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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.

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.

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.

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.

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.

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.

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.

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.

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 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.

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.

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.

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.

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 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 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.

Due to the broadcast nature of radio transmission, both authorized and unauthorized users can access the network, which leads to the increasingly prominent security problems of wireless network. At the same time, it is more difficult to detect and identify users in wireless network environment due to the influence of noise. In this paper, the performance of energy detection (ED), matched filtering (MF) and K-nearest neighbor algorithm (KNN) are analyzed under different noise and uncertain noise separately. The Gaussian noise, α-stable distribution noise and Laplace distribution noise models are simulated respectively under the different uncertainty of noise when the false alarm probability is 0.01. The results show that the performance of the detectors is significantly affected by different noise models. In any case, the detection probability of KNN algorithm is the highest; the performance of MF is much better than ED under different noise models; KNN is not sensitive to noise uncertainty; MF has better performance on noise uncertainty which makes ED performance decline fleetly.

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.

Multi-view learning reveals the latent correlation between different input modalities and has achieved outstanding performances in many fields. Recent approaches aim to find a low-dimensional subspace to reconstruct each view, in which the gross residual or noise follows either Gaussian or Laplacian distribution. However, the noise distribution is often more complex in practical applications, and a deterministic distribution assumption is incapable of modeling it. Additionally, referring to time-changed data, e.g., videos, the noise is temporal smooth, preventing us from processing the data with the whole input, as have generally been done in many existing multi-view learning methods. To tackle these problems, a novel online multi-view subspace learning is proposed in this paper. Particularly, our proposed method not only estimates a transformation for each view to extract the correlation among various views, but also introduces a Mixture of Gausssians (MoG) model into the multi-view data, successfully exploiting numbers of Gaussian Distributions to adaptively fit a wider range of the complex noise. Furthermore, we further design a novel online Expectation Maximization (EM) algorithm, being capable of efficiently processing the dynamic data. Experimental results substantiate the effectiveness and superiority of our approach.

The purpose of this publication is to compare the accuracy of a new algorithm based on the Naive Bayesian classifier using the Laplace distribution and named the Laplace Naive Bayes classifier with the Gaussian Naive Bayesian classifier. The t-pairs test statistically proves that the observed difference in the accuracy of the two algorithms on a selected database of English film reviews is not the result of a random order of examples used for training and testing, but of the different classification behaviour of the algorithms.

In this article, it was observed that the noise in some real-world applications, such as wind power forecasting and direction of the arrival estimation problem, does not satisfy the single noise distribution, including Gaussian distribution and Laplace distribution, but the mixed distribution. Therefore, combining the twin hyperplanes with the fast speed of Least Squares Support Vector Regression (LS-SVR), and then introducing the Gauss–Laplace mixed noise feature, a new regressor, called Gauss-Laplace Twin Least Squares Support Vector Regression (GL-TLSSVR), for the complex noise. Subsequently, we apply the augmented Lagrangian multiplier method to solve the proposed model. Finally, we apply the short-term wind speed data-set to the proposed model. The results of this experiment confirm the effectiveness of our proposed model.

In this paper, we exploit the properties of mean absolute error (MAE) as a loss function for the deep neural network (DNN) based vector-to-vector regression. The goal of this work is two-fold: (i) presenting performance bounds of MAE, and (ii) demonstrating new properties of MAE that make it more appropriate than mean squared error (MSE) as a loss function for DNN based vector-to-vector regression. First, we show that a generalized upper-bound for DNN-based vector- to-vector regression can be ensured by leveraging the known Lipschitz continuity property of MAE. Next, we derive a new generalized upper bound in the presence of additive noise. Finally, in contrast to conventional MSE commonly adopted to approximate Gaussian errors for regression, we show that MAE can be interpreted as an error modeled by Laplacian distribution. Speech enhancement experiments are conducted to corroborate our proposed theorems and validate the performance advantages of MAE over MSE for DNN based regression.

