non-Gaussian Data Research Articles

Shrinkage estimation of gene interaction networks in single-cell RNA sequencing data

BackgroundGene interaction networks are graphs in which nodes represent genes and edges represent functional interactions between them. These interactions can be at multiple levels, for instance, gene regulation, protein-protein interaction, or metabolic pathways. To analyse gene interaction networks at a large scale, gene co-expression network analysis is often applied on high-throughput gene expression data such as RNA sequencing data. With the advance in sequencing technology, expression of genes can be measured in individual cells. Single-cell RNA sequencing (scRNAseq) provides insights of cellular development, differentiation and characteristics at the transcriptomic level. High sparsity and high-dimensional data structures pose challenges in scRNAseq data analysis.ResultsIn this study, a sparse inverse covariance matrix estimation framework for scRNAseq data is developed to capture direct functional interactions between genes. Comparative analyses highlight high performance and fast computation of Stein-type shrinkage in high-dimensional data using simulated scRNAseq data. Data transformation approaches also show improvement in performance of shrinkage methods in non-Gaussian distributed data. Zero-inflated modelling of scRNAseq data based on a negative binomial distribution enhances shrinkage performance in zero-inflated data without interference on non zero-inflated count data.ConclusionThe proposed framework broadens application of graphical model in scRNAseq analysis with flexibility in sparsity of count data resulting from dropout events, high performance, and fast computational time. Implementation of the framework is in a reproducible Snakemake workflow https://github.com/calathea24/ZINBGraphicalModel and R package ZINBStein https://github.com/calathea24/ZINBStein.

Functional Principal Component Analysis for Continuous Non-Gaussian, Truncated, and Discrete Functional Data.

Mobile health studies often collect multiple within-day self-reported assessments of participants' behavior and well-being on different scales such as physical activity (continuous scale), pain levels (truncated scale), mood states (ordinal scale), and the occurrence of daily life events (binary scale). These assessments, when indexed by time of day, can be treated and analyzed as functional data corresponding to their respective types: continuous, truncated, ordinal, and binary. Motivated by these examples, we develop a functional principal component analysis that deals with all four types of functional data in a unified manner. It employs a semiparametric Gaussian copula model, assuming a generalized latent non-paranormal process as the underlying generating mechanism for these four types of functional data. We specify latent temporal dependence using a covariance estimated through Kendall's bridging method, incorporating smoothness in the bridging process. The approach is then extended with methods for handling both dense and sparse sampling designs, calculating subject-specific latent representations of observed data, latent principal components and principal component scores. Simulation studies demonstrate the method's competitive performance under both dense and sparse sampling designs. The method is applied to data from 497 participants in the National Institute of Mental Health Family Study of Mood Spectrum Disorders to characterize differences in within-day temporal patterns of mood in individuals with the major mood disorder subtypes, including Major Depressive Disorder and Type 1 and 2 Bipolar Disorder. Software implementation of the proposed method is provided in an R-package.

Separation of earthquake and hydrology signals from GRACE satellites data via independent component analysis: a case study in the Sumatra region

SUMMARY The Gravity Recovery and Climate Experiment (GRACE) satellites have observed mass migrations caused by megathrust earthquakes. Extracting earthquake-related signals from GRACE data is still a challenge due to the interference from non-earthquake sources such as terrestrial hydrology. Instead of reducing hydrological signals by potentially biased hydrological models, in this study we apply a model-free technique of independent component analysis (ICA), to separate earthquake and non-earthquake signals from non-Gaussian GRACE data. We elucidate the principles and mechanisms of ICA for the separation of earthquake and hydrology signals, employing simulated data to demonstrate the process. Our findings demonstrate that both spatial ICA and temporal ICA are highly effective in discerning earthquake related to 2004 Mw 9.2 event and hydrological signals from GRACE data in the Sumatra region. This stands in stark contrast to principal component analysis, which often encounters challenges with signal intermingling. The utility of ICA is evident in its ability to distinctly delineate coseismic and post-seismic behaviours associated with megathrust events, including the 2004 Sumatra, the 2010 Maule, and the 2011 Tohoku earthquakes. ICA effectively mitigates the potential for misestimation of earthquake signals, an issue that can carry substantial implications. Therefore, employing ICA facilitates the accurate extraction of earthquake-related data from satellite gravity observations—a critical process for refining earthquake source parameters and understanding Earth's rheological properties, especially when non-earthquake signals are significant and cannot be disregarded.

