Laplace Distribution Research Articles

Differential Privacy via Haar Wavelet Transform and Gaussian Mechanism for Range Query.

Range query is the hot topic of the privacy-preserving data publishing. To preserve privacy, the large range query means more accumulate noise will be injected into the input data. This study presents a research on differential privacy for range query via Haar wavelet transform and Gaussian mechanism. First, the noise injected into the input data via Laplace mechanism is analyzed, and we conclude that it is difficult to judge the level of privacy protection based on the Haar wavelet transform and Laplace mechanism for range query because the sum of independent random Laplace variables is not a variable of a Laplace distribution. Second, the method of injecting noise into Haar wavelet coefficients via Gaussian mechanism is proposed in this study. Finally, the maximum variance for any range query under the framework of Haar wavelet transform and Gaussian mechanism is given. The analysis shows that using Haar wavelet transform and Gaussian mechanism, we can preserve the differential privacy for each input data and any range query, and the variance of noise is far less than that just using the Gaussian mechanism. In an experimental study on the dataset age extracted from IPUM's census data of the United States, we confirm that the proposed mechanism has much smaller maximum variance of noises than the Gaussian mechanism for range-count queries.

Structural health monitoring data cleaning based on Bayesian robust tensor learning

Data cleaning is necessary to obtain high-quality monitoring data, which consists of two parts: anomaly identification and repair. A high-quality dataset is always used to repair data anomalies in current research, which may be difficult to obtain in practice because of the large number of data anomalies existing in real data. Besides, outliers, baseline shifts, and noise in dataset can lead to wrong missing restoration results. Therefore, the anomaly identification and location are needed to remove outliers, baseline shifts, and noise to regard the repair of these anomalies as missing data restoration. However, the identification of multiple data anomalies is time consuming and challenging. To this end, we propose a Bayesian robust tensor learning model to reconstruct the monitoring data tensor by extracting potential data features of the spatiotemporal tensor. A mixture of Laplace distribution with generalized inverse Gaussian distribution is used to model noise to enhance robustness. The tensor formed from the data containing anomalies can be input directly into the model without the identification and location of the data anomalies in advance. All data anomalies are repaired during the tensor reconstruction process. In addition, outliers, baseline shifts, and noise do not interfere with the missing restoration results owing to the robustness of the model. Furthermore, different cleaning strategies are proposed for dynamic and static response data. The data performance is verified using the real monitoring data of a concrete box girder bridge. The results show that the proposed method exhibits good data-cleaning performance when dealing with data containing a large number of data anomalies. In addition, as the abnormal rate increases, the proposed method still exhibits good data-cleaning performance.

Robust AOA-Based Target Localization for Uniformly Distributed Noise via ℓp-ℓ1 Optimization.

This paper addresses the problem of robust angle of arrival (AOA) target localization in the presence of uniformly distributed noise which is modeled as the mixture of Laplacian distribution and uniform distribution. Motivated by the distribution of noise, we develop a localization model by using the -norm with as the measurement error and the -norm as the regularization term. Then, an estimator for introducing the proximal operator into the framework of the alternating direction method of multipliers (POADMM) is derived to solve the convex optimization problem when . However, when , the corresponding optimization problem is nonconvex and nonsmoothed. To derive a convergent method for this nonconvex and nonsmooth target localization problem, we propose a smoothed POADMM estimator (SPOADMM) by introducing the smoothing strategy into the optimization model. Eventually, the proposed algorithms are compared with some state-of-the-art robust algorithms via numerical simulations, and their effectiveness in uniformly distributed noise is discussed from the perspective of root-mean-squared error (RMSE). The experimental results verify that the proposed method has more robustness against outliers and is less sensitive to the selected parameters, especially the variance of the measurement noise.