Introduction Time-of-flight (TOF) positron emission tomography (PET) scanners can provide significant benefits by improving the noise properties of reconstructed images. In order to achieve this, the timing response of the scanner needs to be modelled as part of the reconstruction process. This is currently achieved using Gaussian TOF kernels. However, the timing measurements do not necessarily follow a Gaussian distribution. In ultra-fast timing resolutions, the depth of interaction of the γ-photon and the photon travel spread (PTS) in the crystal volume become increasingly significant factors for the timing performance. The PTS of a single photon can be approximated better by a truncated exponential distribution. Therefore, we computed the corresponding TOF kernel as a modified Laplace distribution for long crystals. The obtained (CTR) kernels could be more appropriate to model the joint probability of the two in-coincidenceγ-photons. In this paper, we investigate the impact of using a CTR kernel vs. Gaussian kernels in TOF reconstruction using Monte Carlo generated data. Materials and methods The geometry and physics of a PET scanner with two timing configurations, (a) idealised timing resolution, in which only the PTS contributed in the CTR, and (b) with a range of ultra-fast timings, were simulated. In order to assess the role of the crystal thickness, different crystal lengths were considered. The evaluation took place in terms of Kullback–Leibler (K-L) distance between the proposed model and the simulated timing response, contrast recovery (CRC) and spatial resolution. The reconstructions were performed using STIR image reconstruction toolbox. Results Results for the idealised scanner showed that the CTR kernel was in excellent agreement with the simulated time differences. In terms of K-L distance outperformed the a fitted normal distribution for all tested crystal sizes. In the case of the ultra-fast configurations, a convolution kernel between the CTR and a Gaussian showed the best agreement with the simulated data below 40 ps timing resolution. In terms of CRC, the CTR kernel demonstrated improvements, with values that ranged up to 3.8% better CRC for the thickest crystal. In terms of spatial resolution, evaluated at the 60th iteration, the use of CTR kernel showed a modest improvement of the peek-to-valley ratios up to 1% for the 10-mm crystal, while for larger crystals, a clear trend was not observed. In addition, we showed that edge artefacts can appear in the reconstructed images when the timing kernel used for the reconstruction is not carefully optimised. Further iterations, can help improve the edge artefacts.

International audience; In this paper, we propose to enhance learned image compression systems with a richer probability model for the latent variables. Previous works model the latents with a Gaussian or a Laplace distribution. Inspired by binary arithmetic coding , we propose to signal the latents with three binary values and one integer, with different probability models. A relaxation method is designed to perform gradient-based training. The richer probability model results in a better entropy coding leading to lower rate. Experiments under the Challenge on Learned Image Compression (CLIC) test conditions demonstrate that this method achieves 18 % rate saving compared to Gaussian or Laplace models.

We present a Bayesian filter for state space models with Laplace-distributed observation noise that is robust to heavy-tailed and outlier-ridden univariate time-series data. We analytically derive a closed-form expression of the exact posterior for a Laplace likelihood conditioned on a Gaussian prior. Posterior statistics are propagated forward in time by a proxy Gaussian density. The forward Kullback-Leibler divergence from the posterior to the Gaussian is minimized by matching their moments. The proposed method supports both linear and non-linear systems, and has a fast recursive structure analogous to the Kalman filter that enables online inference. Results show that the new method outperforms existing approximate inference methods, especially in challenging scenarios where the system’s parameters are uncertain.

Low-rank Multiview Subspace Learning (LMvSL) has shown great potential in cross-view classification in recent years. Despite their empirical success, existing LMvSL-based methods are incapable of handling well view discrepancy and discriminancy simultaneously, which, thus, leads to performance degradation when there is a large discrepancy among multiview data. To circumvent this drawback, motivated by the block-diagonal representation learning, we propose structured low-rank matrix recovery (SLMR), a unique method of effectively removing view discrepancy and improving discriminancy through the recovery of the structured low-rank matrix. Furthermore, recent low-rank modeling provides a satisfactory solution to address the data contaminated by the predefined assumptions of noise distribution, such as Gaussian or Laplacian distribution. However, these models are not practical, since complicated noise in practice may violate those assumptions and the distribution is generally unknown in advance. To alleviate such a limitation, modal regression is elegantly incorporated into the framework of SLMR (termed MR-SLMR). Different from previous LMvSL-based methods, our MR-SLMR can handle any zero-mode noise variable that contains a wide range of noise, such as Gaussian noise, random noise, and outliers. The alternating direction method of multipliers (ADMM) framework and half-quadratic theory are used to optimize efficiently MR-SLMR. Experimental results on four public databases demonstrate the superiority of MR-SLMR and its robustness to complicated noise.

Differential privacy is a mathematical technique that provides strong theoretical privacy guarantees by ensuring the statistical indistinguishability of individuals in a dataset. It has become the de facto framework for providing privacy-preserving data analysis over statistical datasets. Differential privacy has garnered significant attention from researchers and privacy experts due to its strong privacy guarantees. However, the accuracy loss caused by the noise added has been an issue. First, we propose a new noise adding mechanism that preserves \((\epsilon ,\delta )\)-differential privacy. The distribution pertaining to this mechanism can be observed as a generalized truncated Laplacian distribution. We show that the proposed mechanism adds optimal noise in a global context, conditional upon technical lemmas. We also show that the generalized truncated Laplacian mechanism performs better than the optimal Gaussian mechanism. In addition, we also propose an \((\epsilon )\)-differentially private mechanism to improve the utility of differential privacy by fusing multiple Laplace distributions. We also derive the closed-form expressions for absolute expectation and variance of noise for the proposed mechanisms. Finally, we empirically evaluate the performance of the proposed mechanisms and show an increase in all utility measures considered, while preserving privacy.