A novel class of non-Gaussian system performance assessment and controller parameter tuning methods

Traditional variance-based control performance assessment (CPA) and controller parameter tuning (CPT) methods tend to ignore non-Gaussian external disturbances. To address this limitation, this study proposes a novel class of CPA and CPT methods for non-Gaussian single-input single-output systems, denoted as data Gaussianization (inverse) transformation methods. The idea of quantile transformation is used to transform the non-Gaussian data with the goal of maximizing mutual information into virtual Gaussian data. In addition, optimal system data for the virtual loop are mapped back to the actual non-Gaussian system using quantile inverse transformation. Furthermore, a CARMA model-based recursive extended least square algorithm and a CARMA model-based least absolute deviation iterative algorithm are used to identify virtual Gaussian and non-Gaussian system process models, respectively, while implementing the CPT. Finally, a unified framework is proposed for the CPA and CPT of a non-Gaussian control system. The simulation results demonstrate that the proposed strategy can provide a consistent benchmark judgment criterion (threshold) for different non-Gaussian noises, and the tuned controller parameters have good performance.

Unsupervised outlier detection using random subspace and subsampling ensembles of Dirichlet process mixtures

Probabilistic mixture models are recognized as effective tools for unsupervised outlier detection owing to their interpretability and global characteristics. Among these, Dirichlet process mixture models stand out as a strong alternative to conventional finite mixture models for both clustering and outlier detection tasks. Unlike finite mixture models, Dirichlet process mixtures are infinite mixture models that automatically determine the number of mixture components based on the data. Despite their advantages, the adoption of Dirichlet process mixture models for unsupervised outlier detection has been limited by challenges related to computational inefficiency and sensitivity to outliers in the construction of outlier detectors. Additionally, Dirichlet process Gaussian mixtures struggle to effectively model non-Gaussian data with discrete or binary features. To address these challenges, we propose a novel outlier detection method that utilizes ensembles of Dirichlet process Gaussian mixtures. This unsupervised algorithm employs random subspace and subsampling ensembles to ensure efficient computation and improve the robustness of the outlier detector. The ensemble approach further improves the suitability of the proposed method for detecting outliers in non-Gaussian data. Furthermore, our method uses variational inference for Dirichlet process mixtures, which ensures both efficient and rapid computation. Empirical analyses using benchmark datasets demonstrate that our method outperforms existing approaches in unsupervised outlier detection.

Denoising diffusion delensing: reconstructing the non-Gaussian CMB lensing potential with diffusion models

ABSTRACT Optimal extraction of cosmological information from observations of the cosmic microwave background (CMB) critically relies on our ability to accurately undo the distortions caused by weak gravitational lensing. In this work, we demonstrate the use of denoising diffusion models in performing Bayesian lensing reconstruction. We show that score-based generative models can produce accurate, uncorrelated samples from the CMB lensing convergence map posterior, given noisy CMB observations. To validate our approach, we compare the samples of our model to those obtained using established Hamiltonian Monte Carlo methods, which assume a Gaussian lensing potential. We then go beyond this assumption of Gaussianity, and train and validate our model on non-Gaussian lensing data, obtained by ray-tracing N-body simulations. We demonstrate that in this case, samples from our model have accurate non-Gaussian statistics beyond the power spectrum. The method provides an avenue towards more efficient and accurate lensing reconstruction, which does not rely on an approximate analytical description of the posterior probability. The reconstructed lensing maps can be used as an unbiased tracer of the matter distribution, and to improve delensing of the CMB, resulting in more precise cosmological parameter inference.

