Laplace Distribution Research Articles

Granger Causal Inference Based on Dual Laplacian Distribution and Its Application to MI-BCI Classification.

Granger causality-based effective brain connectivity provides a powerful tool to probe the neural mechanism for information processing and the potential features for brain computer interfaces. However, in real applications, traditional Granger causality is prone to the influence of outliers, such as inevitable ocular artifacts, resulting in unreasonable brain linkages and the failure to decipher inherent cognition states. In this work, motivated by constructing the sparse causality brain networks under the strong physiological outlier noise conditions, we proposed a dual Laplacian Granger causality analysis (DLap-GCA) by imposing Laplacian distributions on both model parameters and residuals. In essence, the first Laplacian assumption on residuals will resist the influence of outliers in electroencephalogram (EEG) on causality inference, and the second Laplacian assumption on model parameters will sparsely characterize the intrinsic interactions among multiple brain regions. Through simulation study, we quantitatively verified its effectiveness in suppressing the influence of complex outliers, the stable capacity for model estimation, and sparse network inference. The application to motor-imagery (MI) EEG further reveals that our method can effectively capture the inherent hemispheric lateralization of MI tasks with sparse patterns even under strong noise conditions. The MI classification based on the network features derived from the proposed approach shows higher accuracy than other existing traditional approaches, which is attributed to the discriminative network structures being captured in a timely manner by DLap-GCA even under the single-trial online condition. Basically, these results consistently show its robustness to the influence of complex outliers and the capability of characterizing representative brain networks for cognition information processing, which has the potential to offer reliable network structures for both cognitive studies and future brain-computer interface (BCI) realization.

Alternative modified Cholesky decomposition of the precision matrix of longitudinal data

The correlation matrix might be of scientific interest for longitudinal data. However, few studies have focused on both robust estimation of the correlation matrix against model misspecification and robustness to outliers in the data, when the precision matrix possesses a typical structure. In this paper, we propose an alternative modified Cholesky decomposition (AMCD) for the precision matrix of longitudinal data, which results in robust estimation of the correlation matrix against model misspecification of the innovation variances. A joint mean-covariance model with multivariate normal distribution and AMCD is established, the quasi-Fisher scoring algorithm is developed, and the maximum likelihood estimators are proven to be consistent and asymptotically normally distributed. Furthermore, a double-robust joint modeling approach with multivariate Laplace distribution and AMCD is established, and the quasi-Newton algorithm for maximum likelihood estimation is developed. The simulation studies and real data analysis demonstrate the effectiveness of the proposed AMCD method.

A study of China’s financial market risks in the context of Covid-19, based on a rolling generalized autoregressive score model using the asymmetric Laplace distribution

A study of China’s financial market risks in the context of Covid-19, based on a rolling generalized autoregressive score model using the asymmetric Laplace distribution

Analysis and Modeling of Geodetic Data Based on Machine Learning

Abstract This paper underscores the significance of earth deformation observation in analyzing earth tide curves and predicting earthquakes, positioning it as a cornerstone of Earth observation technology. We delve into the critical task of detecting and diagnosing anomalies in geodetic data. Utilizing Python for data preprocessing, our approach identifies missing values, categorizes them by their spatial occurrence, and employs spline interpolation and autoregressive prediction methods for data imputation. This process ensures the integrity of the dataset for subsequent analysis and modeling, reinforcing the precision and reliability of geodetic data analysis in Earth science research. For problem I To expand the data set, we propose three models. Model I: Adding gaussian noise to the data. Model II: Resample the data. Model III: Using machine learning methods to learn the internal laws of the data and predict itself to generate new data. For each model, we discuss its advantages and disadvantages. Finally, we structurally fuse the three models to complete data enhancement. For problem II To extract the noise, we use DB4 wavelet transform to denoise the data set and extract the noise. Then we make descriptive statistics on the noise distribution, and use Laplace distribution to fit the probability distribution of noise, and finally get the accurate noise distribution. For problem III We start from the time domain and frequency domain to extract the features of the data. First, 17 features are extracted in the time domain, then the discrete fourier transform algorithm is used to transform the data into frequency domain data, and 13 are extracted. Therefore, we encode each data as a feature vector with a length of 30. We first use the decision tree as the baseline model to establish the recognition model to select the features. Logistic Regression, KNN, Naive Bayes and SVM are used to establish the recognition model. Finally, we use the Voting ensemble learning method to fuse the model, achieving an accuracy of 86% on the test set.