The aim of this paper is to generalize the statistical models of nonnegative matrix factorization (NMF) and multichannel NMF (MNMF). For the NMF and its multichannel extensions, various statistical models have been proposed to improve the model flexibility in the literature on signal separation. However, few studies have been done on the generalization which includes the model based on the Laplace distribution. Thus, we propose the generalized models of the NMF and the MNMF, which include the models based on the Gaussian distribution and the two types of the Laplace distributions, using the Bessel function distribution. To estimate unknown model parameters, we derive the update rules based on the majorization-minimization algorithm. The performances of the proposed NMF and MNMF are evaluated in fitting synthetic data and music signal separation, respectively.

The functionals defining the entropies of Shannon, Tsallis, Renyi are considered. It is shown that the entropies of Tsallis and Renyi, defined for continuous random variables, approach the Shannon entropy as the parameter of extensiveness approaches unity. The concept of relative parametric sensitivity, defined by the elasticity of entropy with respect to the parameters of the distribution law, is introduced. The expressions for the parametric elasticity of the Shannon entropy for the normal distribution, Laplace distribution, Cauchy distribution, logistic distribution, loglogistic distribution, Rayleigh distribution, exponential distribution, lognormal distribution, Pareto distribution, Weibull distribution, gamma distribution are obtained. The general expressions for determining the elasticity with respect to the extensivity parameter for the entropies of Tsallis and Renyi are obtained. The general conditions for determining the elasticity for the entropies of Tsallis and Renyi with respect to one of the parameters of the distribution density are obtained.

One of the key issues of noisy speech enhancement technique is to achieve appropriate statistical distributions to model the clean speech and noise signals accurately. Most of the existing algorithms try to employ a sole model assumption in transform domain, which, however, has been proven to being contrary with the fact. To address this problem, the statistical properties of clean speech as well as several noise signals are analyzed using actual data in Discrete cosine transform (DCT) domain, and the study indicates the statistic of clean speech DCT coefficients tending to fall somewhere in between the Gaussian and Laplacian distribution. Based on the results, a novel speech enhancement algorithm is proposed using Gaussian-Laplacian combination model, whose core is employing a linear combination of Gaussian and Laplacian distribution to model the statistic of clean speech DCT coefficients. The corresponding weights of either distribution to the combination model are adaptively adjusted in terms of the probability of each hypothesis, which is estimated based on a soft decision technique by using Bayesian theorem. Through a number of objective and subjective tests, we compare the performance of the proposed algorithm with other recent model based approaches and have found that our algorithm is superior to the related approaches at all testing environments.

This paper introduces two graph-based semi-supervised embedding methods for generic classification and recognition tasks. These proposed methods combine the merits of Flexible Manifold Embedding, non-linear graph based embedding, and adaptive loss function. The adaptive loss function seems to be a good choice for reaching a flexible and adaptive regressor in the sense that the effect of outliers is reduced and the regression function is more elastic and more robust. Furthermore, unlike label propagation approaches, the proposed methods provide a data embedding into a space whose dimension is not limited to the number of classes. The adaptive loss function combines the merits of the l 1, 2 and l2 matrix norms . It can handle the Laplacian distribution of outliers and the Gaussian distribution of samples with small loss. We provide extensive experiments on eight benchmark datasets in order to study the performance of the proposed methods. These experiments show that the proposed methods can be more discriminative than other state-of-the-art methods .

Using the estimates of the accuracy of the normal approximation to distributions of Poisson-binomial random sums from [I. G. Shevtsova, Theory Probab. Appl., 62 (2018), pp. 278--294], we obtain mom...

This paper presents a novel wavelet method based on a prior model of wavelet coefficient with a generalized bivariate distribution function. The generalized bivariate prior model extends the isotropic bivariate prior model and anisotropic prior model, and can well characterize the joint parent-child statistical properties of many categories of images. Owing to the fact that the variance of wavelet coefficients are quite different for different scales and positions, we propose a local adaptive marginal variance estimation method based on this newly generalized prior model to improve the accuracy of marginal variance by considering the distribution of wavelet coefficients as the combination of Gaussian distribution and Laplacian distribution. By simulation experiments from three aspects, the results of the proposed method outperform the mainstream isotropic wavelet methods and anisotropic wavelet methods, and the practicality of the proposed method is verified by real noise image experiments.