On the efficiency of misspecified Gaussian inference in nonlinear regression: Application to time-delay and Doppler estimation

Nonlinear regression plays a crucial role in various engineering applications. For the sake of mathematical tractability and ease of implementation, most of the existing inference procedures are derived under the assumption of independent and identically distributed (i.i.d.) Gaussian-distributed data. However, real-world situations often deviate from this assumption, with the true data generating process being a correlated, heavy-tailed and non-Gaussian one. The paper aims at providing the Misspecified Cramér–Rao Bound (MCRB) on the Mean Squared Error (MSE) of any unbiased (in a proper sense) estimator of the parameters of a nonlinear regression model derived under the i.i.d. Gaussian assumption in the place of the actual correlated, non-Gaussian data generating process. As a special case, the MCRB for an uncorrelated, i.i.d. Complex Elliptically Symmetric (CES) data generating process under Gaussian assumption is also provided. Consistency and asymptotic normality of the related Mismatched Maximum Likelihood Estimator (MMLE) will be discussed along with its connection with the Nonlinear Least Square Estimator (NLLSE) inherent to the nonlinear regression model. Finally, the derived theoretical findings will be applied in the well-known problem of time-delay and Doppler estimation for GNSS.

Principal Variables Analysis for Non-Gaussian Data

Principal variables analysis (PVA) is a technique for selecting a subset of variables that capture as much of the information in a dataset as possible. Existing approaches for PVA are based on the Pearson correlation matrix, which is not well-suited to describing the relationships between non-Gaussian variables. We propose a generalized approach to PVA enabling the use of different types of correlation, and we explore using Spearman, Gaussian copula, and polychoric correlations as alternatives to Pearson correlation. We compare performance in simulation studies varying the form of the true multivariate distribution over a range of possibilities. Our results show that on continuous non-Gaussian data, using generalized PVA with Gaussian copula or Spearman correlations provides a major improvement in performance compared to Pearson. On ordinal data, generalized PVA with polychoric correlations outperforms the rest by a wide margin. We apply generalized PVA to a dataset of 102 clinical variables measured on individuals with X-linked dystonia parkinsonism (XDP), a neurodegenerative disorder involving symptoms of both dystonia and parkinsonism. We find that using different types of correlation yields substantively different sets of principal variables; for example, parkinsonism-related metrics appear more explanatory than dystonia-related metrics on the observed data. Supplementary materials for this article are available online.

A comparison of PCA and ICA in geochemical pattern recognition of soil data: The case of Cyprus

Multivariate analysis of soil geochemistry is a powerful tool for differentiating lithological units and detecting geochemical dispersion halos related to mineralization or contamination. While univariate analysis can effectively identify lithological units with pronounced variations, it may fail to differentiate between subtler variations in lithologies. Traditional multivariate techniques such as principal component analysis (PCA) have limitations, including difficulties in understanding the individual contributions of each variable and an inability to work with non-Gaussian data. Independent component analysis (ICA) has emerged as a potential alternative, as it can effectively identify independent components of non-Gaussian data. In this study, we compared the effectiveness of PCA and ICA in relating multivariate soil geochemistry to parent lithology using the Soil Geochemical Atlas of Cyprus and associated digital geological maps. Both PCA and ICA were able to differentiate between the ultramafic units within the Troodos Ophiolite (TO) and the Circum-Troodos Sedimentary Succession (CTSS). However, ICA was more effective than PCA in identifying pillow lavas, providing a clear separation in the scores for IC4 and IC5. Furthermore, both PCA and ICA were able to separate the sheeted dykes from the cumulate mafic units within the TO. The gabbro unit is closely defined by IC2 scores. In contrast, PCA failed to provide factors that effectively delineated the Mamonia Terrane from other units, especially the TO, while ICA was able to provide a distinct separation in IC4 and IC5 scores. Separation between the CTSS and Quaternary units was weakly observed in IC2 scores. These findings demonstrate that there is a difference in the effectiveness of PCA and ICA in identifying different lithological units and emphasize the need for a careful selection of multivariate methods to differentiate between subtle differences in soil geochemistry relating to variations in parent lithology.