Distribution of Money on Connected Graphs with Multiple Banks

This paper studies an interacting particle system of interest in econophysics inspired from a model introduced in the physics literature. The original model consists of the customers of a single bank characterized by their capital, and the dynamics consists of monetary transactions in which a random individual x gives one coin to another random individual y, the transaction being canceled when x is in debt and there are no more coins in the bank. Using a combination of numerical simulations and heuristic arguments, physicists conjectured that the distribution of money (the random number of coins owned by a given individual) at equilibrium converges to an asymmetric Laplace distribution in the large population limit when the money temperature is large. We prove and extend this conjecture to a more general model including multiple banks and interactions among customers across banks. More importantly, we assume that customers are located on a general undirected connected graph (as opposed to the complete graph in the original model) where neighbors are interpreted as business partners, and transactions occur along the edges, thus modeling the flow of money across a social network. We first derive an exact expression for the distribution of money for all population sizes and money temperatures, then prove its convergence to an asymmetric Laplace distribution in the large population limit.

Laplacian Gradient Consistency Prior for Flash Guided Non-Flash Image Denoising.

For flash guided non-flash image denoising, the main challenge is to explore the consistency prior between the two modalities. Most existing methods attempt to model the flash/non-flash consistency in pixel level, which may easily lead to blurred edges. Different from these methods, we have an important finding in this paper, which reveals that the modality gap between flash and non-flash images conforms to the Laplacian distribution in gradient domain. Based on this finding, we establish a Laplacian gradient consistency (LGC) model for flash guided non-flash image denoising. This model is demonstrated to have faster convergence speed and denoising accuracy than the traditional pixel consistency model. Through solving the LGC model, we further design a deep network namely LGCNet. Different from existing image denoising networks, each component of the LGCNet strictly matches the solution of LGC model, giving the network good interpretability. The performance of the proposed LGCNet is evaluated on three different flash/non-flash image datasets, which demonstrates its superior denoising performance over many state-of-the-art methods both quantitatively and qualitatively. The intermediate features are also visualized to verify the effectiveness of the Laplacian gradient consistency prior. The source codes are available at https://github.com/JingyiXu404/LGCNet.

Smoothing denoising method of spatial filtering image based on Contourlet transform

This paper studies the threshold denoising problem of image in Contourlet transform domain, and proposes a smoothing denoising method of spatial filtering image based on Contourlet transform. This method combined with Contourlet transform can highlight the advantages of image edge information and direction information. The image is decomposed by using the Contourlet transform algorithm, and then the spatial filtered image is obtained. Based on the threshold estimation of the Contourlet coefficient of the Treelet difference variable and the soft threshold denoising of the high-frequency coefficient, the spatial filtered image denoising is realized. Then, aiming at the unsmooth denoising problem of the spatial filtered image, the spatial filtered image filtering smoothing processing method based on generalized Laplace distribution is used to solve the unsmooth denoising problem of the spatial filtered image. The experimental results show that this method has strong smoothing denoising ability and obvious application effect.

Tests for skewness parameter of skew log Laplace distribution

Laplace probability density function with additional shape parameter that regulates the degree of skewness is a skew Laplace distribution. The various forms of skew Laplace distribution are found in the literature, the distributions defined by Mc Gill (1962), Holla and Bhattacharya (1968), Lingappaiah (1988), Fernandez and Steel (1998). The skew log Laplace distribution is the probability distribution of a random variable whose logarithm follows a skew Laplace distribution. In this paper, the classical optimum tests for skewness parameter of skew log Laplace distribution (SLLD) derived from Lingappaiah (1988) distribution are discussed. Uniformly most powerful test, uniformly most powerful unbiased test and Wald’s sequential probability ratio test for skewness parameter are compared. The exact likelihood ratio test and Neyman structure test for testing skewness parameter when scale parameter is known are derived. Finally, the underreported income of Road Transport Company is analysed on the basis of the tests derived in this paper.

