagrange optimization is then used to solve the optimal $\lambda $ -QP pair for each sub-frame layer in frame layer 1 (L1) to frame layer 4 (L4). For frame layer 0 (L0), additional L0 frames are adaptively inserted according to inter-frame dependency of the previous L0 frame, and the optimal $\lambda $ -QP pair for each sub-frame in L0 is solved using Lagrange optimization. When peak signal to noise ratio is used as quality criterion, the proposed algorithm achieves 7.0% and 7.0% bit-rate reduction with lowdelay_P_main (LDP) and lowdelay_main (LDB) configurations respectively; when structural similarity is used as quality criterion, the proposed algorithm achieves 13.3% and 13.1% bit-rate reduction with LDP and LDB configurations respectively.

The purposes of this paper is to introduce the ability of the Bayesian quantile regression method in overcoming the problem of the nonnormal errors using asymmetric laplace distribution on simulation study. Method: We generate data and set distribution of error is asymmetric laplace distribution error, which is non normal data. In this research, we solve the nonnormal problem using quantile regression method and Bayesian quantile regression method and then we compare. The approach of the quantile regression is to separate or divide the data into any quantiles, estimate the conditional quantile function and minimize absolute error that is asymmetrical. Bayesian regression method used the asymmetric laplace distribution in likelihood function. Markov Chain Monte Carlo method using Gibbs sampling algorithm is applied then to estimate the parameter in Bayesian regression method. Convergency and confidence interval of parameter estimated are also checked. Result: Bayesian quantile regression method results has more significance parameter and smaller confidence interval than quantile regression method. Conclusion: This study proves that Bayesian quantile regression method can produce acceptable parameter estimate for nonnormal error.

Concrete CT test is an effective method used to analyze micro-structural damage evolution of concrete under loading condition. In the CT images of different stress states in the same section, gray value changes in the original number matrix of CT image are the key to investigate microscopic damage evolution of concrete. However, the slight changes in the gray value of CT images before peak stress cannot be intuitively observed with the naked eye. This study mainly aims to develop a coupled method based on statistical method and X-ray CT to extract indicators of such subtle changes. Different cross-sectional CT images of concrete at different stress stages were obtained by uniaxial static compression CT test. The theory of the grey level co-occurrence matrices (GLCM) was then adopted to analyze micro-damage evolution and crack properties. That is, the GLCM of the CT images were calculated, and four statistical features including contrast, energy, correlation, and homogeneity were extracted, which can be extended into quantitative analysis of micro-damage hidden in the CT images of concrete. Results exhibited that the damage growth area, concentrating between the 45th and 225th concrete cross-section, could be predicted by the distribution of four statistical features (i.e., contrast, energy, correlation and homogeneity) at the fourth scan, and it is consistent with the crack location in the fifth scan. The contrast, energy and homogeneity follow the Gaussian distribution under five scan stages, and the correlation follows the Laplace distribution. In addition, the changes of bandwidth in color heat map illustrating the GLCM before and after specimen failure show that the bandwidth is positively correlated with micro-structural damage of concrete specimen.

Quantile regression is a regression method that modelling a relationship between quantile of variable response and one or more variable predictors. Quantile regression has advantages that linear regression does not have; it is robust against outliers and can model heteroscedasticity data. The parameters of quantile regression can be estimated using the Bayesian method. The Bayesian method is a data analysis tool derived based on the Bayesian inference principle. Bayesian inference is the process of studying data analysis inductively with the Bayes theorem. To estimate regression parameters with Bayesian inference, it is necessary to find the posterior distribution of the regression parameters where the posterior distribution is proportional to the product of the prior distribution and its likelihood function. Since the calculation of the posterior distribution analytically is difficult to do if more parameters are estimated, the Markov Chain Monte Carlo (MCMC) method is proposed. The use of the Bayesian method in quantile regression has advantages, namely the use of MCMC has the advantages of obtaining sample parameter values from an unknown posterior distribution, using computationally efficient, and easy to implement. Yu and Moyeed (2001) introduced Bayesian quantile regression using the likelihood function of errors with an Asymmetric Laplace Distribution (ALD) and found that minimizing parameter estimates in quantile regression is the same as maximizing the likelihood function of errors with an Asymmetric Laplace Distribution (ALD). The method used to estimate quantile regression parameters is Gibbs sampling from the ALD, which is a combination of the exponential and normal distributions. To find the parameters of the regression model by sampling the posterior distribution found in this thesis. The results obtained from Gibbs sampling are a sample sequence of estimated parameters. After obtaining the sample sequences, the sample lines are averaged to obtain an estimated regression parameter. The case study in this thesis discusses the effect of risk factors from motor vehicle insurance customers on the size of claims submitted by customers.