AGBM: An Adaptive Gradient Balanced Mechanism for the End-to-End Steering Estimation

End-to-end steering estimation is one of the important deep regression tasks. However, driving datasets are always imbalanced on the distribution of the steering value, which makes end-to-end learning models perform bad steering estimation accuracy, especially for the sharp steering value estimation, which finally, leads to an imbalanced training problem. In this paper, the essential reason for the imbalanced training problem for the deep regression task is first revealed as the gradient disharmony. To hedge the gradient disharmony, a general gradient balanced mechanism for the deep regression task is proposed in this paper. Meanwhile, the general gradient balanced mechanism is applied for the steering estimation task. According to the steering distribution, a novel Adaptive Gradient Balanced Mechanism (AGBM) is proposed to hedge the gradient disharmony based on the Laplace distribution with the general gradient balanced mechanism theory. Further, a new loss function embedding with AGBM is proposed to balance the gradient disharmony for the training process. AGBM makes the process of gradient hedging with adaptive style without parameters finetuning. Based on five driving datasets, four experiments are conducted for demonstrations. The experiment results show the AGBM enables the end-to-end models to perform the best accuracy of the steering estimation, including the sharp steering samples. Meanwhile, AGBM is generalized with different end-to-end learning models.

Correlated continuous-time random walk with stochastic resetting

It is known that the introduction of stochastic resetting in an uncorrelated random walk process can lead to the emergence of a stationary state, i.e. the diffusion evolves towards a saturation state, and a steady Laplace distribution is reached. In this paper, we turn to study the anomalous diffusion of the correlated continuous-time random walk considering stochastic resetting. Results reveal that it displays quite different diffusive behaviors from the uncorrelated one. For the weak correlation case, the stochastic resetting mechanism can slow down the diffusion. However, for the strong correlation case, we find that the stochastic resetting cannot compete with the space-time correlation, and the diffusion presents the same behaviors with the one without resetting. Meanwhile, a steady distribution is never reached.

A structure noise-aware tensor dictionary learning method for high-dimensional data clustering

With the development of data acquisition technology, high-dimensional data clustering is an important yet challenging task in data mining. Despite advances achieved by current clustering methods, they can be further improved. First, many of them usually unfold the high-dimensional data into a large matrix, consequently resulting in destroying the intrinsic structural property. Second, some methods assume that the noise in the dataset conforms to a predefined distribution (e.g., the Gaussian or Laplacian distribution), which violates real-world applications and eventually decreases the clustering performance. To address these issues, in this paper, we propose a novel tensor dictionary learning method for clustering high-dimensional data with the coexistence of structure noise. We adopt tensors, the natural and powerful tools for the generalizations of vectors and matrices, to characterize high-dimensional data. Meanwhile, to depict the noise accurately, we decompose the observed data into clean data, structure noise, and Gaussian noise. Furthermore, we use low-rank tensor modeling to characterize the inherent correlations of clean data and adopt tensor dictionary learning to adaptively and accurately describe the structure noise instead of using the predefined distribution. We design the proximal alternating minimization algorithm to solve the proposed model with the theoretical convergence guarantee. Experimental results on both simulated and real datasets show that the proposed method outperforms the compared methods for high-dimensional data clustering.

A Novel Physical Mechanism to Model Brownian Yet Non-Gaussian Diffusion: Theory and Application.

In the last years, a few experiments in the fields of biological and soft matter physics in colloidal suspensions have reported “normal diffusion” with a Laplacian probability distribution in the particle’s displacements (i.e., Brownian yet non-Gaussian diffusion). To model this behavior, different stochastic and microscopic models have been proposed, with the former introducing new random elements that incorporate our lack of information about the media and the latter describing a limited number of interesting physical scenarios. This incentivizes the search of a more thorough understanding of how the media interacts with itself and with the particle being diffused in Brownian yet non-Gaussian diffusion. For this reason, a comprehensive mathematical model to explain Brownian yet non-Gaussian diffusion that includes weak molecular interactions is proposed in this paper. Based on the theory of interfaces by De Gennes and Langevin dynamics, it is shown that long-range interactions in a weakly interacting fluid at shorter time scales leads to a Laplacian probability distribution in the radial particle’s displacements. Further, it is shown that a phase separation can explain a high diffusivity and causes this Laplacian distribution to evolve towards a Gaussian via a transition probability in the interval of time as it was observed in experiments. To verify these model predictions, the experimental data of the Brownian motion of colloidal beads on phospholipid bilayer by Wang et al. are used and compared with the results of the theory. This comparison suggests that the proposed model is able to explain qualitatively and quantitatively the Brownian yet non-Gaussian diffusion.