A flexible Bayesian hierarchical quantile spatial model for areal data

This article introduces a new class of nested models that extends the literature standard combination of spatial autoregressive model for areal data with parametric quantile regression by considering an asymmetric Laplace distribution for the random errors. In addition to being more flexible, the new proposed model can incorporate a hierarchical structure, allowing it to deal with clustered data. Such an approach produces a robust statistical method for modeling the quantiles of areal data distributed in a geographically hierarchical setting. The proposed non-hierarchical model is evaluated using a wellknown house pricing dataset and a simulation study. In addition, its hierarchical version is applied to a real dataset of math scores related to public high schools within the metropolitan area of Rio de Janeiro, Brazil.

A minimum entropy criterion for distribution selection for measurement uncertainty analysis

This paper presents a minimum entropy criterion for selecting the best probability distribution among a set of candidate distributions based on available information for measurement uncertainty analysis. We consider two cases that are most commonly encountered in practice: A and B. In Case A, the available information is a series of observations. In Case B, the available information is the maximum permissible error according to manufacturer’s specification. Three candidate distributions are considered in Case A: the scaled and shifted z-distribution (i.e. normal distribution), the scaled and shifted t-distribution, and the Laplace distribution. Five candidate distributions are considered in Case B: rectangular, triangular, quadratic, raised cosine, and half-cosine. According to the proposed minimum entropy criterion, the scaled and shifted z-distribution is the best distribution in Case A, and the raised cosine distribution is the best distribution in Case B. A case study is presented to demonstrate the effectiveness of the proposed minimum entropy criterion.

Uncertainty-aware MR-based CT synthesis for robust proton therapy planning of brain tumour

Background and purposeDeep learning techniques excel in MR-based CT synthesis, but missing uncertainty prediction limits its clinical use in proton therapy. We developed an uncertainty-aware framework and evaluated its efficiency in robust proton planning. Materials and methodsA conditional generative-adversarial network was trained on 64 brain tumour patients with paired MR-CT images to generate synthetic CTs (sCT) from combined T1-T2 MRs of three orthogonal planes. A Bayesian neural network predicts Laplacian distributions for all voxels with parameters (μ, b). A robust proton plan was optimized using three sCTs of μ and μ±b. The dosimetric differences between the plan from sCT (sPlan) and the recalculated plan (rPlan) on planning CT (pCT) were quantified for each patient. The uncertainty-aware robust plan was compared to conventional robust (global ± 3 %) and non-robust plans. ResultsIn 8-fold cross-validation, sCT-pCT image differences (Mean-Absolute-Error) were 80.84 ± 9.84HU (body), 35.78 ± 6.07HU (soft tissues) and 221.88 ± 31.69HU (bones), with Dice scores of 90.33 ± 2.43 %, 95.13 ± 0.80 %, and 85.53 ± 4.16 %, respectively. The uncertainty distribution positively correlated with absolute prediction error (Correlation Coefficient: 0.62 ± 0.01). The uncertainty-conditioned robust optimisation improved the rPlan-sPlan agreement, e.g., D95 absolute difference (CTV) was 1.10 ± 1.24 % compared to conventional (1.64 ± 2.71 %) and non-robust (2.08 ± 2.96 %) optimisation. This trend was consistent across all target and organs-at-risk indexes. ConclusionThe enhanced framework incorporates 3D uncertainty prediction and generates high-quality sCTs from MR images. The framework also facilitates conditioned robust optimisation, bolstering proton plan robustness against network prediction errors. The innovative feature of uncertainty visualisation and robust analyses contribute to evaluating sCT clinical utility for individual patients.