Sparse representation based on over-complete dictionaries is a hot issue in the field of computer vision and machine learning. In probability theory, over-complete dictionary can be learned by non-parametric Bayesian techniques with Beta Process. However, traditional probabilistic dictionary learning method assumes noise follows Gaussian distribution, which can only remove Gaussain noise. In order to remove outlier or complex noise, we propose a dictionary learning method based on non-parametric Bayesian technology by assuming the noise follows Laplacian distribution. Because the non-conjugacy of Laplacian distribution makes the calculation of posteriors of latent variables more complicate, thus we utilize a superposition of an infinite number of Gaussian distributions to substitute for L1 density function. The weights of mixture Gaussian distribution are controlled by an extra hidden variable. Then the Bayesian inference is applied to learn all the key parameters in the proposed probabilistic model, which avoids the processing of parameter setting and fine tuning. In the experiments, we mainly test the performance of different algorithms in removing salt-and-pepper noise and mixture noises. The experimental results show that the PSNRs of our algorithm are higher 2-4 dB at least than other classic algorithms.

Запропоновано спеціалізований програмний калькулятор, який надає можливість генерувати засобами ЕXCEL псевдовипадкові числа, розподілені згідно з розподілом Лапласа, розподілом Коші, розподілом найменшого значення, розподілом найбільшого значення, логістичним розподілом, розподілом Чампернауна, подвійним експоненційним розподілом, експоненційним розподілом, розподілом Ерланга порядку m, нормованим розподілом Ерланга порядку m, узагальненим розподілом Ерланга другого порядку, розподілом Вейбулла, зсуненим розподілом Вейбулла, гіперекспоненційним розподілом другого порядку, розподілом модулю n-вимірного випадкового вектору, розподілом Релея, розподілом Релея-Райса, розподілом Максвела, логарифмічно нормальним розподілом, розподілом Парето, розподілом модулю нормально розподіленої випадкової величини, параболічним розподілом, лівобічним зрізаним нормальним розподілом, правобічним зрізаним нормальним розподілом, двобічним зрізаним нормальним розподілом, розподілом арксинусу, розподілом Симпсона (трикутним розподілом), розподілом Бірнбаума-Сандерса, степеневим розподілом, c2-розподілом Пірсона, c-розподілом Пірсона, t-розподілом Стьюдента, F-розподілом Фішера-Снедекора, Z-розподілом Фішера. Наведено основні відомості про структуру спеціалізованого програмного калькулятора. Вказані законі розподілу включено до переліку рекомендованих Міжнародною Спілкою Електрозв’язку для використання в процесі моделювання розповсюдження радіохвиль (документ МСЭ-R P.1057-5 (12/2017).Запропоновано спеціалізований програмний калькулятор для отримання значень функції нормального розподілу для різних форм її подання. Калькулятор реалізовано в середовищі Excel.

The Laplace distribution of random processes was observed in numerous situations that include glasses, colloidal suspensions, live cells, and firm growth. Its origin is not so trivial as in the case of Gaussian distribution, supported by the central limit theorem. Sums of Laplace distributed random variables are not Laplace distributed. We discovered a new mechanism leading to the Laplace distribution of observable values. This mechanism changes the contribution ratio between a jump and a continuous parts of random processes. Our concept uses properties of Bernstein functions and subordinators connected with them.

This article introduces a Bayesian neural network estimation method for quantile regression assuming an asymmetric Laplace distribution (ALD) for the response variable. It is shown that the posterior distribution for feedforward neural network quantile regression is asymptotically consistent under a misspecified ALD model. This consistency proof embeds the problem from density estimation domain and uses bounds on the bracketing entropy to derive the posterior consistency over Hellinger neighborhoods. This consistency result is shown in the setting where the number of hidden nodes grow with the sample size. The Bayesian implementation utilizes the normal-exponential mixture representation of the ALD density. The algorithm uses Markov chain Monte Carlo (MCMC) simulation technique - Gibbs sampling coupled with Metropolis-Hastings algorithm. We have addressed the issue of complexity associated with the afore-mentioned MCMC implementation in the context of chain convergence, choice of starting values, and step sizes. We have illustrated the proposed method with simulation studies and real data examples.

In this paper we reveal that each subordinated Brownian process, leading to subdiffusion, under Poissonian resetting has a stationary state with the Laplace distribution. Its location parameter is defined only by the position to which the particle resets, and its scaling parameter is dependent on the Laplace exponent of the random process directing Brownian motion as a parent process. From the analysis of the scaling parameter the probability density function of the stochastic process, subject to reset, can be restored. In this case the mean time for the particle to reach a target is finite and has a minimum, optimal for the resetting rate. If the Brownian process is replaced by the L\'evy motion (superdiffusion), then its stationary state obeys the Linnik distribution which belongs to the class of generalized Laplace distributions.