On the Conditional Value at Risk Based on the Laplace Distribution with Application in GARCH Model

In this article, the Laplace distribution is employed in lieu of the well-known normal distribution for finding better scalar values of risk. Explicit formulas for value-at-risk (VaR) and conditional value-at-risk (CVaR) are studied and used to manage the risk involved in a stock movement by using the GARCH model. Numerical simulations are given for a variety of stocks in equity markets to uphold the findings.

Optimizing cut-off grade considering grade estimation uncertainty - A case study of Witwatersrand gold-producing areas

Due to grade estimation uncertainty, two statistical errors can occur. The Type I error is where material is classified as ore and mined, despite the true value being below the break-even grade. This material is dilution. The Type II error is where the material is estimated to be below the cut-off grade and is classified as waste, although the true grade is actually above the break-even grade. This material is not mined and is lost. The uncertainty was assumed to follow a normal distribution in a previous study. For this study, estimated block values are compared to those determined after mining (the best estimate of the true grade). This actual data from four mines shows that the uncertainty follows a Laplace distribution. There is no single solution regarding adjusting the cut-off grade away from the break-even grade, considering estimation uncertainty, that could be applied to all gold mines. However, adjusting the cut-off grade downwards (up to 22% for one mine) is noted when optimizing the profit considering grade uncertainties. This type of adjustment could open up significant mining areas and extend the life of the mine.

Clustering of bivariate satellite time series: A quantile approach

AbstractClustering has received much attention in statistics and machine learning with the aim of developing statistical models and autonomous algorithms which are capable of acquiring information from raw data in order to perform exploratory analysis. Several techniques have been developed to cluster sampled univariate vectors only considering the average value over the whole period and as such they have not been able to explore fully the underlying distribution as well as other features of the data, especially in presence of structured time series. We propose a model‐based clustering technique that is based on quantile regression permitting us to cluster bivariate time series at different quantile levels. We model the within cluster density using asymmetric Laplace distribution allowing us to take into account asymmetry in the distribution of the data. We evaluate the performance of the proposed technique through a simulation study. The method is then applied to cluster time series observed from Glob‐color satellite data related to trophic status indices with aim of evaluating their temporal dynamics in order to identify homogeneous areas, in terms of trophic status, in the Gulf of Gabes.

Hyperspectral image denoising by low-rank models with hyper-Laplacian total variation prior

The total variation (TV) regularized low-rank models have emerged as a powerful tool for hyperspectral image (HSI) denoising. TV, defined by the ℓ1-norm of gradients, is assumed that gradients obey the Laplacian distribution from the statistics point of view. By investigating the histogram of HSI’s gradients, we find that gradients in real HSIs are actually distributed as the hyper-Laplacian distribution with the power parameter q=1/2. Taking this prior into account, a hyper-Laplacian spectral-spatial total variation (HTV), defined by the ℓ1/2-norm of gradients, is proposed for HSI denoising. Furthermore, by incorporating HTV as the regularizer, a low-rank matrix model and a low-rank tensor model are proposed. The two models can be solved by the augmented Lagrange multiplier algorithm. To validate the effectiveness of HTV, we formulate baseline models by replacing HTV with ℓ1-norm and ℓ0-norm based TV regularizations, and it is revealed that our proposed HTV outperforms them. Furthermore, compared with several popular HSI denoising algorithms, the experiments conducted on both the simulated and real data demonstrate the superiority of proposed models.

Median regression models for clustered, interval-censored survival data - An application to prostate surgery study.

Genitourinary surgeons and oncologists are particularly interested in whether a robotic surgery improves times to Prostate Specific Antigen (PSA) recurrence compared to a non-robotic surgery for removing the cancerous prostate. Time to PSA recurrence is an example of a survival time that is typically interval-censored between two consecutive clinical inspections with opposite test results. In addition, success of medical devices and technologies often depends on factors such as experience and skill level of the medical service providers, thus leading to clustering of these survival times. For analyzing the effects of surgery types and other covariates on median of clustered interval-censored time to post-surgery PSA recurrence, we present three competing novel models and associated frequentist and Bayesian analyses. The first model is based on a transform-both-sides of survival time with Gaussian random effects to account for the within-cluster association. Our second model assumes an approximate marginal Laplace distribution for the transformed log-survival times with a Gaussian copula to accommodate clustering. Our third model is a special case of the second model with Laplace distribution for the marginal log-survival times and Gaussian copula for the within-cluster association. Simulation studies establish the second model to be highly robust against extreme observations while estimating median regression coefficients. We provide a comprehensive comparison among these three competing models based on the model properties and the computational ease of their Frequentist and Bayesian analysis. We also illustrate the practical implementations and uses of these methods via analysis of a simulated clustered interval-censored data-set similar in design to a post-surgery PSA recurrence study.