A Laplace-based model with flexible tail behavior

The proposed multiple scaled contaminated asymmetric Laplace (MSCAL) distribution is an extension of the multivariate asymmetric Laplace distribution to allow for a different excess kurtosis on each dimension and for more flexible shapes of the hyper-contours. These peculiarities are obtained by working on the principal component (PC) space. The structure of the MSCAL distribution has the further advantage of allowing for automatic PC-wise outlier detection – i.e., detection of outliers separately on each PC – when convenient constraints on the parameters are imposed. The MSCAL is fitted using a Monte Carlo expectation-maximization (MCEM) algorithm that uses a Monte Carlo method to estimate the orthogonal matrix of eigenvectors. A simulation study is used to assess the proposed MCEM in terms of computational efficiency and parameter recovery. In a real data application, the MSCAL is fitted to a real data set containing the anthropometric measurements of monozygotic/dizygotic twins. Both a skewed bivariate subset of the full data, perturbed by some outlying points, and the full data are considered.

Bayesian variable selection and estimation in binary quantile regression using global-local shrinkage priors

In this paper, we construct a Bayesian hierarchical model with global-local shrinkage priors for the regression coefficients, which includes the horseshoe prior and normal-gamma prior. This model is used for high-dimensional quantile regression models with dichotomous response data. We have developed an efficient sampling algorithm to generate posterior samplings for making posterior inference. We use a location-scale mixture representation of the asymmetric Laplace distribution. We assess the performance of the proposed methods through Monte Carlo simulations and two real-data applications in terms of parameter estimation and variable selection. Numerical results demonstrate that the proposed methods perform comparably with existing Bayesian methods under a variety of scenarios.

A novel robust multivariate Laplace distribution-based distributed consensus information fusion

The multivariate Laplace distribution (MLD) possesses a lesser set of prior parameters compared to the Student’s t-distribution, thus simplifying the modeling of measurement data subject to heavy-tailed noise corruption. This short communication introduces a novel distributed hybrid consensus fusion algorithm utilizing the MLD modeling and variational Bayesian approach. Firstly, we formulate the measurement noise associated with individual sensor nodes and introduce the hierarchical Gaussian form of MLD to enhance the feasibility of variational inference. Secondly, we opt for the inverse-Wishart (IW) distribution to serve as the conjugate prior to the positive definite matrix, enabling adaptive parameter learning within the MLD context. As a result, a novel hierarchical Gaussian state–space model (SSM) is established for multi-sensors network. Subsequently, we derive a novel local information filter to acquire higher precision information and innovation vector–matrix pairs, outperforming the Student’s t distribution-based filter. These derivations are grounded in the new hierarchical SSM, and the vector–matrix pairs aid in information fusion across the network. Lastly, we present a distributed robust consensus filter for fusing local information using a hybrid consensus approach that devoid of a central fusion node. Simulation results demonstrate that the proposed consensus filter can obtain better estimation accuracy under the interference of heavy-tailed measurement outliers.

A Robust Dissimilarity Distribution Analytics With Laplace Distribution for Incipient Fault Detection

Incipient faults with small magnitudes are usually masked by the data outliers and ambient noise, and thus the robustness should be taken into consideration when developing monitoring models for them. In this study, a robust dissimilarity distribution analytics (RDDA) method is proposed for incipient industrial fault detection. The probabilistic model of the RDDA method is formulated with Laplace distribution, and thus it is more robust to the disturbance when compared with the Gaussian distribution based monitoring models. Using the variational inference, the maximum likelihood estimations of the latent variables and model parameters in the RDDA method can be derived. After that, a monitoring strategy is established based on the obtained results with both static and dynamic statistics, which are designed using the dissimilarity between the distributions of different data sets. Since the missing data problem is also considered, the proposed RDDA method is more suitable for practical industrial applications. The proposed method is applied to identify the operation status of a deaerator. Experimental results illustrate that the proposed method can be established using the historical data with missing values, and it can accurately detect the incipient faults with small magnitude.