A probabilistic model for the distribution of narrowband short-time Fourier transform (STFT) coefficients, having frequency resolution of 5 Hz and below, is proposed for speech signals. An important application is to model speech to depict the perceptual stability of human listening capability which allows humans to perceive speech reliably under a wide range of acoustic conditions. Representation of speech with high spectral resolution finds applications in the design of digital hearing aids and cochlear implants for people with hearing disability. While speech is generally considered as a non-stationary signal over segment lengths longer than 20–30 ms, the perceptual stability of human auditory system motivates the need for an invariant representation of speech over long segment lengths. Computer modelling shows that STFT coefficients of speech with high-frequency resolution fit reasonably accurately to Laplace distribution (LD). Parameters of the corresponding LD are estimated using maximum-likelihood estimation. Cramer–Rao bound for the estimated parameters and root mean square error for the fitted distribution confirm the validity of the fitted distribution.

Probabilistic neural network (PNN) is a single-pass feed-forward neural network with the capability of providing nonlinear decision boundaries. In this work, we propose the modifications to the existing PNN approach. Contributions are threefold: First, symmetric Laplace distribution has been used instead of Gaussian distribution in the pattern layer of the PNN approach. Second, a new weight coefficients’ estimation method has been introduced between the pattern and summation layers of PNN. Third, a novel convex fitness function has been designed for the bat algorithm to obtain an optimal smoothing parameter vector. The designed fitness function maintains a balance between the sensitivity and specificity values. The performance of the proposed algorithm “bat algorithm-based weighted Laplacian probabilistic neural network” is compared with that of conventional PNN, weighted PNN, extreme learning method, optimal pruned extreme learning machine and K-nearest neighbor. Experimental evaluation is carried out on eleven benchmark data sets using different performance measures such as accuracy, sensitivity, specificity and Youden’s index. The comparative performance evaluation of the proposed model has been made with the PNN, and other standard algorithms outperform the compared approaches in terms of classification accuracy. Friedman test is used for statistical evaluation of the proposed approach.

Recently sparse coding based on regression analysis has been widely used in face recognition research. Most existing regression methods add an extra constraint factor to the coding residual to make the fidelity term in the $$l_{2}$$ loss approach the Gaussian or Laplace distribution. But the essence of these methods is that only the fidelity term of $$l_{1}$$ loss or $$l_{2}$$ loss is used. In this paper, weighted Huber constrained sparse coding (WHCSC) is used to study the robustness of face recognition in occluded environments, and alternating direction method of multipliers is used to solve the problem of model minimization. In WHCSC, we propose a sparse coding with weight learning and use Huber loss to determine whether the fidelity is a $$l_{2}$$ loss or $$l_{1}$$ loss. For the WHCSC model, the two kinds of classification modes and the two kinds of weight coefficients are further studied for the intra-class difference and the inter-class difference in the face image classification. Through a large number of experiments on a public face database, WHCSC shows strong robustness in face occlusion, corrosion and illumination changes comparing to the state-of-the-art methods.

This paper presents a new approach to a robust Gaussian process regression, creating a non-parametric Bayesian regression estimate robust to outliers. Most existing approaches replace an outlier-prone Gaussian likelihood with a non-Gaussian likelihood induced from a heavy tail distribution, such as the Laplace distribution and Student-t distribution. However, the use of a non-Gaussian likelihood would incur the need for a computationally expensive Bayesian approximate computation in the posterior inferences. The proposed approach models an outlier as a noisy and biased observation of an unknown regression function, and accordingly, the likelihood contains bias terms to explain the degree of deviations from the regression function. We introduce two bias models that handle the bias terms differently, treating a bias as an unknown and fixed quantity or treating a bias as a random quantity. We entail how the biases can be estimated accurately with other hyperparameters by a regularized maximum likelihood estimation. Conditioned on the bias estimates, the robust GP regression can be reduced to a standard GP regression problem with analytical forms of the predictive mean and variance estimates. Therefore, the proposed approach is simple and very computationally attractive. It also gives a very robust and accurate GP estimate for many tested scenarios. For the numerical evaluation, we perform a comprehensive simulation study to evaluate the proposed approach with the comparison to the existing robust GP approaches under various simulated scenarios of different outlier proportions and different noise levels. The approach is applied to data from two measurement systems, where the predictors are based on robust environmental parameter measurements and the response variables utilize more complex chemical sensing methods that contain a certain percentage of outliers. The utility of the measurement systems and value of the environmental data are improved through the computationally efficient GP regression and bias model.