RG-Flow: a hierarchical and explainable flow model based on renormalization group and sparse prior

Flow-based generative models have become an important class of unsupervised learning approaches. In this work, we incorporate the key ideas of renormalization group (RG) and sparse prior distribution to design a hierarchical flow-based generative model, RG-Flow, which can separate information at different scales of images and extract disentangled representations at each scale. We demonstrate our method on synthetic multi-scale image datasets and the CelebA dataset, showing that the disentangled representations enable semantic manipulation and style mixing of the images at different scales. To visualize the latent representations, we introduce receptive fields for flow-based models and show that the receptive fields of RG-Flow are similar to those of convolutional neural networks. In addition, we replace the widely adopted isotropic Gaussian prior distribution by the sparse Laplacian distribution to further enhance the disentanglement of representations. From a theoretical perspective, our proposed method has complexity for inpainting of an image with edge length L, compared to previous generative models with complexity.

Quantile hidden semi-Markov models for multivariate time series.

This paper develops a quantile hidden semi-Markov regression to jointly estimate multiple quantiles for the analysis of multivariate time series. The approach is based upon the Multivariate Asymmetric Laplace (MAL) distribution, which allows to model the quantiles of all univariate conditional distributions of a multivariate response simultaneously, incorporating the correlation structure among the outcomes. Unobserved serial heterogeneity across observations is modeled by introducing regime-dependent parameters that evolve according to a latent finite-state semi-Markov chain. Exploiting the hierarchical representation of the MAL, inference is carried out using an efficient Expectation-Maximization algorithm based on closed form updates for all model parameters, without parametric assumptions about the states’ sojourn distributions. The validity of the proposed methodology is analyzed both by a simulation study and through the empirical analysis of air pollutant concentrations in a small Italian city.Supplementary InformationThe online version contains supplementary material available at 10.1007/s11222-022-10130-1.

Spatially enhanced electric field induced second harmonic (SEEFISH) generation for measurements of electric field distributions in high-pressure plasmas

The spatial resolution of the ps electric field induced second harmonic (EFISH) generation has been enhanced by using non-collinear pump laser beam arrangements. The pump laser beam (1064 nm, nominal pulse duration 150 ps, pulse energy 20–40 mJ) is separated into two coaxial or crossing beams, overlapping only near the focal point. The spatially enhanced EFISH (SEEFISH) signal is generated over a shorter beam overlap region compared to the collinear beam arrangement. Blocking of either of the two beams results in a complete suppression of the signal. The signal is spatially isolated from the ‘conventional’ EFISH signal and measured by a photomultiplier detector. Measurements of a known Laplacian field generated between two parallel cylinder electrodes in ambient air shows that SEEFISH improves the spatial resolution of the measurements by up to a factor of 2, such that the measurement results agree with the Laplacian field distribution. The spatial resolution is improved further by reducing the focal distance of the lens. The magnitude of the SEEFISH signal is significantly lower compared to that of the single-beam EFISH and decreases rapidly as the beam crossing angle is increased, due to the phase mismatch. This approach has a significant potential for measurements of electric field distributions in high-pressure plasmas, with an additional benefit of removing the stray second harmonic signal from the optical access windows.

Bayesian bridge and reciprocal bridge composite quantile regression

Bayesian regularized composite quantile regression approaches with bridge and reciprocal bridge penalties are adopted to conduct variable selection in composite quantile regression. Two MCMC algorithms were developed for posterior inference using the normal-exponential mixture representation of the asymmetric Laplace distribution. Gamma prior is placed on the regularization parameter. The parameters of the Gamma prior are treated as unknowns and estimate them along with the other parameters. The performance of the proposed composite quantile regression methods over the existing methods is then illustrated via simulated studies and a real data set.