Fast Bayesian Functional Regression for Non-Gaussian Spatial Data

Functional generalized linear models (FGLM) have been widely used to study the relations between non-Gaussian response and functional covariates. However, most existing works for FGLM assume independence among observations and therefore they are of limited applicability for correlated data. A particularly important example is spatial functional data, where we observe functions over spatial domains, such as the age population curve or temperature curve at each areal unit. In this paper, we extend FGLM by incorporating spatial random effects. Especially, we study the relationship between the non-Gaussian response variable and functional covariates that are spatially observed. However, such model has computational and inferential challenges. The high-dimensional spatial random effects cause the slow mixing of Markov chain Monte Carlo (MCMC) algorithms. Furthermore, spatial confounding can lead to bias in parameter estimates and inflate their variances. To address these issues, we propose an efficient Bayesian method using a sparse reparameterization of high-dimensional random effects. We also study the average coverage probabilities of the credible intervals of functional parameters. We apply our methods to simulated and real data examples, including malaria incidence data and US COVID-19 data. The proposed method is fast while providing accurate functional estimates.

A grid-based methodology for the assessment of time-dependent building damage at large scale

Leakage into underground constructions can result in time-dependent settlements in soft clays. In urban areas with spatial variability in geologic stratification, groundwater conditions and soil compressibility, differential settlements may occur, causing damage to buildings. Current methods for damage assessment that rely on 1D formulations for settlement prediction are not representative for drawdown-induced settlements in heterogeneous environments. Thus, in this paper, we propose a stand-alone approach to integrating spatially distributed, non-Gaussian settlement data into early-stage building damage assessments at a district scale. Deformations computed using a 2D coupled hydro-mechanical finite element model with an advanced constitutive model were then employed to get the time-dependent settlements computed as a 3D grid (along x- and y-directions) over a large area. Building damage was then calculated from these green-field simulations with typically used damage parameters for each building-specific settlement profile and comparing these with damage criteria. The approach was applied to 215 buildings in central Gothenburg, Sweden by simulating scenarios of 10 kPa and 40 kPa pore pressure drawdown in the lower (confined) aquifer. Several scenarios were studied, and the correlation between damage parameters and damage criteria was assessed. Finally, a sensitivity study on grid resolution was performed, as well as a validation against observed damage data. The proposed methodology offers an effective way for early-stage damage assessments at a large area for non-Gaussian settlements so that further investigations and mitigation measures can be targeted to the buildings and locations at the highest risk for damage.

Non-Gaussian normative modelling with hierarchical Bayesian regression

Abstract Normative modelling is an emerging technique for parsing heterogeneity in clinical cohorts. This can be implemented in practice using hierarchical Bayesian regression, which provides an elegant probabilistic solution to handle site variation in a federated learning framework. However, applications of this method to date have employed a Gaussian assumption, which may be restrictive in some applications. We have extended the hierarchical Bayesian regression framework to flexibly model non-Gaussian data with heteroskdastic skewness and kurtosis. To this end, we employ a flexible distribution from the sinh-arcsinh (SHASH) family, and introduce a novel reparameterisation and a Markov chain Monte Carlo sampling approach to perform inference in this model. Using a large neuroimaging dataset collected at 82 different sites, we show that the results achieved with this extension are equivalent or better than a warped Bayesian linear regression baseline model on most datasets, while providing better control over the parameters governing the shape of distributions that the approach is able to model. We also demonstrate that the attained flexibility is useful for accurately modelling highly nonlinear relationships between aging and imaging derived phenotypes, which shows that the extension is important for pushing the field of normative modelling forward. All methods described here are available in the open-source pcntoolkit.