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).

The paper introduces a recursive procedure to invert the multivariate Laplace transform of probability distributions. The procedure involves taking independent samples from the Laplace transform; these samples are then used to update recursively an initial starting distribution. The update is Bayesian driven. The final estimate can be written as a mixture of independent gamma distributions, making it the only methodology which guarantees to numerically recover a probability distribution with positive support. Proof of convergence is given by a fixed point argument. The estimator is fast, accurate and can be run in parallel since the target distribution is evaluated on a grid of points. The method is illustrated on several examples and compared to the bivariate Gaver–Stehfest method.

The analysis of the statistical characteristics of the geomagnetic field generated in the numerical geodynamo models has shown that the distribution of the spherical harmonic coefficients in some cases is not Gaussian but, instead, has much in common with the Laplace distribution. The shape of the corresponding histograms depends on the time scale, which allows interpreting the obtained data in terms of a mixture of Gaussian distributions. The similar effects associated with the intermittence were observed in the experiments in a turbulent fluid flow. Hence, the behavior of secular variations in the magnetic field of the Earth should perhaps be described in terms of a mixture of several Gaussian stationary processes corresponding to switching between the different regimes of geodynamo generation.

International audience; We provide a new and simple characterization of the multivariate generalized Laplace distribution. In particular, this result implies that the product of a Gaussian matrix with independent and identically distributed columns by an independent isotropic Gaussian vector follows a symmetric multivariate generalized Laplace distribution.

Most of Voice activity detection (VAD) methods are based on statistical model. In these methods, the noise signal is always assumed to satisfy and characterized by Gaussian distribution, while the assumption of noise does not always hold in practice and which causes that these kinds of method fail to distinguish speech from noise at low Signal-noise-ratio (SNR) level in non-stationary noise condition. For going further to improve the robustness of VAD, a enhanced speech based method is proposed. In the proposed method, the Laplacian distribution is used to model the remained noise since we find that the remained noise in enhanced speech satisfy Laplacian distribution; in addition, Gaussian mixture model is used to characterize the Discrete Fourier transform (DFT) coefficients of reconstructed speech in enhanced speech. Experimental results show that the proposed method performs better than the baseline method, especially in low SNR and non-stationary noise conditions.

This article considers the adaptive elastic net estimator for regularized mean regression from a Bayesian perspective. Representing the Laplace distribution as a mixture of Bartlett–Fejer kernels with a Gamma mixing density, a Gibbs sampling algorithm for the adaptive elastic net is developed. By introducing slice variables, it is shown that the mixture representation provides a Gibbs sampler that can be accomplished by sampling from either truncated normal or truncated Gamma distribution. The proposed method is illustrated using several simulation studies and analyzing a real dataset. Both simulation studies and real data analysis indicate that the proposed approach performs well.

We improve bounds of accuracy of the normal approximation to the distribution of a sum of independent random variables under relaxed moment conditions, in particular, under the absence of moments of orders higher than the second. We extend these results to Poisson binomial, binomial, and Poisson random sums. Under the same conditions, we obtain bounds for the accuracy of approximation of the distributions of mixed Poisson random sums by the corresponding limit law. In particular, we construct these bounds for the accuracy of approximation of the distributions of geometric, negative binomial, and Poisson-inverse gamma (Sichel) random sums by the Laplace, variance gamma, and Student distributions, respectively.

Image denoising problem has attracted a large number of researchers. Generally speaking, there are two kinds of image priors considering the source of training sets, external priors and internal priors. The realistic image priors can be obtained from a large number of external example images or the corrupted internal image itself. However, external priors cannot give accurate image representations towards various corrupted images because the total number of example images is limited. While internal priors may bring too much noise along with useful information for denoising, which leads to unexpected denoising results. The most common assumption in denoising problem is that the image noise obeys Gaussian distribution, which is simple and ideal. If there are outliers in the corrupted images, Laplace distribution is more suitable to model the image noise. This paper proposes a denoising model towards image noise in Laplace distribution utilizing both external priors and internal priors. Gaussian Mixture Model (GMM) is used to model external priors and l1 norm is aimed to deal with outliers. Experiments on some publicly available databases show the performance of proposed method, resulting in denoised image of high quality.