Horizontally Partitioned Data Publication with Differential Privacy

In this paper, we study the privacy-preserving data publishing problem in a distributed environment. The data contain sensitive information; hence, directly pooling and publishing the local data will lead to privacy leaks. To solve this problem, we propose a multiparty horizontally partitioned data publishing method under differential privacy (HPDP-DP). First, in order to make the noise level of the published data in the distributed scenario the same as in the centralized scenario, we use the infinite divisibility of the Laplace distribution to design a distributed noise addition scheme to perturb the locally shared data and use Paillier encryption to transmit the locally shared data to the semitrusted curator. Then, the semitrusted curator obtains the estimator of the covariance matrix of the aggregated data with Laplace noise and then obtains the principal components of the aggregated data and returns them to each data owner. Finally, the data owner utilizes the generative model of probabilistic principal component analysis to generate a synthetic data set for publication. We conducted experiments on different real data sets; the experimental results demonstrate that the synthetic data set released by the HPDP-DP method can maintain high utility.

Disentangling the Influence of Data Contamination in Growth Curve Modeling: A Median Based Bayesian Approach

Growth curve models (GCMs), with their ability to directly investigate within-subject change over time and between-subject differences in change for longitudinal data, are widely used in social and behavioral sciences. While GCMs are typically studied with the normal distribution assumption, empirical data often violate the normality assumption in applications. Failure to account for the deviation from normality in data distribution may lead to unreliable model estimation and misleading statistical inferences. A robust GCM based on conditional medians was recently proposed and outperformed traditional growth curve modeling when outliers are present resulting in nonnormality. However, this robust approach was shown to perform less satisfactorily when leverage observations existed. In this work, we propose a robust double medians growth curve modeling approach (DOME GCM) to thoroughly disentangle the influence of data contamination on model estimation and inferences, where two conditional medians are employed for the distributions of the within-subject measurement errors and of random effects, respectively. Model estimation and inferences are conducted in the Bayesian framework, and Laplace distributions are used to convert the optimization problem of median estimation into a problem of obtaining the maximum likelihood estimator for a transformed model. A Monte Carlo simulation study has been conducted to evaluate the numerical performance of the proposed approach, and showed that the proposed approach yields more accurate and efficient parameter estimates when data contain outliers or leverage observations. The application of the developed robust approach is illustrated using a real dataset from the Virginia Cognitive Aging Project to study the change of memory ability.

Single image based 3D human pose estimation via uncertainty learning

In monocular image scenes, 3D human pose estimation exhibits inherent ambiguity due to the loss of depth information and occlusions. Simply regressing body joints with high uncertainties will lead to model overfitting and poor generalization.In this paper, we propose an uncertainty-based framework to jointly learn 3D human poses and the uncertainty of each joint. Our proposed joint estimation framework aims to mitigate the adverse effects of training samples with high uncertainties and facilitate the training procedure. To be specific, we model each body joint as a Laplace distribution for uncertainty representation. Since visual joints often exhibit low uncertainties while occluded ones have high uncertainties, we develop an adaptive scaling factor, named the uncertainty-aware scaling factor, to ease the network optimization in accordance with the joint uncertainties.By doing so, our network is able to converge faster and significantly reduce the adverse effects caused by those ambiguous joints. Furthermore, we present an uncertainty-aware graph convolutional network by exploiting the learned joint uncertainties and the relationships among joints to refine the initial joint localization. Extensive experiments on single-person (Human3.6M) and multi-person (MuCo-3DHP & MuPoTS-3D) 3D human pose estimation datasets demonstrate the effectiveness of our method.

A comprehensive empirical power comparison of univariate goodness-of-fit tests for the Laplace distribution

In this paper, we present the results from an empirical power comparison of 40 goodness-of-fit tests for the univariate Laplace distribution, carried out using Monte Carlo simulations with sample sizes n = 20, 50, 100, 200, significance levels , and 400 alternatives consisting of asymmetric and symmetric light/heavy-tailed distributions taken as special cases from 11 models. In addition to the unmatched scope of our study, an interesting contribution is the proposal of an innovative design for the selection of alternatives. The 400 alternatives consist of 20 specific cases of 20 submodels drawn from the main 11 models. For each submodel, the 20 specific cases corresponded to parameter values chosen to cover the full power range. An analysis of the results leads to a recommendation of the best tests for five different groupings of the alternative distributions. A real-data example is also presented, where an appropriate test for the goodness-of-fit of the univariate Laplace distribution is applied to weekly log-returns of Amazon stock over a recent 4-year period.