Machine learning Based Bearing Fault Classification Using Higher Order Spectral Analysis

In the defense sector, where mission success often hinges on the reliability of complex mechanical systems, the health of bearings within aircraft, naval vessels, ground vehicles, missile systems, drones, and robotic platforms is paramount. Different signal processing techniques along with Higher Order Spectral Analysis (HOSA) have been used in literature for the fault diagnosis of bearings. Bispectral analysis offers a valuable means of finding higher-order statistical associations within signals, thus proving to detect the nonlinearities among Gaussian and non-Gaussian data. Their resilience to noise and capacity to unveil concealed information render them advantageous across a range of applications. Therefore, this research proposesa novel approach of utilizing the features extracted directly from the Bispectrum for classifying the bearing faults, departing from the common practice in other literature where the Bispectrum is treated as an image for fault classification. In this work vibration signalsare used to detect the bearing faults. The features from the non-redundant region and diagonal slice of the Bispectrum are used to capture the statistical and higher-order spectral characteristics of the vibration signal. A set of sixteen machine learning models, viz., Decision Trees, K-Nearest Neighbors, Naive Bayes, and Support Vector Machine, is employed to classify the bearing faults. The evaluation process involves a robust 10-fold cross-validation technique. The results reveal that the Decision Tree algorithm outperformed all others, achieving a remarkable accuracy rate of 100 %. The naive Bayes algorithm also demonstrated the least performance, with an accuracy score of 99.68 %. The results obtained from these algorithms have been compared with those achieved using Convolutional Neural Network (CNN), revealing that the training time of these algorithms is significantly shorter in comparison to CNN.

An integrated approach combining randomized kernel PCA, Gaussian mixture modeling and ICA for fault detection in non-linear processes

Principal component analysis (PCA) and independent component analysis (ICA), as well as their kernel extensions, have been widely applied in the past for industrial fault detection with Gaussian or non-Gaussian process data with linear or non-linear characteristics. Kernel-based techniques lead to computational complexity due to the high dimensionality of the dataset in the feature space. In this work, a randomization approach is used to obtain a low-rank approximation of the high-dimensional kernel matrix. A hybrid machine learning technique is proposed that integrates randomized kernel PCA (RKPCA) with ICA and Gaussian mixture modeling (GMM). The proposed approach, ICA-RKPCA-GMM, addresses the Gaussian and non-Gaussian characteristics of non-linear process data. Another hybrid algorithm combining three basic techniques of ICA, PCA and GMM is also developed (ICA-PCA-GMM). The fault detection performances of the proposed techniques (ICA-RKPCA-GMM and ICA-PCA-GMM) are compared with PCA, ICA, KPCA and combined ICA-PCA techniques by applying the techniques to two benchmark systems. Monitoring performances were evaluated by determining the false alarm rate and fault detection rate for different types of process and sensor faults. The simulation results show that the proposed ICA-RKPCA-GMM approach yields better results than individual ICA, PCA and KPCA techniques, the combined ICA-PCA and the proposed ICA-PCA-GMM technique.