In this paper, we have investigated recently proposed feature extraction technique for texture image representation. In the introduced method, features are extracted via bounded Laplace mixture model (BLMM) in wavelet domain. Due to nature of wavelet coefficients that can be modeled accurately with Laplace distribution, it is proposed to apply classifiers based on this distribution, which leads us to introduce Naive Bayes classifier with Laplace distribution for image categorization. The proposed approach is validated through experiments on different texture image datasets and it has shown very good results as compared to the model based on Gaussian distribution. The generalized Gaussian distribution is a generalization of both Laplace and Gaussian distributions, thus we have introduced also Naive Bayes classifier with generalized Gaussian distribution to achieve better performance as compared to the above two models. The proposed approach is also validated through extensive experiments and it is observed that by taking into account the nature of data, proposed models have very good performance. Classification results are presented by different performance metrics to ensure the effectiveness of proposed algorithms in texture image classification.

Time delay is a very significant measurement in the field of detection and positioning. The accuracy of time delay estimation directly influences the precision and reliability of the positioning. Aiming at improving the accuracy of time delay estimation in positioning, a new method of high precision time delay estimation is proposed in this paper. Assuming the background noise as Gaussian distribution and Laplace distribution, the new method firstly performs auto-correlation and cross-correlation operation on the received signal to suppress noise initially, and then performs fourth order cumulant operation to further suppress noise and finally extract time delay information by cross correlation method. Comparing with generalized cross-correlation method, second-correlation method and high-order cumulant method, the new method can effectively suppress Gaussian noise and Laplace noise interference. The new method can achieve higher precision and obtain high precise detection and positioning. At last the simulation results verify the effectiveness of the new method.

The main purpose of this paper is to give several identities of symmetry for type 2 Bernoulli and Euler polynomials by considering certain quotients of bosonic p-adic and fermionic p-adic integrals on Z p , where p is an odd prime number. Indeed, they are symmetric identities involving type 2 Bernoulli polynomials and power sums of consecutive odd positive integers, and the ones involving type 2 Euler polynomials and alternating power sums of odd positive integers. Furthermore, we consider two random variables created from random variables having Laplace distributions and show their moments are given in terms of the type 2 Bernoulli and Euler numbers.

Accurate and sufficient node location information is crucial for Wireless Sensor Networks (WSNs) applications. However, the existing range-based localization methods often suffer from incomplete and detorted range measurements. To address this issue, some methods based on low-rank matrix recovery have been proposed, which usually assume noises follow single Gaussian distribution or/and single Laplacian distribution, and thus cannot handle the case with wider noise distributions beyond Gaussian and Laplacian ones. In this paper, a novel Anomaly-aware Node Localization (ANLoC) method is proposed to simultaneously impute missing range measurements and detect node anomaly in complex environments. Specifically, by utilizing inherent low-rank property of Euclidean Distance Matrix (EDM), we formulate range measurements imputation problem as a Robust ℓ 2 , 1 -norm Regularized Matrix Decomposition (RRMD) model, where complex noise is fitted by Mixture of Gaussian (MoG) distribution, and node anomaly is sifted by ℓ 2 , 1 -norm regularization. Meanwhile, an efficient optimization algorithm is designed to solve proposed RRMD model based on Expectation Maximization (EM) method. Furthermore, with the imputed EDM, all unknown nodes can be easily positioned by using Multi-Dimensional Scaling (MDS) method. Finally, some experiments are designed to evaluate performance of the proposed method, and experimental results demonstrate that our method outperforms three state-of-the-art node localization methods.

Representation-based classification (RC) methods, such as sparse RC, have shown great potential in face recognition (FR) in recent years. Most previous RC methods are based on the conventional regression models, such as lasso regression, ridge regression, or group lasso regression. These regression models essentially impose a predefined assumption on the distribution of the noise variable in the query sample, such as the Gaussian or Laplacian distribution. However, the complicated noises in practice may violate the assumptions and impede the performance of these RC methods. In this paper, we propose a modal regression (MR)-based atomic representation and classification (MRARC) framework to alleviate such limitations. MR is a robust regression framework which aims to reveal the relationship between the input and response variables by regressing toward the conditional mode function. Atomic representation is a general atomic norm regularized linear representation framework which includes many popular representation methods, such as sparse representation, collaborative representation, and low-rank representation as special cases. Unlike previous RC methods, the MRARC framework does not require the noise variable to follow any specific predefined distributions. This gives rise to the capability of MRARC in handling various complex noises in reality. Using MRARC as a general platform, we also develop four novel RC methods for unimodal and multimodal FR, respectively. In addition, we devise a general optimization algorithm for the unified MRARC framework based on the alternating direction method of multipliers and half-quadratic theory. The experiments on real-world data validate the efficacy of MRARC for robust FR and reconstruction.