Controlling the Flexibility of Non-Gaussian Processes Through Shrinkage Priors

The normal inverse Gaussian (NIG) and generalized asymmetric Laplace (GAL) distributions can be seen as skewed and semi-heavy-tailed extensions of the Gaussian distribution. Models driven by these more flexible noise distributions are then regarded as flexible extensions of simpler Gaussian models. Inferential procedures tend to overestimate the degree of non-Gaussianity in the data and therefore we propose controlling the flexibility of these non-Gaussian models by adding sensible priors in the inferential framework that contract the model towards Gaussianity. In our venture to derive sensible priors, we also propose a new intuitive parameterization of the non-Gaussian models and discuss how to implement them efficiently in Stan. The methods are derived for a generic class of non-Gaussian models that include spatial Matérn fields, autoregressive models for time series, and simultaneous autoregressive models for aerial data. The results are illustrated with a simulation study and geostatistics application, where priors that penalize model complexity were shown to lead to more robust estimation and give preference to the Gaussian model, while at the same time allowing for non-Gaussianity if there is sufficient evidence in the data.

TADRP: Toward Thermal-Aware Data Replica Placement in Data-Intensive Data Centers

With the mushrooming growth of data volumes, data replica placement plays a key role in promoting the energy efficiency and Quality-of-Service (QoS) of data-intensive data centers. The existing data placement strategies mainly focus on storage performance improvement or QoS enhancement in data centers, but ignore the indispensable factor -heat recirculation. To bridge this gap, we propose a thermal-aware data replica placement strategy called TADRP, aiming to improve cooling efficiency and minimize the total power consumption of data-intensive data centers. TADRP leverages an ant colony optimization (ACO) algorithm coupled with Laplacian probability distribution to find a quasi-optimal disk sequence (or Disk Sequence for short), which consists of disks selected from different rack servers to place data replicas. TADRP categorizes disks of Disk Sequence into active and inactive ones, by placing hot and cold replicas on active and inactive disks, respectively. We quantitatively evaluate TADRP in terms of cooling costs, total power consumption, number of power-state transitions, and execution time. We compare TADRP with four alternative solutions, namely, Random, Hadoop, SRS, and CDP-NSGAII. Experimental results show that TADRP can reduce the cooling costs and the total power of the existing solutions by 14.7% -61.7% and 19.2 %-55.1%, respectively, without undesirable I/O performance drops.

Robust multi-mode probabilistic slow feature analysis with application to fault detection

This paper proposes a robust multi-mode dynamic data-driven model to identify complex industrial processes and study its application in detecting incipient faults. To model the process dynamics, a robust multi-mode probabilistic slow feature analysis (RMPSFA) is developed. A multiple switching conditional random field (MSCRF) is utilized to handle the multi-modality in the process. In order to make the model robust to the presence of outliers, a Laplace distribution is utilized to describe noise characteristics. To apply it in fault detection, several informative indices are defined. Then, the practical applicability of the proposed method is studied through an experiment on a hybrid tank pilot plant setup. Moreover, the real-life performance of the proposed model is tested on an industrial example.

On Stein factors for Laplace approximation and their application to random sums

In this article, we consider an integral Stein equation for Laplace distribution and solve it using the semigroup approach. Next, we derive regularity estimates for the solution of the Laplace Stein equation. Finally, we apply these estimates to obtain a convergence rate for Laplace approximation of the random geometric sums. We also compare our rate with the existing literature.

EEG based emotion recognition by hierarchical bayesian spectral regression framework

Spectral regression (SR), a graph-based learning regression model, can be used to extract features from graphs to realize efficient dimensionality reduction. However, due to the SR method remains a regularized least squares problem and being defined in L2-norm space, the effect of artifacts in EEG signals cannot be efficiently resisted. In this work, to further improve the robustness of the graph-based regression models, we propose to utilize the prior distribution estimation in the Bayesian framework and develop a robust hierarchical Bayesian spectral regression framework (named HB-SR), which is designed with the hierarchical Bayesian ensemble strategies. In the proposed HB-SR, the impact of noises can be effectively reduced by the adaptive adjustment approach in model parameters with the data-driven manner. Specifically, in the current work, three different distributions have been elaborately designed to enhance the universality of the proposed HB-SR, i.e., Gaussian distribution, Laplace distribution, and Student-t distribution. To objectively evaluate the performance of the HB-SR framework, we conducted both simulation studies and emotion recognition experiments based on emotional EEG signals. Experimental results have consistently indicated that compared with other existing spectral regression methods, the proposed HB-SR can effectively suppress the influence of noises and achieve robust EEG emotion recognition.