A NONLINEAR REGRESSION MODEL FOR EARLY LOC ESTIMATION OF OPEN-SOURCE KOTLIN-BASED APPLICATIONS

Context. The early lines of code (LOC) estimation in software projects holds significant importance, as it directly influences the prediction of development effort, covering a spectrum of different programming languages, and open-source Kotlin-based applications in particular. The object of the study is the process of early LOC estimation of open-source Kotlin-based apps. The subject of the study is the nonlinear regression models for early LOC estimation of open-source Kotlin-based apps. Objective. The goal of the work is to build the nonlinear regression model with three predictors for early LOC estimation of open-source Kotlin-based apps based on the Box-Cox four-variate normalizing transformation to increase the confidence in early LOC estimation of these apps. Method. For early LOC estimation in open-source Kotlin-based apps, the model, confidence, and prediction intervals of nonlinear regression were constructed using the Box-Cox four-variate normalizing transformation and specialized techniques. These techniques, relying on multiple nonlinear regression analyses incorporating multivariate normalizing transformations, account for the dependencies between variables in non-Gaussian data scenarios. As a result, this method tends to reduce the mean magnitude of relative error (MMRE) and narrow confidence and prediction intervals compared to models utilizing univariate normalizing transformations. Results. An analysis has been carried out to compare the constructed model with nonlinear regression models employing decimal logarithm and Box-Cox univariate transformation. Conclusions. The nonlinear regression model with three predictors for early LOC estimation of open-source Kotlin-based apps is constructed using the Box-Cox four-variate transformation. Compared to the other nonlinear regression models, this model demonstrates a larger multiple coefficient of determination, a smaller value of the MMRE, and narrower confidence and prediction intervals. The prospects for further research may include the application of other data sets to construct the nonlinear regression model for early LOC estimation of open-source Kotlin-based apps for other restrictions on predictors.

Constant false alarm rate detection method in mixed Weibull distribution sea clutter

In maritime surveillance, sea clutter represents a major obstacle to target detection accuracy due to missed or false alarms. Several methods have been proposed to find the most efficient. However, most of these methods have only been tested with simulated data, and their applicability and success in a real environment cannot be guaranteed. This work proposes constant false alarm rate (CFAR) detection techniques tailored to mixed Weibull-distributed sea clutter. The mixed Weibull distribution was used to model a mixture of Gaussian and non-Gaussian data. For CFAR detection in mixed Weibull distribution and Weibull clutter, the log t CFAR (logt-CFAR), Weber Haykin CFAR (WHCFAR), Geometric Mean Order Statistic CFAR (GMOS-CFAR), Trimmed Mean Order Statistic CFAR (TMOS-CFAR), Inclusion/Exclusion CFAR (IE-CFAR), and Weber Haykin Order Statistic CFAR (WHOS-CFAR) algorithms were performed. Both synthetic and actual ice multi-parameter imaging X-band (IPIX) radar data were used for experimental verification of this distribution. Simulation studies demonstrated that the proposed detectors maintain the CFAR property in Weibull-distributed clutter, making them effective in controlled scenarios, and confirming how better is the detection performance compared to other detection methods. This research bridges a critical gap in radar signal processing by significantly improving the detection of targets in challenging clutter environments.

Identification of Causal Structure with Latent Variables Based on Higher Order Cumulants

Causal discovery with latent variables is a crucial but challenging task. Despite the emergence of numerous methods aimed at addressing this challenge, they are not fully identified to the structure that two observed variables are influenced by one latent variable and there might be a directed edge in between. Interestingly, we notice that this structure can be identified through the utilization of higher-order cumulants. By leveraging the higher-order cumulants of non-Gaussian data, we provide an analytical solution for estimating the causal coefficients or their ratios. With the estimated (ratios of) causal coefficients, we propose a novel approach to identify the existence of a causal edge between two observed variables subject to latent variable influence. In case when such a causal edge exits, we introduce an asymmetry criterion to determine the causal direction. The experimental results demonstrate the effectiveness of our proposed method.

Optimizing motion imagery classification with limited channels using the common spatial pattern-based integrated algorithm.

The extraction of effective classification features from electroencephalogram (EEG) signals in motor imagery is a popular research topic. The Common Spatial Pattern (CSP) algorithm is widely employed in this field. However, the performance of the traditional CSP method depends significantly on the choice of a specific frequency band and channel number of EEG data. Furthermore, inter-class variance among these frequency bands and the limited number of available EEG channels can adversely affect the CSP algorithm's ability to extract meaningful features from the relevant signal frequency bands. We hypothesize that multiple Intrinsic Mode Functions (IMFS), into which the raw EEG signal is decomposed, can better capture the non-Gaussian characteristics of the signal, thus compensating for the limitations of the CSP algorithm when dealing with nonlinear and non-Gaussian distributed data with few channels. Therefore, this paper proposes a novel method that integrates Variational Mode Decomposition (VMD), Phase Space Reconstruction (PSR), and the CSP algorithm to address these issues. VMD is used to filter and enhance the quality of the collected data, PSR is employed to increase the effective data channels (data augmentation), and the subsequent CSP filtering can obtain signals with spatial features, which are decoded by Convolutional Neural Networks (CNN) for action decoding. This study utilizes self-collected EEG data to demonstrate that the new method can achieve a good classification accuracy of 82.30% on average, confirming the improved algorithm's effectiveness and feasibility. Furthermore, this study conducted validation on the publicly available BCI Competition IV dataset 2b, demonstrating an average classification accuracy of 87.49%.

Effects of Mindful Eating in Patients with Obesity and Binge Eating Disorder.

Binge eating disorder (BED) is a psychiatric illness related to a high frequency of episodes of binge eating, loss of control, body image dissatisfaction, and suffering caused by overeating. It is estimated that 30% of patients with BED are affected by obesity. "Mindful eating" (ME) is a promising new eating technique that can improve self-control and good food choices, helping to increase awareness about the triggers of binge eating episodes and intuitive eating training. To analyze the impact of ME on episodes of binge eating, body image dissatisfaction, quality of life, eating habits, and anthropometric data [weight, Body Mass Index (BMI), and waist circumference] in patients with obesity and BED. This quantitative, prospective, longitudinal, and experimental study recruited 82 patients diagnosed with obesity and BED. The intervention was divided into eight individual weekly meetings, guided by ME sessions, nutritional educational dynamics, cooking workshops, food sensory analyses, and applications of questionnaires [Body Shape Questionnaire (BSQ); Binge Eating Scale (BES); Quality of Life Scale (WHOQOL-BREF)]. There was no dietary prescription for calories, carbohydrates, proteins, fats, and fiber. Patients were only encouraged to consume fewer ultra-processed foods and more natural and minimally processed foods. The meetings occurred from October to November 2023. To carry out inferential statistics, the Shapiro-Wilk test was used to verify the normality of variable distribution. All variables were identified as non-normal distribution and were compared between the first and the eighth week using a two-tailed Wilcoxon test. Non-Gaussian data were represented by median ± interquartile range (IQR). Additionally, α < 0.05 and p < 0.05 were adopted. Significant reductions were found from the first to the eighth week for weight, BMI, waist circumference, episodes of binge eating, BSQ scale score, BES score, and total energy value (all p < 0.0001). In contrast, there was a significant increase in the WHOQOL-BREF score and daily water intake (p < 0.0001). ME improved anthropometric data, episodes of binge eating, body image dissatisfaction, eating habits, and quality of life in participants with obesity and BED in the short-term. However, an extension of the project will be necessary to analyze the impact of the intervention in the long-term.

Bayesian estimation in generalized linear models for longitudinal data with hyperspherical coordinates

Under the framework of generalized linear models (GLM), the generalized estimating equation (GEE) method is typically applied for longitudinal data analysis. However, there are a series of problems due to the misspecification of the within-subject correlation structure, especially in Bayesian estimation. To handle these difficulties, in this paper, we construct a class of generalized estimating equations for longitudinal data with hyperspherical coordinates (HPC) and propose a Bayesian approach established through empirical likelihood (EL). Additionally, an efficient Markov chain Monte Carlo (MCMC) procedure is developed for the required computation of the posterior distribution. As proved by the simulation studies and an application to a real longitudinal data set, our method not only performs better than traditional empirical likelihood estimation and Bayesian estimation with partial autocorrelations (PAC) but also is suitable for non-Gaussian data.